Insights into Data...Insights into Data v Contents Overview NCTM Principles and Standards for School...

Data Analysis andProbability

Insightsinto Data

TEACHER’S GUIDE

Mathematics in Context is a comprehensive curriculum for the middle grades.It was developed in 1991 through 1997 in collaboration with the Wisconsin Centerfor Education Research, School of Education, University of Wisconsin-Madison andthe Freudenthal Institute at the University of Utrecht, The Netherlands, with thesupport of the National Science Foundation Grant No. 9054928.

The revision of the curriculum was carried out in 2003 through 2005, with thesupport of the National Science Foundation Grant No. ESI 0137414.

National Science FoundationOpinions expressed are those of the authorsand not necessarily those of the Foundation.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, thethistle logo, Mathematics in Context, and the Mathematics in Context logo areregistered trademarks of Encyclopædia Britannica, Inc.

All rights reserved.

No part of this work may be reproduced or utilized in any form or by any means,electronic or mechanical, including photocopying, recording or by any informationstorage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-966-5

Printed in the United States of America

1 2 3 4 5 13 12 11 10 09

Wijers, M., de Lange, J., Bakker, A., Shafer, M. C., & Burrill, G. (2010). Insightsinto data. In Wisconsin Center for Education Research & Freudenthal Institute(Eds.), Mathematics in context. Chicago: Encyclopædia Britannica, Inc.

The Teacher’s Guide for this unit was prepared by David C. Webb, ElaineMcGrath, Monica Wijers, Dédé de Haan, and Els Feijs.

The Mathematics in Context Development TeamDevelopment 1991–1997

The initial version of Insights into Data was developed by Monica Wijers and Jan de Lange. It was adapted for use in American schools by Mary C Shafer and Gail Burrill.

Wisconsin Center for Education Freudenthal Institute StaffResearch Staff

Thomas A. Romberg Joan Daniels Pedro Jan de LangeDirector Assistant to the Director Director

Gail Burrill Margaret R. Meyer Els Feijs Martin van ReeuwijkCoordinator Coordinator Coordinator Coordinator

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie NiehausLaura Brinker James A, Middleton Nina Boswinkel Nanda QuerelleJames Browne Jasmina Milinkovic Frans van Galen Anton RoodhardtJack Burrill Margaret A. Pligge Koeno Gravemeijer Leen StreeflandRose Byrd Mary C. Shafer Marja van den Heuvel-PanhuizenPeter Christiansen Julia A. Shew Jan Auke de Jong Adri TreffersBarbara Clarke Aaron N. Simon Vincent Jonker Monica WijersDoug Clarke Marvin Smith Ronald Keijzer Astrid de WildBeth R. Cole Stephanie Z. Smith Martin KindtFae Dremock Mary S. SpenceMary Ann Fix

Revision 2003–2005

The revised version of Insights into Data was developed by Arthur Bakker and Monica Wijers. It was adapted for use in American schools by Gail Burrill.

Wisconsin Center for Education Freudenthal Institute StaffResearch Staff

Thomas A. Romberg David C. Webb Jan de Lange Truus DekkerDirector Coordinator Director Coordinator

Gail Burrill Margaret A. Pligge Mieke Abels Monica WijersEditorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie KuijpersBeth R. Cole Anne Park Peter Boon Huub Nilwik Erin Hazlett Bryna Rappaport Els Feijs Sonia PalhaTeri Hedges Kathleen A. Steele Dédé de Haan Nanda QuerelleKaren Hoiberg Ana C. Stephens Martin Kindt Martin van ReeuwijkCarrie Johnson Candace UlmerJean Krusi Jill VettrusElaine McGrath

Cover photo credits: (left, middle) © Getty Images;(right) © Comstock Images

Illustrationsx (left) Jason Millet; xviii (top right), 2 (top left and right) Christine McCabe/© Encyclopædia Britannica, Inc.; 11, 18 Holly Cooper-Olds; 28 ChristineMcCabe/© Encyclopædia Britannica, Inc.; 46, 47 Holly Cooper-Olds;59 © Encyclopædia Britannica, Inc.

Photographsxvii PhotoDisc/Getty Images; xviii (bottom right) Victoria Smith/HRW;1 (top) © Getty Images; (bottom) Lynn Betts, USDA Natural ResourcesConservation Service; 3 © Corbis; 6 (left to right) Ron Dahlquist;Mark E. Gibson/ Corbis; 12 © Corbis; 16Victoria Smith/HRW Photo;17 (top) Sam Dudgeon/HRW Photo; (bottom) © Bettmann/Corbis;29 Dennis MacDonald/Alamy; 32Victoria Smith/ HRW Photo;35 © PhotoDisc Getty Images; 38Victoria Smith/HRW Photo;39 © PhotoDisc/Getty Images; 44 © Corbis; 46 Amos Morgan/PhotoDisc/Getty Images; 53 (left, middle) PhotoDisc/Getty Images; (right) Siede Preis/PhotoDisc/ Getty Images; 54 PhotoDisc/Getty Images; 55 Siede Preis/PhotoDisc/Getty Images; 56 George K. Peck; 58 Stephanie Friedman/HRW

Insights into Data v

Contents

OverviewNCTM Principles and Standards for School Mathematics viiMath in the Unit viiiData Analysis and Probability Strand: An Overview xStudent Assessment in Mathematics in Context xivGoals and Assessment xviMaterials Preparation xviii

Student Material and Teaching NotesStudent Book Table of ContentsLetter to the Student

Section Patterns in DataSection Overview 1ABean Sprout Experiment: Conducting an Experiment; Collecting Data 1Living in Cities: Reading Scatter Plots; Identifying Trends in Data 3Summary 8Check Your Work 8

Section Selecting SamplesSection Overview 11ACollecting “Fair” Data: Modeling by Simulation; Displaying Data 11Biased Samples: Identifying Bias 16Random Numbers: Simulating Events 18Summary 20Check Your Work 20

Section Interpreting GraphsSection Overview 22ADifferent Impressions: Conducting a Survey; Recognizing

Misleading Graphs 22Summary 30Check Your Work 30

Section Using DataSection Overview 32AExploring Growth: Displaying Data; Finding Measures of

Central Tendency 32

D

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B

A

vi Insights into Data Contents

Contents

Presenting the Bean Sprout Data: Communicating Results; Drawing Conclusions 38

Summary 40Check Your Work 41

Section Correlating DataSection Overview 44AGrowing Babies: Translating Among Different Mathematical

Representations; Describing Correlations 44Summary 50Check Your Work 50

Section Lines That Summarize DataSection Overview 53AEgg Hunt: Translating Among Different Mathematical Representations;

Using Lines of Best Fit 53Gone Fishing: Making Scatter Plots; Drawing Lines of Best Fit 59Summary 60Check Your Work 61

Additional Practice 64

Assessment and SolutionsAssessment Overview 70Quiz 1 72Quiz 2 74Unit Test 77Quiz 1 Solutions 82Quiz 2 Solutions 83Unit Test Solutions 84

Glossary 88

Blackline MastersLetter to the Family 90Student Activity Sheets 91

F

E

Overview

Insights into Data andthe NCTM Principlesand Standards forSchool Mathematicsfor Grades 6–8The process standards of Problem Solving, Reasoning and Proof,Communication, Connections, and Representation are addressedacross all Mathematics in Context units.

In addition, this unit specifically addresses the following PSSMcontent standards and expectations:

Data Analysis and Probability

In grades 6–8 all students should:

• formulate questions and collect data about a characteristicshared by two populations or different characteristics withinone population;

• select, create, and use appropriate graphical representations ofdata, including histograms, box plots, and scatter plots;

• find, use, and interpret measures of center and spread, includingmean and interquartile range;

• discuss and understand the correspondence between data sets andtheir graphical representations, especially histograms, stem-and-leafplots, box plots, and scatterplots;

• use observations about differences between two or more samples tomake conjectures about the populations from which the sampleswere taken;

• make conjectures about possible relationships between twocharacteristics of a sample on the basis of scatter plots of thedata and approximate lines of fit; and

• use conjectures to formulate new questions and plan new studiesto answer them.

Overview Insights into Data vii

Overview

Prior Knowledge

This unit assumes students can do the followingwith understanding:

• add, subtract, multiply, and divide rationalnumbers in all representations: decimals,fractions, and percents;

• read, interpret, make, and use a variety ofgraphical representations such as number lineplots, bar graphs, histograms, pie graphs, scatterplots, box plots, and stem-and-leaf plots, asintroduced in the units Picturing Numbers andDealing with Data;

• understand and use statistical measures such asmean, median, and mode, as introduced andused in the units Picturing Numbers and Dealingwith Data;

• perform random sampling, as introduced in theunit Dealing with Data;

• understand and use chance, as introduced in theunits Take a Chance and Second Chance;

• make and use equations of a straight line andunderstand the meaning of slope and y-intercept,as introduced in the unit Graphing Equations;

• use metric measurements, as introduced in theunit Made to Measure; and

• use rules to compute area and volume, asintroduced in the unit Reallotment.

Students should already have developed a criticalapproach to analyzing graphical representations tosome extent.

Math in the Unit

The unit Insights into Data follows the DataAnalysis unit Dealing with Data and the unit onprobability Second Chance. Insights into Dataencourages students to think critically aboutrepresenting and analyzing data. It builds ongraphical representations of data and numericalmeasures of data introduced and used in the unitsPicturing Numbers and Dealing with Data.

Students conduct their own experiment in whichthey measure and record the growth of mungbeans watered with tap water and three othersolutions. Later in the unit, they present their ownbean sprout data. They organize, depict, anddescribe these data and present the findings onplant growth in a report. Students have to decide onthe type of graphs they want to use to make theirfindings clear to their classmates. They use toolssuch as stem-and-leaf plots, box plots, coordinatesystems, and histograms. Students consider appro-priate scales for labeling their graphs, the numericalmeasures of center, the information each type ofgraph provides, and the factors that might haveaffected their results.

Students study plant growth histograms fromanother experiment as well and describe datanumerically with such terms as mean, mode,maximum, and minimum.

Students study patterns in data. They do so, forexample, in a scatter plot in the context of percapita income data connected to data on percentof population living in urban areas per state.Students are pre-formally introduced to terms likecluster and outliers. Later in the unit, patterns inscatter plots are connected to correlation andregression. This is done as an informalintroduction in the context of the dimensions ofbird eggs. Students describe correlation as “strongpositive linear correlation,” “no apparentcorrelation,” and “weak correlation.”

Math in the Unit

viii Insights into Data Overview

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4. Match each of the descriptions of correlationbelow to a corresponding scatter plot.

i. linear, negative, and moderate

ii. linear positive, and strong

iii. no apparent correlation

iv. linear, positive, and weak

The term regression is not used; however, studentsare informally introduced to this concept. In thecontext of the length and width of birds eggs, theydraw lines in scatter plots that “seem to best fit thepoints.” In this way, they summarize the pattern indata and make it visible. They think about criteriafor drawing “a good line.” Equations are used todescribe the lines, the slope, and the y-interceptand are connected to what they mean in terms ofthe context. A data set on the growth rate of fish isused as an example of data where a curve betterdescribes the data than a straight line.

A magazine for health foods and organic healingwants to establish that large doses of vitaminsimprove health. The editor asks readers who haveregularly taken vitamins in large doses to writeto the magazine and describe their experiences.Of the 2,754 readers who reply, 93% report somebenefit from taking large doses of vitamins.

In this unit, students work with the concepts of arandom sample and bias. These concepts wereinformally introduced and used in the unitsDealing with Data and Second Chance. Studentslook at the differences between a survey, a sample,and a simulation. They reason about severalsamples and why they might be biased.

By studying a variety of graphs, students discoverways in which graphic displays can give differentimpressions or even be misleading.

When students have finished this unit, they can:

• understand graphical representations of datasuch as: scatter plots, box plots, number lineplots, stem-and-leaf plots, and histograms;

• identify the misrepresentation of data, and theyare able to correct it (if possible);

• reason about (random) sampling and describepossible causes for bias in the process ofsampling and survey results;

• describe how to choose a fair or good sampleof a population;

• collect data through survey, experiment, andsimulation;

• represent data graphically and describe data withstatistical measures (mean, median, and mode);

• describe a correlation in a scatter plot inpre-formal terms like: weak, moderate, strong,positive, negative, linear, and non-linear;

• draw straight lines that summarize data and usethe equations of these lines to predict outcomes;they can also describe the meaning of the slopeand the y-intercept of the line in terms of thecontext; and

• draw conclusions based on data and representa-tions of data.

Overview

x Insights into Data Overview

One thing is for sure: our lives are full of uncertainty.We are not certain what the weather will be tomor-row or which team will win a game or how accuratea pulse rate really is. Data analysis and probabilityare ways to help us measure variability and uncer-tainty. A central feature of both data analysis andprobability is that these disciplines help us makenumerical conjectures about important questions.

The techniques and tools of data analysis andprobability allow us to understand general patternsfor a set of outcomes from a given situation such astossing a coin, but it is important to remember thata given outcome is only part of the larger pattern.Many students initially tend to think of individualcases and events, but gradually they learn to thinkof all features of data sets and of probabilities asproportions in the long run.

The MiC Approach to Data Analysisand Probability

The Data Analysis and Probability units in MiCemphasize dealing with data, developing anunderstanding of chance and probability, usingprobability in situations connected to data analysis,and developing critical thinking skills.

The strand begins with students’intuitive understanding of thedata analysis concepts of most,least, and middle in relationto different types of graphicalrepresentations that showthe distribution of data and

the probability concepts of fairness and chance.As students gradually formalize these ideas, theyuse a variety of counting strategies and graphicalrepresentations. In the culminating units of thisstrand, they use formal rules and strategies forcalculating probabilities and finding measuresof central tendency and spread.

Throughout this development, there is a constantemphasis on interpreting conclusions made bystudents and suggested in the media or othersources. In order for students to make informeddecisions, they must understand how informationis collected, represented, and summarized, and theyexamine conjectures made from the informationbased on this understanding. They learn aboutall phases of an investigative cycle, starting withquestions, collecting data, analyzing them, andcommunicating about the conclusions. They areintroduced to inference-by-sampling to collectdata and reflect on possible sources of bias. Theydevelop notions of random sampling, variationand central tendency, correlation, and regression.Students create, interpret, and reflect on a widerange of graphical representations of data and relatethese representations to numerical summaries suchas mean, mode, and range.

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TheDataAnalysis and Probability Strand:An Overview

Overview

Overview Insights into Data xi

Organization of the Strand

Statistical reasoning based on data is addressed inall Data Analysis and Probability units. Students’work in these units is organized into two substrands:Data Analysis and Chance. As illustrated in thefollowing map of the strand, the three core unitsthat focus on data analysis are Picturing Numbers,Dealing with Data, and Insights into Data. The twounits that focus on probability are Take a Chanceand Second Chance. The sixth core unit in thisstrand, Great Predictions, integrates data analysisand probability.

Data Analysis

In the units of the DataAnalysis substrand,students collect,depict, describe, andanalyze data. Usingthe statistical toolsthey develop, theymake inferences anddraw conclusionsbased on data sets.

The substrand beginswith Picturing Numbers.Students collect dataand display them in tabular and graphical forms,such as histograms, number line plots, and piecharts. Measures of central tendency, such as themean, are used informally as students interpretdata and make conjectures.

In Dealing with Data, students create and interpretscatter plots, box plots, and stem-and-leaf plots, inaddition to other graphical representations. Themean, median, mode, range, and quartiles are usedto summarize data sets. Students investigate datasets with outliers and make conclusions about theappropriate use of the mean and median.

Sampling is addressed across this substrand, butin particular in Insights into Data, starting withinformal notions of representative samples,randomness, and bias. Students gather data usingvarious sampling techniques and investigate thedifferences between a survey and a sample. Theycreate a simulation to answer questions about asituation. Students also consider how graphicalinformation can be misleading, and they areintroduced informally to the concepts of regressionand correlation.

123

Pathways through the Data Analysisand Probability Strand

(Arrows indicate prerequisite units.)

GreatPredictions

PicturingNumbers

SecondChance

Insightsinto Data

Dealingwith Data

Take aChance

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1 2 3 4 5 6 7 8 9Number of Babies in Litter

1 2 3 4 5 6 7 8 9ABCDEFGHIJKLMNOPQR

Overview

xii Insights into Data Overview

In Great Predictions, students learn to recognizethe variability in random samples and deepen theirunderstanding of the key statistical concepts ofrandomness, sample size, and bias. As the capstoneunit to the Data Analysis and Probability strand,data and chance concepts and techniques areintegrated and used to inform conclusions aboutdata.

Chance

Beginning with the concept of fairness, Take aChance progresses to everyday situations involvingchance. Students use coins and number cubes toconduct repeated trials of an experiment. A chanceladder is used as a model throughout the unit torepresent the range from impossible to certainand to ground the measure of chance as a numberbetween 0 and 1. Students also use tree diagramsto organize and count, and they use benchmarkfractions, ratios, and percents to describe theprobability of various outcomes and combinations.

The second probability unit, Second Chance, furtherdevelops students’ understanding of fairness andthe quantification of chance. Students make chancestatements from data presented in two-way tablesand in graphs.

Students also reason about theoretical probabilityand use chance trees as well as an area model tocompute chances for compound events. They useinformation from surveys, experiments, and simu-lations to investigate experimental probability.Students also explore probability concepts suchas complementary events and dependent andindependent events.

These concepts are elaborated further in the finalunit of the strand, Great Predictions. This last unitdevelops the concepts of expected value, featuresof independent and dependent events, and the roleof chance in world events.

Sure to Happen

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Overview

Overview Insights into Data xiii

Critical Reasoning

Critical reasoning about data and chance is a themethat exists in every unit of the Data Analysis andProbability strand. In Picturing Numbers, studentsinformally consider factors that influence datacollection, such as the wording of questions on asurvey, and they compare different graphs of thesame data set. They also use statistical data to buildarguments for or against environmental policies.

In Take a Chance, students use their informalknowledge of fairness and equal chances as theyevaluate decision-making strategies.

In Dealing with Data, students explore how thegraphical representation of a data set influences theconjectures and conclusions that are suggested bythe data. They compare advantages and disadvan-tages of various graphs and explore what you learnfrom using different measures of central tendency.

Throughout the curriculum, students are asked toview representations critically. Developing a criticalattitude is especially promoted in Insights into Data,when students analyze graphs from mass media.

In Second Chance, students explore the notionof dependency (for instance, the relation of genderand wearing glasses) and analyze statements aboutprobabilities (for instance, about guessing duringa test).

In Great Predictions, students study unusual samplesto decide whether they occurred by chance or forsome other reason (pollution, for instance). Theyexplore how expected values and probability canhelp them make decisions and when this informa-tion could be misleading.Dry SpellMay EndSoon 3.1

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xiv Insights into Data Overview

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Student Assessment inMathematics in ContextAs recommended by the NCTM Principles and Standards for SchoolMathematics and research on student learning, classroom assessmentshould be based on evidence drawn from several sources. An assessmentplan for a Mathematics in Context unit may draw from the followingoverlapping sources:

• observation—As students work individually or in groups, watchfor evidence of their understanding of the mathematics.

• interactive responses —Listen closely to how students respond toyour questions and to the responses of other students.

• products —Look for clarity and quality of thought in students’solutions to problems completed in class, homework, extensions,projects, quizzes, and tests.

Assessment Pyramid

When designing a comprehensive assessment program, the assessmenttasks used should be distributed across the following three dimensions:mathematics content, levels of reasoning, and difficulty level. TheAssessment Pyramid, based on Jan de Lange’s theory of assessment,is a model used to suggest how items should be distributed acrossthese three dimensions. Over time, assessment questions should“fill” the pyramid.

Overview

Overview Insights into Data xv

Levels of Reasoning

Level I questions typically address:

• recall of facts and definitions and

• use of technical skills, tools, and standardalgorithms.

As shown in the pyramid, Level I questions are notnecessarily easy. For example, Level I questions mayinvolve complicated computation problems. Ingeneral, Level I questions assess basic knowledgeand procedures that may have been emphasizedduring instruction. The format for this type ofquestion is usually short answer, fill-in, or multiplechoice. On a quiz or test, Level I questions closelyresemble questions that are regularly found in agiven unit substituted with different numbersand/or contexts.

Level II questions require students to:

• integrate information;

• decide which mathematical models or tools touse for a given situation; and

• solve unfamiliar problems in a context, basedon the mathematical content of the unit.

Level II questions are typically written to elicit shortor extended responses. Students choose their ownstrategies, use a variety of mathematical models,and explain how they solved a problem.

Level III questions require students to:

• make their own assumptions to solve open-endedproblems;

• analyze, interpret, synthesize, reflect; and

• develop one’s own strategies or mathematicalmodels.

Level III questions are always open-ended problems.Often, more than one answer is possible and thereis a wide variation in reasoning and explanations.There are limitations to the type of Level III prob-lems that students can be reasonably expected torespond to on time-restricted tests.

The instructional decisions a teacher makes as heor she progresses through a unit may influence thelevel of reasoning required to solve problems. If amethod of problem solving required to solve aLevel III problem is repeatedly emphasized duringinstruction, the level of reasoning required to solvea Level II or III problem may be reduced to recallknowledge, or Level I reasoning. A student who doesnot master a specific algorithm during a unit butsolves a problem correctly using his or her owninvented strategy may demonstrate higher-levelreasoning than a student who memorizes andapplies an algorithm.

The “volume” represented by each level of theAssessment Pyramid serves as a guideline for thedistribution of problems and use of score pointsover the three reasoning levels.

These assessment design principles are usedthroughout Mathematics in Context. The Goalsand Assessment charts that highlight ongoingassessment opportunities—on pages xvi and xviiof each Teacher’s Guide—are organized accordingto levels of reasoning.

In the Lesson Notes section of the Teacher’s Guide,ongoing assessment opportunities are also shownin the Assessment Pyramid icon located at thebottom of the Notes column.

Assessment Pyramid

5a

5b

Determine whetherrepresentations of dataare appropriate.

Analyze representationsof data.

Overview

xvi Insights into Data Overview

Goals and AssessmentIn the Mathematics in Context curriculum, unit goals, organized accordingto levels of reasoning described in the Assessment Pyramid on page xiv,relate to the strand goals and the NCTM Principles and Standards forSchool Mathematics. The Mathematics in Context curriculum is designedto help students demonstrate their understanding of mathematics in

each of the categories listed below. Ongoingassessment opportunities are also indicated ontheir respective pages throughout the teacherguide by an Assessment Pyramid icon.

It is important to note that the attainment ofgoals in one category is not a prerequisite to theattainment of those in another category. In fact,students should progress simultaneously toward

several goals in different categories. The Goals and Assessment table isdesigned to support preparation of an assessment plan.

Level I:

Conceptual

and Procedural

Knowledge

Ongoing Unit

Goal Assessment Opportunities Assessment Opportunities

1. Represent data graphically Section B p. 15, #8b Quiz 2 #1b, 2band find a given point on Section C p. 29, Activity Test #1abc, 2c, 3aba graph. Section D p. 38, Activity

Section F p. 59, #16b

2. Describe data numerically. Section B p. 15, #7b Quiz 2 #2aSection C p. 29, Activity Test #2a, 3aSection D p. 37, #6b

p. 38, Activity

3. Describe the relationship Section A p. 6, #10 Quiz 1 #1abbetween two variables. Section E p. 49, #10b Quiz 2 #3ab

Section F p. 57, #13b Test #3cp. 59, #16c

4. Identify the degree of Section E p. 47, #6 Quiz 2 #3abcorrelation between variables. p. 48, #8 Test #2d, 3cd

5. Describe a linear Section F p. 56, #10abcrelationship with an p. 57T, Extensionequation of a line. p. 58, #14ab

6. Design, conduct, and Section B p. 13, #4 Quiz 1 #2aanalyze ways of gathering Section C p. 29, Activitydata: surveys, simulations, Section D p. 38, Activityexperiments.

7. Use random samples in Section B p. 13, #4 Quiz 1 #2agathering data. Section C p. 29, Activity

Overview

Overview Insights into Data xvii

Level II:

Reasoning,

Communicating,

Thinking,

and Making

Connections

Ongoing Unit


8. Analyze and interpret Section A p. 5, #5ab, 6 Quiz 1 #1bcrepresentations of data. Section B p. 19, #16 Quiz 2 #1ab

Section C p. 25, #5b Test #1c, 3eSection D p. 33, #3

p. 38, ActivitySection E p. 45, #3

9. Draw conclusions based Section A p. 7, #15 Quiz 2 #2cbased on given data sets Section B p. 15, #8c Test #2cd, 3cfand representations of data. Section C p. 27, #8b

p. 29, ActivitySection D p. 37, #6a

p. 38, ActivitySection E p. 47, #6

Level III:

Modeling,

Generalizing,

and Non-Routine

Problem Solving

Ongoing Unit


10. Recognize possible bias Section B p. 17, #11a-din sample surveys. Section C p. 29, Activity

11. Determine whether Section C p. 25, #5a Quiz 2 #1abrepresentations of data Section C p. 27, #8a Test #2b(numerical and visual) are p. 31,appropriate. For Further Reflection

Section D p. 36, #5c

12. Generate appropriate Section A p. 10, Quiz 1 #2bquestions for analyzing data For Further Reflection Test #2e, 3fsets and representations Section C p. 29, Activityof data. p. 31,

For Further ReflectionSection D p. 38, Activity

Overview

The following items are the necessary materialsand resources to be used by the teacher andstudents throughout the unit. For further details,see the Section Overviews and the Materialssection at the top of the Hints and Commentscolumn of each teacher page. Note: Some contextsand problems can be enhanced through the use ofoptional materials. These optional materials arelisted in the corresponding Hints and Comments:

Student Resources

Quantities listed are per student

• Letter to the Family

• Student Activity Sheets 1–11

Teacher Resources

Quantities listed are per class.

• Cola (one fluid ounce)

• Lemon-lime soda (one fluid ounce)

• Petri dish

• Quart jars (four)

• Salt (one teaspoon)

• Tap water (one pint)

Student Materials

Quantities listed are per pair of students, unlessotherwise noted.

• Blank paper (one sheet per student)

• Blue and red marker (one of each per pair ofstudents)

• Centimeter ruler with millimeter markings

• Chicken egg (one per group of students)

• Colored self-stick notes (four)

• Completed Student Activity Sheet 1

• Drawing paper (two sheets per student)

• Glue or tape

• Graph from a newspaper or magazine (one perstudent)

• Graph paper (10 sheets per student)

• Graphing calculator

• Mung bean seeds (10)

• Paper bag or box

• Paper towels (three sheets)

• Scissors

• Teaspoon

• Thread or string (one segment)

xviii Insights into Data Overview

Materials Preparation

StudentMaterial

andTeaching

Notes

StudentMaterial

andTeaching

Notes

Teachers Matter

v Insights into Data Teachers Matter

ContentsLetter to the Student vi

Section A Patterns in DataBean Sprout Experiment 1Living in Cities 3Summary 8Check Your Work 8

Section B Selecting SamplesCollecting “Fair” Data 11Biased Samples 16Random Numbers 18Summary 20Check Your Work 20

Section C Interpreting GraphsDifferent Impressions 22Summary 30Check Your Work 30

Section D Using DataExploring Growth 32Presenting the Bean Sprout Data 38Summary 40Check Your Work 41

Section E Correlating DataGrowing Babies 44Summary 50Check Your Work 50

Section F Lines That Summarize DataEgg Hunt 53Gone Fishing 59Summary 60Check Your Work 61

Additional Practice 64

Answers to Check Your Work 70

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Teachers Matter

Teachers Matter Insights into Data vT

Dear Student,

Welcome to Insights into Data. Do you look at the graphs innewspapers to see if they make sense? Numbers and graphsare used to describe situations all around you: sports, grades,sales, marketing, taxes, and even car ratings.

In this unit, you will learn how to use numbers and graphs tohelp you make decisions and draw conclusions. You will alsostudy surveys and how they are conducted. You will grow mungbeans in soda, salt water, and tap water to see which is the bestsolution for growing sprouts. You will even learn to use lines tohelp you investigate the relationship between two things, suchas the length and width of birds’ eggs. (Do you think birds’ eggsare mostly round?)

Look for graphs and numerical information in newspapers andmagazines to develop your own insights into data.

Sincerely,

TThe Mathematics in Context Development Team

1A Insights into Data Teachers Matter

Teachers MatterA

Section Focus

Students set up a one-week experiment to collect plant growth data. They plantbean sprouts in different solutions; this is an experiment that simulates theeffects of different chemicals on crop growth. Later in the unit (Section D),these data are analyzed and presented, and students compare the variability inthe lengths of the sprouts over seven days.

Students investigate statements about data represented in a graph. Theyexplore a scatter plot with census data that compare states by per capitaincome and the percent of the population that live in cities. Students analyzeunderlying patterns in the data. They find that it may be necessary to collectadditional data in order for them to make appropriate inferences.

Pacing and Planning

Additional Resources: Additional Practice, Section A, Student Book pages 64 and 65

Day 1: Bean Sprout Experiment Student pages 1–3

ACTIVITY Problem 1 Set up a one-week experiment to measurethe height of ten bean sprouts in differentsolutions.

Day 2: Living in Cities Student pages 3–5

INTRODUCTION Problems 2 and 3 Discuss census data for cities and ruralareas and explain the term per capitaincome.

CLASSWORK Problems 4–7 Read and interpret a scatter plot thatcompares states by per capita income andpercent of population that lives in citiesand review the concept of the mean.

HOMEWORK Problems 8 and 9 Evaluate statements made about relativeposition of states on the scatter plot.

Day 3: Living in Cities (Continued) Student pages 6–10

INTRODUCTION Problems 10 and 11 Use the concepts of clusters and outliersto interpret data in a scatter plot.

CLASSWORK Problems 12–15 Group states geographically to identitytrends in scatter plot data.

HOMEWORK Check Your Work and Student self-assessment: Create andFor Further Reflection interpret scatter plots.

Teachers Matter Section A: Patterns in Data 1B

Teachers Matter A

Materials

Student Resources

Quantities listed are per student.

• Letter to the Family

• Student Activity Sheet 1–3

Teachers’ Resources

Quantities listed are per class.

• Cola, one fluid ounce

• Lemon-lime soda (one fluid ounce)

• Petri dish

• Quart jars (four)

• Salt, one teaspoon

• Tap water (one pint)

Student Materials


• Blue and red marker (one of each per pairof students)

• Metric ruler with millimeter markings

• Mung bean seeds (10)

• Paper towels (three sheets)

• Scissors

• Teaspoon

• Thread or string (one segment)

*See Hints and Comments for optional materials.

Learning Lines

In this section, students set up an experimentto collect their own data on plant growth. Anexperiment is one way to collect data. Sampling,conducting a survey, or doing a simulation areother ways of collecting data. Students exploreand compare different data-collection methods inthis unit. In the context of data gathered by theU.S. Census Bureau, students think about reasonsfor collecting certain data. Data collected andstudied in this section are data on the length ofbean sprouts and data on per capita income andpercent of population living in cities for all U.S.states. The terms population and sample,introduced in the unit Dealing with Data,are revisited here.

Patterns in Scatter Plots

Students analyze a scatter plot that shows the resultsof research conducted by the U.S. Census Bureauabout per capita income in each state and thepercentage of people who live in urban areas inthat state. They look for patterns in the data, try toexplain these patterns, and draw conclusions.

Scatter plots were introduced and used in theunit Dealing with Data. A scatter plot can beused to investigate the relationship between twoquantitative variables. The variable you beginwith (sometimes called the independent variable)is generally associated with the horizontal axis.The second variable (sometimes called thedependent variable) is generally placed on thevertical axis. There are often underlying patternsin scatter plots of paired data. Students look forclusters of points, outliers. and trends in thescatter plot.

Identifying the data points as belonging todifferent categories can also help reveal patterns.For example, in the scatter plot on page 4, eachdata point can be circled in a color that indicateswhich region of the country that state is in. In thisway, regional trends can be discovered. Criticalreview of representations of data may uncovermore complex relationships than are initiallynoticed.

Statistics

The meaning of the mean is addressed in thissection connecting the scatter plot with the censusdata. The per capita income per state is a “mean,”and the nationwide per capita income is a “mean”as well. Students reflect on the meaning of this“mean” nationwide per capita income in relationto the per capita income per state.

At the End of the Section: LearningOutcomes

Students will have conducted an experiment andcollected and recorded data from it. They will haveanalyzed representations of data, in particulardata represented in a scatter plot, and should bemore aware of questions that should be askedwhen analyzing data sets and representations ofdata. Students will also be able to describe therelationship between two variables represented ina scatter plot; in their description they can useterms like clusters, patterns, and outliers.

1 Insights into Data

Notes

If your students live in anurban area, you coulddiscuss how safe chemicalsare also needed for lawnsas well as other crops.

If you have difficultylocating mung beans,check the Internet forsources, such as an Asiangrocery. If mung beans arenot available, other types ofbeans will also work.

Patterns inData

A

Vocabulary Building

A class discussion is a good way to start this section to make sure studentsunderstand the vocabulary, such as parasite, pesticide, and germinate.

Interdisciplinary Connection

You may want to arrange for the science teacher in your school or someother knowledgeable person to speak to the class about the problemsagriculturists have had in trying to control parasites with pesticides. Thisperson might address the alternatives available. If you want to replace themung bean experiment on your own, you may want to contact otherteachers, such as the science teacher, for suggestions.

Reaching All Learners

APatterns in Data

Bean Sprout Experiment

Farmers are concerned about parasites damagingtheir crops. Chemical companies develop pesticidesthat kill the parasites, but they have to be careful thatthe chemicals do not harm the crops. In this section,you will collect and examine data, and you will studyhow graphs can help you reach conclusions about adata set. In the experiment that follows, you willexamine the effect of different liquids on the growthof bean sprouts germinated from mung beans.

In the experiment below, you will investigate the answers to thefollowing questions:

• How fast do bean sprouts grow?

• What happens to the growth of bean sprouts if the beans areplaced in different solutions?

You will use the results of this experiment later in the unit.

Before conducting an experiment, researchers hypothesize about,or predict, what will happen in the experiment.

1. Read the description of the activity on page 2 and then make aprediction about the outcome.

mung beans

Section A: Patterns in Data 1T

Hints and CommentsOverviewStudents are introduced to an activity in which theywill investigate the effects of different solutions on thegrowth of bean sprouts.

About the Mathematics

An experiment is introduced as a way to collect data.Students formulate hypotheses about the experiment.

Other data-collection methods such as surveys andsimulations get attention in other sections.

Planning

You may want to read the text on Student Book page 1as a class. Make sure students understand the textbefore moving on to the next page.

Instead of the mung bean experiment, you may haveyour students collect data with another experiment,for example, the length of time one can balance onone leg under different conditions such as smoothsurface, rough surface, eyes open, eyes closed, etc.Students can record the times. Note there is noover-time process in these data.

If you replace the mung bean experiment with anexperiment of your own choice, you will have tochange the activity and the materials accordingly.

Solutions and Samples

1. Answers will vary. Sample student responses:

Bean sprouts grow best in tap water.

Bean sprouts in salt water will not grow.

Bean sprouts in different solutions will growdifferently.


Notes

Prepare for at least fourgroups so you have resultsfor the four differentsolutions. You may be ableto borrow petri dishes froma science teacher. Be sureyou have soaked the beansovernight.

Patterns inData

A

Hands-On Learning

Some students may need a review on reading a metric ruler and on somebasic equivalencies: 10 mm � 1 cm, 10 cm � 1 dm, 10 dm � 1 meter. Anactivity could be a metric “scavenger hunt.” Make a list of 10 metricmeasurements (3mm, 5 cm, 2 dm, etc.) and have pairs of students findobjects in the classroom with these lengths. They could first have toestimate the lengths and then check with a ruler.

Parent Involvement

Students may conduct a similar activity at home, using either differentconcentrations of the solutions or different solutions altogether.


Your group will need the following items:

i.

ii.

iii.

iv.

v.

Directions:

i. Cut three paper towels to fit the bottom of the petri dish.

ii. Put two layers of paper towels in the petri dish and arrangethe mung beans on top. Figure out a way to identify the tenindividual beans so that you will be able to collect data foreach bean as it grows.

iii. Soak the paper towels and beans by adding several teaspoonsof your solution.

iv. Place another paper towel over the beans and dampen it withyour solution. Place the cover on the petri dish. Label the petridish with the name of your group and the type of solution used.

v. At approximately the same time each day, measure the lengthof each sprout in millimeters. Use the table on Student

Activity Sheet 1 to record the lengths of the sprouts and yourobservations about the growth of the beans. Keep track of theprogress of the bean sprouts for seven days. During this time,add more solution as needed to keep the paper towels wet.

• one petri dish

• paper towels

• ten mung beans that have been soakedovernight in tap water

• a metric ruler for measuring in millimeters

You will also need one of the following solutions:

• tap water

• 1 teaspoon of salt per pint of water

• 1 fluid ounce of cola per pint of water

• 1 fluid ounce of lemon-lime soda per pintof water


Hints and CommentsMaterialsStudent Activity Sheet 1 (one per group)cola (one fluid ounce per class);lemon-lime soda (one fluid ounce per class);metric rulers with millimeter markings (one per groupof students);mung beans (10 per group of students);paper towels (three sheets per group of students);petri dishes (one per class);quart jars (four per class);salt (one teaspoon per class);scissors (one pair per group of students);tap water (one pint per class);teaspoons (one per group of students);thread or string (one segment per group of students);

Overview

Students collect and prepare the materials needed forthe mung bean experiment. They conduct theexperiment and measure the lengths of the beansprouts each day for one week.

Planning

Students may work in small groups. Provide a differentsolution for each group of students. You can use quart-size canning jars to hold the solutions. One jar of eachsolution is more than adequate for a class for theduration of the experiment. Each jar should be labeledwith the name and concentration of the solution.You might discuss with the class the reason why onesolution is plain tap water. In a scientific experiment,a control is used. The growth of the mung beans inexperimental solutions is compared to the growth ofmung beans in plain water.Daily measurements of the beans should be recordedfor seven school days on Student Activity Sheet 1.Taking exact measurements is very important. Beansprouts curl when they grow, and students need todecide how to measure them. One suggestion is togently hold string or thread close to the sprout,following its curls, and then measure the length of thethread with a ruler. Students also must decide whatparts of the sprout to measure. For example, will theymeasure from the edge of the seed to the end of theleaves, include the length of the seed, or exclude theleaves? You might have a class discussion about thisbefore students measure the sprouts. Emphasize theneed to have consistent measurements if conclusionsare to be drawn across groups. Since the sprouts arebrittle, students must also decide whether to excludea bean from the experiment if its sprout breaks.The salt solution should be stirred or shaken beforeeach use.


Notes

The measurements shouldbe from seven school days.It is all right to skip theweekend. However, ifstudents skip the weekend,make sure that when theyplot the data, the scale fortime is correct. The days inwhich no measurementswere taken should be onthe axis, but no points canbe plotted. The classshould reach consensus onhow to do themeasurements since beansprouts curl. Onepossibility is to use a stringto follow the twists andthen measure the length ofthe string. They must alsodecide if they will includethe length of the seed andthe length of the leaves.Emphasize the need for allthe groups to be consistentfor reliable data.

Each student should have acopy of the activity sheetwith the measurements forthe project coming later inthe book.

Patterns inData

A

Accommodation

Some students may need a brief review of percentage. A visual of percentbars with the benchmark percents marked on them, as well as a mini-lesson on using a ratio table to change a ratio so it has a denominator of100, should be helpful.

Extension

You may have students investigate examples of other data that are gatheredby the U.S. Census Bureau.


The United States Census Bureau regularly investigates whatpercentage of people live in cities and what percentage live in ruralareas. The information also indicates the movement of people fromcities to rural areas and vice versa.

2. Why do you think it is important to know if people are movingfrom cities to rural areas?

Living in Cities

Bean Length of Bean Sprout (in mm) Observations

NumberDay 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

and/or

Problems

1

2

3

4

5

6

7

8

9

10


Hints and CommentsOverviewStudents consider reasons why the U.S. CensusBureau gathers certain data.


A census originally is a count of the whole populationof a country. Data is collected on every member of thepopulation. In most data collection methods, asample of the population is taken, and the results ofthe sample reflect those of the whole population,given that the sample was randomly taken andrepresentative.

Planning

You may have students discuss the solutions toproblem 2 in class.

Did You Know?

The U.S. census bureau regularly collects data of thepopulation. Every ten years they take a census of thewhole population, and every year they take over100 surveys. A survey is a method of collecting andanalyzing social, economic, and geographic data. Itprovides information about the conditions of theUnited States, states, and counties. The CensusBureau has its own website where a lot of informationand data can be found.


2. Answers will vary. Sample responses:

A change in the number of people who live incities will have an effect on the need for variouscity services.

New schools may have to be built in some areas.

There may be a greater demand for publictransportation from rural areas to cities.


Notes

A warm-up activitymight be helpful beforeanalyzing the scatter plot.Each student couldanonymously record onpaper his or her typicalweekly income (allowanceand/or money earned fromworking). Then calculatethe mean income for astudent. To review percent,they could calculate thepercent of boys and thepercent of girls in the class.

A list of all fifty states andtheir abbreviations ishelpful. They should keep itfor the entire section.

Patterns inData

A

Vocabulary Building

Have students add scatter plot and mean to the vocabulary section of theirnotebooks. Use class discussion to help students who have not hadexperience with scatter plots. Some students may find it helpful to use aruler to read the information that goes with each dot. Reinforce that eachdot represents two pieces of data.

Extension

Students could research what the Census Bureau uses for its definition ofurban.


A Patterns in Data

After the information is collected, the Census Bureau reports thepercentage of urban population by state and the per capita incomefor each state.

3. Explain the meaning of “per capita income for each state.”

The scatter plot here and on Student Activity Sheet 2 shows theinformation collected by the United States Census Bureau aboutper capita income in each state and the District of Columbia and thepercentage of people who live in urban areas in that state. The meannationwide per capita income per state was about $30,000. Each stateis identified as a data point in the plot labeled by its postal code. Forexample, MS represents Mississippi, and DC represents the Districtof Columbia.

40 45 50 55 60 65 70 75 80 85 90 95 100

22,000

Percent Urban

Urban Population and Per Capita Income by State in 2000

24,000

26,000

28,000

30,000

32,000

34,000

36,000

38,000

40,000

42,000

44,000

Pe

r C

ap

ita

In

co

me

(in

do

lla

rs)

MS

ME

MT

MO

MN

MD

MA

NJ

KY

IAKS

MI

IL

GAIN

LASC

TX

RIPA

VA

UT

TNSD ND

NH

DC

CT

NY

NE

NC

OK

NM

AZ

COCA

AK

OR

OH

ID

DE

FL

HI

NV

WV

VT

WY

WA

WI

AR

AL

Source: Per capita income in 2000: Statistical Abstract of the United States,2003, Table 671. Percent urban: US Census Bureau

A


Hints and CommentsMaterialsStudent Activity Sheet 2 (one per student);transparency of Student Activity Sheet 2, optional(one per class);maps of the United States, optional (one per student)

Overview

Students discover what per capita income is and howit is determined. Students explore the scatter plot.Problems about the scatter plot are on the pages thatfollow. If students are not familiar with scatter plots,you may want to spend some time talking about thistype of graph and ask some questions. (See Extensionbelow.)


In explaining the meaning of per capita income,students apply the concept of mean in a new context.The mean was introduced in the units PicturingNumbers and Dealing with Data. Scatter plots are away to graph data in order to investigate whetherrelationships between the two variables exist. Scatterplots are introduced in the unit Dealing with Data.

Planning

The scatter plot can be presented to the class on atransparency and can be discussed in a whole classactivity before students start working on the problemson the next pages. Note: The District of Columbia isincluded in the data. There are 51 data points in thescatter plot.

Comments About the Solutions

3. The per capita income per state allows for acomparison between states. For example, $30,000per capita income in Wisconsin compared to$35,000 per capita income in Minnesota meansthat the average person living in Wisconsinearned an income of $5,000 less per year than aperson living in Minnesota.

You might want to make sure students know whatthe term urban population means.

Extension

To help students gain experience reading scatterplots, you might ask, Which place has the highestpercent of urban population? (District of Columbia)Which state has the lowest percent of urbanpopulation? (Vermont) Which state has the lowest percapita income? (Mississippi) Which place has thehighest per capita income? (Connecticut)


3. Answers will vary. Sample response:

Per capita income for a state is the averageincome per person in that state. Per capitaincome is determined by dividing the totalamount of income earned by all persons living ina certain area by the number of persons who livein the area.


Notes4 Explain that scatter plotsare a way to graph data tosee if relationships existbetween the two dataitems. Ask, Are theretrends you see in the dots?If so, what does thatindicate about the data?

7 Rephrasing the questionmay be helpful. As thepercentage of people livingin cities (urban areas)increases, what happensto the per capita income?

8 Explain that $30,000 isthe mean nationwide percapita income—all theincomes of the U.S. wereadded and divided by thetotal population. It is notfound by adding the percapita incomes of thestates and dividing by 51.So populous states willhave more influence onthe mean nationwide percapita income.

Patterns inData

A

Assessment Pyramid

5ab, 6

Analyze and interpretrepresentations of data.

Intervention

To help students understand the difference between the mean of the meansand the mean nationwide per capita income, it might be helpful to discussthe difference between finding the mean score for each class in eighthgrade on the last test and then finding the mean of the means for all theclasses in eighth grade. Then point out that is different from collecting thetest scores for each eighth grader and then dividing by the total number ofstudents in eighth grade.


APatterns in Data

Use Student Activity Sheet 2 to answer problems 4–15.

4. What does this graph tell you? Write two general statementsbased on the graph.

5. Look at the data point for Utah (UT).

a. What percentage of the people in Utah lived in urban areasin 2000?

b. What other information is shown by this data point?

6. Find the data point for California (CA). Explain what that pointrepresents.

Scott studied the scatter plot of the Census Bureau data on per capitaincome and percentage living in urban areas and said,“The higherthe percentage of people who live in cities, the higher the per capitaincome is for the state.”

7. In general, do you agree with Scott’s statement?

Margaret studied the same scatter plot and made the followingcomment. “I don’t believe that $30,000 is the mean. There areonly 20 states above the mean.”

8. Explain why so few states are above the mean. (Note: $30,000 isthe correct mean.)

Dan, Eliza, and Yolanda also studied the scatter plot of the CensusBureau data. Dan noticed,“Minnesota (MN) has a higher per capitaincome than Georgia (GA).”

9. a. Consider Dan’s statement. What else can you tell aboutMinnesota as compared to Georgia?

A


Hints and CommentsMaterialsStudent Activity Sheet 2 (one per student)

Overview

Students investigate the scatter plot in detail byexplaining what some data points mean and bycomparing data points for states and more in generalby commenting on interpretations of the scatter plot.In doing so, students reflect on the meaning of themean.


The scatter plot can be used to record and findinformation of both variables of each individual datapoint but is more often used to study patterns in dataand especially to investigate whether relationshipsbetween the two variables exist.

Numerical measures to describe data—the mean,the mode, and the median—are introduced in theunit Dealing with Data.

Planning

Students may work on problems 4 through 9aindividually or in small groups. You may want todiscuss problem 8 in class. Note that problem 9 iscontinued on page 6.


4. Telling the story of the graph focuses students’attention on the scatter plot as a whole instead ofon the individual data points.

5. Encourage students to use a map of the UnitedStates to help them analyze the data in the scatterplot. They might consider, for example:

• the size of the cities in each state,

• the geographical features of each state, and

• the regions of the country.

7. In general, this statement seems to fit the data inthe graph, although the relationship is very loose.

8. Populous states will have more influence on themean per capita income for the whole UnitedStates than other states. For instance, the $33,000of California has more influence than the samenumber from Virginia, since the population ofCalifornia is about 5 times that of Virginia.


4. Answers will vary. Sample statements about thegraph:

• Most data points are in the center of the graph.(This is a purely visual statement, nointerpretation.)

• The higher the percentage of urban population,the higher the per capita income. (This is ageneral statement about the context,interpretation.)

• Most states have a per capita income below$32,000. (This is a statement in some detail,interpretation.)

5. a. In Utah, 88% of the people lived in urban areasin 2000.

b. The mean per capita income for Utah wasabout $24,300 in 2000. Since $24,300 cannot beread exactly from the graph, any answerbetween $24,100 and $24,500 may beconsidered correct.

6. The data point for California indicates that about94% of the people lived in urban areas, and theper capita income of the people in the state wasabout $33,000.

7. In general, this statement seems to fit the data inthe graph.

8. More people may live in the 19 states and DC thatare above the mean than in the 32 states that haveper capita income below the mean.

There may be some states that are outliers. Thesestates have an unusually high per capita income,which is throwing off the mean.

9. a. Minnesota has a higher per capita income thanGeorgia, but they have about the same percentof urban population. This is around 72%.


Notes

10 Explain that relativeposition means its positionas compared to the otherstates.

12a It may be helpful forstudents to highlight thevertical line for 85% urbanand 90% urban to isolatethe dots they are studying.

Patterns inData

A

Assessment Pyramid

10

Describe the relationshipbetween two variables.

Vocabulary Building

Range is the difference between the highest and the lowest points. Thisterm was also introduced in the unit Dealing with Data.

Extension

Students might look for other states that have the same per capita incomeor the same percent of urban population as the state in which they live. Forexample, students who live in Utah might compare Utah with New Mexicoin terms of per capita income and compare Utah with Arizona in terms ofpercent of urban population. Make sure students give clear explanations oftheir statements.


Patterns in DataA

Eliza: “Hawaii (HA) and Nevada (NV) must be the same kind ofstates.”

b. Comment on Eliza’s statement. Do you agree?

Yolanda: “Alaska (AK) and Oklahoma (OK) are very different.”

c. Do you agree with Yolanda? Explain your answer.

10. Locate your home state on the scatter plot. Write a statementabout the per capita income in your state and its relativeposition on the graph.

Draw a horizontal line through a per capita income of $35,000.A group, or cluster, of states is above this line.

11. a. What states are in this cluster?

b. Write two sentences describing these states in terms of theirper capita income and the percentage of people in the stateliving in urban areas.

c. What can you say about the states below the line?

Sometimes you can find more information and make new statementsby looking more closely at a graph.

States in which fewer than half of the people live in urban areas havea per capita income that ranges from a little over $22,000 to about$29,500.

12. a. What is the range of the per capita income for states in which85% to 90% of the population live in urban areas?

b. What is the range of the per capita income for states in whichover 90% of the population live in urban areas?



Overview

Students continue to study the U.S. Census Bureaudata in the scatter plot: they investigate statementsthat compare states; they look for patterns in the datalike groups (or clusters) of data points; and they findways to describe the relationship between urbanpopulations and per capita income.


In the statements on pages 5 and 6, states arecompared to each other. There are two variables thatdescribe a state: mean per capita income and urbanpopulation. When comparing items (or states), thereasoning behind a conclusion is more importantthan the conclusion itself. It is important for studentsto learn to justify and explain a statement orconclusion.

When looking for patterns in scatter plots, one canconsider the following:

• If there are clusters (or groups) of data points,the population can be split into subgroups.

• If the points are all scattered, there is norelationship among the data points.

• If the points form a pattern such as a straight ora curved line, there is, to some degree, arelationship among variables. Investigationsabout the strength of correlations that appear toexist in the data are discussed later in this unit.

Planning

Students may continue to work on problems 9b –12individually or in small groups.


11. The states above the line can be seen as forming acluster of states in the upper right part of thescatter plot. This means they are high on bothvariables: per capita income and percent urban.


9. b. Answers may vary, sample answers:

Yes, as far as the per capita income and thepercent urban population are concerned thestates are similar.No. Hawaii and Nevada have almost the sameper capita income and percent of urbanpopulation. However, they are different types ofstates. For example, Hawaii is a collection oftropical islands in the middle of the PacificOcean. Nevada is a Western state with a desertclimate.

c. Yes. Alaska and Oklahoma have the samepercent of urban population, but Alaska has amuch higher per capita income.

10. Answers will vary, depending on the state inwhich the student lives. Sample response:

About 77% of the population in Ohio lives inurban areas and the per capita income is typicalof states in the U.S., at about $29,400.

11. a. Above the line are Maryland (MD), New York(NY), Massachusetts (MA), New Jersey (NJ),Connecticut (CT), and Washington D.C. (D.C.).

b. Sample responses:

The per capita income in the states above theline is over $35,000.

In all these states, the percent living in urbanareas is 85% or more.

c. Sample response:

The states below the line have a per capitaincome lower than $35,000; there is no patternin the percent of the population living in urbanareas for these states. It ranges from about 40%to about 95%.

12. a. The per capita income for these states rangesfrom $24,000 to $43,000.

b. The per capita income for these states rangesfrom $30,000 to $42,000.

7 Insight Into Data

Notes

13 Discuss with studentsthe differences they seebetween the group ofMidwestern states and thegroup of Southern stateswith respect to theirpositions in the scatterplot. (See About theMathematics.)

14 Discuss factors thataffect the high per capitaincome for Washington,D.C. and Maryland andwhy the percent of urbanpopulation is so high.

15 You could also havestudents find the onlymidwestern state that hasa higher per capita incomethan Illinois—Minnesota. Inthe 1993 census, its percapita income was lessthan Illinois.

Patterns inData

A

Assessment Pyramid

15

Analyze representations ofdata and draw conclusions.

Writing Opportunity

You may have students write about the following: Review the scatter plotagain. Write a paragraph about the relationship between per capita incomeand percent of population living in urban areas for the U.S. Be sure toinclude several statements that are more detailed than the statement ofScott (on page 5): “the higher the percent of people who live in cities, thehigher the per capita income is for the state.”

Extension

Students could examine the cost of living in large cities as opposed to ruralareas.


WEST

MIDWEST

SOUTH

NORTHEAST

WA

OR

MT

ID

WY

NV

CA

AZNM

UTCO

KS

NE

SD

NDMN

IA

WI

IL

MO

OK

TX

AR

LAMS

ALGA

TN

KY

SC

NC

VA

FL

INOH

WV

PADC

MI

NYVT

NH

ME

MD

DE

NJ

CT

RI

MA

AK

H I

APatterns in Data

The United States Census Bureau categorizesstates according to their geographical location.The map on this page shows these categories.

A

13. On the scatter plot on Student Activity Sheet 2, circle the dot foreach state in the Midwest in blue and circle the dot for each statein the South in red.

14. a. Washington, D.C. (DC) and Maryland (MD) might be calledoutliers in comparison to the other southern states. Explainwhat it means to be an outlier.

b. What other state might also be an outlier?

15. Explain the position of Illinois (IL) on the scatter plot. Compare itsposition to those of other states in the Midwest.


Hints and CommentsMaterialsStudent Activity Sheet 2 (one per student);blue and red markers (one of each per student)

Overview

Students continue the exploration of the scatter plotand look for clusters of data points and outliers.Students indicate states in the South and in theMidwest on the scatter plot and explain the positionsof representative states from each area. They look forpatterns in the clusters. It is important that studentslook closely at the graph to find more complexrelationships. Critical review of data representations isrequired throughout this unit.


An outlier is a data point that is far from most of thevalues in the sample. The states are now groupedaccording to geographic region. It appears that thedata points for the southern states are located moreor less in a tight cloud along a straight line, describinga relationship. How to describe this linear relationshipis examined in Section E.

Planning

You may have students work in small groups onproblems 13–15.


13. See the scatter plot below.

14. a. Explanations may vary. Sample response:

Maryland and the District of Columbia are wellremoved from other southern states on thescatter plot.

There are no rural areas in the District ofColumbia. One hundred percent of thepopulation lives in the city. Also, the per capitaincome is higher than in any other state.Because it is the nation’s capital city, manyprofessional people earn their living there,which drives up the area’s per capita income.

Maryland is tiny, but it has a large city,Baltimore, and because it is so near the nation’scapital, many people work in government andearn a good living.

b. Answers will vary. Sample answer: Other thanMaryland and DC, there are clear outliers forthe southern states, although Florida (FL) andTexas (TX) seem to have a lower per capitaincome than expected.

15. Explanations may vary. Sample response:

Illinois is well removed from other midwesternstates on the scatter plot because its per capitaincome places it well above neighboring states.

No doubt, this is due to Chicago, the third-largestcity in the country.

35 40 45 50 55 60 65 70 75 80 85 90 95 100

22,000

Percent Urban


24,000

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Per

Cap

ita In

co

me (

in d

ollars

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MS

ME

MT

MO

MN

MD

MA

NJ

KY

IAKS

MI

IL

GAIN

LASC

TX

RIPA

VA

UT

TNSD ND

NH

DC

CT

NY

NE

NC

OK

NM

AZ

COCA

AK

OR

OH

ID

DE

FL

HI

NV

WV

VT

WY

WA

WI

AR

AL


Notes

To help students processthe Summary, you coulddiscuss essentialcomponents of a scatterplot (title, constant intervalon axes, labels on axes, andso on) or have groups ofstudents create a rubric forassessing a scatter plot. Youmight also ask, What kindof data is necessary forcreating a scatter plot?

1 You might specify thatthe question asks forfeatures that would enablea person to drawconclusions or interpret ascatter plot, notcomponents such as a title.

Patterns inData

A

Assessment Pyramid

1

Assesses Section A Goals

Vocabulary Building

Add clusters of data and outlier to the vocabulary section of the students’notebooks. Students could also act out what a cluster of students andoutlier might look like.


Patterns in DataA

Data can be represented in a graph, such as a scatter plot like theone shown here. The data point A represents a car that weighs1,975 pounds and gets 51 miles per gallon.

Some conclusions you draw from a graph may be very obvious.For example, a scatter plot can show if there are clusters of dataor outliers.

Other conclusions may require more complex explanations, such asa description of a typical data point.

Often, careful examination of a graph can raise new questions. Moredata gathering and research may be necessary to answer these newquestions.

1. Describe some features you might look for in a scatter plot.Why might these be important?

01900 2000 2100 2200 2300

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(in

mil

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Vehicle Fuel Economy

A


Hints and CommentsOverviewStudents read the Summary and complete the firstCheck Your Work problem.


A scatter plot (as introduced in Dealing with Data) isa helpful visual tool used to describe data that comein pairs (two variables). Patterns in the scatter plotmay suggest relationships. Sometimes, however, athird (or fourth) variable or factor that is notrepresented in the scatter plot is involved, and thisextra variable may describe the pattern much better.

Students should develop a critical attitude towardrepresenting data and interpreting represented data.They should always ask themselves questions aboutother possible factors or variables that may beinvolved, and they should ask themselves questionsabout how the data were collected. This issue isaddressed in more detail in the next section.


1. You may want to look for clusters (groups) in thedata. There may be something special about thedata points in the clusters. For example, theremay be a common feature for these data, such asa cluster of states that are in the same geographicregion. You may want to look for patterns. Doesthe scatter plot show a trend of some kind? Isthere more than one pattern? If the data areclustered, does each cluster have its own pattern?You can also look for outliers. What characteristicsmake a point an outlier?


Notes

2a Remind students theyneed general statementsso they should look for atrend in the data.

2b You may need to pointout that fuel consumptionmeans miles per gallon offuel.

Patterns inData

A

Assessment Pyramid

2a

2bc


Accommodation

Some students may need a review of constant interval and to be advised toexamine the data to find the minimum and maximum points needed foreach axis before they start the scatter plot for 3a.


2. a. Study the scatter plot for Vehicle Fuel Economy shown in theSummary. What does this graph tell you? Write two generalstatements.

b. Vehicle B weighs 2,100 pounds. Locate the data point for B inthe scatter plot. What can you tell about the fuel consumptionof car B?

c. Is there an outlier in the scatter plot? Explain your answer.

The table below shows the percentage of eighth-grade studentswho scored at or above the basic level in math and science on the2005 National Assessment of Educational Progress in the southernstates of the United States.

Percentage at or above Percentage at or above

State Basic Level in Mathematics Basic Level in Science

Alabama 66 48

Arkansas 64 56

Delaware 72 63

Florida 65 51

Georgia 62 53

Kentucky 64 63

Louisiana 59 47

Maryland 66 54

Mississippi 52 40

North Carolina 72 53

Oklahoma 63 57

South Carolina 71 54

Tennessee 61 55

Texas 72 53

Virginia 75 66

West Virginia 60 57

Source: http://nces.ed.gov/nationsreportcard/states


Hints and CommentsOverviewStudents complete the Check Your Work problems.

Planning

After students complete Section A, you may assign ashomework appropriate activities from the AdditionalPractice for Section A, located on Student Book pages64 and 65.


2. a. You can write different general statements.Three examples:

• The heavier the vehicle is, the fewer milesper gallon it can drive.

• The change in the number of miles pergallon is not constant for a given change inweight.

• If the weights is between 2,000 and 2,100 lb,the fuel economy is about the same, around30 miles per gallon.

b. The fuel consumption of vehicle B is almost30 miles per gallon, which is in the middle.

c. You can write different answers. There seem tobe no apparent outliers if you see all the pointsas lying on a curve. You can also argue that thetwo points on each of the “ends” of the curveare outliers, because they are a little out of thegeneral pattern.

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A

B


Notes3a Students need the listof states and their abbre-viations for labeling thedata points.

3c Drawing the linethrough the points wherethe scores are the same(50, 50) and (60, 60) mayhelp to examine the dataand see whether generallythe states did better inmath or science.

3d Specify that bothof the scores need to beabove 57%. To see howthese states differ from theothers, have them lookback in the section to themap on page 7 or thescatter plot on page 4.

For Further ReflectionStudents might do thiswith a partner and thenreport to the class.

Patterns inData

A

Assessment Pyramid

FFR

3be

3acd


English Language Learners

Understanding what is meant by relationship can be difficult, so it mighthelp to ask, As the percentage of students at or above the basic level in mathincreases, what happens to the percentage in science—does it go up or down?


Patterns in DataA

3. a. Use Student Activity Sheet 3 to make a scatter plot of thepercentage of students at or above the basic level in mathand science. Identify each data point by labeling it with thestate it represents. Write a general statement about thepattern(s) in the data you can observe from the graph.

b. Which state(s) do not seem to fit the pattern?

c. How can you tell from the graph whether, overall, the statesseemed to do better in math or in science? Explain yourreasoning.

d. Circle the group of states whose percentage of studentsscoring at or above the basic level in both math and sciencewas more than 60%. Identify these states. How might thesestates differ from the others?

e. Which state(s) had the most students scoring at or above thebasic level in both math and science? Justify your answer.

Using the scatter plot of Urban Population and Per Capita Income byState in 2000, select two other states that are in the same region asyour state. Write two or more statements that compare your state’sdata point to that of your neighbors. If they are different, tell why.


Hints and CommentsMaterialsStudent Activity Sheet 3

Overview

Students complete the Check Your Work problemsand the For Further Reflection problem.

Planning

After students complete Section A, you may assign forhomework appropriate activities from the AdditionalPractice for Section A, located on pages 64 and 65 ofthe Insights into Data Student Book.


3. The math and science data come fromhttp://nces.ed.gov/nationsreportcard/states


3. a. See sample scatter plot below.

You might suggest that as the percentage at orabove the basic level in math increases, so doesthe percentage at or above the basic level inscience.

b. Texas, North Carolina, and South Carolina havehigh percentages of students at or above thebasic level in math but are in the middle of thestates with respect to science.

c. All states performed better in math than inscience. One way to tell this from the graph isby thinking about the points that would havethe same percentage for both math andscience. The line that would go through point(50, 50) and (60, 60) is the line M � S. Since allstates fall below this line, they all have a higherpercentage of students at or above the basiclevel in math than they do in science.

d. The states with more than 60% of the studentsat or above the basic level in both math andscience are Kentucky, Delaware, and Virginia.These states might receive more funding forboth math and science programs.

e. You might say that Virginia did the best in bothmath and science because they had the highestpercentage of students (75%) who were at orabove the basic level in math and were thehighest (at 66%) in science. Delaware was thenext best state, with both math (72%) andscience (63%) at or above the basic level.

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Math

Scie

nce

Percentage of Students

OK

ARTN

MD

NCTX

VA

SC

DE

GA

AL

FL

LA

WV

KY

MS

For Further Reflection

Answers will vary, depending on what state studentslive in and what states they select. The statementsstudents write will in some way compare per capitaincomes for the states and percent urban population.Note that students may give reasons for differencesbetween states that are based on students’ knowledgeof factors other than income and urban population.You may want to have students do some extraresearch using the Internet. Or you may consult thegeography teacher at your school on this topic.


Teachers MatterB

Section FocusIn this section, students investigate the concepts of sample and simulation. They discussthe process of sampling and how to collect “fair” data. They learn how to select a randomsample of the whole population. They use a simulation and graph the results to investigatehow likely certain outcomes in a random sample are. In doing so, students investigate thevariation in samples. Students investigate possible bias in survey results.

Pacing and Planning

Day 4: Collecting “Fair” Data Student pages 11–15

INTRODUCTION Problems 1–3 Investigate a context involving 20 Texasband students and explore whether thissample of students is representative ofthe state’s urban versus real population.

CLASSWORK Problems 4–6 Use random numbers to simulate thenumber of students who live in cities andcompile individual results to create aclass tally chart.

HOMEWORK Problems 7 and 8 Represent simulation data using ahistogram and compare the results tothe original sample to determine if it isrepresentative.

Day 5: Collecting “Fair” Data (Continued) Student pages 16, 17, 66

INTRODUCTION Problem 9 Identify possible bias in situationsinvolving polling samples.

CLASSWORK Problems 10 and 11 Discuss the band camp sample and otherpolling situations to identify possible bias.

HOMEWORK Additional Practice, Analyze the results from experiments forSection B, page 66 bias.

3

Day 6: Random Numbers Student pages 18–21

INTRODUCTION Problems 12 and 13 Explore features of a set of randomlygenerated numbers.

CLASSWORK Problems 14–16 Investigate random sampling techniques.

HOMEWORK Check Your Work and Student self-assessment: DemonstrateFor Further Reflection understanding of bias and sampling

techniques.

Teachers Matter Section B: Selecting Samples 11B

Teachers Matter B

Materials

Student Resources


• Student Activity Sheets 4 and 5


No resources required

Student Materials

Quantities listed are per pair of students.


Learning Lines

The focus of this section is on how to get goodsamples from a population. The terms populationand random are defined in this section. (Studentshave been informally introduced to these terms inthe unit Dealing with Data.) This section introducesstudents to the use of simulations to gather data toinvestigate how likely it is that certain outcomesoccur in a random sample or in other words: howrepresentative of a population random samplescan be.

Students carry out a simulation by actually drawingrandom samples from a population of “notes” in abox. The notes in the box represent the distributionof the population of Delaware: 80% living in cities(8 notes in the box) and 20% living in rural areas(2 notes in the box). Later students repeat thissimulation using random numbers.

Variation in Samples

Students take a number of random samples andrecord the class results both in a table and ahistogram. By taking a number of samples,students create a sampling distribution. The tableand histogram show the variation in samples.

Students learn that the more samples they add tothe study, the more they begin to approach theexpected 80% figure for urban dwellers. As amathematical rule, the results of the simulationwill vary, but by adding additional trials, studentsfind a cluster of samples that center around16 out of 20 students in the sample (80%) comingfrom cities. The distribution in the collection ofsamples, if larger samples are taken, will approachthe normal distribution or bell curve.

Conclusions from Samples and Bias

Inferences about the population are based on theanalysis of the data from the sample. In a randomsample, each person has the same chance ofbeing selected for the survey. It is important forstudents to recognize that any outcome ispossible but that some outcomes are more likelythan others. There is a range, or cluster, of likelyvalues. In the example in this section, the clusterof likely outcomes centers around 16 studentsfrom cities, which is 80% of the sample. Bias canresult if the selection procedure for determiningthe sample was not objective and impartial. Whenthe selection procedure is biased, taking a largersample does not help. Many sources of bias areinvestigated in this section.


Students recognize possible causes for bias insample surveys. They understand how to set upand carry out a simulation to collect data. Theywill be able to use random samples in gatheringdata and can represent collected data in tablesand graphs. Students will be able to recognize thatin random samples variation in outcomes isnormal. They understand that some outcomes aremore likely then others.

Day 7: Summary

INTRODUCTION Review homework. Review homework from Day 6.Read Summary.

ASSESSMENT Quiz 1 Quiz addressing Section A and B Goals

Additional Resources: Additional Practice, Section B, Student Book page 66


Notes

A discussion on collectingdata is a good way to startthis section. The questionslisted can be used to startthe discussion.

SelectingSamples

B

Vocabulary Building

Make sure students understand the vocabulary in the questions. Discussexactly what is meant by reliable data and list all the ways the class knowsto visually represent data.

Extension

You may ask students to look carefully at media such as TV, newspapers,and magazines for ways in which data are collected and represented.Students’ observations should include whether or not the data collectionresulted in good data, how the data were represented, and whatconclusions could be drawn from the data.


Data can be obtained from organizations such as the United StatesCensus Bureau, and the results can then be graphed. However, it isnot always easy to get accurate data, as you may have seen in theunit Dealing with Data.

Questions such as the following are important in statistics:

• How do you get reliable data?

• What is the best way to visually present the data?

• How do you draw accurate conclusions based on the data?

Looking carefully at graphical representations of data is important.Even graphs based on complete data, such as the scatter plot onpage 4, must be studied carefully before reliable conclusions canbe made.

BSelecting Samples

Collecting “Fair” Data

A summer band camp has middle school students from all 50 statesand Washington, D.C. Twenty students from Delaware are at the camp.

Section B: Selecting Samples 11T

Hints and CommentsOverviewStudents read about data collection, data represen-tation, and drawing conclusions. They are introducedto the context of this section involving a group of20 Delaware band students, for which students explorehow representative this group is for the whole state.

There are no problems on this page for students tosolve.


As described in the three questions about statistics onthis page, a statistical investigation can be split intothree parts: data collection, data representation, anddrawing conclusions. In this section, the focus is onhow to get “good” samples. Determining what is gooddepends on the question to be answered. Often, a“good” sample is a representative sample of a largergroup (the population), this is often a randomlyselected sample. A good (or representative) samplehas the same characteristics as the population. Thismeans you can generalize for the whole populationbased on data collected for the sample.

Planning

Note that on page 12 the information about thecontext is continued.


Notes1 A good strategy forexplaining is to use a ratiotable to show how 80% is16 out of 20.

2 Students should reasonthat one or two or eveneight students are alsounlikely results. If the 20students are typical of therest of the students inDelaware, students mightthink that 12 or 13 is amore reasonable answer.

3 It may help to firstdiscuss ways to choosea random sample ofstudents from a smallergroup, such as a class orschool.

SelectingSamples

B

Vocabulary Building

Random, random sample, bias, and simulation should be entered in yourstudents’ notebooks. To help understand the difference between randomand biased, you could list examples of samples—random and biased—andhave students determine which they think are biased and which arerandom.


The Census Bureau data indicate thatabout 80% of the population in Delawarelives in urban areas, as shown in thescatter plot on page 4.

1. Do you think it is likely that of the20 middle school students fromDelaware at the band camp, 16 livein urban areas? Explain your thinking.

Sue states, “Only eight out of the 20Delaware students in the band camp livein urban areas.”

2. Does this number surprise you? Whatare some possible reasons for therather low number?

Selecting SamplesB

The question to investigate is, “How likely is it that in a randomlyselected sample of 20 middle school students from Delaware, onlyeight of them live in urban areas?” Choosing a random sample isimportant because it helps reduce bias in the sampling process. Asample is biased when it favors certain outcomes or some parts ofthe population over others. Care must be taken so that any memberof the population has an equally likely chance of being chosen in thesample. In statistics, random means that each element of a set has anequal probability of occurring.

3. a. Reflect What is meant by a “randomly selected sample” ofstudents from Delaware?

b. How could someone randomly select 20 middle schoolstudents from Delaware?

Suppose you had a random sample of students from Delaware. Howmany of them do you think would be likely to come from urban areas?To investigate this question, you can create a model, or a simulation,of the situation in Delaware.


Hints and CommentsOverviewStudents continue to investigate the context introducedon page 11 involving a group of 20 Delaware bandstudents. Students explore how representative thisgroup is for the whole state.


A critical problem in statistics is finding a way to specifya range of likely values for some event. In a situationwhere 80% of the population has some characteristic,one may expect that in a sample of 20, there would beabout 16 persons having this characteristic. “About 16”should be interpreted as a number between 14 and18; other numbers are possible but less likely. Theexpectation of 16 assumes that the sample of 20 israndomly selected and representative of the greaterpopulation.

However, this assumption needs to be checkedcarefully before accurate statements can be madeabout expectations. In the unit Dealing with Data,the concepts of sample, population, and randomsampling are informally introduced. Students areexpected to be somewhat familiar with these terms.In the unit Great Predictions, students will moreformally learn about expectations.

Planning

You may have students work on problems 1 and 2in small groups. Address problem 3 and the textsurrounding it in a class discussion.


1. Students may recognize that to have a represen-tative sample that is exactly 80% of 20, or 16 people,is not likely, especially when they are dealing withsuch a small sample.

3. You may want to refer to the unit Dealing with Datawhere the notion of a “good” sample is introducedin an informal way. The sample must be taken in aproper (non-biased, non-selective) way.



Yes. In Delaware about 80% of the populationlives in cities. So one would expect that 80% of thestudents selected from Delaware live in cities.Eighty percent of 20 is 16.

On the other hand, you cannot be sure how thestudents from Delaware were “selected” for theband camp. They may be from one school.

2. Answers will vary, but some students may findthis somewhat surprising. Some possible reasonsmay include the following:

• The 20 students may not be typical of thestudents in the state. Many students may havecome from one school in a rural area.

• Twenty students is a small group to representthe population distribution of the state ofDelaware.

3. a. It means that every student in Delaware hadthe same chance of being selected in thesample. The sampling process is unbiased.(Students may use other wording in theiranswer.)

b. Answers will vary. Since 20 is a very smallnumber and the number of middle schoolstudents in Delaware is high, the samplingprocess can better consist of different roundsof sampling. Sample answer: One could firstmake a list of all counties, take a randomsample of 10, list all middle schools per county,and then randomly select two schools percounty and of each school randomly selectone student.


Notes

Simulation Activity

Ask, Why are 8 of therectangles marked Urbanand 2 marked Rural? Howdoes this represent 80%urban? Are there other waysthis could have been done?

The activity can be donewith partners so one drawsout the rectangles and theother records the results.They could repeat theexperiment, if necessary, toget enough data.

5 Prepare a table inadvance for results of thesurvey on a transparencyor on a poster. List thenames of all your studentsor pairs of students on thetable.

SelectingSamples

B

Assessment Pyramid

4

Design, conduct and analyzeways of gathering data fromsimulations.

Use random samples ingathering data.

Accommodation

Students who struggle with writing will benefit if you prepare the table fromproblem 5 in advance. Fill in student names before making copies.


Student Activity Sheet 4 is divided into ten rectangles.

Urban Urban

Urban Urban

Urban Urban

Urban Rural

Urban Rural

These rectangles represent the percentageof people from urban and rural areas inDelaware, where two out of every tenpeople are from a rural area.

• Cut out the rectangles. Fold them once and put them in a paperbag or box. Shake the container well.

• Take out a rectangle. Record in a table what is written on therectangle and put the rectangle back in the bag or box. Shake thecontainer to thoroughly mix the rectangles. Repeat this 20 times.

4. Explain how this activity has simulated taking a random sampleof 20 students from Delaware.

5. a. Make a table to tally the results for the entire class.

b. Reflect How do your results compare with your classmates’results? Explain any similarities or differences.

c. How many of your classmates have exactly eight studentsfrom Delaware who live in an urban area in their sample?Does this result surprise you? Why or why not?

d. How many of your classmates have exactly 16 students fromDelaware who live in an urban area in their sample? Does thisresult surprise you? Why or why not?

Urban

Rural

Name Number of “Urban”in Sample

Number of “Rural”in Sample


Hints and CommentsMaterialsStudent Activity Sheet 4 (one per student);paper bag or box (one per student)

Overview

Students use a simulation—pulling notes from abox—to simulate taking random samples of 20students from the Delaware population. They studythe samples for likely and less likely outcomes.


A simulation is designed to simulate taking randomsamples of 20 from a population with 80% urban. Thissimulation is a very simple one: ten pieces of paperare in a box: eight have “urban” written on them, 2have “rural.” Students take notes from the box (theyput them back each time!) and record the findings.Sample data will show variability. Data from the wholeclass are collected in problem 5. It will probablybecome clear that the more data collected, the morethe distribution will look like the expected distribution.This is investigated further on pages 14–16.

Planning

Students do the activity and answer problem 4individually. You may want to discuss problem 4 inclass. Problem 5a is a whole class activity. Problems5b–d can be done individually first and thendiscussed in class.


4. It is important that students realize that the notesin the box represent the Delaware populationwith the distribution 80% urban, 20% rural. Youmay want to discuss why they have to put thepaper back each time.

5a. and d.There will be variability in the sample results.Students may already note that the results in thesamples are more like the distribution in thepopulation when more samples are being collected.This is investigated further on the next pages.


4. The notes represent the Delaware population, ofwhich 8 out of 10 (80%) live in cities. Every timeyou take a note, you randomly select a studentliving in either an urban or a rural area. Eachstudent in your class has done the activity. If thereare 25 students in your class, you have taken 25samples. In other words, each one of you has“created” one random sample of 20 band studentsand counted how many of those live in cities.

5. a. and b. Results in the table will vary dependingon students’ data.

b. Sample explanations: Differences may becaused by chance; sample results will showvariability. Similarities may be explained by thefact that all samples are taken from the samepopulation.

c. This number will be very low. It is not unlikelythat no students at all will have this result. Thisis not surprising since you would expectaround 16 students from the sample to live incities, because the sample is random, and 80%of 20 equals 16.

d. This number will be higher but may still berather low. (There is about a 20% chance tohave exactly 16 out of 20 live in cities.) Moststudents will have numbers around 16 so 15 or17 are about equally likely. The rather lownumber of classmates who have exactly 16 maysurprise students.


Notes

Prepare a copy of this tablein advance on a poster,chart paper, or transparency.Number from 1 to 20 in thefirst column; then mark thetallies in the second columnduring class.

6b Each student needs acopy of the table with theresults for making thehistogram of the data onpage 15.

6c Advise students thatthey do not need a specificfraction here, but a relativeanswer such as, “verylikely” or “not very likely.”

SelectingSamples

B

Accommodation

Having a copy of the table prepared ahead of time enables many studentswho need extra time to make tables to keep up.

Extension

Students may want to calculate the experimental or empirical chance forseveral outcomes for the results in the table. This was introduced in theunit Second Chance.


A

Organizing the data may help you see any patterns in the class results.You can use a table like the one below to show the possible numberof rectangles in the sample of 20 that had “urban” on them and a tallyof the students who had each number in their sample.

6. a. What do the three tally marks in the table mean?

b. Use the data collected in problem 5 to make a table like theone above.

c. Based on the results in the table what is your answer to thequestion, “How likely is it that in a randomly selected sampleof 20 middle school students from Delaware, only eight ofthem live in an urban area?”

Selecting SamplesB

Number of “Urban” in Number of Students in Class

Sample of 20 Who Had This Number

….

…

7

8

9

10

11

12

13

14

15

16

17

. . . .

///


Hints and CommentsOverviewStudents rearrange the data collected from allstudents in the class and investigate probability ofcertain outcomes.


As students work on this page, the effect of largersamples will become apparent. In the outcomes,patterns may emerge: 15,16, or 17 “urban” slips inthe sample probably appear more frequently thanthe other numbers.

Students use the results in the table to make astatement about the (experimental) probability ofa certain outcome. The notion of experimentalprobability is introduced in the unit Second Chance.

Planning

Students may work on problem 6 individually.


6. If you have fewer than 25 students in your class,you might complete the experiment a secondtime to obtain a larger sample. Too few trials maynot give enough data for students to see thepatterns emerging.


6. a. The 3 tally marks indicate that 3 studentspicked a rectangle with “Urban” on it exactly 14times.

b. Student answers will vary, depending on theresults for problem 5.

Sample table with results of 24 students

c. Students answers will vary depending on theirresults for 6b. Sample response for the sampledata in 6b:

Based on the results, it is highly unlikely thatonly eight out of 20 students live in an urbanarea. This was not found as a possible result,so the empirical chance in this case is 0.

Number of “Urban” Number of Students in Class

in Sample of 20 Who Had This Number

…

…

7

8

9

10

11 //

12 /

13 ///

14 //

15 //// /

16 ///

17 //// //

….


Notes

7a You will want to make ahistogram of the class dataso you will know what toexpect from the students.Have students save theirhistogram to use foranswering problem 16.

7b If students needprompting, ask which barswere the tallest (frequency)and what did that indicate.

SelectingSamples

B

Assessment Pyramid

8c

7b, 8b

Draw conclusions based ona graphical representation.

Represent data numericallyand as a histogram.

Intervention

For students who might not be familiar with histograms, review theessential features: a bar graph on the horizontal axis, frequency (number ofoccurrences) on the vertical axis, labels on each axis, and a title.


Help students understand what is meant by how likely. One possibility is tomake a cube out of card stock and write the same number on four of thefaces and two different numbers on the other faces. Ask students how likelythey are to get the number that is repeated on four of the faces when theytoss the cube. Is it very likely or not very likely?


A

It is often easier to get a clear picture of the data if you have a graph.

7. a. On Student Activity Sheet 5, use the data from the table youmade in problem 6b to make a histogram.

b. Based on your histogram, write two sentences about thenumber of Delaware students in a sample of 20 students wholive in cities.

c. Based on your data, how likely do you think it is to have 14 to18 Delaware students in a random sample of 20 students wholive in urban areas?

d. What do the results of your simulation tell you about thenumber of Delaware students at the band camp who arelikely to live in urban areas?

8. a. Repeat the simulation you did in the activity on page 13.Collect the class data. Add the new results to the table youmade in problem 6b.

b. Make a histogram using the new table. How does thishistogram compare with the first one?

c. What kind of results do you think you would get if youcontinued to repeat the experiment?

BSelecting Samples

Avera

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Overview

Students graph the data from the table they made inproblem 6 on page 14. They draw conclusions basedon the graph. They repeat the simulation, the datacollection in the table, and the making of the graph toinvestigate the effect of a bigger sample size.

Planning

Note that problem 8a (doing the simulation andcollecting class data) must be done as a class activity.


7. a. Student data can also be recorded in a number-line plot or a line graph (as introduced in theunit Dealing with Data).

8. b. When the data are graphed for large numbersof students (when the experiment is donemany times), it will be clear that a distributionwith the top at 16 will develop.


7. a. Answers will vary, depending on student’sanswers for 6b. Sample histogram for thetable in 6b:

b. Answers will vary. Sample answer (based on thehistogram for 7a): It seems most likely thatthere are between 15 and 17 students fromDelaware in the band camp who live in urbanareas. It is very unlikely that less than 10students in the sample of 20 live in urban areas,that is if the sample is a random one.

c. Answers will vary. Obtaining 14 to 18 urban isvery likely, but the number from one trial toanother will vary.

d. Answers will vary. The results of the simulationmost likely will indicate that from 14 to 18students might live in urban areas.

8. a. Tables will vary depending on the dataaccumulated by the class.

b. Graphs will vary, depending on the dataaccumulated by the class. Sample histogram:

c. The more often the experiment is repeated, themore likely that the result is 16 students out of20 living in urban areas. See the histogram above.

70

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Number of Urban in Sample of 20

Number of Samples with

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Number of “Urban” Number of Students in Class

in Sample of 20 Who Had This Number

7

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18 2


Notes

Ask students to give theiropinion of how bias mayaffect the results of surveysand polls conducted bynewspapers and magazines.

SelectingSamples

B

Vocabulary Building

Discuss these first two paragraphs to make sure students understand thevocabulary and the results of the simulation. You might have them use eachof these words in a sentence to show that they understand them: simulation,random sample, population, and sampling distribution.

Extension

You could have students compare and contrast bias with prejudice orconsider how the results of polls might affect an election and how biasmight be involved.


On pages 13–15 you completed a simulation. By taking a numberof random samples of 20 from a population where 80% live in urbanareas, you created a sampling distribution of those that live in urbanareas. You probably found that having around 14 to 18 out of 20who live in urban areas was a likely result. Your simulation probablyindicated that 8 out of the 20 did not happen very often in any of thesamples and so was a very unlikely result.

If only eight out of the 20 students from Delaware at the band camplive in urban areas, you can conclude that this sample of 20 studentsdid not seem to be typical of the population of Delaware with respectto the living environment. Having only eight of the 20 students fromurban areas could have occurred by chance, but it does not seemvery likely.

Selecting SamplesB

Biased Samples

Many samples are biased because they favor certain outcomesor they favor some parts of the population over others. In suchinstances, there is a systematic error in the way the samplerepresents the population. Consider the following situation:

Some TV stations poll the public. Viewers are urged to callspecific numbers to voice their opinions. Dialing one numberregisters a “yes” vote; dialing another number registers a“no” vote.

9. Mention at least two problems with this type of sampling.


Hints and CommentsOverviewStudents read a text summarizing the simulationactivity and the investigation of the results. Next, biasand underlying factors in sampling techniques arediscussed.


Unlikely results in a sample can occur. These may bedue to chance, but it is also possible that the sampleis biased. At this point in their study of statistics,students should begin to appreciate the need to thinkmore deeply about problems in which statistics areused, developing a critical attitude toward the use ofstatistics.

Planning

You may want to use the text and problem 9 as thebasis for a discussion of bias in sampling.


9. The first problem is that only people wishing tovote will call. This is called a voluntary response.The second problem is that only people with atelephone can respond, and not everyone has aphone. Consider that in 2000 in Arkansas about6% of the households had no telephoneconnection and that many households are usingcell phones rather than the traditional phoneconnected by a phone line. Last, the same personcan call more than once.

17 Insight Into Data

Notes

Help students understandwhat is meant by bias insampling and causes for it,such as voluntary response,asking the question in anincorrect manner, selectinga poor sample, and so on.

11 These four situationsare good for group work.Be sure each group has agood reader to help thosewho may have difficultyunderstanding eachscenario. After discussingpossible reasons thatwould cause bias, haveeach group record theiropinions and share withthe class.

SelectingSamples

B

Assessment Pyramid

11abcd

Recognize possible bias insample surveys.

Extension

In the Reaching All Learners section for page 11, an activity was suggestedin which students investigate media resources for examples of datacollection. You may remind students of this activity and ask them toinvestigate examples of data collection in which bias is involved.


Bias can result when underlying factors about a situation are notconsidered during the selection of a sample.

10. Why might the information collected about students at a bandcamp be biased?

11. Read each of the following survey situations carefully. Explainhow each poll could involve bias.

Selecting Samples B

a. The chief of police in a major U.S. city wants todetermine how the public feels about the department.He prepares a questionnaire and sends police officersout to interview people in randomly selected sectionsof the city.

b. A magazine for health foods and organic healingwants to establish that large doses of vitaminsimprove health. The editor asks readers who haveregularly taken vitamins in large doses to write tothe magazine and describe their experiences. Of the2,754 readers who reply, 93% report some benefitfrom taking large doses of vitamins.

c. A researcher wants to find out how many Americansintend to vacation in the United States in one year.To avoid bias, she selects 27 travel agencies in largecities and interviews every seventh visitor. The resultsof her research are published and titled “RecordNumber of Americans to Foreign Destinations.”

d. Reflect In 1936, the largest poll about the presidentialelection between Franklin Roosevelt and Alf Landonwas taken by a magazine called Literary Digest.The publisher sent out 10 million questionnairesto people listed in telephone books. They alsoused other sources, such as car registrations andsubscriber lists. The magazine received 2.4 millionreplies. As a result of the poll, Literary Digestpredicted that Landon would win by a margin of57% to 43%. However, Roosevelt won the election.Another research group used a much smallersample of 50,000 people and predicted correctlythat Roosevelt would win the election. Give somereasons why the smaller sample gave a betterprediction.


Hints and CommentsOverviewStudents investigate several situations involving bias.


Bias in a sample is a common phenomenon instatistics. It is almost impossible to have a completelyunbiased sample, but one should consider verycarefully what factors might cause a relevant biasbefore collecting data. There are different kinds ofbias addressed in these problems:

• a low response rate

• surveys that are not conducted by an appropriateperson

• depending on voluntary response

• asking the wrong questions, suggesting onlypositive (or negative) responses

• selecting a poor sample (not representative for thewhole population)

Planning

Students may work in small groups on problem 11.


10. Students should explain why the informationabout the student band members could bebiased. You might want to collect students’answers and discuss which reasons cause morebias than others and which type of bias is moreimportant than others.

11. Students should recognize the need for furtherinvestigations in biased situations. Two thingsthey might consider are using a larger sampleand investigating how the sample is chosen.

Did You Know?

Sometimes a biased sample is chosen on purposeto manipulate the outcomes of a survey. In DarrellHuff’s How to Lie with Statistics (New York: Norton &Company, 1954), several examples of the abuse ofstatistics are described. Huff also provides examplesof poor sampling.



• The students may be from the same school.

• The sample of students may not be a randomsample.

• Students often go to camp with their friends, sothey would live in the same place.

• Twenty is a small sample size.


a. Since the police officers are interviewing thepublic about police officers, people may feeluncomfortable relating their true feelings andmay choose not to respond; this would be anexample of bias based on a high non-responserate. Bias may also occur if people answer inways they feel the officer wants, rather thansharing their true feelings.

b. This is an example of bias based on voluntaryresponse. The people who responded tookvitamins regularly, so it is likely that theyconsider their experiences with vitamins to bepositive. Bias may also occur when peopleinterpret words differently. For example, whatdoes “improvement in health” mean? What is “alarge dose” of vitamins? Also, the populationreading the magazine may not berepresentative of the whole population.

c. This is an example of bias resulting from usinga poor sample. Americans who go on vacationsin the United States may not use the services ofa travel agent, while those traveling overseasmight need such a service. Also, travel agenciesin large cities might receive more requests forforeign travel than agencies in small cities.

d. This is an example of bias resulting from asample that was selected by an inappropriatesampling technique. Literary Digest sampledpeople who had telephones or cars. In 1936,few Americans owned these items. LiteraryDigest also used its subscriber list to select pollparticipants. The subscribers may not berepresentative of the whole population. Theother research group probably used a morerepresentative sample of the population.


Notes

12 It might help to explainthat 50 random numbersis similar to putting thenumbers 0, 1, 2, 3, … 9in a hat, drawing one outat random, putting it back,drawing again, andrepeating it 50 times. Ask,What fraction or percentwould you expect to be a0 or 9? How many wouldthis be out of 50?

14 Students can choosetheir movements in thetable. A student mightthink, After I put mypencil on a digit, I willmove down a column toget my numbers. If mycolumn ends, I will go tothe left one column andthen go up that columnuntil I get 20 digits.

SelectingSamples

B

Advanced Learners

Students could use a graphing calculator to generate 20 random numbersand compare the results.

Act It Out

Using a random-number table may not be intuitive. Complete a fewexamples with the class. An effective method is to make a transparency ofthe random-number table and ask a student to come to the overheadprojector to complete the process of determining a set of random numbers.


You simulated taking a random sample of students from Delaware bypulling rectangular pieces of paper out of a box. Simulations are oftendone using a set of random numbers. The set you will use consists ofnumbers from 0 through 9 in random order. You can read randomnumbers from a table or generate them on a computer or calculator.

Suppose you looked at a set of 50 random numbers ranging from 0through 9.

12. How many numbers in the set would you expect to be a 0 or a 9?Why?

The following is a set of 50 random numbers.

13 a. How many numbers in this set are a 0 or a 9? Compare this toyour answer to problem 12.

b. Would you expect to get exactly this many numbers being a0 or a 9 every time you look at a set of 50 random numbers?

Selecting SamplesB

Random Numbers

1 2 6 7 2 4 0 1 7 0

2 7 9 3 7 9 0 4 7 2

1 4 6 2 2 5 6 1 6 4

0 5 7 6 4 6 4 7 3 5

2 7 9 0 4 1 2 0 2 7

Remember the band camp? You can also userandom numbers to simulate the chance thatyou would see only eight students fromDelaware who lived in an urban area.

14. Select a set of 20 random numbers from thetable by arbitrarily choosing one of the rowsor columns and counting out 20 numbers.How many of the 20 numbers that youselected are a 0 or a 9?


Hints and CommentsOverviewStudents are introduced to random numbers. Theyuse random numbers to simulate taking a randomsample of 20 students at the band camp.


On this page, the term random number is introduced.A random number is a number that is chosen with nopreference, thus with as much chance of being chosenas any other number. It is difficult to tell whether agiven set of numbers is random, especially when it isa small set of numbers. One way to see whether a setof numbers is random is to make a histogram andobserve whether the frequencies for each digit balanceout as the set of random numbers increases. Each digitwill occur approximately the same number of times.There is no rule for how large this set should be.

Planning

Students can work on problems 12 and 13 individually.You may want to discuss their answers in class.


13. b. It is normal to expect variability in the results.


12. Each one of these ten numbers would have a10% chance of being the last digit, so there is a20% chance that a 0 or 9 is the last digit. 20% of50 is 10.

13. a. There are six 0's and three 9's. So a total of9 out of 50, or 18%, are 0 or 9. This is lowerthan the expected 20% (problem 12).

b. No, the number is likely to vary.

14. Student answers may vary. Sample responses:

I selected rows 3 and 4 from the table andcounted one 0 and no 9.

I selected the third through sixth column andcounted three 9’s and one 0.


Notes

15 Make sure studentsunderstand the design ofthis simulation.

16 Comparing the newhistogram with thehistogram made from theresults of the activity onpage 13 will help answerthis question.If the results of the twosimulations are verydifferent, you may wantto discuss reasons forthis. Maybe thesimulation of pullingnotes from a box wasbiased because studentsdid not properly mix thenotes. Simulation resultsmay be influenced by thefact that 50 randomnumbers may not beenough to choose from.

SelectingSamples

B

Assessment Pyramid

16

Analyze and interpretrepresentations of data.

Accommodation

For problem 15b, if students need prompting for answering how likely theyare to have 14 to 18 urban students according to their histogram, ask themwhich bars are the tallest in their histogram?


The 20 numbers could represent the 20 students from Delaware at theband camp. The 0s and 9s represent those who lived in rural areas, andthe other eight numbers (1 to 8) represent those living in urban areas.

15. a. Using your set of 20 random numbers from problem 14, howmany students did you have who lived in urban areas? (Thatis, how many of the random numbers in your set were 1 to 8?)

b. Collect the class results for their sets of 20 random numbersand make a histogram of the number of Delaware studentsfrom all of the sets who were from urban areas. Using thehistogram, how likely do you think it would be to have 14 to18 Delaware students in a random sample live in cities?

16. How do your results from the simulation with random numberscompare to the simulation you did with the numbers in the box?

Selecting Samples B

Math HistoryThe United States Census

In the United States, the first prototype of a population pyramidwas published in the Statistical Atlas of the United States Basedon the Ninth Census (1870).

Statistical data about the population in the United States iscollected by the Census Bureau. Fact-finding is one of America’soldest activities. In the early 1600s, a census was taken inVirginia, and people were counted in nearly all of the Britishcolonies that later became the United States.

Following independence, there was an almost immediate needfor a census of the entire nation. The first census was takenin 1790, under the direction of Secretary of State ThomasJefferson. That census, taken by U.S. Marshals on horseback,counted 3.9 million inhabitants.

Nowadays, graphs have fewer mistakes because most of themare made using computer software; however, they all looksomewhat similar. Of course unique graphs still exist. You canfind them in newspapers!

000000

054062

033

070

031

013

004016

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044

189

479

001003

001000

Wyoming

Mississippi

002002

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012015

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088

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Vermont

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084081

106 102

107110

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016

Washington

136141

081088

065114

050145

024086

011038

005010

002003

002001

The total number of living inhabitants in each case, as reportedin the census, is reduced to thousandths, and the number ofthousandths of each sex in each decade of life represented bythe distance measured on the horizontal lines, severally, fromthe perpendicular base line.


Hints and CommentsMaterialscopy of table used for simulation results from page 14,optional (one per group);transparency or chart paper to record class results,optional (one per class)

Overview

Students collect and graph the results of the wholeclass of the random number simulation. Theycompare the results of it to the results of thesimulation with the notes in the box.


Random numbers can be found in random numbertables. Random number tables usually list 5 digitnumbers in rows and columns. Random numbersmay also be generated by a graphing calculator orcomputer software.

Planning

Problem 15b is a class activity.


15. a. Answers will vary depending on students setof random numbers for problem 14. Sampleanswers fitting the sample answers for 14:19 out of 20 students lived in urban areas.16 out of 20 students lived in urban areas.

b. The histogram depends on student results.Sample histogram:

Based on this histogram, it seems very likely thatbetween 14 and 18 students from the sample livein cities: 21 out of the 24 samples have this result.So the experimental chance is 87.5%.

16. Answers will vary, depending on students’ resultson previous problems.

Sample answer: The results differ in detail, butoverall both simulations show that it is likely thatthe number of students in the sample living incities will be around 16, and that 8 of 20 is a veryunlikely result.

120

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Notes

You may want to havestudents put the vocab-ulary words and definitionsfrom the Summary on indexcards so they can reviewthem.

Another possibility is to havegroups of 4 or 5 studentswrite review questions forthe Summary, at least onequestion for eachparagraph. Choose onestudent from each groupto rotate to a new groupand see if the students inthe new group can answerall the questions.

SelectingSamples

B

Assessment Pyramid

1a

1b

Assesses Section B Goals

Parent Involvement

Students could ask their parents if they have been involved in a poll or ifthey have read about a poll recently in a newspaper or magazine.

Extension

Ask groups of students to design a survey that is biased. (It could be thesurvey question, the way the sample is chosen, or the way the data iscollected.) Have groups exchange surveys and decide how to change thesurvey so it is not as biased.


Selecting SamplesB

A population is a group of people or set of objects about which youwant to gather information.

You can collect data by questioning a sample of people from a specificpopulation or by examining a sample of objects from a set that haslike characteristics.

When taking a sample, it is important to do so randomly so that everymember of the population has an equal chance of being selected.

You can also collect data by designing and running an experiment orby carrying out a simulation.

When collecting data from a sample, you should avoid bias. Somepossible causes of bias are:

• incorrectly choosing the sample;

• neglecting to account for the people who do not respond; and

• letting interviewers select the people they want to interview.

A researcher is interested in preferences of middle school students.Your school is willing to participate in a survey for sixth- and seventh-grade students, but not all students can participate. Susan suggestsgiving the survey to all of the students in one class.

1. a. Will this be a fair sample? Explain your thinking.

b. How would you select a random sample of sixth- and seventh-grade students from your school?


Hints and CommentsOverviewStudents read the Summary and complete Check YourWork problem 1.

Planning

After students complete Section B, you may assign ashomework appropriate activities from the AdditionalPractice for Section B, found on Student Book page 66.

Solutions and SamplesAnswer to CheckYourWork1. a. One disadvantage of taking one class as the

sample is that you have all students from eithersixth or seventh grade. The sampling procedurewould be biased because it would leave out animportant part of the population. You mightalso argue that students in one class influenceone another with respect to their preferences,and so the results of the sample will not bereliable.

b. Different procedures are possible; for example:

• Make one list of all sixth- and seventh-gradestudents ordered according to their lastname; then take every fifth student in thesample.

• Randomly select students from each class,for example, by putting all the names in abox and taking out as many as you need foryour sample.


Notes

SelectingSamples

B

Assessment Pyramid

3, 4

2abc

Assesses Section B Goals

Extension

The class could choose a survey question. Part of the class could use abiased sample and graph the results. The other part of the class could usea random sample and graph the results. Then compare the two graphs.


2. a. If you look at a set of 50 random numbers ranging from 0 to 9,how many would you expect to be even?

b. Use the random number set from problem 12 and find howmany of these numbers are even. (Note: Count 0 as an evennumber.)

c. Would you expect to get this many even numbers out of everyset of 50 random numbers?

3. Select one example from this section that you think illustrateseach of the following causes of bias.

• incorrectly choosing the sample

• neglecting to account for the people who do not respond

• letting interviewers select the people they want to interview

4. Reflect Some people think that the larger the sample you take,the less chance you have of bias. Do you agree? Explain youranswer.

Samui chose a random sample of the eighth graders at his schooland found that their favorite sport was basketball. In his reporthe stated, “Eighth graders prefer basketball to any other sport.”Comment on his conclusion.

Hints and CommentsOverviewStudents complete the Check Your Work and ForFurther Reflection problems.

Planning

After students complete Section B, you may assign ashomework appropriate activities from the AdditionalPractice for Section B, found on Student Book page 66.


2. a. You would expect half of the 50 numbers to beeven, so about 25.

b. 28 out of the 50 are even.

c. No. The number will vary, but is most likelyaround 25; in about 90% of the cases, thenumber will be between 20 and 30.

3. As an example for incorrectly choosing thesample, you may have chosen the example fromproblem 11c about the travel agencies or from11d about the political poll.

For “neglecting to account for the people who didnot respond,” you may have chosen the examplefrom problem 11b on the health food magazinesurvey.

For letting interviewers select the people theywant to interview, you may have chosen theexample from problem 11a about the policeofficers interviewing people.

Other examples can be found as well. Discussyour answers with your classmates.

4. It is not true that the larger the sample, the lesschance of bias. If the sample is not takenproperly—for instance, because people withouttelephones cannot be chosen in the sample—alarger sample does not change this. The same biaswill still occur regardless of the sample size.


Students’ answers will vary. Sample answer:

Samui's conclusion will be valid for the eighthgraders at his own school because he selected arandom sample.

If one would only read his conclusion, it wouldseem that this holds for all eight grade students,but you cannot tell based on Samui's survey if thisis the case. Samui's school may not be a typicalschool.



Teachers MatterC

Section FocusThis section introduces the importance of developing a criticalapproach to analyzing graphical representations. Studentsinvestigate several graphs to determine whether they representdata clearly and accurately. They investigate the influence offactors such as: the scaling of the axes, the location of theorigin, and the dimensions of three-dimensional representations.At the end of this section, students design and conduct their ownstatistical survey.

Pacing and Planning

Additional Resources: Additional Practice, Section C, Student Book page 66

Day 8: Different Impressions Student pages 22–25

INTRODUCTION Problems 1 and 2 Discuss a graphical representation thathas no scale and investigate the effect ofa change in scale on a graph.

CLASSWORK Problems 3–5 Draw a graph of world population dataand critique two graphs of worldpopulation growth that do not representequal intervals of time.

HOMEWORK Problem 6 Investigate the effect of scale on graphicalrepresentations.

Day 9: Different Impressions (Continued) Student pages 26–29

INTRODUCTION Review homework. Review homework from Day 8.

CLASSWORK Problems 7–10 Investigate the effect of scale on variousgraphical representations.

ACTIVITY/HOMEWORK Activity, page 29 Design and conduct a statistical survey.

Day 10: Summary Student pages 30 and 31

INTRODUCTION Review homework. Review work on statistical surveys.

CLASSWORK Check Your Work Student self-assessment: Analyze andFor Further Reflection critique the representation of data

in graphs.

Teachers Matter Section C: Interpreting Graphs 22B

Teachers Matter C

Materials

Student Resources


• Student Activity Sheet 6



Student Materials


• Blank paper (one sheet per student)

• Glue or tape

• Graph from a newspaper or magazine(one per student)

• Graph paper (two sheets per student)


Learning LinesConclusions from Graphs

A statistical investigation can be split in threeparts: collecting data, representing data, anddrawing conclusions. This section introducesmisleading graphical representations of data. Inthe unit Dealing with Data and in the previoussections, students have seen and used differenttypes of graphs to represent data. Now the focusis on the (mis)representation of data in suchgraphs and the effect this has on the drawing ofconclusions. Some common methods formisrepresentation are using inconsistent scalingof the axes, not beginning each axis at the origin,and using dimensions for pictographs or barcharts that do not match the data. Each of themethods can make a change in data values seemdramatic or minimal, depending on the desiredeffect.

First, students encounter a graph with noquantitative information in it and, consequently,no scaling of the axes.

Then, students look at a pair of graphs thatpresent the same data, but whose y-axes are scaleddifferently and begin with different numbers.Histograms and pictographs are then explored formisleading representations of data. And, finally,students design their own statistical surveys andrepresent the data graphically.


Students will:

• be able to represent data graphically in acorrect way;

• analyze representations of data;

• determine whether representations of dataare appropriate and accurate;

• know and understand methods used formisrepresentation of data in different typesof graphs;

• describe the effect misrepresentations ofdata can have; and

• be able to design and conduct a statisticalsurvey, represent the data graphically, anddraw conclusions.

RAGE

FEARFEAR


Notes

In the discussions that occurthroughout this section, besure to promote students’critical attitude towardgraphical representations.Prompt students toquestion, Does the displaygraph the data appro-priately? Is the graphcomplete? Is the graphmisleading?

This is a powerful contextand very engaging repre-sentation for students todiscuss. Ask students todescribe any scaryencounters they have hadwith dogs, cats, or otheranimals. Have themdescribe the animal’snon-verbal cues that letthem know when theyneed to back away or takeother precautions.

InterpretingGraphs

C

Act It Out

As a humorous connection to the display shown here, have students act outthe different human faces for the two variables shown here (or some otheremotions that are easy for students to demonstrate).

Intervention

Ask if students recognize any patterns of change as they move horizontallyor vertically from face to face, for example, teeth, ears, hair, nose, and so on.


CInterpreting Graphs

Different Impressions

Graphs are useful for representing information in a clear and conciseway. The graph of rage and fear is a good example of a graph thatconveys information using only two words.

1. a. Reflect Describe what is represented in this graph.

b. Why is this a “good” example?

Many people do not trust the information provided in statisticalcharts. Sometimes the data come from a poorly selected sample,or the data are presented improperly. Some graphs have “mistakes”or “misrepresent” data. When using graphs, it is important to thinkabout how they are constructed and to make sure the graphs do notgive the wrong impressions.

RAGE

FEARFEAR

Source: Data from “Catastrophe Theory” by E.C. Zeeman.Copyright © 1976 by Scientific American, Inc. All rights reserved.

Section C: Interpreting Graphs 22T

Hints and CommentsOverviewStudents explore a graph that uses pictures instead ofnumbers and discuss what determines a good datarepresentation.


This graph does not contain any quantitativeinformation. There is no scaling on the axes. Onlyordering is possible: the more to the right, the angrierthe dog. You cannot say how much angrier.

Planning

You may have students work on problem 1 in smallgroups.

Did You Know?

E. C. Zeeman, author of “Catastrophe Theory”(Scientific American, April 1967), noted that theaggression of dogs can be described by a mathe-matical model that uses the principles of catastrophetheory. The mathematical model is based on theassumption that fear and rage are the controllingfactors of aggressive behavior. These factors can bemeasured by changes in the facial expressions ofdogs. Levels of rage are determined by the size ofthe opening of the dog’s mouth. Levels of fear aredetermined by the degree to which the dog’s earsflatten toward the back of the head. The mathematicalmodel can be used to predict a dog’s behavior.

The theory states that the neutral state of emotion isfound at the origin. If a stimulus increases the dog’srage without introducing fear, aggressive behaviorresults. At high levels of rage, or if the dog isfrightened and a stimulus increases the dog’s rage,the model predicts that the dog may suddenly attack.If the dog progressively becomes more fearful whileat a high level of rage, the dog will eventually stop theattack and retreat.

If a dog’s emotional state is neutral, and a stimulusincreases rage and fear simultaneously, the dog willattack or retreat depending on its state of mind beforethe stimulus was introduced.


1. a. The pictures are the data points in thecoordinate system:

In the lower left corner, the animal is neitherangry nor afraid.

In the upper left corner, the animal is not angrybut is very afraid.

In the upper right corner, the animal has a highlevel of anger and fear.

In the lower right corner, the animal is veryangry but is not afraid.

b. Answers will vary. One possible answer is thatthe graph conveys a lot of information, yet it iseasy to read.


Notes

Some middle schoolstudents may not befamiliar with a numericalgrade point average.Explain how this iscalculated and discusswhat the differencebetween a 2.6 and 3.0 reallymeans.

After discussing problems2a and b, have studentsmark the range of gradepoint average with theirforefinger and thumb (from2.6 to 3.1), and then showhow the same range isdisplayed on the secondgraph using their forefingerand thumb.

InterpretingGraphs

C

Parent Involvement

Ask students to begin collecting graphs that distort and misrepresent data.At the end of the section, students may present their collection of graphs totheir parents, along with a brief paragraph that explains how each graphmisrepresents the data.


A

The graphs on this page give two different impressions of therelationship between the number of hours per week that astudent works at a job and his or her grade point average.

2. a. Why do the graphs provide different impressions?

b. Which graph do you think a high school principal would usewhen talking to parents about their child’s decreasing gradepoint average? What argument do you think the principalwould make?

CInterpreting Graphs

3.1

2824201612840

2.6

3.0

2.9

2.8

2.7

Hours Worked Per WeekG

rad

e P

oin

t A

ve

rag

e

3.2

28242016128400

2.8

2.4

1.6

0.8

2.0

1.2

0.4


Gra

de

Po

int

Av

era

ge


Hints and CommentsOverviewStudents investigate the effect of a change in scaleand a missing origin in a graph.


The effect described on this page is known as the lostorigin. By having the vertical axis start at a value otherthan 0, the changes are magnified. Usually a break inthe axis is used to indicate a missing origin. In somecases, graphs that start at the origin leave muchunused space, as in the second graph on StudentBook page 23.

Example of a graph with a break in the axis:

Planning

You may want students to work on problem 2 in smallgroups.


2. a. Answers will vary. Sample response:

In the first graph, the scale on the vertical axisdoes not start at zero and is marked inincrements of 0.05 for grade point average. Theeffect of not starting at the origin is to make thedrop in grade point average seem very large forstudents who work more than 12 hours a week.

In the second graph, the scale on the verticalaxis is marked in increments of 0.2 for gradepoint average. And because the scale on thevertical axis begins at zero, the drop in gradepoint average does not seem as large as in thefirst graph.

b. Answers will vary. Sample responses:

A high school principal might favor the firstgraph because it seems to provide an argumentagainst high school students working longhours on a job.

The first graph is easier to read and is moreconvincing to support an argument for thereason that working many hours a week is notgood for a student.

For someone who believes that working manyhours a week does not really affect the gradepoint average, the second graph is better.

1930

2.5

3

3.5

4

4.5

1950

Year

Alabama

Po

pu

lati

on

(in

mil

lio

ns)

1970 1990


Notes

3 To make a histogram,students should draw thescale for the horizontal axiscorrectly, leaving blankspaces for years for whichthe population is not givenin the table. However, aline graph that is correctlyscaled on the horizontalaxis seems more appro-priate to use in this case.

4b World population datamay be found on severalwebsites and in almanacs.If an almanac or an Internetsearch is not feasible foryour students, you shouldresearch this informationbefore class.

InterpretingGraphs

C

Intervention

Before students make the graph, discuss how to determine a suitableconstant interval for each axis and why it is important to do so. Also discusswhich data is the independent variable and belongs on the horizontal axis.If time is one of the variables, it is usually placed on the horizontal axis.


The table contains information about the world population forspecific years.

3. Draw a graph to represent the data from the table.

4. a. Based on these data, what would you expect the worldpopulation to be in the current year?

b. Use an almanac or search the Internet to find the worldpopulation in the current year. How does the actual populationfor the current year compare to your answer for part a?

c. What do you think the world population was in 1985? How didyou make your estimate?

d. Which estimate of the world population do you think will bemore accurate: your estimate for 1985 or an estimate for 2025?Explain your reasoning.

Interpreting GraphsC

Year World Population

(in billions)

1630 0.5

1820 1.0

1890 1.5

1930 2.0

1950 2.5

1960 3.0

1968 3.5

1975 4.0

1981 4.5

1987 5.0

1994 5.5

2000 6.0

Source: http://www.ibiblio.org/lunarbin/worldpop/


Hints and CommentsMaterialsgraph paper (one sheet per student)

Overview

Students make a graph that represents the worldpopulation growth and use the graph to makeestimations.


The lost origin was the problem on the previous page.Here the scaling of the axis is addressed. In general,the axes should be scaled linearly, meaning that thedistances between grid marks are the same.

In most tables (and graphs) that represent growthover time, the units of time are at equal intervals.

In a graph over time, you can interpolate andextrapolate to estimate data that were not measured.Interpolation usually gives better estimates thanextrapolation.

Planning

Students may work on problems 3 and 4 in smallgroups.


3. Student’s graphs will vary. The following is anexample of a line graph.

Note: The horizontal axis should not start at 0,and students should have a vertical axis from0 to about 7 billion.

4. a. Answers will vary depending on the currentyear. For 2010, an estimate may be 7 billion.

b. Students may find the actual world populationsize on the Internet. They will compare thisnumber to their answer to part a.

c. Answers may vary. Sample answer: About4.8 billion. I estimated this from the tablebetween 1981 and 1987.

d. Answers may vary. Sample answer: Estimatesfor 1985 will be more accurate because datafrom 1981 and 1987 are known, so you know forsure that the population for 1985 is between4.5 and 5 billion. You cannot be sure about thetype of growth of the population in the future.You can extend the graph but in reality thismay not be how the population grows.

Wo

rld

Po

pu

lati

on

(in

bil

lio

ns)

Year

0.5

0

1630

1680

1730

1780

1830

1880

1930

1980

2030

2080

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7


Notes

After students completeproblem 5, you may wantto have a class discussionabout the lost origin andthe scaling of axes. Alsodiscuss which kind ofgraph best represents thedata—a line graph or a bargraph.

You may find it helpful tomake a transparency ofStudent Activity Sheet 6 touse during class discussion.

InterpretingGraphs

C

Assessment Pyramid

5a

5b



Extension

Have students draw the graph for problem 6 so it looks like there has notbeen a significant change in the average temperature.


ACInterpreting Graphs

Use Student Activity Sheet 6, which shows examples of graphs madeby two students to answer problem 3.

5. a. Do these graphs accurately represent the data? Explain.

b. Compare these graphs to the graph you drew in problem 3.

You may hear that the climate is changing since the average globaltemperature is rising. The graph below shows the average globaltemperature from 1900 to 2002.

Av

era

ge

Glo

ba

l Te

mp

era

ture

(in°F

)

Year

2000–2001

1990–1999

1980–1989

1970–1979

1940–1969

1930–1939

1910–1929

1900–1909

2002

58

57

56

Increase in Global Temperatures 1900–2002

Source: National Oceanic and Atmospheric Administration

6. a. Does the graph give you a good representation of the changein temperature? Explain your thinking.

b. How do the scales of the graph affect your impression?

c. Comment on the statement: The data show that the earth isgetting warmer than ever before.


Hints and CommentsMaterialsStudent Activity Sheet 6 (one per student);transparency of Student Activity Sheet 6, optional(one per class)

Overview

Students interpret graphs on world populationgrowth. They investigate misleading datarepresentation in a graph.


The line graph on Student Activity Sheet 6 illustratesthe effects of incorrect scaling of the horizontal axis.

The bar graph about the same data shows the sameeffect although the graph in itself is correct. In a bargraph, the numbers on the horizontal axis need notbe ordered or scaled. In this case, however, we canargue that this bar graph is not appropriate to showthese data in a sense making way.

The graph for the average global temperatureillustrates the effect of incorrect scaling of thehorizontal axis. The vertical axis is correctly scaled,but the reader should look at the numbers to see ifthe part of the vertical axis that is shown is over-dramatizing the increase in temperatures from 56° to58° F. For the average global temperature, this can beseen as meaning a large difference, while for a dailytemperature, a difference of 2 degrees may be hardlynoticeable. This example illustrates that making adecision for a proper scale for a graph involves morethan mathematical reasons; there may also be reasonsbased on the situation that need to be taken intoaccount.

Planning

Students may use problem 6 to demonstrate theirimproved critical attitude toward data representation.You may want to discuss problem 6 as a class orassign it as homework.

Extension

You may want to draw the graph of problem 6 againusing a consistent scale on the horizontal axis.

DidYou Know?

Data on global surface temperatures are collected byseveral organizations. These data can be representedin several ways, conveying different messages. Oneshould always have a critical attitude towardsgraphical representations of data.



No. In both graphs, the intervals on thehorizontal axes are not equally spaced. Thehorizontal axes are scaled incorrectly. Bothgraphs give the impression that the worldpopulation grows in a linear way.

b. Answers will vary, depending upon students’graphs from problem 3.

6. a. Answers will vary. Possible answers:

No. The change seems to be very large, but thetemperature only rose about one and a halfdegrees over about 100 years.

No, it is difficult to see how the temperaturechanged because the horizontal axis is notproperly scaled.

b. Answers will vary. Sample answer:

Since the horizontal axis is not scaled properly,it seems that the rise of temperature is thesame between 1900 and 1910 as between 2000and 2002, but the time period between 2000and 2002 is much shorter. The rise was fasterafter 2000, so the graph should be steeperthere.

The scaling of the vertical axis makes the risefrom 56.6 to 57.9 seem very large.

c. Answers will vary. Sample comments:

I agree that the data show this. The averageglobal temperature in 2002 was 57.9° F, whichwas the highest ever.

I agree, but the difference between the averageglobal temperature in 1980 and 2002 is only0.4° F.


Notes

7 Make sure studentsunderstand that it is oftenpossible to draw a graph insuch a way that it gives acertain impression that isfavorable to the personmaking or using the graph.

InterpretingGraphs

C

Vocabulary Building

Review the term lost origin (see also page 23T About the Mathematics) andthe effect it has on a graph. By having a graph start at a value other thanzero, the changes in the bars are frequently magnified.


Although the two graphs below represent the same data, they givedifferent impressions.


980

5

10

15

20

25

99 00 01 02 03 04

Percentage of Delayed Flights

Pe

rce

nt

De

lay

ed

Year

98

15

20

25

99 00 01 02 03 04

Percentage of Delayed Flights

Year

Pe

rce

nt

De

lay

ed

7. a. Which graph suggests that fewer flights are delayed?

b. How was this impression achieved?

c. What groups of people might choose to use each graph? Why?

d. Do you think these data were gathered by sampling, or do youthink they represent all flights? Give reasons to support youranswer.


Hints and CommentsOverviewStudents investigate two graphs that represent thesame data but give different impressions.


Graphs can be represented in several ways. Differentrepresentations can be completely correct but stillachieve different impressions. In this case, the secondgraph is correct because a “zigzag” is used to indicatethat the vertical axis does not start at zero. Even inthat case, the graphs give different impressions due tothe “length” and scaling of the vertical axis.

Planning

Problem 7 can be done in small groups and discussedin class.


7. a. The second graph suggest that fewer and fewerflights are delayed..

b. The bars seem to decrease a lot and becomevery small. This impression is caused by thefact that the vertical axis starts at 15 instead ofzero.

c. Answers will vary. Sample responses:

The president of an airline company might usethe second graph to indicate that only a verysmall number of flights is delayed.

Individuals promoting flying over other ways oftransportation might use the second graph togive people the impression that flying does notinvolve much delay.

A president of an airport may use the secondgraph to convince people to fly from thatairport because almost no flights are delayed.

A competing airline company may use the firstgraph to indicate that 15% of the fights are stilldelayed.

d. Answers will vary. The data for these graphswere likely based on all flights from a companyor for a certain airport, since these data areavailable.


Notes

8 Ask students what causesthe misrepresentation ofdata in the graph.

9 Have students comparethe decreases in popula-tion with the decreasesof the pictures. Ask, Whatdo you think is morepersuasive, the decreasesin the population figures orthe decreases in the sizesof the buildings?

InterpretingGraphs

C

Assessment Pyramid

8a

8b

Determine whetherrepresentations of data areappropriate.

Draw conclusions based onrepresentations of data.

Intervention

Review scale and discuss ways to determine if the size of each building wasincreased proportionally to the data.


Interpreting Graphs C

8. The graph represents the percentage of airline seats filled duringthe second quarter of 2003 through the first quarter of 2004.

a. It appears that the first quarter of 2004 had double thepercentage of passengers as in the second quarter of 2003.Is this an accurate description? Explain your answer.

b. Write a statement that accurately describes the changes fromthe second quarter of 2003 to the first quarter of 2004.

Populations of Five Large U.S. Cities

9. a. What message does the graph above tell you?

b. Do you think the pictures express the populations in anaccurate way? Why or why not?

69.5

Second Quarterof 2003

Passenger Load Factor

Third Quarterof 2003

Fourth Quarterof 2003

First Quarterof 2004

70.0 70.5 71.0 71.5 72.0 72.5 73.0

Percent

500,000 540,000 650,000 790,000 900,000

Oklahoma City, OK Charlotte, SC Memphis, TN Indianapolis, IN San José, CA


Hints and CommentsOverviewStudents investigate two graphs that create amisleading picture.


The first graph illustrates how a missing origin on thex-axis can lead to a misleading impression. This graphis comparable to the second graph in problem 7 onpage 26. In the second graph, the absolute decrease(in population) is not directly represented by thedecrease in area of the picture. Pictographs often usepictures with a certain area or volume to representnumerical data. The visual impression may bemisleading if the area or volume does not correctlyrepresent the size of the numerical data.


8. a. No this is not an accurate description. It seemsto be correct if you look only at the size of thebars. But these do not represent the percent-ages correctly since the scaling of the horizontalaxis is misleading: it does not start at zero.

b. Answers may vary.A sample correct statement is:

In the first quarter of 2003 the percent loadfactor was almost 71% and in the first quarterof 2004 it was almost 73%, this is only a verysmall change, it is less than 2%.

9. a. Answers will vary. Sample answer:

The graph shows the cities’ populationsordered from small to large, but it is difficultto find the populations from the pictures. Youcannot really tell, for example, how much largerthe population of San José is compared to thepopulation of Oklahoma City from looking atthe size of the houses in the graph.

b. No, the graph does not express the populationin an accurate way. The buildings that representthe population in the cities are not drawnproportionally to the numbers. The numbersare correct.

28 Insights Into Data

Notes

10d Discuss with studentsthe drawings of the secondand fourth box. Point outthat the price is doubledand the amount is doubled.Ask, Do these drawingsshow this? Is the volumeof the fourth packagedouble the volume of thesecond package?

InterpretingGraphs

C

Intervention

Use a model to help students understand the effect of doubling eachdimension of a rectangular prism. Eight cereal boxes or tissue boxes of thesame size work well.

First double the width by putting two boxes next to each other; then askhow many times the volume has increased. Then double the depth (younow have 4 boxes) and ask how many times the volume has increased.

Finally use all eight boxes to model doubling all three dimensions. Then askhow this model relates to the size of the cereal packages.



10. a. How was the $209 computed?

b. What data are used to make this graph?

c. The last package has $838 written above it. The amount $838is 4 � $209.40 rounded to the nearest whole number. Explainhow this fits the data.

d. Is the size of the last package four times the size of the first?Carefully explain your answer.

$838

5 10 15 20

packages packages packages packages

per month per month per month per month

$628

$419

$209

Eating Cereal Is a Good Investment

Average cost of a 16 oz box

of cereal is $3.49.

Annual cost


Hints and CommentsMaterialsgraph paper (one sheet per student)

Overview

Students explore the effect of volume in therepresentation of data.


In pictographs, where three-dimensional objectsrepresent quantities, attention is not always given tothe scaling of all three dimensions of the depictedobjects.

To make the volume of the object twice as large, youcan either double one of the dimensions (height,width, or length), or have each dimension 3��2 � 1.26times as long. Note that if only one dimension isenlarged, the shape of the object is deformed.

Planning

Have students work on problem 10 individually.


10. a. 5 � 12 � $3.49 = $209 (rounded). This is theamount of money you pay for cereal in oneyear when you buy 5 packages per month.

b. The data used to make this graph are theaverage cost of a box of cereal, the number ofpackages per month, and the number ofmonths in a year.

c. Explanations may vary. Sample response:Twenty packages per month is four times five,and five is written with the first package ofcereal. So the amount for the last packageshould be four times the amount for the first;this is calculated correctly. The numbers areaccurate.

d. Answers will vary. Sample student responses:

Yes, the last package is four times as wide asthe first.

No, the volume of the last package is more thanfour times the volume of the first package.


Notes

You might discuss theprocess of planning thesurvey with the class. Asthey design their surveys,students need to thinkahead to the analysis of thedata and to how the datawill be used to answerquestions. They mayconsider questions such as,What should be measured?How should it be measured?How many individualsshould be included in thesurvey? How should theseindividuals be selected?

InterpretingGraphs

C

Assessment Pyramid

Activity

Writing Opportunity

You may want students to write a report that summarizes their experiencesin designing and completing their surveys.

Promoting Student Self-Assessment

It is important to discuss your expectations for students’ work (what is to beincluded, how should it be presented) and when the survey results are due.Students may also include a self-critique of the study, noting any possiblebias.


Design and carry out a statistical survey. Pay attention to thefollowing aspects.

• What are you going to investigate?

• How do you write a good questionnaire?

• What is a good sample for your survey?

• How will you graphically represent your data?

• What conclusions can you make from your data?

• Do any new questions arise from your data?

• What further investigations might be necessary?

• Is there possible bias in the way you intend to carry out yoursurvey? If so, can you eliminate it?

➞

➞

Design, conduct, andanalyze ways of gatheringdata. Use random samplesin gathering data. Recognizepossible bias in samplesurveys.


Hints and CommentsOverviewStudents design and carry out a statistical survey.They graph their data, draw conclusions, andformulate questions for further research. They reflecton their sampling method and think about possiblecauses for bias.


A statistical investigation can be split in three parts:data collection, data representation, and drawingconclusions. In the activity, students do all three parts.

Planning

You may want students to work on the activity insmall groups. You might give students one week tocomplete their surveys. Have students hand in areport and/or present their results to the class. Thisactivity can be used as an informal assessment.


Activity

Surveys will vary. Students should present their datagraphically and write about the conclusions they havedrawn.


Notes

To process the Summary,one possibility is to haveexamples of graphs thatcontain misrepresentationson posters on your wallsand have students movearound the room in smallgroups and identify thetype of misrepresentation.

InterpretingGraphs

C

Assessment Pyramid

1

Assesses Section C Goals


For problem 1 in Check Your Work, it may be helpful to discuss reasons foran accurate graph before you have students write their answer.



In order to make accurate conclusions from data, the data must bereliable and presented appropriately.

Data can be presented in many different ways.

• picture graphs • histograms

• line graphs • scatter plots

• bar graphs

When you see a graph, you should look carefully to make sure thatthe graph is a fair one that accurately tells the story of the data. Datamay be misrepresented if one or more of the following occurs:

• the graph’s axes are scaled improperly;

• origins on the graph are excluded;

• three-dimensional pictures are used inappropriately;

• numbers that should not be compared are compared;

• pictures that do not fit the numbers are used.

1. Why is it important for a graph to be an accurate representationof data?

Richard is a member of a neighborhood football club. His father andhis brother John were members as well.

They recorded the number of club members for 10 different years.

1983 45 2003 701984 41 2005 671985 53 2006 801995 60 2007 752002 68 2008 70


Hints and CommentsOverviewStudents read the Summary, which reviews the maintopics of this section, and complete the first CheckYour Work problem.

Solutions and SamplesAnswers to Check Your Work

1. Different answers are possible. It is important fora graph to be an accurate representation of datato reveal relationships between the two variablesso that proper conclusions can be drawn.


Notes

3 You may wish to havestudents do this with apartner or small group.Then each group couldshare their graph withthe whole class, sayingwhether or not it is anaccurate representation.It may be necessary tobring in a supply ofpapers or magazinesthat have graphs.

InterpretingGraphs

C

Assessment Pyramid

2a

FFR

2b, 3

Assesses Section C Goals

Intervention

If any students have difficulty with problem 2, suggest they reread theSummary. Ask, Which of the misrepresentations listed is shown?


Richard graphs these data in the following way.

2. a. Does Richard’s graph represent the data accurately? Explain.

b. Draw another graph that you think accurately representsthe data.

3. Cut out a graph from a newspaper or magazine. Include thecaption or article that accompanies the graph. Attach the graphwith the article or caption to a sheet of paper. Write a paragraphthat explains how the graph presents the information in thecaption or article. Do you think the graph is a good representationof the data or not? Explain your reasoning.

Make a list of all of the ways you think a graph can be misleading.Make another list of things you should watch for when you arelooking at graphs in the media.

1983

10

20

30

40

50

60

70

80

90

1984 1985 1995 2002 2003 2005 2006 2007 2008

YearN

um

be

ro

fC

lub

Me

mb

ers


Hints and CommentsMaterialsgraphs from newspapers or magazines (one perstudent);blank paper (one sheet per student);glue or tape (one per student)

Overview

Students complete the Check Your Work and ForFurther Reflection problems.

Planning

It is useful to have some examples of misleadinggraphs available for Check Your Work problem 3 incase students weren’t able to find any.

After students complete Section C, you may assign ashomework appropriate activities from the AdditionalPractice section for Section C, located on StudentBook page 66.


2. a. No, because the horizontal axis should beordered like a number line. Richard ordered theyears evenly on the horizontal axis, but thereare more years between 1985 and 1995 thanbetween 1983 and 1984, and you cannot seethis on Richard’s graph. So between 1985 and1995, there seems to be a rather steep increase,steeper than between 2002 and 2003. Butactually, the first increase should be spread outover ten years, although you do not know whathappened exactly in the years in between.

b. You might make a line graph in which the yearsare placed on the axis with the right scale.

3. Your answer will be different from your classmates’answers. Have one of your classmates commenton your work and the other way around. Discussboth your articles or captions and paragraphs.


Answers will vary. Sample answer:

A list of ways that graphs can be misleading:

• One axis or both axes will not start at zero(although this is not made clear).

• Axis may be improperly scaled by having partsmissing, or by having distances not equal.

• Pictures may not fit the numbers because, forexample, the area or the volume is not inproportion to the numbers, or pictures are usedinappropriately.

• A pictograph may be very difficult to read becauseof all kinds of extra “art.”

A list of things you should watch for when you arelooking at graphs in the media:

• How are the data gathered?

• What is the sample and the sample size?

• What questions have been posed?

• Are data left out?

• Is the data represented in an appropriate andcorrect way?


Teachers MatterD

Section FocusThis section emphasizes the use of graphs and numbers to displaythe results of the bean sprouts experiment that was conducted inSection A. Students analyze their data from the experiment. Theyselect appropriate plots to summarize the data and use histogramsand box plots to compare information. They also study andanalyze other plant growth data. They review the use of mean,median, and mode.

Pacing and Planning

Additional Resources: Additional Practice, Section D, Student Book pages 67 and 68

Day 11: Exploring Growth Student pages 32–35

INTRODUCTION Problems 1 and 2 Read and interpret a histogramrepresenting plant growth and reviewthe mean, median, and mode.

CLASSWORK Problem 3 Match plant growth graphs with statementsabout the plant growth and create graphsfor statements with no matching graph.

HOMEWORK Problem 4 Compare three histograms that representthe heights of a set of plants after 10 days,12 days, and 14 days.

Day 12: Presenting the Bean Sprout Data Student pages 36–38


CLASSWORK Problems 5 and 6 Interpret plant height data represented asa line graph and a box plot.

ACTIVITY/HOMEWORK Activity, page 38 Write a report summarizing the beansprout experiment conducted in Section A.

Day 13: Presenting the Bean Sprout Data (Continued) Student pages 39–43

INTRODUCTION Problem 7 Make a class histogram of bean sprout data.

CLASSWORK Problem 8 Create box plots for the heights of beansprouts grown in different solutions andcompare histogram and box plotrepresentations.

HOMEWORK Check Your Work and Student self-assessment: Use graphs toFor Further Reflection draw conclusions from data sets.

Teachers Matter Section D: Using Plant Growth Data 32B

Teachers Matter D

Materials

Student ResourcesNo resources required

Teachers’ ResourcesNo resources required

Student MaterialsQuantities listed are per pair of students, unlessotherwise noted.

• Centimeter ruler

• Colored self-stick notes (four)

• Completed Student Activity Sheet 1

• Graph paper (four sheets per student)

* See Hints and Comments for optional materials.

Learning LinesRepresentation of Data: Graphsand Statistics

In this section, several plots and graphs are usedto represent data about the growth rate of beansprouts in various solutions. Box plots andhistograms were introduced in the unit Dealingwith Data and used in Section B. Most of theconcepts from statistics (measures and graphs)that appear in this section were introduced in theunit Dealing with Data.

Students make histograms and box plots thatshow the lengths of sprouts grown in varioussolutions. A histogram best presents clusters andgaps in the data. A box plot provides a summaryof the five major data points: the minimum, thefirst quarter, the median, the third quarter, andthe maximum.

They discuss using a plot over time (usually a linegraph) to show the data points as they changefrom day to day. There are many ways to make aplot over time: plot the growth of a representative

member of the population; plot the range ingrowth per day; plot the mean or median lengthper day; or plot the growth of each individual inthe sample.

A plot over time allows you to look for trends inthe data.

Drawing Conclusions

Students study and interpret differentrepresentations of the same data. They do so inthe context of plant growth, connecting differenttypes of graphs to different conclusions andstatements about the plant data. In doing so theyalso revisit the use of one-number summaries:mean, mode, and median.

For the plant growth data collected from the beangrowth experiment, students summarize theresults and draw conclusions.


Students will:

• be able to represent data graphically in differentways, depending on the type of data and thestatements they want to make;

• be able to describe data numerically using one-number summaries such as mean, mode, andmedian in combination with summaries such asrange that tell about the spread of the data;

• know how to analyze representations of data;

• draw conclusions based on given data sets andrepresentations of data; and

• be able to connect different representations ofdata to statements about these data.

Box Plots for Final Length of Sprouts in Various Solutions

Q1 M Q3

Q1 MQ3

Q1 M Q3

Q1 M Q3

0 10

2 2.5 3

20 30 40

45 72.5 90

453528

34 41.5 50

50 60 70 80 90 100

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Notes

A brief discussion ongraphing could help youget information onstudents’ prior knowledgebefore starting this section.Ask, What are some reasonsfor making a graph? Whatdifferent kinds of graphs doyou know how to make?What other ways can yousummarize data? (Mean,median, and mode areexamples.)

1a Ask why it is importantto read the labels on agraph carefully. Additionalquestions may be used tomake sure they are readingthe graph correctly. Whatis the height of the tallestplant? How many plantsreached that height? Whatis the range of the mostcommon plant heights?

Using PlantGrowth Data

D

Act It Out

This activity could be used to review the vocabulary mean, median, andmode that are used on the next page. Have five students of varying heightsline up in front of the room from shortest to tallest. Then ask which studentis the median height? Is there a mode? Have one more student join thegroup and position himself or herself so they are still lined up from shortestto tallest. Ask how they would determine the median height now that thereare six students (average the middle two). Is there a new mode? Anotherpossibility is to have students calculate the mean height of the six students.


Graphs of data can help you get a better picture of how the dataare distributed. Graphs can help you interpret the data and makestatements about an experiment.

The graph below represents the height of plants after a seven-dayperiod.

1. a. Study the graph. What does the bar at 7 millimeters (mm)represent?

b. Write three statements about how tall the plants in the studygrew.

DUsing Data

Exploring Growth

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Section D: Using Plant Growth Data 32T

Hints and CommentsMaterialstransparency of the bar graph on Student Bookpage 32, optional (one per class)

Overview

Students study and interpret a bar graph representingdata on plant growth.


In the bar graph, data on three variables arerepresented: the heights, the frequency, and thegrowth period of the plants. For different growthperiods, you would need different graphs. Anotherway to represent growth would be to use a line graphover time. Since so many plants are involved,however, a line graph would be very difficult to read.

Planning

Students may work on problem 1 individually. Atransparency of the graph may be helpful during aclass discussion.


1. b. Students may use the mean and the mode intheir statements; these concepts have beentaught in earlier units. The mode is the mostfrequent data point. In this case, it is the heightwith the tallest bar (11 mm).


1. a. The bar at 7 millimeters represents the 16 plantsthat grew to a height of 7 millimeters by the endof seven days.

b. Answers will vary. Sample student responses:

The shortest height of any plant that grew was1 mm.

The tallest height was 21 millimeters.

Most of the heights were between 4 and15 mm.

Twenty-five plants grew to a height of11 mm.

More plants grew to 11 mm than to any otherheight.

Two hundred and fifteen plants are beingstudied.

The mean plant height is about 10.6 mm.


Notes

One way to see whichheight represents the meanis by using the graph. Havestudents draw a horizontalline so that the portion ofthe graph above the linewould fill in the partsbelow. This strategy isintroduced in the unitPicturing Numbers and iscalled the Compensationstrategy. See the graphbelow.


D

Assessment Pyramid

3

Interpret representationsof data.

Accommodation

Give students a copy of this page to cut apart the graphs and thedescriptions. Match the ones that can be matched and have them makegraphs for the missing ones. Students could work together in a group andput their results on a poster.

Vocabulary Building

Have students add median and mode to their notebooks and make sure allstudents know how to find the median and mode.


Jo Mei, says “The mean height of the plants at the end of theexperiment is about 10 mm.”

Jorge, says “The mean height of the plants at the end of theexperiment is about 15 mm.”

2. a. Who do you think is right, Jo Mei or Jorge?

b. Which number is most easily found in the graph: the mean,the median, or the mode of the height values?

c. Reflect Explain how you might use the information in thegraph to find the median height of the plants.

Akir, Kari, Viviana, and Marja were studying the growth of plants. Afterseven days, they each graphed their data. Then they wrote statementsabout the growth of their plants. Unfortunately, some of the work wasmisplaced, and the rest was mixed together. On this page you see whatis left of their work.

3. For each student, match the statement with a graph. If a student’sgraph is missing, make an appropriate graph.

Using Data D

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Hints and CommentsMaterialsgraph paper (several sheets per student);centimeter rulers (one per student);transparency of graph on Student Book page 32,optional (one per class);photocopies of graphs and statements on StudentBook page 33, optional (one per student);scissors, optional (one pair per student)

OverviewStudents connect statements about the plant data tothe data on graphs and vice versa. The one-numbersummaries of the mean, the mode, and the medianare revisited.

See more Hints and Comments on page 102.


2. a. Jo Mei’s estimate seems to be closer to themean plant height, which turns out to beabout 10.6 mm. Jorge’s estimate of 15 mm istoo high for the mean.

b. The mode is most easily found in the graph.The most frequent plant height is 11 mm.

c. The median is the data point in the middle ofthe distribution. The following method can beused to find the median: First count to find thetotal number of plants in the study (215). Thencount from left or right until you find plantnumber 108. This plant has the median length.It is in the bar with plants of 11 mm, so 11 mmis the median height.

3. Students must make appropriate graphs torepresent the data for Viviana and Marja.Possible graphs are shown below.

Kari: “The data cluster around 12 mm, exceptfor two plants that are at 20 mm.”

Akir: “Almost all of the data are centered around10 mm.”

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Viviana: “My plants grew rather well. I had noreally short ones and just a few that grew reallytall.”

Marja: “The data are spread out from 4 mm to18 mm with almost the same number of plantsfor each height.”

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Notes

Have students carefullystudy all three histograms.Ask why they think thescaling of both axes is thesame on all three graphseven though there is a lotof blank space on the graphfor 10 days. Students couldwork in small groups toanalyze the histograms andthen have a whole classdiscussion.


D

Intervention

To build students’ sense of measurement, you might want to havecentimeter rulers available so that students can draw out the heights ofsome of the plants represented here. They can then compare the heights ofplants in these histograms with the heights of their bean plants.


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A

The histograms shown below and on the next page represent thefrequency of heights of a group of plants at 10 days, 12 days, and14 days.

Using DataD


Hints and CommentsOverviewTwo histograms are presented showing the heights ofplants after different growth periods.

There are no problems on this page for studentsto solve.


Notes

4a and b If students arehaving difficulty gettingstarted, suggest theydiscuss the range of theheights of the plants ineach histogram. Makesure they understand thatthe height of each bar isindicating frequency or thenumber of plants, not theheight of the plants.


D

Making Connections

Remind students that this third histogram represents Day 14 of the sameset of plants. Ask students to locate the plants that were over 150 mm onDay 14. Then ask, What do you think the heights of these plants were onDay 12? On Day 10?


ADUsing Data

4. a. Write down what each histogram tells you about the plantheights.

b. Describe the growth pattern of these plants.

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Hints and CommentsOverviewStudents interpret and compare three histogramsshowing the heights of plants after different growthperiods. Two of the graphs are presented on page 34;the other is on page 35.


The growth of a large group of plants can bepresented in a series of histograms as shown on thesepages. If the scaling of the axis is the same for allhistograms, statements can be made and conclusionscan be drawn about the growth process of the wholegroup. The growth of individual plants cannot betracked.

It can be important to distinguish between bar graphsand histograms. Here are two important differences:

Note: There are also cases in which it is not completelyclear if a graph is a bar graph or a histogram such ason page 32 of the Student Book.

Planning

Students may work in small groups on problem 4. Besure to discuss their answers before proceeding to thenext problem.



Plant height at 10 days: The heights of theplants varied. Some of the plants did not growmuch at all, while others grew to a height of58 millimeters. Most of the plants grew tobetween 20 and 30 mm.

Plant height at 12 days: The heights of theplants were more spread out than at 10 daysand ranged from 0 mm to 107 mm.Most of the plants grew to heights of between30 and 70 mm.

Plant height at 14 days: The heights rangedfrom nearly 0 mm to 159 mm, with one outlierthat grew to a height of 180 mm. With a fewexceptions, namely, at 25 mm, 65 mm, 85 mm,110 mm, 125 mm, and 150 mm, there were oneor two plants at many heights.

b. Answers will vary. Sample response:

With time, the heights of the plants becomemore varied. After 10 days, the plants rangedin height from about 10 to 50 mm, with mostplants at around 30 mm. After 12 days, mostof the plants were between 30 and 70 mm tall,and after 14 days, most of the plants werebetween 50 and 140 mm tall.

Bar Graph Histogram

for categorical data: only for rational data:each category has each category is anits own bar, which interval and each barrepresents a frequency represents the frequencyof data points in that of data points withincategory that category

bars are represented bars are representedseparately adjacently


Notes

Before students answer 5a,ask them why there areonly three points on thegraph.

5c If prompting is needed,ask students what infor-mation has been lost thatthey could see on thehistogram.


D

Assessment Pyramid

5c


Making Connections

Ask students to locate the mean heights for each day (as shown on this linegraph) on the histograms on the previous pages.


Using DataD

Josh decided to make a line graph of the mean plant height for all ofthe plants for certain days.

5. a. How did he find the height for day 10? What does this pointmean?

b. When did the plants seem to grow the most? How can yousee this on the graph?

c. Comment on the advantages and disadvantages of usingJosh’s graph to describe the growth of the plants.

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Hints and CommentsOverviewStudents investigate a line graph showing the meanplant height over time.


A line graph with time on the horizontal axis canshow a growth process over time. This line graphshows the mean plant height over time. This gives ageneral impression of the type of growth. It does notshow individual plant growth. Because noinformation is available on the height of plants onother days than at the start and on the 10th, 12th, and14th day, the data points are connected with linesegments.

Planning

Students may work on problem 5 in small groups. Youmay want to discuss problem 5c in class.


5. a. He took the mean of all plant heights on day 10.He could have done so from the histogram andcalculated the mean height. This point meansthat on day 10 the mean plant height was30 mm.

b. The plants seem to grow most between day 12and day 14. In the graph, the line is steepest inthis time span.

c. Answers may vary. Sample response:

An advantage is that the growth pattern of theplants over time is visible. A disadvantage isthat all information on the spread of theheights and the individual plant heights is lost.


Notes

Discuss the vocabulary fora box plot: minimum, firstquartile, median, thirdquartile, and maximum.Be sure students know howto find each of these andwhere they are located ina box plot.


D

Assessment Pyramid

6a

6b

Draw conclusions based onrepresentations of data anddescribe data numerically.

Intervention

Box plots may be new for some students. Provide them with a couple ofdata sets (an odd number of items and an even number of items) topractice finding the minimum, first quartile, median, third quartile, andmaximum. Remind them that first they must order the data from least togreatest, and if two items are in the middle, they must average them.


Box plots can be also be used to compare the height of the plants.

Remember that a box plot is a graph in which the data are groupedinto four groups of roughly equal size. To draw a box plot, you needfive numbers from your data:

• the lowest number, or minimum

• the middle of the lower half, or first quartile (Q1)

• the median

• the middle of the upper half, or third quartile (Q3)

• the highest number, or maximum

6. a. Explain what these box plots tell you about the growth ofthe plants from the 12th to the14th day.

b. What can you tell about the growth of the plants from thebox plots that you cannot tell from the line graph? Fromthe histograms?

Using Data D

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Hints and CommentsOverviewStudents interpret and compare box plots showing theheight of plants after different growth periods. Theycompare the different graphs representing the samedata set.


Box plots were introduced and used in the unitDealing with Data, and they are revisited here. Boxplots can be helpful to compare large groups of data.They are often used to compare the center and spreadof different samples, especially when the samples arelarge.

Box plots do not show all characteristics of a data set.They do not show the mean, and in this case they donot reveal that there is no apparent center of the data.This is visible in the histograms on pages 34 and 35.

Planning

Students may work on problem 6 in small groups.You may want to discuss problem 6b in class.


6. a. The median has gone up from about 50 mm toalmost 100 mm. The largest plant height hasincreased from 108 to 180 mm. Note that in theupper box plot, the outlier of 180 mm is notincluded. The right whisker ends at a height of160 mm. The smallest plants have not grown orhave only grown a little. The spread of the datahas therefore increased a lot.

b. Answers may vary. Sample response:

Box plots show the spread of the data on plantheight; they show how consistent the growth is;if the plot is more compact, the growth is moreconsistent. A histogram preserves individualdata. A box plot provides a summary.


Notes

Decide ahead of time howyou want your class tocomplete this activity. Youcould have each group thatcollected data for one ofthe four solutions put alltheir results on a posterand report to the class onthe results. There may besome parts you want eachstudent to do, such asmake at least one graphand write a description ofwhat that graph represents.

A discussion about variousways to show the growth ofthe sprouts over time willbe helpful before they startthat graph.


D

Assessment Pyramid

Represent data graphicallyand numerically.

Generate questions foranalyzing data.

Accommodation

Chunk the activity so certain parts of it are due over a period of two or threedays. To help students write the statements about the mean or median, youmay need to be more specific with the directions, such as, Find the mean ormedian for each of the 7 days and graph these points. Then write thestatement. Pairs of students could do the mean, and other pairs could dothe median.

Writing Opportunity

The bean-growth experiment reports are an excellent way for students towork on their writing skills. Students could apply techniques they havelearned from language arts, speech, or English classes and work as a groupto organize and present their reports to the class.


Presenting the Bean Sprout Data

In the bean sprout growth experiment you conductedon page 2, you collected data and kept a record of thelengths of the bean sprouts and your observations.

Prepare a report for the class on the results of yourexperiment. Your report should include the following:

• the data you collected;

• a graph of the final length of each sprout;

• a written description of what the graph showsabout the final sprout lengths;

• a graph of the growth of the sprouts over time;

• a written description of what the graph showsabout the growth of the sprouts over time;

• some statements about the data in which you usethe mean, median, or mode (select the one you feelis best for your data);

• a statement about how the growth of the beansvaried;

• your conclusions about how the bean sprouts grewin the solution you used;

• a list of the things that affected the results of yourexperiment; and

• a list of the things you would change if you were torepeat the experiment.

Activity

➞

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Hints and CommentsMaterialsgraph paper (several sheets per student)

Overview

Students write a report summarizing the experimentthey conducted in Section A. They use statistical tools,such as graphs and numerical measures, to describetheir results.


Statistical tools are often used to describe the resultsof an experiment. It is important to be able to usethese tools correctly.

Planning

Students may work in small groups on this activity.Students may write their reports during class, or youmight assign them as homework.

Comments About the Activity

The design of an experiment is an importantstatistical process. In this experiment, the beans werethe subjects. Water was the treatment for the controlgroup. The other treatments were the differentsolutions.

Note: If you replaced the bean growth experiment inSection A with another experiment, you need tochange the activity on page 38 accordingly. Studentsmight discuss whether the experiment contained biasand whether they could design the experiment inanother way to reduce the bias that may haveresulted.

Solutions and SamplesActivity

Reports will vary. The final lengths will vary with thesolution. For example, in one experiment, the finallengths of bean sprouts grown in tap water were 44,57, 95, 95, 85, 70, 45, 75, 40, and 90 mm. These dataillustrate that, in some instances, data are distributedin such a way that there is no apparent center of thedata. This fact is obscured by the box plot but isrevealed by the stem-and-leaf plot, where there is acluster of lengths at the low end and a cluster at thehigh end, with few lengths in the middle. Studentsmay select various ways to plot the final lengths: anumber-line plot, a box plot, and a stem-and-leaf plotare some examples. Students may also choose a varietyof ways to plot the lengths of the sprouts over time.For example, they may use several graphs or differentcolors.

The mode does not give significant information aboutthe growth of the beans over time. In one given dataset, there may be no mode, one mode, or more thanone mode. The mean and median give some indicationof the center of the data. The mode doesn’t have to beat the center.


Notes

Students will create a classhistogram of self-sticknotes on which they havewritten the final length ofeach bean sprout. Use adifferent color note foreach of the four solutions.Prepare a number line onthe board ahead of time,choosing an appropriatescale for the range of thedata. Each group will need10 self-stick notes, one foreach plant.

8b Each group could put acopy of their box plot on atransparency. Then the boxplots could be comparedby overlapping them on anoverhead projector.


D

Accommodation

Have available in the classroom directions for making a box plot for thosewho have difficulty remembering all the steps required to make one.Specifically for 8a, you may want to give each student a paper with thenumber line drawn on it, marked with the common scale. Direct them tofirst write their 10 data items in order from least to greatest and identify theminimum, first quartile, median, third quartile, and maximum on thissheet. Last draw the box plot above the number line, not directly on theline.


ADUsing Data

Your teacher will give you some self-stick notes. Write the final lengthof each bean sprout on a separate self-stick note. Put your notes onthe number line your teacher has drawn on the board to make ahistogram. Use the finished histogram on the board to answer thefollowing questions.

7. a. What is the average length of the bean sprouts?

b. How did the bean sprouts in your solution compare to thebean sprouts in other solutions? What conclusions can youdraw about the effects of your solution on the way the beansprouts grew?

The histogram gives a general picture of the growth of the beansprouts in each solution.

Suppose you want to compare the effects of the different solutionsmore directly. One way to compare solutions is with box plots.

8. a. Make a box plot for the final lengths of the bean sprouts inyour solution. As a class, decide on a common scale for thebox plots. Why is a common scale necessary?

b. Below your box plot, write the type of solution your groupused. Put your box plot on the board together with thosefrom other groups. How do the box plots help you comparethe different solutions?

c. What is the difference between the information you can seeon a histogram and the information you can see in a box plot?


Hints and CommentsMaterialscentimeter ruler (one per student);colored self-stick notes (four notes per group ofstudents)

Overview

Groups compare the growth of their beans by meansof a histogram and box plots.


Box plots are very useful when comparing data fromdifferent groups, which is the case here. Individualdata, however, are lost in a box plot. Since there areonly very few data in each group, it is possible tocompare the individual data (for instance, in anumber-line plot or a stem-and-leaf plot).

Students learned how to draw a box plot in Dealingwith Data. This is revisited on Student Book page 37.In problem 6 on that page students investigate twobox plots on plant height.

Planning

You might want to work on problem 8 as a wholeclass.


7. A different color self-stick note should be used foreach of the four solutions. When deciding on thelength and scale of the number line, take intoaccount the size of the self-stick notes and therange of the data (the shortest and longest lengthsrecorded).

8. a. A suitable scale for the box plot will probablybe from 0 to10 units beyond the length of thelongest sprout.

c. Box plots offer a quick way to see how growthwas affected by each solution. The ends of thewhiskers mark the range of growth. If thewhisker on one end is long, there was at leastone sprout that showed inconsistent growthcompared to the others in the set. This is calledan outlier.



The average length of the bean sprouts can beestimated from the histogram by looking at thepattern of the final sprout lengths andapproximating the balance point.

b. Answers will vary. Sample response (consideringthe results of a sample experiment):

The sprouts grown in tap water grew the most.Sprouts grown in cola and lemon-lime sodagrew more than those grown in a salt solutionbut less than those grown in tap water. Thesprouts grown in a salt solution grew the least.

8. a.–b. Answers will vary. Sample response:

Using the results from a sample experiment,the box plots for the final lengths of the beansprouts are as follows.

By using a common scale, the informationfrom the box plots can be compared. Forexample, the box representing the middle 50%of the final sprout lengths in one solution canbe compared with the boxes of other plots.

c. Answers will vary. Sample response:

A histogram provides a general picture of theset of data yet preserves individual data. Youcan detect gaps or clusters in the data. A boxplot provides a summary of the data and can beused to compare sets of data. Box plots alsoshow how consistent the growth was. If the plotis more compact, the growth is moreconsistent. However, individual values are lost.

Box Plots for Final Length of Sprouts in Various Solutions

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Notes

If you have postedexamples of each kind ofgraph on your walls thatthe students have made inthis section, you could askwhat is the advantage ofeach graphicalrepresentation as youdiscuss the Summary.


D

Intervention

If students had difficulty making box plots, they may need to review how tofind the minimum, first quartile, median, third quartile, and maximum onsets with 12, 13, 14, or 15 data items and how to draw the box. They shouldunderstand that the box represents the middle 50% of the data.


Using Data

A data set can be graphed in different ways.

A histogram is a general picture of the data. It allows you to see howthe data are distributed.

A box plot is a graph in which the data are grouped into four groupsof approximately equal size. A box plot gives you a summary of thefive values:

• the lowest number, or minimum

• the middle of the lower half, or first quartile (Q1)

• the median

• the middle of the upper half, or third quartile (Q3)

• the highest number, or maximum

D

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Hints and CommentsOverviewStudents read and review vocabulary in the Summary.


Notes

1 You may have studentscut out a histogram andactually balance thehistogram on a pencil tosee how the balance pointis the same as the mean.


D

Assessment Pyramid

2

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Assesses Section D Goals

Intervention

Some students may need to practice estimating the median in a histogramby counting off an equal number of frequencies until they meet in themiddle.


These five values divide the data in four groups, each of whichcontains about 25% of the data. A box plot shows an overall pictureof the data but does not allow you to see details. Box plots areparticularly useful for comparing several data sets.

A line graph (graph over time) shows change over a period of time—for example, the change in the length of the bean sprouts from day today. The graph allows you to look for trends in the data.

A description of the data and what you can see in the graph can makeinterpreting the graph easier. Statements about the center of the datausing mean or median and about the spread using range or quartilescan also give insights into the data.

1. Can you estimate the mean, median, or mode from each of thesetypes of graphs? Explain.

• a histogram • a box plot

2. When is it helpful to use box plots?

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Hints and CommentsOverviewStudents read and review the Summary and completethe Check Your Work problems.

Planning

After students complete Section D, you may assign ashomework appropriate activities from the AdditionalPractice for Section D, located on pages Student Book67 and 68.


1. In a histogram, the mode is easily seen since it isrepresented by the tallest bar. Note that the modeis not the height of the bar, but the datarepresented by the bar. It is difficult to estimatethe mean from a histogram; it is kind of the“balance” point of the distribution. The mediancan be found in a histogram by finding thehalfway point of the number of data. You can findthis by counting in the bars. If, for example, thereare 15 values, the eighth one represents themedian. You can also count from both ends at thesame time until you meet in the middle; this is themedian.

In a box plot, the median is drawn in the box asone of the summary points. There is no way tofind the mean and the mode from just the boxplot.

2. Different answers are possible. Box plots are veryuseful if you want to compare groups of data or ifyou want a summary of key points in the data.


Notes

Before having studentsstart this problem, it maybe helpful to discuss theterm cost of living and howthis might influence thevalues shown in this table.


D

Assessment Pyramid

3ac

3b


Accommodation

With problem 3b, some students may find it difficult to use 25 data items tomake a box plot as found in Group 1. One possibility is to have them onlymake a box plot for Group 2. Another possibility is to have them make adouble box plot but only use the data from this page.


The table contains the starting salaries of teachers in secondaryschools in different countries in 2001. The salaries are given indollars and adjusted for different money values among countries.The countries have been coded by grouping the continents inwhich they are located: I—North America, Europe, Australia, andNew Zealand; II—South America, Africa, and Asia.

3. a. What salary differences between the two groups of continentsdo you expect to find?

b. Make box plots to represent the starting salaries of teachers inthe two continent groups.

c. How would you describe the differences in starting salaries inthe two continent groups?

Country Starting Salary Continent Group

(in dollars)

Argentina 11,000 II

Australia 28,000 I

Austria 25,000 I

Belgium (Fl.) 31,000 I

Belgium (Fr.) 30,000 I

Brazil 17,000 II

Chile 12,000 II

Czech Republic 12,000 I

Denmark 30,000 I

Egypt 2,000 II

England 23,000 I

Finland 23,000 I

France 24,000 I

Germany 43,000 I

Greece 20,000 I

Hungary 8,000 I

Iceland 23,000 I

Indonesia 1,000 II

Ireland 24,000 I

Source: Organization for Economic Cooperation and Development

Using DataD


Planning

The data students need to make the box plots arespread over this page and the next.


3. a. Your answers may vary, but you may noticethat the set of continents in Set I seems toconsist primarily of countries that have a highlevel of industry, education, and generally goodeconomic conditions, compared to thecontinents in Set II that have countries wherethe standard of living is low and industrialgrowth is not yet in place. It might bereasonable to conclude that teachers in Set Iwould earn more than teachers in Set II.

b. Your box plot may look a little different if youchose to use another scale.

c. Both the median and range of the yearly salariesin the group that includes North America, Europe,and Australia/New Zealand are much larger thanin the other groups of continents ($24,000 and$44,000 as opposed to $11,000 and $24,000).Almost all of those in Set II are below the mediansalaries for Set I. The United States data point isaround the third quartile (Q3) of the first group,and this is higher than any of the countries in thesecond group.

5,0000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000

Salary (in dollars)

Box Plots of Salary (in dollars)

Co

nti

ne

nt

I

II



Notes

For Further ReflectionReflective questions aremeant to summarize anddiscuss important concepts.


D

Assessment Pyramid

FFR


Extension

You may wish to have students write a paragraph discussing the advantagesand disadvantages of a histogram, box plot, or line plot.


Country Starting Salary Continent Group

(in dollars)

Italy 25,000 I

Korea (South) 25,000 II

Malaysia 14,000 II

Netherlands 29,000 I

New Zealand 18,000 I

Norway 29,000 I

Paraguay 14,000 II

Peru 6,000 II

Philippines 11,000 II

Portugal 20,000 I

Scotland 22,000 I

Slovakia 5,000 I

Spain 31,000 I

Sweden 23,000 I

Switzerland 49,000 I

Thailand 6,000 II

Tunisia 21,000 II

United States 29,000 I

Uruguay 6,000 II

Write a paragraph that helps another class to prepare to do the beansprout experiment. Make sure you include any changes that youwould make that might make it easier to collect data that gives youinformation you want.

Hints and CommentsOverviewStudents complete the For Further Reflectionproblem.

Planning

After students complete Section D, you may assign ashomework appropriate activities from the AdditionalPractice section, for section D, located on StudentBook pages 67 and 68.


3. a. Your answers may vary, but you may noticethat the set of continents in Set I seems toconsist primarily of countries that have a highlevel of industry, education, and generally goodeconomic conditions, compared to thecontinents in Set II that have countries wherethe standard of living is low and industrialgrowth is not yet in place. It might bereasonable to conclude that teachers in Set Iwould earn more than teachers in Set II.

b. Your box plot may look a little different if youchose to use another scale.

c. Both the median and range of the yearly salariesin the group that includes North America, Europe,and Australia/New Zealand are much larger thanin the other groups of continents ($24,000 and$44,000 as opposed to $11,000 and $24,000).Almost all of those in Set II are below the mediansalaries for Set I. The United States data point isaround the third quartile (Q3) of the first group,and this is higher than any of the countries in thesecond group.


Answers will vary depending on students’ experienceswith the bean sprout experiment.

You may have students share their paragraphs in classand write one paragraph summarizing the advicefrom the class as a whole.


5,0000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000

Salary (in dollars)

Box Plots of Salary (in dollars)

Co

nti

ne

nt

I

II


Teachers MatterE

Section FocusIn this section, students investigate the relationship between twovariables in scatter plots, tables, and three-dimensional plots.They determine if there is a strong, weak, positive, negative,linear, non-linear, or no apparent correlation between the twovariables. They also look for linear patterns in scatter plots.Students draw conclusions based on scatter plots andcorrelations and learn that a relationship between two variablesis not necessarily a cause-effect relationship.

Pacing and Planning

Additional Resources: Additional Practice, Section E, Student Book page 68

Day 14: Growing Babies Student pages 44–47

INTRODUCTION Problems 1 and 2 Interpret general patterns in a table of datathat compares the length of a baby’s bodyto the circumference of their head.

CLASSWORK Problems 3–5 Compare a scatter plot and a three-dimensional graph to data represented ina table.

HOMEWORK Problems 6 and 7 Investigate and extend the patterns in thedata for growing babies and analyze limitson extending these patterns.

Day 15: Growing Babies (Continued) Student pages 48–49


CLASSWORK Problems 8–12 Introduce strong, weak, positive, andnegative correlation of data in a scatterplot and discuss the relationship betweencorrelation and cause-effect relationships.

HOMEWORK Activity, page 49 Collect data on two variables and determineif there is a correlation between them.

Day 16: Summary Student pages 50–52


REVIEW Check Your Work Student self-assessment: Interpret theFor Further Reflection strength of correlation in scatter plots.

Teachers Matter Section E: Correlating Data 44B

Teachers Matter E

Materials

Student ResourcesNo resources required

Teachers’ ResourcesNo resources required

Student MaterialsNo resources required


Learning Lines

Correlation in Scatter Plots

In Section A, students were introduced tostudying patterns in data represented in scatterplots. A scatter plot can be used to describe therelationship between two quantitative variables.In this section, students continue studyingpatterns in two variable data in differentrepresentations. In doing so, students areinformally introduced to the idea of linearcorrelation. Correlation is a measure of the linearassociation between the two variables.

Students are not expected to learn how tocompute the strength of the correlation (calledthe correlation coefficient) in this unit. An informal,intuitive understanding of the strength and typeof the correlation is sufficient.

If the cloud of points forms a tight pattern, thecorrelation is referred to as strong. If the pointsform a loose pattern, the correlation is referred toas weak. When the data in the scatter plot form a“cloud” of points that are evenly spread across anarea, there is no correlation between the databeing graphed. If the cloud of points in a scatterplot slopes upward and to the right, there is apositive correlation between the variables. If thecloud of points slopes downward and to the right,there is a negative correlation between thevariables.

In Section F, students’ intuitive understanding isextended by including straight lines to describe alinear relationship.

Correlation and Cause-Effect

Students learn that the existence of a correlationbetween two variables, however, does not meanthat a cause-effect relationship exists. Sometimesa third variable can be found that explains therelationship. Reasoning alone is sometimesenough to recognize a relationship and tellwhether there is a strong correlation and/or acause-effect relationship. At other times, moredata is needed. Students should be able tocontribute examples of strong cause-effectrelationships.


Students will:

• be able to analyze two variable data in tables,graphs, and three dimensional plots;

• be able to describe the relationship betweentwo variables;

• be able to identify the degree of correlationbetween two variables; and

• know that a strong correlation between twovariables does not imply a cause-effectrelationship.

a b

c d


Notes

To engage students,you may want to have abrief discussion aboutcoincidence versus cause-effect relationships andhow you can determine thedifference.

Give students time to studythe table. Then let them tellyou what they learned fromthe graph. If necessary, askmore specific questions,such as how many babieshad a head circumferenceof 33 cm.

CorrelatingData

E

Vocabulary Building

You might use a tape measure to demonstrate how doctors measure thecircumference of a baby’s head and review the definition of circumference.

Intervention

You might ask some additional questions to help students who haveproblems understanding the meaning of the data in the table:

• How many babies with a head circumference of 34 cm had a body lengthof 50 cm? (Nine babies.)

• What is the most frequent combination of body length and headcircumference in the table? (Body length of 49 cm and headcircumference of 34 cm occurs 10 times.)


In 1962, researchers studied 100 newborn babiesto see whether there was a relationship betweenthe length of the body and the circumference ofthe head. The table shows the results of thisstudy.

Most tallies appear along the diagonal thatextends from the lower left to the upper rightcorner of the graph.

ECorrelating Data

Growing Babies


47

39

38

37

36

35

34

33

32

48 49 50 51 52 53 54 55 56

Body Length (in cm)

Cir

cu

mfe

ren

ce o

f th

e H

ead

(in

cm

)

III

I IIIIII

I IIII

IIII

IIII

IIIIIIIII

I

IIII

IIIIIIII

III

IIIIIIII

IIII

IIIII

IIIIII

I

I

I

III

I

I I

I

I

I

I

I

I I

II

Section E: Correlating Data 44T

Hints and CommentsOverviewStudents interpret general patterns in a table.


The illustration shows that the distinction between agraph and a table is not always very clear. The tableused here can almost be interpreted as a scatter plot.However, the table depicts three variables: bodylength, head circumference, and the frequency withwhich certain combinations of length andcircumference occur.

Note: Students may notice that the horizontal scale isabove the table instead of below, which is morecommon in graphs.


Notes

Have students share theirstrategies for interpretingthis graph.

1 Explain that in relation tothe diagonal means wherethe tallies are— above,below, or on the diagonal.

Make sure students answerproblems 1 and 2 for thetable on page 44 of theStudent Book.

3a You may need toexplain what is meant by(0, 0).

3b If needed, ask if theycan tell from the scatterplot how many babies hada body length of 52 cmand head length of 34 cm.

4 Remind students tocarefully examine thenumbers on both axes ofthe graph to answer this.

CorrelatingData

E

Assessment Pyramid

3


Extension

You might ask students to indicate which babies seem to be different fromthe others and why. Students might suggest that the baby with a bodylength of 53 centimeters and a head circumference of 33 centimeters has asmall head compared to other babies of that length.


1. In relation to the diagonal, where do you find the babies withlarge heads relative to their lengths?

2. As the body lengths increase, what happens to the head sizes ofthe babies?

The scatter plot displays a picture of the information in the table.

3. a. Explain why the axes do not start at (0, 0).

b. What information seems to get lost in the scatter plot whencompared with the table?

4. Are there babies for whom body lengths are almost equal tohead sizes?

Correlating Data E

46 48 50 52 54 56 5830

32

34

36

38

40

Body Length (in cm)

Cir

cu

mfe

ren

ce

of

the

Head

(in

cm

)


Hints and CommentsOverviewStudents compare the table on Student Book page 44with a scatter plot of the same data.


In a scatter plot, two variables can be depicted.Sometimes a small number is used beside a point toindicate that this point represents more than oneoccurrence of a certain combination of variables. Butin this example, information about which pointsrepresent more than one baby is lost.

Planning

Students may work on problems 1–4 in small groups.


4. Suggest that students carefully examine thenumbers along both axes of the graph. Refer toSection C where the scaling of the axes and thelost origin were discussed. A scatter plot drawnwith both axes starting at the same number (forinstance, 30) and having the same scale willclearly show that head circumference is signif-icantly smaller than body length.


1. Above the diagonal. The data for babies with largeheads in relation to their lengths can be found onthe upper left side of the table.

2. As the baby’s length increases, the headcircumference also increases.


If the axes started in (0,0), the axes wouldbecome longer. In the lower left part in theplot, no points would be located since bodylengths start around 45 cm and headcircumferences around 30 cm.

b. In the scatter plot, it is not clear which pointsrepresent more than one baby.

4. No. There are no babies who have the same headcircumference and body length.


Notes

5b Students could workwith a partner to discussthe advantages anddisadvantages before theywrite their answer.

The cartoon has studentsreexamine the graph onpage 44 and see if it lookslike there is a correlationbetween a baby’s bodylength and circumference.

CorrelatingData

E

Accommodation

A three-dimensional graph may be new for many students, so give themtime to study the graph. You could have them tell a partner how todetermine the height of each bar (whether the bar represents 1 baby,2 babies, 3 babies, and so on) and then check page 44 to see if they arecorrect.

Two students could read the parts of Juan and Brenda on this page and thenext to assist those who have difficulty reading.


To show body lengths, head circumferences, and the number ofbabies studied, a three-dimensional graph can be drawn.

5. a. What information does the purple bar in the graph represent?Check this information against the data in the table.

b. What are the advantages and disadvantages of athree-dimensional graph?

Juan and Brenda have a discussion about the data.

Correlating DataE

Body Length (in cm)

32 3

4 3

6 3

8

Head

Circu

mfe

rence

(in

cm

)

4850

5254 56


I think you can say that thereis a relationship between ababy’s body length and thecircumference of its head.

I’m not sure about that.The tallies seem to be justspread over the chart.


Hints and CommentsOverviewStudents read information from a three-dimensionalgraph, which represents the data for body length andhead circumference. Students read the first part of acartoon in which Juan and Brenda discuss therelationship between a baby’s body length and thecircumference of its head.


Three-dimensional graphs can be used to graph dataon three variables simultaneously. This graph is athree-dimensional histogram.

Planning

Students may work on problem 5 in small groups.After students have completed problem 5, you maywant to discuss the three different representations ofdata covered thus far.


5. Students might make connections to top views,side views, hidden views, and so on, from theirwork in the Geometry strand units. You might askstudents to make a top view of the graph.


5. a. The purple bar represents three babies, eachhaving a body length of 52 centimeters and ahead circumference of 34 centimeters. Thisinformation can be found in the table onstudent book page 44 in the corresponding cell:it shows three tally marks.


The advantages are that the graph preservesmost of the information and offers visualimpressions. The disadvantages are that thegraph is difficult to interpret, and some barsmay be hidden from view by taller bars.


Notes

6 Emphasize that it isimportant to give thereason why they agreewith Brenda or Juan.

7a Direct students tolook at the scatter plot onpage 45 to help determinea reasonable circumference.

CorrelatingData

E

Assessment Pyramid

6

7

Draw conclusions based onrepresentations of data.


Accommodation

To help students relate Brenda and Juan’s comments to the scatter ploton page 45, suggest that they copy it and write numbers next to the datapoints, indicating the number of babies represented by each point.

Intervention

If necessary, have students review the definition of outlier from Section A—a point outside the cluster in a scatter plot.


ACorrelating Data

6. Do you agree with Brenda or Juan? Explain.

7. a. What do you think the head circumference is for a baby with abody length of 55 centimeters (cm)? How did you determineyour answer?

b. A one-year-old child has a height of 92 cm. What can you tellabout the circumference of the child’s head?

E

You can say that the longerthe baby, the larger thecircumference of its head.

Yes, I agree that you can find those exceptions.I can even find a baby that is 53 cm long andhas a head circumference of 33 cm, but thisseems to be an outlier!

If you look at the table,most babies fit in ageneral pattern.

Okay, but the relationship you describedis not a strong one, and I think when thebabies start growing, this relationship willget even weaker or disappear.

That’s not true for all babies. Lookat the table. I see babies with bodylengths of 47 centimeters (cm) andof 52 cm, and both have a headcircumference of 34 cm.


Hints and CommentsOverviewStudents read the second part of the cartoon in whichJuan and Brenda discuss the relationship between ababy’s body length and the circumference of its head.

They interpret the data and describe the generalpattern of the relationship between a baby’s bodylength and the circumference of its head.


Looking for a general pattern in the scatter plotinformally introduces the concepts of linearcorrelation between two variables and the line of bestfit. This ties in with the activities connected to thescatter plot in Section A and prepares for theproblems in the rest of Section E and in Section F.

Planning

Problems 6 and 7 may be assigned as homework.Discuss students’ answers briefly in class.


6. Individual data can detract from the generalpattern without actually disturbing it. Therelationship is moderately strong and could getweaker. As a person grows, his or her body lengthchanges more rapidly than head circumference.

7. a. Students can imagine or draw a line of best fitand use it to estimate the best-fitting headcircumference. The line of best fit is introducedin Section F. In this case one should be carefulwith drawing the line. Not all data are shown inthe graphs. Some points in the graph refer tomore than one baby. In fact, these pointsshould be "weighted" to better represent thepattern in the data.

b. The data were collected from newborn babies;the relationship between the variables may notbe the same for older children.


6. Answers will vary. Sample student responses:

I can see a pattern, so I agree with Juan. ButBrenda makes some very good points to thinkabout. I think Brenda is right when she says thatthe relationship will get weaker as the babiesgrow. As we get taller, our heads don’t seem to getthat much bigger.

7. a. Estimates will vary. Sample estimate:

The head circumference would be about 38.5centimeters. One method is to draw a diagonalline as shown on the scatter plot below (known asthe line of best fit). This line can be used toestimate the head circumference.

b. Answers will vary. Sample student responses:

The table or graph does not provide thisinformation, so you cannot tell anythingabout it.

From the graph, I can tell that the baby’s headcircumference should be greater than 40 cm.Since I don’t know how the relationshipbetween a baby’s length and head circum-ference continues after a length of 58 cm,I cannot approximate the baby’s headcircumference.

46 48

(55, 38.5)

50 52 54 56 5830

32

34

36

38

40

Body Length (in centimeters)

Cir

cu

mfe

ren

ce o

f th

e H

ead

(in

cen

tim

ete

rs)


Notes

The terms strongcorrelation and weakcorrelation are informallydefined. Explain that thereare many arrangements fora weak correlation—thosethat are moderately strongdown to those that showalmost no correlation.

9 In all graphs, you canfind points for which xexactly equals 4, and findthe corresponding y-value,but that is not the focus ofthis problem. Studentsshould use the shape ofthe cloud, or an imaginaryline that summarizes it, toestimate what y-value bestfits a data point with anx-value of 4.

CorrelatingData

E

Assessment Pyramid

8

Identify the degree ofcorrelation betweenvariables.

Vocabulary Building

In the vocabulary section of their notebook, have students sketch anexample of each type of correlation: strong, weak, or no correlation.

Extension

You may want students to refer to the scatter plot on Student Activity Sheet2 from Section A and ask them to describe the correlation between thepercent of urban population by state and per capita income. (The scatterplot shows a moderate to weak correlation between percent of urbanpopulation and per capita income.)


Scatter plots are often made to investigate the relationship betweentwo variables. The points on a scatter plot may look like a “cloud.”When the cloud is long and slender, there is a strong correlationbetween the two variables. When the correlation is strong, knowingsomething about one of the variables helps you know how the othervariable will behave. If the cloud of points is very scattered or in acircle, no underlying relationship exists between the variables.Knowing something about one of the variables cannot tell youabout the other. In this case, there is no correlation between thetwo variables.

8. Describe the correlation in the scatter plots. Indicate whetherthere seems to be no correlation, a weak correlation, or astrong correlation.

9. For which of the above scatter plots can you best predict thevalue of y when x � 4? Explain your reasoning.

Correlating DataAE

0 1

1

2

3

4

5

6

2 3 4 5 6 0 1

1

2

3

4

5

6

2 3 4 5 6

0 1

1

2

3

4

5

6

2 3 4 5 60 1

1

2

3

4

5

6

2 3 4 5 6

y y

y y

x x

x x

a b

c d


Hints and CommentsMaterialsStudent Activity Sheet 2, optional (one per student)

Overview

Students learn the meaning of linear correlation andhow the strength of it can be read from a scatter plot.


This unit introduces a rather informal notion of linearcorrelation. The correlation coefficient, which is ameasure for the strength of the correlation, is notintroduced in this unit. Students use every day termslike weak and strong to informally describe thestrength of the correlation. In this section, only linearcorrelation is discussed. However, there can also benon-linear correlation, such as quadratic or hyperboliccorrelation. In Section F, an example of a non-linearrelationship is informally discussed.

Planning

Students may work individually or in small groups onproblems 8 and 9.


8. There are many gradations between a weakcorrelation and a strong correlation. Studentsmight say that scatter plot b shows a moderatelystrong correlation, closer to strong than to weak.


8. Scatter plots c and d show no correlation.

Scatter plots b shows a weak correlation.

Scatter plots a shows a strong correlation.

9. You can best predict the value of y when x � 4 inscatter plot a because the points form a tightcloud that resembles a straight line. To make aprediction of the value of y when x � 4, draw oneline vertically from 4 to the cloud of points andanother line from the cloud left to the verticalaxis, as shown below. A possible value for y is1.5 or 2.

0 1

1

2

3

4

5

6

2 3 4 5 6

y

x

a


Notes

10a It may help to askstudents which directionsthey move for locating apositive number (right orup) and which directionsfor locating a negativenumber (left or down).

11 If students are havingdifficulty, it may help torephrase each case, suchas, As a person’s heightincreases, does his or herpulse rate increase ordecrease, or is there norelationship?

Students could discusseach case with a partner orsmall group and then sharewith the whole class.

CorrelatingData

E

Assessment Pyramid

10b


Vocabulary Building

Students could sketch an example of a scatter plot with a positive correlationand one with a negative correlation in their notebooks. Also have studentsadd cause-effect relationship and an example.


Students may ask the physical education teacher or the biology teacher forinformation about some of the cases described in problem 11, such as therelationship between training time and pulse rate. However, theexplanation of the presence or absence of correlation should remain theprimary focus.


In the diagrams for problem 8, you can see that a correlation can beweak or strong. Correlations can also be positive or negative.

10. a. What do you think is meant by the phrase “negativecorrelation”?

b. Which of the scatter plots on the previous page show anegative correlation?

11. For each of the following cases, decide whether there is nocorrelation, a strong correlation, or a weak correlation betweenthe two variables mentioned. If there is a correlation, is it positiveor negative?

• a person’s height and pulse rate

• the number of hours of sports training per week and pulse rate

• the height of a dinosaur and the length of its tail

• results on a math test and a science test

• temperature outside in the summer and kilowatt hours ofelectricity used

• number of children per household and number of televisionsets per household

• number of hours students study and their grade pointaverages

In a cause-effect relationship between two variables, a change in onevariable directly causes a change in the other.

12. a. If you expect a strong correlation between the two variables,can you be sure that a cause-effect relationship exists? Whyor why not?

b. Do you think there is a cause-effect relationship in any of thesituations in problem 11? Explain your reasoning.

Collect some data about one of the cases in problem 11 and usethe data to check your answer about the correlation. Be sure tothink carefully about how you will select your sample.

Correlating Data E


Hints and CommentsOverviewStudents are introduced to the concept of “negativecorrelation.” They decide what kind of correlation willexist between variables in different situations, andthey think about possible cause-effect relationships.In the activity, they collect data on different situationsto check their statements about correlation.


Solutions and Samples10. a. Answers may vary. A negative correlation exists

if the values of one variable increase while thevalues of the other decrease. In a negativecorrelation, the cloud of points in a scatter plotwill slant from the upper left to the lower right.Using similar reasoning, a positive correlationexists if the values of one variable increasewhile the values of the other also increase. In apositive correlation, the cloud of points willslant from the lower left to the upper right.

b. Scatter plots a and b show a negative correlation.

11. Answers will vary. Students may offer differentperspectives that are valid. For some of theexamples, students can carry out an experimentor survey. Sample responses:

• a person’s height and pulse rate: no correlation

• the number of hours of sports training perweek and pulse rate: negative, weak correlation

• the height of a dinosaur and the length of itstail: positive, strong correlation.

• results on a math test and a science test:positive, weak correlation

• temperature outside and kilowatt hours ofelectricity used: positive, strong correlation

• the number of children per household and thenumber of television sets per household: nocorrelation

• the number of hours students study and theirgrade point averages: positive, strong correlation

12. a. No. Even though there may be a strongcorrelation between two variables, you cannotbe sure that one variable actually causes theother. It is reasonable to expect that sportstraining does affect pulse rate. But while gradeson math and science tests may be a function ofattendance or effort, understanding one subjectdoes not cause one to understand the other.

b. Answers will vary. Sample response:Yes, the temperature outside in the summerand the kilowatt hours of electricity used havea cause-effect relationship.

Activity

Answers will vary. Sample response:

The results of investigating the relationship betweena person’s height and pulse rate are as follows.

There is no correlation.

40

4'8"

4'9"

4'10"

4'11"

5'0"

5'1"

5'2"

5'3"

5'4"

5'5"

5'6"

5'7"

5'8"

45 50 55 60 65 70 75 80 85 90 95 100 105

He

igh

t

Pulse (in beats/min)

Height/Pulse

5'7"/40

5'5"/55

5'2"/60

5'5"/65

5'6"/70

5'3"/72

5'6"/78

5'4"/100

5'6"/45

5'4"/58

5'1"/60

5'4"/66

5'0"/70

5'2"/75

5'3"/92

5'3"/103

5'2"/50

5'7"/60

4'8"/60

5'8"/70

5'7"/72

5'6"/78

5'1"/95

4'9"/105

Height/Pulse Height/Pulse


Notes

As you process theSummary with the class,you could have the studentssketch a scatter plot for eachlinear correlation: strong,weak, no correlation,positive, or negative.Another possibility is tohave a group of 6 or 7students model with theirbodies what each corre-lation would look like.Use removable painter’stape on the floor to markthe x-axis and the y-axis;then have students modela strong correlation (standin a straight line), a weakcorrelation, and no corre-lation (stand at randomplaces in the quadrant).

CorrelatingData

E

Assessment Pyramid

1

Assesses Section E Goals

Parent Involvement

Students could explain to their parents about correlations and cause-effectrelationships and have the parents share with them an example of a cause-effect relationship.


Correlating Data

A scatter plot shows information about two variables that are pairedin some way and can help you see whether a relationship existsbetween the variables. By looking for a trend in the scatter plot,you can see that a relationship exists.

If the points on a scatter plot are close to forming a straight line, thereis a strong correlation. If the points are scattered all over and do notfollow a pattern, there is a weak correlation.

Correlations can also be positive or negative. A negative correlationexists if the values of one variable increase while the values of theother decrease.

A scatter plot with a strong correlation doesnot imply that a cause-effect relationshipbetween the variables exists. Other kindsof analysis must be done to determinewhether a change in one variable causesa change in the other.

E

1. Explain how a scatter plot can help you understand correlation.

0 1

1

2

3

4

5

6

2 3 4 5 6

y

x


Hints and CommentsOverviewStudents read the Summary and complete problem 1from Check Your Work

Planning

After students complete Section E, you may assign ashomework appropriate activities from the AdditionalPractice for Section E, located on Student Book page 68.


1. Answers will vary, but you can say that if the pointsin a scatter plot form a tight cloud that is almosta straight line, there is a strong linear correlation.If the points are a scattered more widely, thecorrelation is weaker; if points are in a cloud thatis nearly a circle, no correlation exists. If pointsare in a tight cloud that curves, there may anothertype of correlation, such as quadratic.


Notes

You might also ask studentsto identify any outliers theysee in these scatter plots.

CorrelatingData

E

Assessment Pyramid

2ab, 3



Be sure students understand what is meant by miles per gallon andmaximum number of eggs per hatch.


19000

2000 2100 2200 2300

10

20

30

40

50

60

Mil

es p

er

Ga

llo

n

Weight (in lbs)

Vehicle Fuel Economya

14

12

10

8

6

4

2

00 20 40 60 80 100

Ma

xim

um

No

. o

f E

gg

s p

er

Ha

tch

Size of Bird (in cm)

Birds of Maineb

5034 36 38 40 42 44 46 48

52

54

56

58

60

62

Cir

cu

mfe

ren

ce

of

He

ad

(in

cm

)

Shoe Size (in cm)

Shoe Size

and Head Circumferencec

2. a. Describe the correlation in each of the three scatter plots.

b. In which of these plots might there be a cause-effectrelationship?

3. Find an example of two variables that have a strong correlationbut do not have a cause-effect relationship.




3. One classic example of a strong correlationbetween two phenomena without a causalrelationship is the number of storks found andthe number of babies born in Denmark. Statisticaldata on a regional level shows a strong correlationbetween the two, where obviously there cannotbe a causal relationship.


2. a.

• The data points form a tight cloud that seemsto follow a straight line, so the correlation isstrong. As the weight of the vehicle increases,the miles per gallon decrease; so thecorrelation is negative

• The data points are in a somewhat random,almost circular pattern, so there seems to beno correlation or relationship between the sizeof a bird and the number of eggs hatched.Some large birds hatch lots of eggs and somehatch few eggs; the same thing is true for smallbirds.

• The points are in a rather wide cloud, so thereseems to be a weak correlation. A person with alarger shoe size seems to have a larger headcircumference, so the correlation is positive.On a closer look, it seems as if two clusters ofdata exist. Within these clusters, the data are ina somewhat circular pattern.

b. It seems that the heavier the vehicle, the morefuel it would need to operate, so this examplemight be a case of cause and effect. The graphof the birds of Britain does not even show arelationship, so it would be unwise to searchfor a cause and effect relationship. The shoesize and head circumference plot do show aweak relationship, but having big feet does notcause a large head; both are functions of howbig the person is to start.

3. Your answers will all differ. Compare the exampleyou found to the examples of some of yourclassmates. If a strong correlation exists without acause-effect relationship, often there is anothercommon feature that helps explain thecorrelation. For example, the correlation betweenthe number of schools in a city and the number ofshopping centers may be very strong, but both arefunctions of the population of the city. One doesnot cause the other. Another example might bethe relation between scoring points and makingfouls in a basketball game. The correlation mightbe strong, but this does not mean that makingfouls will increase the number of points. Both arefunctions of how much playing time a player had.


Notes

4i Explain that linear, inthis case, means close toforming a line.

CorrelatingData

E

Assessment Pyramid

4, FFR


Parent Involvement

Students might find it interesting to share the For Further Reflectionproblem with their family members to see if they can think of anysituations where the apparent correlation has to do with another variable.


Correlating Data

4. Match each of the descriptions of correlation below to acorresponding scatter plot.

i. linear, negative, and moderate

ii. linear, positive, and strong

iii. no apparent correlation

iv. linear, positive, and weak

For small children, foot size can be correlated with the ability to read;the larger the foot, the better they can read. But both reading and footsize are a function of age. Find another example of two data sets withstrong correlation but where the relationship is really due to anothercommon factor.

E

a b

c d



Planning

After students complete Section E, you may assign forhomwork appropriate activities from the AdditionalPractice for Section E, located on Student Book page 68.


4. scatter plot a with statement iiscatter plot b with statement iiiscatter plot c with statement iscatter plot d with statement iv


Answers will vary. Sample response:

To carry out the experiment you first decide on thepopulation. If the class is the population, you cangather data from the whole population. If all studentsat school are the population, you will need to take arandom sample. A random sample can be taken byrandomly selecting 5 students from each grade level.

All students from the group in the experimentmeasured height and foot length. These data can begraphed in a scatter plot, with height on the horizontalaxis and foot length on the vertical. The pattern in thescatter plot will show what correlation exists. A famousexample can be found in the Hints and Commentscolumn on page 51T.

Teachers MatterF


Section FocusStudents study the relationship between the lengths and widths of bird eggs by makingscatter plots for various families of birds. They draw lines of best fit on the scatter plotsin order to predict widths for given lengths or vice versa. This leads to an exploration ofthe slope of the line of best fit. They also compare data sets using the best-fit lines.

Pacing and Planning

Day 17: Egg Hunt Student pages 53–55

INTRODUCTION Problems 1 and 2 Use length and width of two birdeggs to sketch and compare their shapes.

CLASSWORK Problems 3–7 Make a scatter plot of the length and widthof warblers’ eggs, draw a best-fit line, andinvestigate its slope.

HOMEWORK Problem 8 Use the best-fit line to predict the width of awarbler’s egg given its length and comparethis width to the actual width.

Day 18: Egg Hunt (Continued) Student pages 56–58


CLASSWORK Problems 9–13 Compare a line of best fit for the warbleregg with the lines drawn for problem 4and use the equation of this line to findthe width for a given length.

HOMEWORK Problems 14 and 15 Compare a line of best fit for all bird eggsto the line for the warbler eggs, and drawconclusions about the shapes of eggs.

Day 19: Egg Hunt (Continued) Student pages 58–63


CLASSWORK Activity, page 58 Compare dimensions of a typical chickenProblems 16–19 egg to the eggs of the birds of Britain.

Make and interpret a scatter plot fortwo-variable fish data.

HOMEWORK Check Your Work Student self-assessment: Compare dataFor Further Reflection sets using best fit lines and investigate

the slopes of these lines.

Additional Resources: Additional Practice, Section F, Student Book page 69

Teachers Matter F

Teachers Matter Section F: Lines That Summarize Data 53B

Materials

Student Resources

Quantities listed are per student, unless otherwisenoted.

• Student Activity Sheets 7–9 (one of each pergroup of students)

• Student Activity Sheets 10 and 11



Student Materials


• Chicken egg (one per group of students)

• Drawing paper (two sheets per student)

• Graph paper (two sheets per student)

• Graphing calculator

• Ruler


Learning Lines

Finding Lines of Best Fit in Scatter Plots

In Sections A and E students have explored patternsin two variable data represented in scatter plots,and they have described the type and strength ofthe linear correlation between the two variables.In this section, students use best-fit lines to furtherinvestigate the correlation between the twovariables.

A linear pattern in a scatter plot can be summarizedby a line through the data. Different lines may beused including an “eyeball” line drawn freehand tocapture the trend, a least squares linear regressionline based on the mean of the data, and a medianfit line based on finding medians as representativepoints. In this section, students only use the“eyeball” line.

Students discuss criteria to decide if a line of bestfit they drew is a “good” line. These criteria caninclude things like: a line going through as manypoints as possible or a line that has as many pointsabove as below it, and so on.

Students are expected to be able to draw a line byeye and to find the equation of this line, as theylearned in Graphing Equations, by finding theslope and the y-intercept.

Using Lines of Best Fit

Students use the lines of best fit to summarize thepattern, to check existing values, to predictunknown values, and to compare data sets. Theaccuracy of the predictions made using a best fitline depends on how the data are scatteredaround the line. When the data are very close tothe line, you are more confident about thepredictions based on that line.

Other Models of Fit for Scatter Plots

Some relationships between variables cannot besummarized with straight lines. The pattern in thescatter plot of the data in that case is clearly not alinear one, but a pattern seems to exist anyway. Acurved model may then be a good one to use.Growth, for example, is often associated with curves.

Note that statistical software and graphingcalculators have the capability of fitting a lot ofdifferent regression models on scatter plots. Everymodel will always generate a graph on the scatterplot even if no relationship between the variablesexists. Often such a model does not really fit thedata or is meaningless in the context. Therefore itis important to always have students first decidewhether a relationship will exist and what kind ofrelationship it will be.


Students will:

• be able to describe the relationship betweentwo variables;

• be able to identify the degree of correlationbetween variables;

• be able to draw a line of best fit in a scatter plotand be able to write the equation of this line;

• know how to use a line of best fit to checkexisting values, to predict unknown values,and to compare different values;

• be able to interpret the slope of a line of best fitin terms of the context;

• know informal criteria for drawing a good lineof best fit; and

• know that sometimes a curved model can betterbe used to summarize a pattern in data than astraight line.


Notes

You might begin this sectionby asking the class whatinformation they learnedfrom the scatter plots theystudied earlier in the unit(pages 4 and 44). In thissection, they will look for alinear correlation in scatterplots and use the equationsfor lines of best fit topredict values.

2 Be sure students notethat the measurements arein millimeters (10 mm =1 cm). Drawing twoperpendicular lines of thecorrect length may helpin sketching the egg.

2 Students should draw astandard egg shape andapply the dimensionsgiven in the table.[Note: Be sure studentskeep their Student ActivitySheets for problems onpage 54.)

Lines ThatSummarizeData

F


Explain that dimensions means length and width.

Accommodation

Highlighting the eider and the whitethroat on the activity sheets is helpful.


FLines That SummarizeData

Egg Hunt

Marisol noticed that a bird had recentlylaid eggs in a nest outside her window.The eggs appeared small, and shewondered whether they had the sameshape as chicken eggs.

1. a. Do you think all bird eggs havethe same shape?

b. How can you check youranswer?

Throughout this section, you will usescatter plots to study the relationshipbetween the lengths and widths ofvarious bird eggs.

On Student Activity Sheets 7, 8, and 9 you will find tables with dataon 96 birds of Britain. Since there are many birds, they are classifiedinto families. Among the data, you will find the average lengths andwidths of the eggs of each bird.

2. a. Draw an eider’s egg using the dimensions found in the table.

b. Draw a lesser whitethroat’s egg.

c. How are the eggs different? What might account for thedifferences?

Quail egg Chicken egg

Section F: Lines That Summarize Data 53T

Hints and CommentsMaterialsStudent Activity Sheets 7–9 (one of each per group ofstudents);drawing paper (two sheets per student)

Overview

A new context about the shape of eggs is introduced,and students think about the question, “Do all birdeggs have the same shape?” Students draw some eggs,using dimensions from a table.


In this section, students will use scatter plots to studycorrelation and use lines of best fit to predict thewidth of an egg if the length is given.

Planning

You may want to discuss problem 1 as a whole class.Students can work in small groups on problem 2.


1. b. Trying to find differences in lengths and widthsfor bird eggs from a table is often difficult. Inthis section, students will learn that a scatterplot is useful in determining such differences.


1. a. Answers will vary depending on the students’knowledge of bird eggs. Most eggs have thesame shape. In the list of 96 birds from Britainon Student Activity Sheets 7–9, for example,only the egg of the common sandpiper isnearly round.

b. Answers will vary. Sample responses:

Measure the lengths and widths of a variety ofbird eggs, real or in photos.

Refer to tables in an encyclopedia or to booksabout birds.

2. a. Eider egg: length 77.6 mm; width 51.9 mm

b. Lesser whitethroat egg: length 16.5 mm;width 12.6 mm

c. Although the eggs are about the same shape,the eider egg is bigger. The eggs are fromdifferent bird families.

77.6 mm51.9

mm


Notes

3a Highlight all thewarblers in the table.There should be 11 ofthem. Also have thestudents make a list of allthe warblers and recordthe length and width of theegg for each. They willneed this data later in thesection.

3b Instruct students tolook for a trend in the datapoints. As the lengthincreases, what happens tothe width?

4a Students could use apiece of spaghetti toexplore different positionsfor a line before theydecide where to draw theirline.

After problem 5, have aclass discussion about howto draw a line that fits bestand how to use this line todescribe the relationshipbetween the variables.


F

Accommodation

For problem 3a, some students may need a discussion about the axesbefore they make the scatter plot. Remind them that they do not have tobegin at (0, 0) for a scatter plot. Discuss how to determine an appropriateminimum value for the x- and y-axis and how to choose the constantinterval for each scale.

Intervention

For problem 4, have students place a ruler over the scatter plot and moveit until the edge seems to cut through the middle of the cloud of points.The line does not need to pass through any data points.


Lines That Summarize DataF

You have discovered that eggs are different sizes.But what about their shapes? Is there a relationshipbetween the lengths and widths of bird eggs? Toanswer these questions, you will begin by lookingat the eggs of one family, the warbler.

3. a. Find all the warblers in the table on Student

Activity Sheets 7, 8, and 9, and make a scatterplot of the lengths and widths of the warblereggs (put length on the horizontal axis andwidth on the vertical).

b. Describe the relationship between the lengthsand widths of the warbler eggs. Is there anycorrelation?

c. What do you expect the width of a warbler eggwill be if its length is 15 mm? How did you findyour answer?

The points in the scatter plot lie almost on a straight line. You mighthave used this information to find an answer to problem 3c. It ispossible to summarize the pattern, or relationship, by drawing a linethat seems to best fit these points. This line will probably not gothrough all the points, but the points should lie close to the line.

4. a. On Student Activity Sheet 10, draw a straight line that seemsto “fit” these points. Use your line to predict the width of awarbler egg that is 15 mm long.

b. Does your line give you the same answer that you predicted inproblem 3c?

5. a. Use your line to determine what happens to the width of anegg if the length increases by 2 mm. Explain how you did this.Include a drawing with your answer.

b. Describe what will happen to the width of an egg if the lengthincreases by 1 mm.


Hints and CommentsMaterialsStudent Activity Sheets 7–10 (one per student);rulers (one per student);graph paper (one sheet per student)

Overview

Students draw a scatter plot and describe therelationship between the lengths and widths of eggs.They draw a straight line in the scatter plot and usethis line to further investigate this relationship.


From looking at the data in the table, it is hard to tellwhether there is a relationship between widths andlengths of bird eggs. A scatter plot is helpful because itmakes the relationship visual. Sketching a line that fitsthe data is a skill that students should develop in thisunit. They should recognize that you can use severalstrategies to draw a best-fit line. There are manypossible definitions for a line of best fit. These includea line that goes through as many points as possible ora line that has as many points above the line as belowthe line. No formal definitions are used here.

Planning

Students may work on problems 3–5 in small groups.



3. a.

b. The points seem to form a straight line. Thereis a strong, positive correlation. The wider theegg, the longer it is.

c. The expected width is approximately 11.5 mm.

Strategies will vary. Sample student response:

I sketched the line made by the points in thescatter plot. Then I drew a vertical line from alength of 15 mm to the line in the scatter plot.At the point of intersection, I drew a horizontalline to the vertical axis, and that would be aboutthe width of the egg.

4. a. Graphs may vary. See the sample graph in thesolution for problem 5 below. The egg wouldbe about 11.5 mm wide.

b. Answers will vary depending upon students’answers to problem 3c.

5. a. If the length increases by 2 mm, the widthincreases by 1.5 mm. One strategy is shown inthe graph below.

13119

12

13

14

15

16

17

11

1015 17 19 21 23

Length (in mm)

Wid

th (

in m

m)

Select a length (14 mm is used in this example).Draw a horizontal line from the line thatsummarizes the data two units to the right(representing the increase of 2 mm). From thispoint, draw a vertical line to the line thatsummarizes the data. This line represents theincrease in the width of the egg.

b. If the length of an egg increases by 2 mm, thewidth increases by 1.5 mm.

If the length of an egg increases by 1 mm, thewidth increases by 0.75 mm.

13119

12

13

14

15

16

17

11

1015 17 19 21 23

Length (in mm)

Wid

th (

in m

m)

1.5 mm

2 mm


Notes

6 Discuss various criteria.The line could go throughas many points as possibleor have as many pointsabove as below.

7ab Encourage students todiscuss the criteria andhow the criteria would beapplied for different lines.

8a Be sure studentsunderstand that they arelooking for a point on theirline that goes with length16.5 mm, not a dot on thescatter plot.

8c Encourage students tocheck their line to see if itpredicts the correct widthfor other kinds of warblers.


F


For problem 6, be sure students understand what is meant by criteria—rules for judging. Give some examples of criteria, such as criteria forchoosing a new pair of shoes or new car.

Extension

After completing problem 6, if students have completed the unit GraphingEquations, you might discuss the slope of the line they drew.


Lines That Summarize Data F

In problem 4, you looked at a scatter plot and drew a line thatsummarized the relationship between the variables. Your line mayhave been very different from the lines drawn by your classmates.

6. How can you decide whether a line drawn to summarize the datain a scatter plot is a good line? What criteria would be helpful inmaking your decision?

7. a. Use your criteria to decide whether or not you drew a good line.

b. How does your line compare with the line drawn by aclassmate?

There are many ways to find a good line. Different criteria are usedfor different purposes. One criterion is how well your line predicts avalue for something you already know.

8. a. Look at the graph of your line from problem 4. What does itpredict for the width of a lesser whitethroat warbler egg thatis 16.5 mm long?

b. Now look at the data on Student Activity Sheets 7–9. What isthe actual value for the width of the lesser whitethroat warbleregg? How well did your line predict this value?

c. Overall, does the line you drew seem to predict values wellwhen you compare it with the actual data? Explain.

Ostrich egg Emu egg

Chicken egg Quail egg


Hints and CommentsOverviewStudents continue with the investigation of the scatterplot and the line of best fit that describes the data.


There are many possible criteria for deciding what agood line of best fit is. Thinking of possible criteria ismore important than formalizing them. When a line ischosen, it can be used to make predictions.

Planning

Students may work in small groups on problems 6–8.


6. Students might think that a good line connects alldata points. This could result in a zigzag pattern,which is not a line. Some students might connectthe first and last points in the data set. However,the line through these points may not summarizethe data.



The line should have a slope that is in the samedirection as the cloud of points and shouldsummarize the points without necessarily goingthrough each point in the plot.

7. a. and b.Answers will vary. Some students may feel it ishard to tell whether their line is a good line. Otherstudents may measure how far off each point isfrom the line. Still others may count the numberof data points on the line.

8. a. Answers will vary. Students should find thepoint on the line where the length is 16.5 mmand estimate the width of the egg at that pointon the line. For the line drawn in problem 4,the width would be about 12.6 mm.

b. The actual width is 12.6 millimeters. Thedifference will vary, depending on the locationof students’ lines.

c. Answers will vary, but generally the line is agood predictor of egg width (as confirmed bychecking the length/width measurements ofother birds’ eggs).


Notes

If your students are notfamiliar with slope, explainthat it means the steepnessof the line.

9b If prompting is needed,advise students to go upvertically from 16.5 mmuntil they reach the line.Ask, What width is this?

10a Students will needtheir list of warblers withthe egg lengths and widthsin order to calculate themean length and the meanwidth. Then they need tocheck to see if these valuesfor L and W satisfy theequation width � 0.7 �length � 1.1.


F

Assessment Pyramid

10abc

Describe a linearrelationship with anequation of a line.

Intervention

Students need to calculate the mean of the lengths and the widths fromthe data in the table first in problem 10a. Students have seen and used thistype of equation in the Algebra unit Graphing Equations. If students haveproblems substituting the values in the equation, you may want to do thisproblem as a whole class.

For problem 11, if the concept of slope needs reviewing, explain that theslope is the number multiplied by the variable L or the ratio of the verticalchange over the horizontal change. Draw a right triangle underneath theline to illustrate this.


Lines That Summarize DataF

In problem 4, you drew a line to predictthe width of the warbler egg. One way toget a line is to draw the line so that it goesthrough the point that is the mean of thelengths and the mean of the widths for theeggs. The slope of the line should reflectthe direction of the trend in the points.

Student Activity Sheet 11 shows a line for the warbler eggs thatcontains the point (mean length, mean width).

9. a. How is this line different from the line you drew in problem 4?

b. Use this new line to find the width of the lesser whitethroatwarbler egg that is 16.5 mm long. How close is the predictionto the width given in the table?

c. Overall, how well does the line seem to predict the widths ofthe warbler eggs?

One equation for the relationship between the length and width ofthe warbler eggs that goes through the mean is:

width = 0.7 � length � 1.1

10. a. Show that this line contains the point (mean length, meanwidth).

b. Use this equation to find the width of the lesser whitethroatwarbler egg that is 16.5 mm long.

c. How close is the width you found to the width given in thetable?

11. a. What is the slope of the line for the warbler eggs in question 10?

b. How can you find this number from the graph?

c. What does the slope tell you about the relationship betweenthe length and the width of warbler eggs?


Hints and CommentsMaterialsStudent Activity Sheet 11 (one per student).

Overview

Students compare a line of best fit for the warblereggs that includes the point (mean length, meanwidth) with the lines they drew in problem 4. Theyuse the equation of this line to estimate the width fora given length, and they interpret the slope in terms ofthe data.


Different lines of best fit can be drawn. One is the linethrough the means of the data; another one is themedian fit line. In this unit, students do not need toknow or use formal definitions of lines of best fit.

The slope, y-intercept form of the equations of lines isintroduced and used in the Algebra unit GraphingEquations. Students must be able to express themeaning of the slope in terms of the data.

Technology

Besides having students draw lines of best fit by“eyeballing,” you may have students draw lines usinga graphing calculator or statistical software. Graphingcalculators and statistical software usually have thecapability of using regression in scatter plots, so youcan have the program draw a line or polynomial thatbest fits a set of data in a scatter plot. Note that such aline can always be drawn, even if there is norelationship between the variables. Therefore, it isimportant to have students first decide whether arelationship will exist and what kind of relationship itwill be before having them draw lines with thegraphing calculator or computer.


9. b. Students can use the graph to find an estimatefor the width. Accuracy will, of course, beaffected by the scale used on the axes of thescatter plot.

11. Have students refer to their solution to problem 5(page 54). The slope is described in terms of thedata in problem 5b.


9. a. Answers will vary. It is likely that students’ linesand the least squares linear regression line areclose since the correlation is quite strong.

b. Students may read the prediction from thegraph; it is about 12.5 mm. This is only 0.1millimeter less than the width given in thetable. For this length, the line seems to predictthe width quite well.

c. For most of the eggs, the predictions madefrom the line seem quite accurate.

10. a. The mean length is 17 mm, the mean width is12.9 mm. Substituting this in the equationgives: 0.7 x 17 + 1.1 �13. This is not exactly12.9, but this can be due to rounding, so theline will go approximately through the pointbut not exactly.

b. Replace L with 16.5.

W � 0.7(16.5) + 1.1

W � 11.55 + 1.1

W � 12.65

The width is 12.65 mm.

c. According to the table, the width of the lesserwhitethroat warbler egg is 12.6 mm. The widthfound using the equation is only 0.05 mm morethan the width in the table, which is very close.

11. a. Students may use either the graph or theformula to find the slope, which is 0.7. In theformula, the slope is the number by which thex is multiplied, or the coefficient of x.

b. Answers may vary. Sample response:

In the graph, this is the increase of y (width),(vertical direction) if x (length) increases(horizontally) by 1.

The slope of the line can be rounded to 0.7,or 7—

10, to simplify the work. From any point onthe line, draw a path that is 10 mm horizontallyand then 7 mm vertically as shown below. Thenew location should also be on the line thatsummarizes the data. This process can berepeated with any point on the line.

c. If the length of an egg increases by 10 mm,the width increases by 7 mm; or if the lengthincreases by 1 mm, the width increases by0.7 mm.10

11

12

13

14

15

13 14 15 16 17 18 19 20

Length (in mm)

Wid

th (

in m

m)

width� 0.7

� length� 1.1 (w

arbler eggs)


Notes

12b Have studentsconsider if it is reasonableto have egg lengths thatare less than 10 mm. Theycould also look in the datato find the smallestlengths.

13a Students should lookat their data list for all thewarblers to find theminimum and maximumlengths for their eggs.Then find these lengths inthe scatter plot.

13b If this question isdifficult for any students,ask what happens to thepoints as the length of theegg increases. Do thepoints still fit a straight lineas closely?


F

Assessment Pyramid

13b, Extension

Describe the relationshipbetween two variables anddescribe a linear relationshipwith an equation of a line.

Advanced Learners

These students could draw a line that seems to fit the data for this scatterplot (Birds of Britain). Give them a copy of this page: they could draw theline and find the slope. Then check on the next page to see if the slope oftheir line matches the one given (0.7).

Accommodation

For problem 13, you may want to make a transparency of the graph forclass discussion. Blank transparencies can be placed on top of the StudentBook for students to circle warbler egg points.



12. a. What does the other number in the equation represent on thegraph (the number that is not the slope)?

b. Does this number make sense in terms of egg length andwidth? Why or why not?

c. What length does the equation predict for eggs that are1 mm wide?

d. Do you think the equation is useful for very small numbers?Why or why not?

The scatter plot shows the data for 96 birds of Britain.

13. a. Where in the scatter plot are the data for the warbler eggs?

b. What happens to the relationship between length and width asthe bird eggs get larger?

0

10

20

30

40

50

60

20 40 60 8010 30 50 70 90 100

Wid

th(i

nm

m)

Length (in mm)

Birds of Britain


Hints and CommentsMaterialstransparency of graph on page 57, optional (one perclass);blank transparencies, optional (several per class)

Overview

Students interpret the y-intercept of a line of best fitin terms of the data. They find the general location ofthe warbler eggs on the scatter plot of all the birdeggs. Then they reason about the relationship as itappears in this scatter plot.


More important than knowing that different lines ofbest fit exist is to know what the numbers in theequation of a line mean in terms of the data. They-intercept does not always have a clear meaning.For example, in the equation of the line through themeans of the data, the y-intercept is 1.1. In terms ofthe data, this would seem to mean that an egg withlength 0 mm has a width of 1.1 mm. This indicatesthat the equation is not useful to predict widths forsmall numbers.

Planning

Students may work in small groups or individually.Discuss the solutions to problems 11 (on page 56)and 12 as a class before proceeding to problem 13.


12. a. Note that the y-intercept is where the graphof a line crosses the y-axis; the y-axis is thevertical line for which x = 0. If the axes do notstart at zero (which should be indicated by azig zag), the reader needs to be careful becausethen it is not possible to read the y-interceptfrom the graph.

Extension

You may have students draw a best-fit line for thescatter plot of all the birds, find its equation, andformulate the relationship in terms of the followingstatement: If the length increases by ___ millimeters,the width increases by ___ millimeters.


12. a. The other number is the y-intercept. It is thelocation on the vertical axis at which the linethat summarizes the data crosses the verticalaxis.

b. This number gives the width of the egg whenthe length is 0. It is meaningless in thissituation. If the data were about somethingelse, the y-intercept might be a meaningfulnumber.

c. Sample solution using the formula:

Replace Width with 1.1 � 0.7 � Length � 1.1�0.1 � 0.7 � LengthL � – 1–7

The length is a negative number, which doesnot make sense.

d. No. Reasons will vary, sample reason: for verysmall numbers the influence of the 1.1 is verybig, so the equation is not very accurate.

13. a. The warbler eggs are in the group of eggswhose lengths range from 13 to 21 mm, asshown below.

b. As the bird eggs get larger, the relationship isnot as strong as the relationship between thelengths and widths of warbler eggs. The warblereggs seem to fit a straight line more closelythan do the dimensions of the larger eggs.

0

10

20

30

40

50

60

20 40 60 8010 30 50 70 90 100

Wid

th(i

nm

m)

Length (in mm)

Birds of Britain


Notes

14a Reviewing thestandard equation fora line y = i � sx fromGraphing Equations mayhelp students rememberthat lines are parallel ifthey have the same slope.

Activity

Discuss strategies formeasuring the length andwidth of an egg before theydo this so the measure-ments are as consistent aspossible. Be sure they donot use a tape measure andfollow the curve of the egg.


F

Assessment Pyramid

14ab

Describe a linearrelationship with anequation of a line.

Accommodation

If some students have not had enough experience graphing lines to answer14a and b, you could just model this for them to show that two lines withthe same slope are parallel.

Writing Opportunity

Have students write a brief report in their notebooks in response to theActivity on this page.


Students can ask their biology teacher about the breeding and nestinghabits of certain birds and whether these habits affect egg size or shape.


Line That Summarize DataF

An equation for the line for the scatter plot of the 96 birds of Britainon page 57 is:

width = 0.7 � length � 1.6

14. a. Compare the equation above to the equation for the warblers:width = 0.7 � length � 1.1. What can you tell about the twolines from the equations?

b. Sketch this line on Student Activity Sheet 11. How do the linescompare?

c. If you make predictions based on the line for all eggs, whatwill happen to your predictions as the egg lengths increase?

15. What can you tell about the egg shapes of birds from differentfamilies?

For this activity, you will need chicken eggs and either a tapemeasure or a ruler.

Measure some chicken eggs at home. Record the lengths andwidths. In class, collect everyone’s data. Find the typical lengthand width for a chicken egg.

How does the typical chicken egg fit in the graph of the birds ofBritain?


Hints and CommentsMaterialsStudent Activity Sheet 11 (one per student);chicken eggs, (one per group of students);tape measure or a ruler (one per each group ofstudents)

Overview

Students compare a line of best fit for all bird eggs tothe line that best fits the warbler eggs. They answerthe question from the beginning of this section aboutthe shapes of bird eggs. They measure chicken eggsand add the data to the scatter plot.


If only part of the data is used to make a line fittingthe data, this line will differ from the line that bestfits all data points. In some cases, it may be moremeaningful to use only part of the data. For instance,in the scatter plot from Section A, a pattern mayappear for all states in the Midwest. In that case,these data points can be selected and a line can befit through them.

Planning

The activity is optional and may be omitted orassigned as homework if time is a concern. Studentsmay work on problems 14 and 15 in small groups orindividually.

Comments About the Solutions14. b. The solution graph show for this problem was

adjusted to fit this page.

Solutions and Samples14. a. Answers may vary. Sample response:

The lines are parallel. Since the y-intercept forthe line that summarizes the data for all 96birds is 1.6, the line starts slightly higher on thevertical axis.

b.

See answer to problem 14a.


As egg length increases, points are fartherfrom the line, and there will be more error inpredicting the width of an egg.

15. Answers will vary. Students may simply note thatthe sizes of eggs from different bird families mayvary greatly. The proportion of length and width isvery similar between families.

Activity

Answers will vary, depending on class data and onwhat one considers “typical.” Record the data of thelengths and widths of chicken eggs from each personin the class on a transparency or on the chalkboard.From these data, students can determine themeasurements of a typical chicken egg.

10

11

12

13

14

15

13 14 15 16 17 18 19 20

Length (in mm)

Wid

th (

in m

m)

width� 0.7

� length� 1.1 (warbler e

ggs)

width� 0.7

� length� 1.6 (a

ll eggs)


Notes

16b Have students thinkabout the scaling of theaxes first.

16b Making the scatterplot for this large data settakes quite a bit of time,so it is a good idea to givestudents additional timeto complete this outsideof class.

17a Using a piece ofspaghetti will give thestudents an opportunityto try to find a line thatwill fit the points. Studentsshould approximate a lineof best fit.


F

Assessment Pyramid

16bc

Represent data raphically.


Accommodation

If time does not permit students to make the scatter plot, just give them acopy of the scatter plot for Bluegill Growth and let them use it to answerproblems 16c, 17, 18, and 19. The goal of this page is for students to seethat a curve fits the data better than a line. You will need to make thescatter plot or have a student do it.


Graphs of growth often result in curves because livingthings do not grow at a constant rate throughout theirlifetimes.

Researchers in fisheries are interested in studying thegrowth rates of fish. One study on bluegills comparedthe lengths of fish at the beginning of the year withtheir lengths at the end of the year.

The table contains the results of this research.

16. a. How much did the fish that was initially 161 mmgrow in one year?

b. Graph the data from the table.

c. What pattern do you see in the scatter plot fromproblem 16b?

17. a. Draw a straight line to represent the data in thescatter plot.

b. Estimate how long a 140-mm fish will be afterone year.

c. Is a straight line a good model to represent thedata in the graph? Explain your reasoning.

18. a. Sketch a model on the graph that betterrepresents the data.

b. Using this model for the growth, describe whatis happening to the growth of the fish as thelength changes.

c. If a fish is now 110 mm long, predict how longit will be in one year.

19. Describe the difference between using a straightline and using the curved line you created inproblem 18 to predict growth.


Gone FishingBluegill Growth

Initial Length after

Length x 1 Year y

(in mm) (in mm)

48 69

52 71

51 69

53 75

68 101

71 107

69 100

75 104

101 138

107 138

100 130

104 140

138 160

132 157

130 156

140 161

160 173

157 168

156 172

161 178

173 176

168 174

172 173

178 178


Hints and CommentsMaterialsgraph paper, optional (one sheet per student);rulers, optional (one per student)

Overview

Students study data in which a nonlinear relationshipbetween variables occurs. Students decide whether aline or some other model best describes the data onthe growth of the fish.


As previously stated, in each scatter plot a line of bestfit can be drawn. Computer graphing programs willalways draw a best-fit line if asked. Often this linedoes not really fit the data or is meaningless in thecontext. The example used here shows data for whicha curve fits better than a line. In cases like this, where“real-world” data are used, it is very important to finda contextual reason why a curve fits better than astraight line. In this case, the model needs to showthat there is a limit to how long bluegills can grow.A curve that gets flatter for larger initial lengths showsthis limit, whereas a straight line does not.



16. a. The fish grew 17 mm, to a length of 178 mm.

b. See the scatter plot below.


Small fish grew faster than fish that were longinitially. A curve may fit the pattern.

17. a. Students answers will vary. Sample line(see graph):

b. Using the straight line model from problem17a, if a fish is initially 140 mm long,it will grow to about 157 mm.

c. No. A straight line does not summarize thepattern in the data well. It is hard to find aline that fits.


Bluegill Growth

Len

gth

aft

er

1 Y

ear

(in

mm

)

0 25 50 75 100 125 150 175 20060

70

80

90

100

110

120

130

140

150

160

170

180


Bluegill Growth

Len

gth

aft

er

1 Y

ear

(in

mm

)

0 25 50 75 100 125 150 175 20060

70

80

90

100

110

120

130

140

150

160

170

180

18. a. Sketches may vary. Sample drawing:


As the fish get longer, the rate of growth slowsdown.

c. The fish should be about 140 mm at the end ofone year.


The curved line goes close to many points, so themodel is rather good.


Bluegill Growth

Len

gth

aft

er

1 Y

ear

(in

mm

)

0 25 50 75 100 125 150 175 20060

70

80

90

100

110

120

130

140

150

160

170

180


Notes

For the Summary, have acouple of scatter plots todisplay so students canpractice filling in the blanksof the bolded statement.


F

Parent Involvement

Have parents review the section with their child to relate the Check YourWork problems to the problems in Section F.


Lines That Summarize Data

If the relationship between two variables appears to be linear, a linecan be found to describe the relationship.

Straight lines can be used to predict unknown values and to checkexisting values over the range of the data in the problem. If the trendseems to be linear, predicting an unknown value between data pointswill probably give an answer close to the truth.

The slope of the line can be expressed in terms of the data. This canlead to statements such as the following:

When the — increases by — , the — — by — .

F

13119

12

13

14

15

16

17

11

10

15 17 19 21 23

Length (in mm)

Wid

th (

in m

m)


Hints and CommentsOverviewStudents read the Summary to review Section F goals.


Notes

1. Ask students to providea few examples to supportthe completion of thisstatement on page 60.You may need to referback to page 54, problem5ab, for examples ofappropriate statements.


F

Assessment Pyramid

1

Assesses Section F Goals

Intervention

For problem 1, some students may need help connecting slope to thebolded statement in the Summary on page 60. A visual of a line drawn ona coordinate grid, with a right triangle drawn under it showing verticalchange over horizontal change, may be helpful. Use an example of a linewith a positive slope and a line with a negative slope.



Bluegill Growth

Le

ng

th a

fte

r 1

Ye

ar

(in

mm

)

0 25 50 75 100 125 150 175 20060

70

80

90

100

110

120

130

140

150

160

170

180

You can draw a line that seems to capture the trend in the data. Youcan check how well your line seems to summarize the relationshipbetween the variables by checking how much the predictions madeusing the line would vary from the actual values in the data. A curvemay sometimes be used to describe a relationship that cannot bedescribed by a straight line. This is usually the case in the context ofgrowth, since growth rates are not constant.

1. Explain how you can find the slope of a line from the statementlike the one bolded in the Summary.


Hints and CommentsOverviewStudents read the Summary and complete Check YourWork problem 1.

Planning

After students complete Section F, you may assign ashomework appropriate activities from the AdditionalPractice for Section F, located on Student Book page 69.


1. You can find the slope by dividing the verticalincrease (or decrease) by the correspondinghorizontal increase.


Notes

2a Make sure studentshave graph paper availableto complete this problem.Encourage students to usea ruler or straightedgewhen drawing the line ofbest fit.


F

Assessment Pyramid

2ac

2ab



If writing a sentence about the “relationship” in problem 2a is difficult, havestudents describe what happens to the grams of carbohydrates as thenumber of calories increases.

Intervention

For problem 2b, you may need to remind students that the standard formfor the equation of a line is y � i + sx to help them write the equation fortheir line, where i is the y-intercept and s is the slope. The y-intercept isthe point where the line crosses the y-axis. It is a good idea to have thisstandard form on a poster on your walls.


Lines that Summarize Data

The table shows the number of calories and grams of carbohydratesfor standard size servings of different kinds of fruit.

2. a. Make a graph of the data with the number of calories on thehorizontal axis and the grams of carbohydrates on the verticalaxis. Write a sentence about the relationship you can observebetween calories and carbohydrates in fruit.

b. Draw a line that seems to represent the relationship youobserved in the data. Write the equation for this line.

c. Describe what the slope and y-intercept each mean in terms ofthe data.

F

Fruit Calories Carbohydrates (grams)

Apple, raw 23��4 in. diameter 80 21

Apricot, 3 raw 60 3

Banana, raw 105 27

Cherries, 10 50 10

Grapefruit, 1��2 raw, white 40 10

Grapes, 10 seedless 35 10

Cantaloupe, 5 in. diameter 95 22

Orange, 25��8 in. diameter 60 15

Peach, raw, 21��2 in. diameter 35 10

Strawberries, whole, 1 cup 45 10

Tomatoes, 1 whole 25 5

Watermelon, 4 � 8 in. wedge 155 35

Home and Garden Bulletin, No. 72, U.S. Department of Agriculture


Hints and CommentsOverviewStudents continue to work on the Check Your Workproblems.


2. a.

You might say that as the calories in fruitincrease, so does the number of grams ofcarbohydrates. Another answer might bethat most of the fruit has less than 20 g ofcarbohydrates, but that would not be astatement about the relationship with thenumber of calories.

b. One line might be the one drawn in thesolution for problem 2a above. The equationfor this line is:

Carbohydrate = 1 � 0.2 � Calories

The y-intercept can be read from the graph, theslope can be found by using the points at theend of the line segment to find the verticalchange (39) and the horizontal change (170);the slope then is vertical change divided byhorizontal change. So slope is 39____

170 , which isabout 0.2.

c. For this example, the slope would be 0.2,which means that for an increase of 10 calories,the number of grams of carbohydrates fruitincreases by 2. The y-intercept is 1 andindicates that if a type of fruit has 0 calories,you would still expect it to have about onegram of carbohydrates.

Calories

Carbohydrates vs Food Energy in FruitC

arb

oh

yd

rate

s (

in g

)

2000246810121416182022242628303234363840

40 60 80 100 120 140 160 180 200


Notes

3 Boxes of cereal are onepossibility for comparingcalories and grams ofcarbohydrates.


F

Assessment Pyramid

FFR

2de, 3


Intervention

This could be a lot of information for students to process. You may needto summarize as a class the different ways that were used in this unit todescribe data before students complete the For Further Refection problem.


d. Will your graph work to predict the grams of carbohydrates ina cup of raisins if you know that they have 435 calories? Whyor why not?

e. Use both the graph of your line and your equation to predictthe number of carbohydrates in a banana. Do the twopredictions agree? How far off was your prediction?

3. Find the relationship between calories and grams of carbohydratesfor another food group. How does it compare to the relationshipin fruit? Share your findings with a classmate.

You have looked at several different ways to describe data in this unit.Choose one of them and describe a situation in which you would usea graphical representation to display the data.


Planning

After students complete Section F, you may assign forhomework appropriate activities from the AdditionalPractice for Section F, located on Student Book page 69.


2. d. The graph will probably not work because itwas made using a range of calories from 25 to155, and 435 is far beyond that range. Even ifyou made the graph over on another scale, itis so far away from the given range that youcannot be sure that the trend you see in thedata still holds.

e. In the table you can find that a banana has105 calories. You can substitute this value inthe equation of the line and calculateCarbohydrate � 1 � 0.2 � 105. So the bananahas 22 grams of carbohydrate.

If you look at the graph, you find that theamount of carbohydrates for a banana of 105calories is about 24 gram.

The difference is caused by the rounding of thevalue of the slope of the line.

3. Your answers will depend on the kind of foodyou choose. You might look at grain products,fish, meat, dairy products, or vegetables. Have aclassmates check your graph and your conclusions.


Student answers will vary.


Additional Practice SolutionsStudent Page

64 Insights into Data Additional Practice Solutions Student Page

Additional Practice

Section Patterns in DataA

Mr. Flores surveyed his high school sophomore class to see if therewas a relationship between the number of hours the students studyper week and their grade point averages (GPA). He created the scatterplot shown below to display the data.

1. What is the GPA of the student who studies six hours per week?

2. Do you agree with each of the statements below? Explain youranswer.

a. The student who studied the least has the lowest GPA.

b. If you study at least nine hours a week you will have a GPAof 3.5.

3. Are there any outliers? If so, describe their locations.

Hours of Study

Comparison of GPA to Hours of Study

2 4 6 8 10 12 14 16

1

2

3

4

0

GPA

Section A. Patterns in Data

1. 2.0

2. Answers and explanations will vary. Sampleresponses:

a. Yes. One student who did not study at allhad the lowest grade point average.

b. No. This was not always true. Five studentswho studied at least nine hours a week hadgrade point averages under 3.5.


No. The points are all fairly close together.

Yes. One student studied 11 hours a week, butstill had a grade point average of 2.0.

Possible outliers are circled below.

Additional Practice Solutions

Additional Practice Solutions Insights into Data 64T

Hours of Study


GP

A

0 2 4 6 8 10 12 14 16

1

2

3

4



The table below shows the percentage of students who scored at orabove the basic levels in math and science on the 2005 NationalAssessment of Educational Progress in states from the Northeast andNorthwest. (Not every state administers the test.)

4. a. Do states with a high average percentage of students at orabove the basic level in math also have a high percentage ofstudents at or above the basic level in science?

b. What is the difference between the two regions, Northwest(NW) and Northeast (NE) in terms of their test scores? Can youthink of any reasons for this difference?

c. In which subject are the scores better? Explain how you cansee this in the graph.

d. The data might almost be grouped into three clusters, one atthe bottom left, one at the top right, and the cluster of pointsin the middle. What can you say about the states in the clusterat the bottom left and those at the top right?

Percentage at or above Percentage at or aboveState

Basic Level in Math Basic Level in ScienceRegion

IL 68 58 NW

IN 74 62 NW

MI 68 66 NW

MN 79 71 NW

MO 68 66 NW

ND 81 77 NW

OH 74 67 NW

WI 76 70 NW

ME 74 72 NE

VT 78 76 NE

MA 80 72 NE

NH 77 76 NE

RI 63 58 NE

CT 70 63 NE

Source: National Assessment of Educational Progress, National Center for Education Statistics,U.S. Department of Education

Section A. Selecting Samples(continued)


If you make a scatter plot, this is easier tosee. The scatter plot shows that in generalyou could say the higher the percentagethat score above basic level in math, thehigher the percentage that also score abovebasic level in science.

4. b. Answers will vary. Sample answer:

The scatter plot shows that the percentagesfor the NW states are more evenly spreadout than the ones for the NE states. In theNE states, there are two clusters—onecluster in the lower left (RI and CT) with lowpercentages and one cluster in the upperright (VT, ME, NH, and MA) with highpercentages for both subjects. Lower scoresmay be caused by the fact that these stateshave big cities, although MA has Boston andit did well.

4. c. The scores are better for math. This can beseen, for example, if you draw a linethrough the points for which thepercentages for both subjects are equal. Theline that goes through (50, 50), (60, 60), andso on, is the line M = S. All points are belowor to the right of this line; this means thatmath scores are higher.

4. d. For the cluster in the bottom left, the scoresin both subjects are low. There are twostates from the NE and one from the NW inthis cluster. For the states in the upper right,the scores on both subjects are high. Thereare three states from the NW and four fromthe NE in this cluster.



5050

52

54

56

58

60

62

64

66

68

70

72

74

76

78

52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82

Math

Scie

nce

Percentage of Students

80

RI IL

CTIN

WI

MIMO

ND

ME

NH

MA

MN

OH

VT



The graph on the left indicates that the priceof erasers increased from 1970 to 2005.

Additional Practice

Section Interpreting GraphsC

2005

50¢

2000

45¢

1990

35¢

1980

30¢

1970

25¢

1. Do you think the graph represents thedata accurately? Explain your answer.

2. Draw a picture that accurately representsthe difference between the prices of theerasers from 1970 to 2005.

A taste test was conducted between two leading soft drinks. Testingbooths were set up at two different shopping centers in the townwhere the soda is made. A total of 425 people stopped by the boothsto take part in the taste test.

The results of the study were used in an advertising campaign:

“Three out of five people prefer the taste of Bingo Pop overother leading brands.”

1. Is this statement reliable or fair? Explain why or why not.

Two weeks later, the Bingo Pop company decided to do another tastetest to see if its advertisement campaign was effective. The boothswere set up in the same locations. Again, 425 people stopped by thebooths to take part in the taste test. This time, the results showed thatfour out of five people preferred the taste of Bingo Pop. The companyput out another ad:

“More and more people enjoy the taste of Bingo Pop every day.”

2. Do you think this is an accurate statement? Why or why not?

3. How could you ensure that the two surveys above were conductedaccurately?

Section Selecting SamplesB

Section B. Selecting Samples

1. Answers and explanations will vary. Sampleresponse:

No, the statement is not reliable. Only twobrands were tested, but Bingo Pop claims to bebetter than all leading brands.

2. Answers and explanations will vary. Sampleresponse:

No, this sample may include even more BingoPop drinkers, and those persons who don’t likeBingo Pop or who were in a hurry may haveavoided the booth.


• Include more brands for testing.

• Move the taste test to a new location thesecond time.

• Don’t have all the testing at only onelocation.

Section C. Interpreting Graphs1. Answers and explanations will vary. Sample

responses:

No, the eraser on top is twice as expensive asthe one at the bottom, but it looks more thantwice as big.

2. Pictures will vary. Sample picture:



2005

50¢

2000

45¢

1990

35¢

1980

30¢

1970

25¢



Additional PracticeAdditional Practice

Recall that in Section A of Additional Practice, Mr. Flores conducteda survey about the number of hours students study and their GPAs.Mr. Harrison and Ms. Simmons conduct the same survey in theirclasses. The results from the two classes are shown in the tables.

Section Using DataD

1. a. Explain the advantages and/or disadvantages of using thefollowing types of graphs to display the data:

• histogram • box plot • scatter plot

b. Reflect If you wanted to compare the two graphs of data,which graph would you use? Explain.

2. Make a scatter plot of the combined data from both classes.

Mr. Harrison’s Class Results

Hours SpentGPA

Studying

0 0

1 1

1 0

2 1

3 1

4 1.5

5 1

5 2

6 2

7 2.5

9 3

10 4

11 3.5

12 3

12 4

13 3.5

14 3.5

15 3.5

16 4

Ms. Simmons’s Class Results

Hours SpentGPA

Studying

1 0.5

2 0.5

4 1

4 1.5

5 1.5

6 2.5

7 2

8 2.5

8 3.5

10 2.5

10 3

11 3

11 4

12 3.5

13 3

13 4

14 4

15 4

16 3.5

Section D. Using Plant GrowthData

1. a. Explanations will vary. Sample explanations:

• Histogram: shows the distribution ofvalues in a sample and can be used toestimate the mean; a two-dimensionalhistogram can only represent thedistribution of one variable, such as GPA.

• Box plot: shows the median and themiddle 50% of the data; cannot findpatterns based upon the distribution ofvalues and cannot represent more thanone variable.

• Scatter plot: shows possible patterns orcorrelation between two variables; noaverage value can be taken from the plot.


A scatter plot would be best because it canrepresent both variables in the same graph.Different colors can be used for the differentclasses.

2.

Hours of Study


GP

A

0 2 4 6 8 10 12 14 16

1

2

3

4





Additional Practice

Section Correlating DataE

When you look carefully at your scatter plot, you can distinguish acluster of data in the upper-hand right corner.

3. Describe this cluster. What does this pattern say about thestudying habits of the students in the two classes?

1. Study the scatter plots shown below. Indicate whether or notthere is a correlation for each plot. If there is a correlation, indicatewhether it is weak or strong.

2. What data could be represented by each of the above scatter plots?

3. Describe some data that show a strong correlation but does nothave a cause-effect relationship.

0 1

1

2

3

4

5

6

2 3 4 5 6 0 1

1

2

3

4

5

6

2 3 4 5 6 0 1

1

2

3

4

5

6

2 3 4 5 6

y y y

x x x

a b c

Section D. Using Plant GrowthData (continued)

3. Answers will vary. Some students may say thatthe graph is evidence that those students whostudied 11 or more hours per week had thehighest grades (GPA = 3.0 or higher). Otherstudents may say that to get a better GPA itdoes not makes much difference whether astudent studies 11 hours or more than 11 hours.The GPA seems to be the same for all numbersof hours of study between 11 and 16.

Section E. Correlating Data1. a. weak correlation (not linear)

b. strong negative correlation

c. strong correlation (y-value is constant)


On each graph, if x is the price of an item andy represents the sales of that item:

(graph a) demand for some items may fall asthe price increases, but at some point theseitems become so valuable that demand goesback up. Antiques are like this.

(graph b) in general, when the price is low,sales are high. When the price increases, salesdecrease.

(graph c) some necessary items like food andwater may be sold at any price because peoplealways need a certain amount.


The easiest examples to offer are ones thatare functions of time. An example of a strong(negative) correlation is: Each week duringthe summer, the number of refrigerators soldincreased but the number of car accidentsdecreased.



Number of Refrigerators Sold

Nu

mb

er

of

Ca

rs I

nv

olv

ed

in

an

Accid

en

t



Additional Practice

Section Lines That Summarize DataF

The relationship between the numbers of hours spent studyingper week and GPA in Mr. Flores’s class can be described by a lineas shown above.

1. a. Estimate the slope of the line.

b. How would you describe the slope in terms of the relationshipbetween number of hours spent studying and GPA?

2. What GPA would you expect of a student who studied 9 hours?

3. a. What criteria do you need to decide if the line above is drawnaccurately?

b. Is the line on the above graph an accurate line? Explain youranswer.

Hours of Study

0

1

2

3

4

2 4 6 8 10 12 14 16

GPA



Section F. Lines ThatSummarize Data

1. a. Answers will vary. Some students may usethe first point and the last point to calculatethe vertical change and the horizontalchange. The vertical change is 3.5 and thehorizontal change is 16. The slope is about0.2.


If the number of hours of study increases by16, the grade point average increases by 3.5.

Or: If the number of hours of studyincreases by 1, the GPA increases by about0.2.

2. 2.5. This can be found in the graph.

3. a. Answers will vary. Sample responses:

If there are no outliers, can we minimize thedistance from the line to each point?

If there are outliers, can we find a line inwhich one-half of the points are above theline and one-half of the points are below theline? Can the line be drawn so that all points(with the exception of the outliers) clusteraround the line?

b. Answers will vary, depending on the criteriastudents mentioned for 3a. Sample answer:

No. There are too many points above the line.

Assessment Overview

70 Insights into Data Assessment Overview

Assessment OverviewAssessment Overview

Unit assessments in Mathematics in Context include two quizzes and a Unit Test.Quiz 1 is to be used anytime after students have completed Section B. Quiz 2 canbe used after students have completed Section E. The Unit Test addresses manyof the major goals of the unit. You can evaluate student responses to theseassessments to determine what each student knows about the content goalsaddressed in this unit.

PacingEach quiz is designed to take approximately 25 minutes to complete. The UnitTest is designed to be completed during a 45-minute class period. For moreinformation on how to use these assessments, see the Planning Assessmentsection on the next page.

Goals Assessment Opportunities Problem Level

• Represent data graphically and find a given Quiz 2 Problems 1b, 2bpoint on a graph. Test Problems 1abc, 2c, 3ab

• Describe data numerically. Quiz 2 Problems 2aTest Problems 2a, 3a

• Describe the relationship between two Quiz 1 Problems 1abvariables. Quiz 2 Problems 3ab

Test Problem 3c I

• Identify the degree of correlation between Quiz 2 Problems 3abvariables. Test Problems 2d, 3cd

• Design, conduct, and analyze ways of Quiz 1 Problem 2agathering data: surveys, simulations,experiments.

• Use random samples in gathering data. Quiz 1 Problem 2a

• Analyze and interpret representations of data. Quiz 1 Problems 1bcQuiz 2 Problems 1abTest Problems 1c, 3e

II

• Draw conclusions based on given data Quiz 2 Problem 2csets and representations of data. Test Problems 2cd, 3cf

• Determine whether representations of data Quiz 2 Problems 1ab(numerical and visual) are appropriate. Test Problem 2b

• Generate appropriate questions for analyzing Quiz 1 Problem 2bIII

data sets and representations of data. Test Problems 2e, 3f

Assessment Overview

Assessment Overview Insights into Data 71

About the MathematicsThese assessment activities assess the majority of the goals forInsights into Data. Refer to the Goals and Assessment Opportunitiessection on the previous page for information regarding the goals thatare assessed in each problem. Some of the problems that involvemultiple skills and processes address more than one unit goal. Toassess students’ ability to engage in non-routine problem solving(a Level III goal in the Assessment Pyramid), some problems assessstudents’ ability to use their skills and conceptual knowledge innew situations. For example, in the hats problem on the Unit Test(problem 2), students must generate appropriate questions foranalyzing a new data set.

Planning AssessmentThese assessments are designed for individual assessment; however,some problems can be done in pairs or small groups. It is importantthat students work individually if you want to evaluate each student’sunderstanding and abilities.

Make sure you allow enough time for students to complete the prob-lems. If students need more than one class session to complete theproblems, it is suggested that they finish during the next mathematicsclass, or you may assign select problems as a take-home activity.Students should be free to solve the problems their own way.Students should be allowed to use a calculator when completingthese assessments.

If individual students have difficulties with any particular problems,you may give the student the option of making a second attemptafter providing him or her a hint. You may also decide to use one ofthe optional problems or Extension activities not previously done inclass as additional assessments for students who need additional help.

ScoringSolution and scoring guides are included for each quiz and theUnit Test. The method of scoring depends on the types of questionson each assessment. A holistic scoring approach could also be usedto evaluate an entire quiz.

Several problems require students to explain their reasoning or justifytheir answers. For these questions, the reasoning used by students insolving the problems as well as the correctness of the answers shouldbe considered in your scoring and grading scheme.

Student progress toward goals of the unit should be considered whenreviewing student work. Descriptive statements and specific feedbackare often more informative to students than a total score or grade.You might choose to record descriptive statements of select aspectsof student work as evidence of student progress toward specific goalsof the unit that you have identified as essential.

72 Insights into Data Quiz 1 Mathematics in Context

Insights into Data Quiz 1 Page 1 of 2

Name ____________________________________________ Date ______________________

Use additional paper as needed.

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1. The scatter plot shows the use of coal in each state per onetrillion British Thermal Units (BTUs) in 1999 and the death rateper 1,000 people (in 2000) in that state. Each state is identifiedas a data point in the plot, labeled by its postal code.

a. What does this graph tell you? Write two statements basedon the graph.

01 2 3 4 5 6 7 8 9 10 11 12

100

200

300

400

500

600

700

800

900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

Use o

f C

oal (p

er

trillio

n B

TU

)

Death Rate (per 1,000 people)

AL

PA

WV

AK

CA

AZVA

AR

OKCO

CT

HI ID

NM

NH

WA

MN

IL

DEOR

VTMA

SD

NJ

MD

MERI

KY

IA

IN

TX

OH

KS

NC

ND

WIWY

SC

MS

D.C.

MO

TN

MTNE

NYNV

MI

LA

FL

GA

UT

Mathematics in Context Insights into Data Quiz 1 73

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Page 2 of 2 Insights into Data Quiz 1

b. Look at the data point for Arkansas (AR). What is the death rateper 1,000 inhabitants in this state? What other information doesthis data point show?

c. What can you tell about the data points for Alaska (AK), Utah (UT), and Texas (TX)?

2. Most estimates on left-handedness of people hover around 10%.

a. How could you use the set of 50 random numbers, ranging from 0 to 9, to simulate the percentage of left-handed people in a sample of 50 people?

b. How many people in the sample would be left-handed, using thesimulation from part a? Is this a likely result? Why or why not?

6 5 7 1 0 3 9 7 3 8

0 6 6 5 7 7 3 9 6 8

1 3 0 6 2 0 9 4 6 6

2 9 5 7 3 8 2 9 9 5

7 1 4 7 4 6 2 3 3 5



Name ____________________________________________ Date ______________________


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This graph shows the dramatic effect of a change in diet on thelevel of cholesterol. The decrease of cholesterol in patient AH isparticularly spectacular.

1. a. Mention at least two features of the graph that may lead tothe misinterpretation stated above.

3

0 1 2 3 4 5 6 7 14 15 16

4

5

6

7

8

9

10

Changeof Diet

Days

AH

EM

Level

of

Ch

ole

ste

rol

(in

mg

pe

r m

L)


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Page 2 of 3 Insights into Data Quiz 2

b. Sketch a better graph to represent this data.

2. This is a list of the points scored by National Football Leagueteams who played last Sunday: 27, 23, 20, 24, 14, 38, 13, 19, 16, 21, 10, 24, 34, 28, 36, 14, 20, 31.

a. Find the range, median, first quartile, and third quartile forthis set of data.

b. Make a box plot of this data set.

c. What does your box plot tell you about the numbers?

0 1 2 3 4 5 6 7

Days

Level

of

Ch

ole

ste

rol

(in

mg

pe

r m

L)



Name ____________________________________________ Date ______________________


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3. This scatter plot shows the hand and arm spans of 404 people,both men and women.

a. Is there a weak correlation, no correlation, or a strongcorrelation? Explain.

b. If there is any correlation, is this positive, or negative?

140

135

130

125

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

145

150

155

160

165

170

175

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185

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195

200

Hand Span (in cm)

Arm

Sp

an

(in

cm

)

Mathematics in Context Insights into Data Unit Test 77

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Page 1 of 5 Insights into Data Unit Test

1. The scatter plot below shows the coal consumption for each state in1999 in trillions of British Thermal Units (BTUs) and the death rate foreach state (in 2000).

Draw a horizontal line through a coal-consumption of 500 trillion BTU.

a. List the states that are above this line.

01 2 3 4 5 6 7 8 9 10 11 12

100

200

300

400

500

600

700

800

900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

Use

of

Co

al

(pe

r tr

illi

on

BT

U)

Death Rate (per 1,000 people)

AL

PA

WV

AK

CA

AZVA

AR

OKCO

CT

HI ID

NM

NH

WA

MN

IL

DEOR

VTMA

SD

NJ

MD

MERI

KY

IA

IN

TX

OH

KS

NC

ND

WIWY

SC

MS

D.C.

MO

TN

MTNE

NYNV

MI

LA

FL

GA

UT

78 Insights into Data Unit Test Mathematics in Context

Insights into Data Unit Test Page 2 of 5

Name ____________________________________________ Date ______________________


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b. Shade or mark the states you listed in part a on the mapabove.

c. Which state(s) would you label an outlier and why?

WEST

MIDWEST

SOUTH

NORTHEAST

WA

OR

MT

ID

WY

NV

CA

AZNM

UTCO

KS

NE

SD

NDMN

IA

WI

IL

MO

OK

TX

AR

LAMS

ALGA

TN

KY

SC

NC

VA

FL

INOH

WV

PADC

MI

NYVT

NH

ME

MD

DE

NJ

CT

RI

MA

AK

H I


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2.

A company that sells hats wants to open a new shop featuring western-style hats. The company wonders what size hats to stock. The tables below list the heights and head circumferences incentimeters for 36 different adults.

a. What do you consider the typical head size? What does thisnumber tell you?

The company has decided to stock the sizes that will fit the middle 50%of the adult population.

b. What kind of plot can you use to find the sizes within the middle50% range?

c. What sizes will the company stock?

d. Investigate whether a relationship exists between head size andheight. Is the result useful for the company?

e. What should the company find out before using the data to helpthem when ordering hats?

Height (in cm)

Head Circumference (in cm)

174

55

190

58

176

57

183

60

180

60

182

58

193

58

182

59

168

57

172

56

189

59

163

58

Height (in cm)


168

57

190

61

198

59

167

57

186

60

169

55

174

57

163

56

175

60

180

62

176

52

175

54

Height (in cm)


178

50

184

62

196

60

185

58

174

58

185

59

171

56

179

61

172

56

185

57

180

56

176

60


Insights into Data Unit Test Page 4 of 5

Name ____________________________________________ Date ______________________


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3. Manatees are large mammals (10–13 feet long) that livein the waters off the coast of Florida. Manatees are indanger of extinction. In recent years, more and moremanatees have been found dead on the beaches. Thedata below are numbers of dead manatees that havebeen recorded at different times.

13, 21, 24, 16, 24, 20, 15, 34, 33, 33, 39, 43, 50, 47

a. Look carefully at these data. Describe what the datatell you about the number of manatee deaths. Youmay want to use a plot of the data.

Because the bodies of the dead manatees had wounds on them that might have been caused by propellers frommotorboats, a conservation group checked the number of powerboats registered at the times the data wererecorded. Then the group looked for a connection to thenumber of manatee deaths. The data they gathered are in the table on the right.

b. Make a scatter plot to show the relationship betweenthe number of powerboats registered and thenumber of manatee deaths.

Power Boat

Registrations

(in thousands)

Manatee

Deaths

447

460

481

498

513

512

526

559

585

614

645

675

711

719Source: Decisions Through Data by David Moore, 1992 COMAP, Inc.

13

21

24

16

24

20

15

34

33

33

39

43

50

47

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c. Describe the correlation in the scatter plot. Does it have a strong,weak, or no correlation? If there is a correlation, is it positive ornegative?

d. Draw a straight line to represent the data in the scatter plot.What does this line show?

e. Estimate the number of manatee deaths for each number ofpowerboats listed below. Explain how you determined youranswers.

i. 450,000 powerboats registered

ii. 700,000 powerboats registered

f. What would you advise the Florida Department of Conservationto do about the problem of manatee deaths?


Insights into Data Quiz 1Solution and Scoring Guide


Possible student answer Problem levelSuggested numberof score points

1. a. Possible student answers: 2 I

• There is no clear pattern.

• Most states use less than 500 trillion BTU coal.

• The death-rate is not related to the use of coal.

b. The death rate for Arkansas per 1,000 inhabitants 2 I/IIis 11. They use about 265 trillion BTU coal in thisstate.

(A range of 250–280 BTU should be acceptable.)

c. They are all outliers. Alaska and Utah both have 2 IIlow death rates and low coal consumption; Texas has a very high coal consumption but a relativelylow death rate.

2. a. You could assign one of the numbers from 0 to 9 2 Ito mean “left-handed”; all the others mean “right-handed.”

b. Student answers should be appropriate according 2 IIIto the number(s) they choose to represent left-handedness.

For example, assume a “9” represents a left-handed person. In this sample, there are six 9’s. This would mean 6 out of 50 people are left-handed. This is close to 10%, which would be 5 out of 50.

Total score points 10

(1 point for eachcorrect statement)

Insights into Data Quiz 2Solution and Scoring Guide


Possible student answer Problem levelSuggested numberof score points

1. a. There is no zero on the y-axis; the x-axis is not 2 II/IIIscaled properly, (There is a break between day 7 (1 point for eachand 14 that is not to scale.) Both of these features statement)make the drop in cholesterol seem even moredramatic

b. Sample graph: 3 I/II/III

2. a. range: from 10 (minimum) to 38 (maximum), so 28 4 I

first quartile: 16

median: 22

third quartile: 28

b. 4 I

c. Student responses should show some connection 2 IIbetween the numerical data and box plot(and problem context).

Possible answer: Half the scores lie between 16 and 28, so most teams score about 20 points (or between two and four touchdowns).

3. a. A weak correlation, since the points are rather 2 Ispread out. They suggest an oval cloud ratherthan a line.

b. It is a positive correlation. The cluster of points 1 Islopes upward.


0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5

AH

change of diet

EH

Days

Seru

m C

ho

lesto

rol C

on

cen

trati

on

(in

mg

per

mL)

6 7 8 9 10 11 12 13 14 15 16

10 15 20 25 30 35 40

Possible student answer

Insights into Data Unit TestSolution and Scoring Guide


Problem levelSuggested numberof score points

(no points if2 or more states

are missing.)

WEST

MIDWEST

SOUTH

NORTHEAST

WA

OR

MT

ID

WY

NV

CA

AZNM

UTCO

KS

NE

SD

NDMN

IA

WI

IL

MO

OK

TX

AR

LAMS

ALGA

TN

KY

SC

NC

VA

FL

INOH

WV

PADC

MI

NYVT

NH

ME

MD

DE

NJ

CT

RI

MA

AK

H I

1. a. Tennessee, Florida, Missouri, North Carolina, 2 IGeorgia, Michigan, Illinois, Alabama, Kentucky,West Virginia, Pennsylvania, Ohio, Indiana, Texas

b. 2 I

c. Texas is somewhat of an outlier since it is the 2 I/IIonly state that does not share a border with oneof the other states.

2. a. Answers will vary. Sample responses: 2 I

• The mode would be a good measure here.It tells you that 57 cm, 58 cm, and 60 cm arethe most frequent head circumferences in thetable. People who sell hats are interested inthe mode (fitting the greatest number ofcustomers).

• The mean is 58 cm. It indicates the averagehead circumference in the table.

• The median head size is 58 cm. This meansthat 58 is the point that separates the givenset of head circumferences into two groups.

b. A box plot since it indicates the spread of themiddle 50% of the data. 1 III

c. Answers will vary. Sample response: 2 I/II

The box plot indicates that 50% of the headcircumferences are between 56 cm and 60 cm,so the company will stock sizes between 56 cmand 60 cm.



Possible student answer Suggested numberof score points

Problem level

d. The scatter plot below indicates a weak but 3 I/IIpositive correlation. Answers about whether the result is useful to the company will vary. One possible answer is that since the correlation is weak, the relationship between height and head circumference is not an issue the company will consider when ordering hats.

e. The company needs to consider whether bias 2 IIIresulted from the way the data were collected. The company should be aware that they are assuming that the sample is representative of the population in the town in which the shop is located.

Height (in cm)

Head

Cir

cu

mfe

ren

ce

(in

cm

)

160 165 170 175 180 185 190 195 20050

52

54

56

58

60

62

64

Possible student answer



Problem levelSuggested numberof score points

3. a. Answers will vary. Sample student responses: 3 I

• In general, the number of manatee deaths seems to be increasing over time.

• The range in the number of manatee deaths is between 13 and 50.

• The median is equal to 28.5.

• The mean is equal to 29.4.

• A box plot of the data:

A box plot shows that at least 50% of the data points fall between 20 and 39 deaths.

b. 2 I

c. There seems to be a moderately strong, positive 1 I/IIcorrelation.

d. 2 I

The scatter plot shows that as the number of power boats increases, so does the number of manatee deaths.

100 20 30

Deaths

Minimum Median Maximum

40 50

Power Boat Registrations

(in thousands)

Man

ate

e D

eath

s

400 450 500 550 600 650 700 750 80010

20

30

40

50

60

Power Boat Registrations

(in thousands)

Man

ate

e D

eath

s

400 450 500 550 600 650 700 750 80010

20

30

40

50

60



Possible student answer Suggested numberof score points

Problem level

e. 4 II

i. Estimates will vary. Sample estimate:

About 14 manatees would be killed. Since 13 manatees were killed when there were 447 thousand power boat registrations, it seemslikely that 14 or 15 manatees might be killed when slightly more power boats are registered.

ii.Estimates will vary. Sample estimate:

About 45 manatees would be killed. Seven hundred thousand registrations is a little more than halfway between 675 and 711. An estimatemight be a little less than halfway between thecorresponding 43 and 50 manatee deaths.

f. Answers will vary. Sample student responses: 2 II/III

The Florida Department of Conservation could require that people who register their power boats take a course about manatees, their habits, the general areas in which they live, and the dangers of power boats to their safety.

The Florida Department of Conservation could recommend a restriction on the number of power boat registrations.

The Florida Department of Conservation could recommend banning power boats in areas in which manatees live.


(2 points for each correct estimate)

Glossary

88 Insights into Data Glossary

GlossaryThe glossary defines all vocabulary words indicatedin this unit. It includes the mathematical terms thatmay be new to students, as well as words having todo with the contexts introduced in the unit. (Note:The Student Book has no Glossary. Instead, studentsare encouraged to construct their own definitions,based on their personal experiences with the unitactivities.)

The definitions below are specific for the use of theterms in this unit. The page numbers given are fromthe Student Books.

bias (p. 12) an error introduced into samplingwhen elements have not been randomly selected;bias appears in sampling when certain outcomesare favored unfairly.

box plot (p. 37) a graphic way of showing asummary of data using five major data points: themedian, the first and third quartile, and extremes(minimum and maximum) of the data; a box plotmakes it easy to see how the data are spread outand where they are concentrated. A box plot alwaysincludes a scale line.

cause-effect relationship (p. 49) a relation-ship in which a change in one variable directlycauses a change in another variable

cluster (p. 6) a group of nearby points on a plot

correlation (p. 48) a relationship among variables

data point (p. 4) a point in a graph determinedby an x-value and a corresponding y-value (thecoordinates)

equation of a line (p. 57T)y � (slope) x � (intercept)

histogram (p. 15) a bar graph that shows afrequency distribution made up of rectangles whosewidths represent class intervals and whose heightsrepresent the number of data values in the class

mean (p. 5) a one-number value that describes a“center” for a set of data that is determined byadding the values for the set of data and dividing thesum by the number of data values

median (p. 33) the middle value of a set ofnumbers written in increasing order

mode (p. 33) the most frequently occurring valuein a set of numbers

line graph (p. 41) a graph that displays thechange of a variable over time

outlier (p. 7) a point that lies outside the clusterin a scatter plot

population (p. 12) a complete set of outcomesor individuals that are of interest in a particularquestion

quartile (p.37) medians of each half of a data set

random number (p. 18) a number chosen bychance in order to eliminate bias due to personalchoice

random sample (p. 12) a sample chosen bychance, a sample in which every member of thepopulation has an equally likely chance of beingchosen in the sample

sample (p. 12) a group taken from a larger popu-lation

scaling (p. 22A) labeling the x-axis by markinga point O (origin), separating the line into equalparts, and labeling the separate points with theintegers, for example: 1, 2, 3 (on the right-handside) or –1, –2, –3 (on the left-hand side); the y-axisis scaled in the same manner, but on a vertical line.

scatter plot (p. 4) a graph that shows therelationship between two variables, representedby points on the diagram

simulation (p. 12) a replicable experiment usedto model some phenomenon

slope (p. 56) the steepness or the slant of a line;it is determined by the ratio of the coordinates oftwo points on a line.

BlacklineMasters

BlacklineMasters

90 Insights into Data Letter to the Family

Letter to the Family

Dear Family,

Soon your child will begin the Mathematics in Context unit Insights intoData. Below is a letter to your child that opens the unit, describing theunit and its goals.

You can help your child relate the concepts in this unit to his or herown life by together looking through newspaper or magazine articlesthat contain numerical data. Discuss how information is presented in many different ways. In this unit, students will learn to thinkcritically about how numbers are used tojustify claims or to draw conclusions. Yourchild will also learn how to make his or herown statements based on data.

Look for surveys in magazines, in the mail, oron the Internet and discuss them with yourchild. Find graphs in articles and have yourchild explain the messages conveyed by the graphs. Do you use graphs or surveys in your work? Your child might share youruse of graphs or surveys with the rest of the class.

Enjoy working with your child to develop“insights into data.”

Sincerely,

The Mathematics in ContextDevelopment Team

Insights into Data

Dear Student,

Welcome to Insights into Data. Do you look at the graphs in

newspapers to see if they make sense? Numbers and graphs

are used to describe situations all around you: sports, grades,

sales, marketing, taxes, and even car ratings.In this unit, you will learn how to use numbers and graphs to

help you make decisions and draw conclusions. You will also

study surveys and how they are conducted. You will grow mung

beans in soda, salt water, and tap water to see which is the best

solution for growing sprouts. You will even learn to use lines to

help you investigate the relationship between two things, such

as the length and width of birds’ eggs. (Do you think birds’ eggs

are mostly round?)Look for graphs and numerical information in newspapers and

magazines to develop your own insights into data.Sincerely,

The Mathematics in Context Development Team

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Student Activity Sheet 1Use with Insights into Data, page 2.

Name ________________________________________

Student Activity Sheet 1 Insights into Data 91

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Day 1

Day 2

Day 3

Day 5

Day 4

Day 6

Day 7

Len

gth

of

Bean

Sp

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t (i

n m

m)

Bean

Nu

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s

an

d/o

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Pro

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ms

1 2 3 4 5 6 7 8 9 10

Student Activity Sheet 2Use with Insights into Data, pages 4–7.

Name ________________________________________

92 Insights into Data Student Activity Sheet 2

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.40 45 50 55 60 65 70 75 80 85 90 95 100

22,000

Percent Urban


24,000

26,000

28,000

30,000

32,000

34,000

36,000

38,000

40,000

42,000

44,000

Per

Cap

ita In

co

me (

in d

ollars

)

MS

ME

MT

MO

MN

MD

MA

NJ

KY

IAKS

MI

IL

GAIN

LASC

TX

RIPA

VA

UT

TNSD ND

NH

DC

CT

NY

NE

NC

OK

NM

AZ

COCA

AK

OR

OH

ID

DE

FL

HI

NV

WV

VT

WY

WA

WI

AR

AL


Name ________________________________________


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50

40

42

44

46

48

50

52

54

56

58

60

62

64

66

68

70

52 54 56 58 60 62 64 66 68 70 72 74 76 78 80

Percentage At or Above Basic Level in Mathematics

Perc

en

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to

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Basic

Levelin

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✄

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✄ ✄

✄

✄

✄

✄

Urban Urban

Urban Urban

Urban Urban

Urban Rural

Urban Rural


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34

56

78

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1112

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1516

1718

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Student Activity Sheet 7Use with Insights into Data, pages 53,

54, and 55.

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Name of BirdBird

FamilyNesting Grounds

Egg

Width

(in mm)

Egg

Length

(in mm)

Days

to

Hatch

Barn owl

Black cap

Black restart

Blackbird

Blue titmouse

Brambling

Bullfinch

Camion crow

Canada goose

Chaffinch

Chiff-chaff

Chough

Cirl bunting

Coal titmouse

Collared dove

Common sandpiper

Corn bunting

Crested titmouse

Crossbill

Dotterel

Dunlin

Eider

Fieldfare

Firecrest

Gadwall

Garden warbler

Garganey

Goldcrest

Golden plover

Goldeneye

Goldfinch

Goosander

Grasshopper warbler

Great titmouse

Green sandpiper

Owl

Warbler

Chat

Thrush

Titmouse

Finch

Finch

Crow

Goose

Finch

Warbler

Crow

Bunting

Titmouse

Pigeon

Sandpiper

Bunting

Titmouse

Finch

Plover

Sandpiper

Duck

Thrush

Warbler

Duck

Warbler

Duck

Warbler

Plover

Duck

Finch

Duck

Warbler

Titmouse

Sandpiper

Hollow trees, barns

Bushes, low trees

Holes, crevices

Trees, bushes

Holes

Trees, bushes

Bushes, conifers

Tall trees

Islets, marshes

Bushes, trees, hedges

Wood verges, hedgerows

Crevices, cliff ledges

Bushes, hedges

Holes, burrows

Trees, bushes

Lake shores

Grassy hollows

Tree holes

Conifers

Moors, scree

Moors, marshes

Near sea islets

Woodland trees

Conifers

Reedbeds

Shrubs, brambles

Dry ground, vegetation

Tall conifers

Boggy moors

Tree holes, burrows

Trees

Trees, holes

Moors, marshes

Tree holes

Trees on marshy ground

39.2

19.6

19.4

28.6

15.4

19.5

20.2

43.5

85.7

19.3

15.5

39.4

20.9

14.7

31.9

36.3

23.8

16.0

22.1

41.1

34.7

77.6

28.8

13.5

51.8

20.1

45.3

13.6

51.8

58.4

17.0

66.4

18.1

17.3

39.1

30.8

14.7

14.4

21.0

11.9

14.6

15.1

30.1

58.2

14.6

12.0

27.9

15.9

11.6

24.0

36.0

17.7

12.4

16.1

28.9

24.7

51.9

20.9

10.3

37.5

14.8

33.3

10.3

35.9

43.2

12.8

46.4

13.8

13.5

28.0

30

14-15

13

13-14

13-15

12-13

13

17-19

28-30

12-13

13-14

19-21

11-13

14-16

15-16

21-22

12-13

15-18

12-13

27-28

17

25-26

13-14

14-15

26

12

21-23

16

27

30

12-13

32

13-15

13-14

20-22

Student Activity Sheet 8Use with Insights into Data, pages 53,54, and 55.

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Name of BirdBird


Egg

Width

(in mm)

Egg

Length

(in mm)

Days

to

Hatch

Greenfinch

Grey wagtail

Greylag goose

Hawfinch

Hooded crow

House sparrow

Jackdaw

Jay

Lapwing

Lesser whitethroat

Linnet

Little owl

Little ringed plover

Long-eared owl

Long-tail titmouse

Magpie

Mallard

Marsh titmouse

Mistle thrush

Nightingale

Pied wagtail

Pintail

Pochard

Raven

Redpoil

Redwing

Reed bunting

Reed warbler

Restart

Ring ouzel

Ringed plover

Robin

Finch

Wagtail

Goose

Finch

Crow

Sparrow

Crow

Crow

Plover

Warbler

Finch

Owl

Plover

Owl

Titmouse

Crow

Duck

Titmouse

Thrush

Chat

Wagtail

Duck

Duck

Crow

Finch

Thrush

Bunting

Warbler

Chat

Thrush

Plover

Chat

Bushes, hedges

Holes, hedges

Reedbeds

Deciduous trees

Trees, bushes

Holes, crevices

Holes, crevices

Bushes, trees

Fields, moors

Thickets, hedgerows

Bushes, thickets

Tree holes

Mud, gravel pits

Abandoned nests

Trees, thickets

Tall trees, hedges

Ground cover

Tree holes

Tall trees

Ground cover

Holes, niches

Moors, marshes

Reeds, banks

Ledges, crevices, trees

Scrub growth

Bushes, trees

Reeds, marshes

Reeds, bushes

Tree holes

Bushes, small trees

Beaches, mud

Hollows, holes

20.2

19.0

85.3

24.5

43.5

22.5

33.7

31.6

47.1

16.5

17.7

33.6

29.8

40.9

13.6

34.1

58.4

16.1

31.2

21.0

20.4

51.8

61.3

49.7

16.9

25.8

19.3

18.3

18.7

30.4

35.7

19.4

14.5

14.5

58.0

17.5

30.1

15.7

25.2

23.0

33.7

12.6

13.3

28.1

22.1

32.7

10.9

24.2

39.5

12.2

22.3

15.6

15.1

37.0

43.7

33.4

12.6

19.2

14.3

13.6

13.8

21.5

25.9

14.8

13-14

12-14

28-29

14

17-19

13

17-18

16-17

24-26

10-11

12-14

28

23-26

27-28

12-13

17-18

28

14

13-14

13

12-14

22-23

24-26

20-21

10-11

13

13-14

11-12

13-14

14

24-27

13-14

Student Activity Sheet 9Use with Insights into Data, pages 53,

54, and 55.

Name ________________________________________


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Name of BirdBird


Egg

Width

(in mm)

Egg

Length

(in mm)

Days

to

Hatch

Rock dove

Rook

Scaup

Sedge warbler

Shelduck

Short-eared owl

Shoveler

Siskin

Skylark

Snow bunting

Song thrush

Stock dove

Stonechat

Tawny owl

Teal

Temminck's stint

Tree sparrow

Turtle dove

Twite

Wheatear

Whinchat

Whitethroat

Wigeon

Wood sandpiper

Wood warbler

Wood-pigeon

Woodlark

Yellow wagtail

Yellowhammer

Pigeon

Crow

Duck

Warbler

Duck

Owl

Duck

Finch

Lark

Bunting

Thrush

Pigeon

Chat

Owl

Duck

Sandpiper

Sparrow

Pigeon

Finch

Chat

Chat

Warbler

Duck

Sandpiper

Warbler

Pigeon

Lark

Wagtail

Bunting

Cliff holes

Tall trees

Moors, islands

Reeds, thickets

Sandy burrows

Moors, marshes

Moors, marshes

Conifers

Open ground

Stony ground

Bushes, trees

Tree holes

Grassy hollows

Hollow trees

Dry ground, vegetation

Islets

Holes, crevices

Trees, hedges

Ground cover

Holes, burrows

Grassy hollows

Scrub, bushes

Islets, ground cover

Boggy moors

Woodland ground

Trees, bushes

Wood verges

Grassy hollows

Grassy hollows

39.3

41.0

63.2

17.7

65.8

40.1

51.8

16.4

24.1

22.4

27.3

37.9

18.9

48.2

45.6

28.0

19.3

30.7

17.2

21.2

19.2

18.1

53.9

38.3

16.1

40.1

21.6

19.1

21.2

29.1

28.3

43.5

13.1

47.6

31.8

37.0

12.3

16.8

16.8

20.4

29.0

14.4

38.7

33.5

20.4

14.0

23.0

12.9

15.9

14.8

13.8

38.2

27.0

12.6

28.7

16.3

14.3

15.9

17

16-18

24-25

12-13

28-29

26-27

23-25

11-12

14

10-13

12-13

16-17

14-15

28-29

21-22

13-14

13-14

13-15

12-13

14

13-14

11-13

22-23

21-24

13

15-17

13-15

13-14

12-14

Student Activity Sheet 10Use with Insights into Data, pages 54,55, and 56.

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.13119

12

13

14

15

16

17

11

1015 17 19 21 23

Length (in mm)

Wid

th (

in m

m)

Student Activity Sheet 11Use with Insights into Data,

pages 56 and 58.

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DJo Mei, says “The mean height of the plants at the end of theexperiment is about 10 mm.”

Jorge, says “The mean height of the plants at the end of theexperiment is about 15 mm.”

2. a. Who do you think is right, Jo Mei or Jorge?

b. Which number is most easily found in the graph: the mean, the median, or the mode of the height values?

c. Reflect Explain how you might use the information in thegraph to find the median height of the plants.

Akir, Kari, Viviana, and Marja were studying the growth of plants. Afterseven days, they each graphed their data. Then they wrote statementsabout the growth of their plants. Unfortunately, some of the work wasmisplaced, and the rest was mixed together. On this page you see whatis left of their work.

3. For each student, match the statement with a graph. If a student’sgraph is missing, make an appropriate graph.

Using Data D

Hints and Comments(continued from page 33T)

About the MathematicsThe relation between one-number summaries andgraphs is addressed. Understanding the relationshipbetween statements and graphs is an important ability.Problems about connecting graphs and statements canalso be found in the algebra unit Ups and Downs.

PlanningStudents may work individually on problems 2 and 3.

Comments About the Solutions2. a. It is possible to calculate the exact mean, but

estimations are sufficient at this point.

3. Other solutions may be possible. It can be arguedthat graph A also fits the statements of Vivianaand Akir. If students choose one of these options,make sure they correctly graph the remainingstatements.

Section E: Correlating Data 103

E


11. If students are having difficulty formulatingcorrelations in general terms, encourage them toexpress the correlations in their own words.

12. It is important that students understand that astrong correlation does not necessarily imply theexistence of a cause-effect relationship.



Correlation between variables does not mean there isa cause-effect relationship between them. Sometimesa third variable can be found that explains therelationship. A famous example, taken from AMathematician Reads the Newspaper by John A.Paulos (New York: Basic Books, 1995), says thatchildren with longer arms seem to perform better onreading tasks. Here it is obvious that arm length doesnot cause better reading. What causes both is the ageof the child.

PlanningStudents may work individually or in small groups onproblems 10–12. After they have completed problem10, see whether they know what is meant by strong,weak, positive, and negative correlations.

In the diagrams for problem 8, you can see that a correlation can beweak or strong. Correlations can also be positive or negative.

10. a. What do you think is meant by the phrase “negativecorrelation”?

b. Which of the scatter plots on the previous page show anegative correlation?

11. For each of the following cases, decide whether there is nocorrelation, a strong correlation, or a weak correlation betweenthe two variables mentioned. If there is a correlation, is it positiveor negative?

• a person’s height and pulse rate

• the number of hours of sports training per week and pulse rate

• the height of a dinosaur and the length of its tail

• results on a math test and a science test

• temperature outside in the summer and kilowatt hours ofelectricity used

• number of children per household and number of televisionsets per household

• number of hours students study and their grade pointaverages

In a cause-effect relationship between two variables, a change in onevariable directly causes a change in the other.

12. a. If you expect a strong correlation between the two variables,can you be sure that a cause-effect relationship exists? Whyor why not?

b. Do you think there is a cause-effect relationship in any of thesituations in problem 11? Explain your reasoning.

Collect some data about one of the cases in problem 11 and usethe data to check your answer about the correlation. Be sure tothink carefully about how you will select your sample.

Correlating Data E CorrelatingData



F Lines That Summarize DataF

You have discovered that eggs are different sizes.But what about their shapes? Is there a relationshipbetween the lengths and widths of bird eggs? Toanswer these questions, you will begin by lookingat the eggs of one family, the warbler.

3. a. Find all the warblers in the table on Student

Activity Sheets 7, 8, and 9, and make a scatterplot of the lengths and widths of the warblereggs (put length on the horizontal axis andwidth on the vertical).

b. Describe the relationship between the lengthsand widths of the warbler eggs. Is there anycorrelation?

c. What do you expect the width of a warbler eggwill be if its length is 15 mm? How did you findyour answer?

The points in the scatter plot lie almost on a straight line. You mighthave used this information to find an answer to problem 3c. It ispossible to summarize the pattern, or relationship, by drawing a linethat seems to best fit these points. This line will probably not gothrough all the points, but the points should lie close to the line.

4. a. On Student Activity Sheet 10, draw a straight line that seemsto “fit” these points. Use your line to predict the width of awarbler egg that is 15 mm long.

b. Does your line give you the same answer that you predicted inproblem 3c?

5. a. Use your line to determine what happens to the width of anegg if the length increases by 2 mm. Explain how you did this.Include a drawing with your answer.

b. Describe what will happen to the width of an egg if the lengthincreases by 1 mm.



3. Some students may draw scatter plots that beginat (12, 10) since the warbler eggs all have lengthsgreater than 12 millimeters and widths greaterthan 10 millimeters. Students need to recognizethat they should scale the axes so that the data forthe warbler eggs can be plotted. You may want toremind students that many plots of real data donot begin at the origin because there are no datathere, and a large area of the plot would be empty.Students could think of such plots as a selectedview of one portion of the graph.

5. a. Note: The scales of the axes are different. Thiscauses the horizontal increase of the length of2 mm to seem smaller than the verticalincrease of the width of 1.5 mm.

b. It is not necessary to have students write anequation for their line.

Section F: Lines That Summarize Data 105

F


17. c. Students should realize that the line would be apoor predictor for points that are far away fromthe line. Students should also notice that withthe straight line the data points in the middleare above the line and data points at the endsare below the line.

18. a. Graphing large scatter plots may be too timeconsuming, since it is not a goal in this section.When fitting lines or curves in a scatter plot, it isimportant to have a contextually soundexplanation for the kind of curve or line that fits.


Technology

Drawing lines can be done by eye. When data areentered in the graphing calculator or the computer, thecalculator or computer can draw the best-fitting line.

Planning

Students may work on problems 16–19 in small groupsor individually. When students have completed theseproblems, you may want to summarize the mostimportant results in a class discussion.

Graphs of growth often result in curves because livingthings do not grow at a constant rate throughout theirlifetimes.

Researchers in fisheries are interested in studying thegrowth rates of fish. One study on bluegills comparedthe lengths of fish at the beginning of the year withtheir lengths at the end of the year.

The table contains the results of this research.

16. a. How much did the fish that was initially 161 mmgrow in one year?

b. Graph the data from the table.

c. What pattern do you see in the scatter plot fromproblem 16b?

17. a. Draw a straight line to represent the data in thescatter plot.

b. Estimate how long a 140-mm fish will be afterone year.

c. Is a straight line a good model to represent thedata in the graph? Explain your reasoning.

18. a. Sketch a model on the graph that betterrepresents the data.

b. Using this model for the growth, describe whatis happening to the growth of the fish as thelength changes.

c. If a fish is now 110 mm long, predict how longit will be in one year.

19. Describe the difference between using a straightline and using the curved line you created inproblem 18 to predict growth.


Gone FishingBluegill Growth

Initial Length after

Length x 1 Year y

(in mm) (in mm)

48 69

52 71

51 69

53 75

68 101

71 107

69 100

75 104

101 138

107 138

100 130

104 140

138 160

132 157

130 156

140 161

160 173

157 168

156 172

161 178

173 176

168 174

172 173

178 178


Insights into Data...Insights into Data v Contents Overview NCTM Principles and Standards for School...

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Transcript of Insights into Data...Insights into Data v Contents Overview NCTM Principles and Standards for School...