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MATHEMATICS 7

iMATHEMATICS 7: A TEACHING RESOURCE

MATHEMATICS 7:

A TEACHING RESOURCE

Please note that all attempts have been made to identify and acknowledge information from externalsources. In the event that a source was overlooked, please contact English Program Services, Nova ScotiaDepartment of Education, [email protected]

© Crown Copyright, Province of Nova Scotia, 2005

Prepared by the Department of Education

Revised and Reprinted 2010

Contents of this publication may be reproduced in whole or in part provided theintended use is for non-commercial purposes and full acknowledgement is given to theNova Scotia Department of Education.

Cataloguing-in-Publication Data

Main entry under title.

Mathematics 7: a teaching resource / Nova Scotia.Department of Education. English Program Services.

ISBN: 0-88871-947-7

1. Curriculum planning – Nova Scotia. 2. Mathematics – Study and teaching –Nova Scotia. I. Nova Scotia. Department of Education. English Program Services.

510.07109716--dc22 2005

Website References

Website references contained within this document are provided solely as aconvenience and do not constitute an endorsement by the Department of Educationof the content, policies, or products of the referenced website. The Department doesnot control the referenced websites and subsequent links, and is not responsible forthe accuracy, legality, or content of those websites. Referenced website content maychange without notice.

School boards and educators are required under the Department’s Public SchoolPrograms’ Internet Access and Use Policy to preview and evaluate sites beforerecommending them for student use. If an outdated or inappropriate site is found,please report it to [email protected].

iiiMATHEMATICS 7: A TEACHING RESOURCE

ACKNOWLEDGMENTS

The Department of Education gratefully acknowledges the contributionsof the following individuals to the preparation of the mathematics gradesprimary–9 teaching resources.

Todd Adams—Halifax Regional School BoardRay Aucoin—South Shore Regional School BoardGeneviève Boulet—Mount Saint Vincent UniversityAnne Boyd—Strait Regional School BoardDarryl Breen—Strait Regional School BoardMargaret Ann Cameron—Strait Regional School BoardNancy Chisholm—Nova Scotia Department of EducationFred Cole—Chignecto-Central Regional School BoardDavid DeCoste—Saint Francis Xavier UniversityDarlene Fitzgerald—Halifax Regional School BoardBrenda Foley—Chignecto-Central Regional School BoardCindy Graham—Chignecto-Central Regional School BoardRobin Harris—Halifax Regional School BoardBetsy Hart—South Shore Regional School BoardJocelyn Heighton—Chignecto-Central Regional School BoardDoug Holland—Annapolis Valley Regional School BoardMarjorie Holland—Halifax Regional School BoardPaula Hoyt—Halifax Regional School BoardDonna Karsten—Nova Scotia Department of EducationShelley King—Annapolis Valley Regional School BoardAnne MacIntyre—Halifax Regional School BoardKenrod MacIntyre—Halifax Regional School BoardRichard MacKinnon—Private ConsultantRon MacLean—Cape Breton-Victoria Regional School BoardSharon McCready—Nova Scotia Department of EducationJanice Murray—Halifax Regional School BoardFlorence Roach—Annapolis Valley Regional School BoardEvan Robinson—Mount Saint Vincent UniversitySherene Sharpe—South Shore Regional School BoardMartha Stewart—Annapolis Valley Regional School BoardMonica Teasdale—Strait Regional School BoardMarie Thompson—Halifax Regional School BoardCindy Wallace—Cape Breton-Victoria Regional School BoardPat Ward—Halifax Regional School BoardApril Weaver—Strait Regional School BoardBrenda White—Strait Regional School Board

ACKNOWLEDGMENTS

vMATHEMATICS 7: A TEACHING RESOURCE

CONTENTS

Introduction ............................................................................................... 1Section 1: Instructional Planning ............................................................... 3

Overview ............................................................................................... 3Section 1A ............................................................................................. 7Section 1B ........................................................................................... 41

Section 2: Language and Mathematics ..................................................... 93Grade 7 Mathematics Language ....................................................... 107

Section 3: Assessment and Evaluation ..................................................... 109Introduction ...................................................................................... 109Assessments for Learning ................................................................... 109Assessments of Learning .................................................................... 110Alignment.......................................................................................... 111Planning Process: Assessment and Instruction ................................... 112Assessing Students’ Understanding .................................................... 114Questioning ....................................................................................... 115Assessment Techniques ...................................................................... 134

Section 4: Technology ............................................................................. 141Integrating Information Technology and Mathematics ..................... 141Using Software Resources in Mathematics Classrooms...................... 145

Appendices ............................................................................................. 157Appendix A: SCOs by Strand ........................................................... 159Appendix B: The Process Standards .................................................. 162Appendix C: Large Frayer Model ..................................................... 163Appendix D: The Concept Definition Map...................................... 164Appendix E: What Is Mental Math? ................................................. 165Appendix F: Recommended Resource List for Grade 7.................... 167Appendix G: Mathematics Resources for Grade 7............................ 170Appendix H: Outcomes Framework to Inform Professional

Development for Mathematics Teachers ........................................ 173Appendix I: Outcomes Framework to Inform Professional

Development for Teachers of Reading in the Content Areas ......... 176Bibliography ........................................................................................... 179

CONTENTS

1MATHEMATICS 7: A TEACHING RESOURCE

SECTION 1: INSTRUCTIONAL PLANNING

INTRODUCTION

Atlantic Canada Mathematics Curriculum: Grade 7 is the key resource forteachers of mathematics at this level. The document is central to daily, unit,and yearly planning. It is through appropriate planning that teachers will beable to address all of the outcomes for the grade 7 mathematics curriculum.It is important to emphasize, however, that the presentation of specificcurriculum outcomes in strands does not represent a suggested teachingsequence. Because teachers might wish to address certain outcomes beforeothers, a great deal of flexibility exists in the approach to the instructionalprogram.

Mathematics 7: A Teaching Resource is intended to provide support forteachers as they design their grade 7 mathematics program. In Section 1:Instructional Planning, sample yearly plans show possible ways to cluster andsequence outcomes. Unit plans and lesson plans give teachers clear examplesof how to structure lessons to meet the needs of students within theirclassrooms. Section 2: Language and Mathematics gives teachers strategiesand examples to use as they challenge students to think and reason aboutmathematics and communicate their ideas to others orally or in writing in aclear and convincing manner. Strategies were chosen to promote vocabularydevelopment, problem solving, and reflection in the mathematics classroom.Section 3: Assessment and Evaluation emphasizes that the way we assess haschanged. It is no longer sufficient to test only a student’s recall of a topic. It isalso important to have students show that they understand and can apply theideas in mathematics and that they are capable of using manipulatives andcalculators to aid in this understanding. Section 4: Technology identifies theunderpinnings of technological competence in the school program and howit applies to the mathematics classroom.

Sixty minutes every day is a minimum requirement in grade 7 forinstructional time allotted to mathematics. Included in this time frame is fiveminutes daily for Mental Math and Estimation. Teachers are encouraged touse Mathematics 7: A Teaching Resource in conjunction with Atlantic CanadaMathematics Curriculum: Grade 7 to ensure that all the dimensions of thegrade 7 mathematics program and the full range of prescribed curriculumare addressed in the instructional time allocated for mathematics.

INTRODUCTION




Overview

To support effective instructional planning, this section includes examples ofyearly plans, unit plans, and lesson plans.

Yearly Plan

A yearly plan is the clustering of outcomes into appropriate units to help youdevelop a cohesive mathematics program. Yearly plans encourage you todevelop time lines, which help ensure that students have opportunities toachieve all outcomes. The examples of yearly plans are meant to besuggestions only. Italicized outcomes indicate that they are of secondaryimportance at this time but will be fully addressed elsewhere in the plan.

You may wish to develop your own yearly plan. Listed in Appendix A are thespecific curriculum outcomes (SCOs) for this grade level. On a large piece ofpaper, decide how you would like to section the school year into intervals oftime, e.g., months or terms. Colour-code the SCOs by strand and cut eachSCO into strips. You may want to duplicate copies of the SCOs that will berevisited throughout the year. Cluster the colour-coded SCOs into unitswithin the time frame you have chosen. Mapping out the year in this waymakes it easier to visualize how your program will evolve.

Unit Plan

A cluster of outcomes from the yearly plan becomes a unit of work. Unitplans for this document were created to align with the template that follows.You should be aware of the units that depend on the concepts of other units,so those dependent units can be taught in proper sequence. Italicizedoutcomes indicate that they are of secondary importance at this time but willbe fully addressed elsewhere.

Unit Plan Template

Unit Plan Title

Purpose or description of unit

• Identify how this unit relates to your yearly plan.• Identify the outcomes that will be focused on and the reasoning behind why you

have chosen to cluster outcomes in this way. Comments can be made onconnections, big ideas, the time of year, and/or the preceding unit.

4 MATHEMATICS 7: A TEACHING RESOURCE


Main Focus

• What mathematical concepts and understandings are the focus of the unit?• What will students be able to do and understand at the completion of the unit?

(What skills and knowledge are required for the students to achieve identifiedoutcomes and desired results?)

• How will you differentiate instruction?

Assessment

• The assessment is the why, what, and how of collecting evidence of studentlearning.

• Why are you assessing?• What are the purposes of the assessments used in your unit?• What evidence needs to be collected to demonstrate student learning and

understanding?• What methods will be used?• Have you designed daily or unit assessments that allow students to demonstrate

their knowledge and understanding in a variety of ways?• Have you included questions that encourage students to apply their learning to

non-routine problems?– More information on the types of questions we should be asking our students

can be found in the Assessment section of this document.• How will you use the data to inform instruction and improve student achievement?

Communication

• What opportunities will you provide for students to communicate about theirlearning? (“The ability to read, write, listen, think creatively, and communicateabout problems will develop and deepen students’ understanding ofmathematics.” National Council of Teachers of Mathematics, Curriculum andEvaluation Standards, page 78)

• How will you model communication?• Have you identified the new vocabulary words for this unit?

Technology Strategies

• Can technology be used to engage students’ mathematics learning and make theoutcomes more accessible to students with diverse learning styles and needs?

• Is technology an effective tool/method to address the outcomes?• What classroom arrangement will allow optimal student use of technology?• How will you ensure that students have the technological skills to complete the

activity?• Remember that technology is another tool to enhance the delivery of curriculum

and should be used as an effective tool to engage, motivate, and enhance studentlearning.

Materials

• List the materials and resources needed.

Summary of Lessons

• Identify the lesson and time frame.• Give a brief description of the lessons.• Identify SCOs being addressed in the lesson.



Lesson Plan

From the unit plan, lesson plans can be developed. The length of time it willtake to complete these lessons will depend on the concept involved. Samplelesson plans are included in this document and were created following thetemplate that is shown below. It is important to note that a valid way toapproach the creation of a lesson plan is to first develop the assessments thatwill guide the teaching and then plan lessons that will provide the learningexperiences students need. Italicized outcomes indicate that they are ofsecondary importance at this time but will be fully addressed elsewhere in theplan.

Lesson Plan Template

Lesson Plan Title

Lesson Summary

• Give a brief description of the lesson.

Specific Curriculum Outcomes

• List SCOs.• What knowledge and skills will the student acquire upon the completion of this

unit? (Desired results)

Assessment

• What evidence will be collected to demonstrate student learning andunderstanding?

• What methods will be used to ensure that students have the opportunity todemonstrate what they know and can do in a variety of ways?

• Will students be involved in the assessment process? How?• How will you use the data to inform instruction?

Communication

• What strategies, methods, and tools will be used to support students whenintroducing, developing, and/or reinforcing the mathematical language being usedin this lesson?

Technology

• In what ways will/can technology enhance this lesson?

Materials

• List the materials and resources needed.

Mental Math

• Identify the strategy (or strategies) to be used.• Describe the activity.

Development

• Describe the development of the lesson and the activities that make up the lesson.



Extensions

• What ways can this lesson be extended?

Follow-up Activities

• Follow-up activities are any activities at home or school that engage students infurther thinking related to the concept(s) they are learning about. These activitiesshould be aligned with your targets, assessment, and instruction.

Comments

• Note any additional information to be provided to the students.



Section 1A

The following section contains a sample of a yearly plan, a unit plan on datamanagement and probability, and five lesson plans.

Sample Yearly Plan 1

September–October

Data Management and Probability (approximate time: six weeks)

This topic does not require a great deal of prior knowledge, integrates wellwith various other subjects (e.g., Interactions with Ecosystems and Solutionsin the science curriculum), and provides a way for students to experiencesuccess early in the year.

Outcomes from strands A, B, and C are introduced here only as they applyto data management and probability. They will be dealt with in depth inNovember and March.

GCO Grade 7 SCOs

By the end of grade 7, students will be expected to

F: Data Management F1 communicate through example the distinction between biassedand unbiassed sampling, and first- and second-hand data

F2 formulate questions for investigation from relevant contexts

F3 select, defend, and use appropriate data collection methods andevaluate issues to be considered when collecting data

F4 construct a histogram

F5 construct appropriate data displays, grouping data whereappropriate and taking into consideration the nature of data

F6 read and make inferences for grouped and ungrouped data displays

F7 formulate statistics projects to explore current issues from withinmathematics, other subject areas, or the world of students

F8 determine measures of central tendency and how they areaffected by data presentations and fluctuations

F9 draw inferences and make predictions based on the variability ofdata sets, using range and the examination of outliers, gaps, andclusters

G: Probability G1 identify situations for which the probability would be near 0, , , , and 1

G2 solve probability problems, using simulations and by conductingexperiments

G3 identify all possible outcomes of two independent events, usingtree diagrams and area models

G4 create and solve problems, using the numerical definition ofprobability

G5 compare experimental results with theoretical results

141

234



GCO Grade 7 SCOs


G: Probability G6 use fractions, decimals, and percentages as numerical expressionsto describe probability

A: Number Sense A7 apply patterning in renaming numbers from fractions and mixednumbers to decimal numbers

A10 illustrate, explain, and express ratios, fractions, decimals, andpercentage in alternative forms

A11 demonstrate number sense for percentage

B: Operation Sense B8 estimate and determine percentage when given the part and thewhole

B9 estimate and determine the percentage of a number

B10 create and solve problems that involve the use of percentage

C: Patterns and Relations C1 describe a pattern, using written and spoken language and tablesand graphs

C7 interpolate and extrapolate number values from a given graph

C9 construct and analyze graphs to show how change in onequantity affects a related quantity

Mid–End of October

Number Sense (approximate time: two weeks)

Students have been introduced to decimals previously, but the percentageand fraction skills are new. It is necessary that students cover divisibility rules,LCM (least common multiple), and GCF (greatest common multiple) beforefractions because we want students to see the connections between LCM andGCF for fractions.

Students need to understand exponents before doing order of operations fordecimals and integers. Scientific notation is a practical application ofexponents.

GCO Grade 7 SCOs


A: Number Sense A1 model and use power, base, and exponent to represent repeatedmultiplication

A2 rename numbers among exponential, standard, and expandedforms

A3 rewrite large numbers from standard form to scientific notationand vice versa

A4 solve and create problems involving common factors and greatestcommon factors (GCF)

A5 solve and create problems involving common multiples and leastcommon multiples (LCM)

A6 develop and apply divisibility rules for 3, 4, 6, and 9



November

Number Sense (approximate time: four weeks)

GCO Grade 7 SCOs



A8 rename single-digit and double-digit repeating decimals tofractions through the use of patterns, and use these patterns tomake predictions

A9 compare and order proper and improper fractions, mixednumbers, and decimal numbers


B: Operation Sense B1 use estimation strategies to assess and justify the reasonablenessof calculation results for integers and decimal numbers

B2 use mental math strategies for calculations involving integers anddecimal numbers

B3 demonstrate an understanding of the properties of operationswith decimal numbers and integers

B4 determine and use the most appropriate computational method inproblem situations involving whole numbers and/or decimals

B5 apply the order of operations for problems involving whole anddecimal numbers

December

Fractions (approximate time: three weeks)

GCO Grade 7 SCOs


A: Number Sense A9 compare and order proper and improper fractions, mixednumbers, and decimal numbers


B: Operation Sense B6 estimate the sum or difference of fractions when appropriate

B7 multiply mentally a fraction by a whole number and vice versa



January–February

Integers (approximate time: five weeks)

We chose integer work in January because this is a very important topic, andstudents will be rested after the break. It is important to allow students ampletime with the manipulatives to ensure a firm foundation for integeroperations. Operations with integers should be introduced into subsequentunits. Using your time for mental math to practise proficiency withoperations and integers allows you to move onto the next unit after fiveweeks.

GCO Grade 7 SCOs


A: Number Sense A12 represent integers (including zero) concretely, pictorially, andsymbolically, using a variety of models

A13 compare and order integers




B11 add and subtract integers concretely, pictorially, and symbolicallyto solve problems

B12 multiply integers concretely, pictorially, and symbolically to solveproblems

B13 divide integers concretely, pictorially, and symbolically to solveproblems

B14 solve and pose problems that use addition, subtraction,multiplication, and division of integers

B15 apply the order of operations to integers

Mid February

Percentage (approximate time: three weeks)

GCO Grade 7 SCOs


A: Number Sense A10 illustrate, explain, and express ratios, fractions, decimals, andpercentage in alternative forms


B: Operation Sense B8 estimate and determine percentage when given the part and thewhole





End of February–First Week of March

Geometry (approximate time: two weeks)

GCO Grade 7 SCOs


E: Geometry E1 recognize, name, describe, and construct polygons

E2 predict and generate polygons that can be formed with atransformation or composition of transformations of a givenpolygon

E3 make and apply generalizations about the properties of regularpolygons

E4 make and apply generalizations about tessellations of polygons

Mid March–April–Early May

Patterns and Algebra (approximate time: six weeks)

GCO Grade 7 SCOs



A: Number Sense A8 rename single-digit and double-digit repeating decimals tofractions through the use of patterns and use these patterns tomake predictions

C: Patterns and Relations C2 summarize simple patterns, using constants, variables, algebraicexpressions, and equations and use them in making predictions

C3 explain the difference between algebraic expressions andalgebraic equations

B: Operation Sense B16 create and evaluate simple variable expressions by recognizingthat the four operations apply in the same way as they do fornumerical expressions

B17 distinguish between like and unlike terms

B18 add and subtract like terms by recognizing the parallel withnumerical situations, using concrete and pictorial models

C: Patterns and Relations C4 solve one- and two-step single-variable linear equations, usingsystematic trial

C5 illustrate the solution for one- and two-step single-variable linearequations, using concrete materials and diagrams

C6 graph linear equations, using a table of values


C8 determine if an ordered pair is a solution to a linear equation




May

Geometry (approximate time: four weeks)

This is a good change of pace for students, because this topic involves moreopportunities for students to use concrete materials and explore angles in oureveryday lives. It may be possible for students do some work outdoors on thisunit, since the weather should be improving at this time of year.

GCO Grade 7 SCOs


D: Measurement D5 demonstrate an understanding of the relationships amongdiameter, radii, and circumference of circles and use therelationships to solve problems

E: Geometry E5 construct polyhedra using one type of regular polygonal face, anddescribe and name the resulting Platonic Solids

E6 construct semi-regular polyhedra and describe and name theresulting solids, and demonstrate an understanding about theirrelationships to the Platonic Solids

E7 make and apply generalizations about angle relationships

E9 make and apply informal deductions about the minimumsufficient conditions to guarantee that a given triangle is of aparticular type

E10 make informal deductions about the minimum sufficientconditions to guarantee that a given quadrilateral is of a particulartype, and to understand formal definitions of the variousmembers of the quadrilateral family

June

Measurement (suggested time: two weeks)

The weather should be fine by now, and this unit lends itself to opportunitiesto go outside to do measurement with concrete materials.

GCO Grade 7 SCOs


D: Measurement D1 identify, use, and convert among the SI units to measure, estimate,and solve problems that relate to length, area, volume, mass, andcapacity

D2 apply concepts and skills related to time in problem situations

D3 develop and use rate as a tool for solving indirect measurementproblems in a variety of contexts

D4 construct and analyze graphs of rates to show how change in onequantity affects a related quantity



Unit 1: Data Management

Purpose

This is a four- to five-week unit in September–October of yearly plan 1. Itprimarily addresses all outcomes in Data Management, but connections toother outcomes are noted.

In the assessment section of this overview is a project that can be used as themain assessment for the unit.

Outcomes

GCO Grade 7 SCOs







F6 read and make inferences for grouped and ungrouped datadisplays




C: Patterns and Relations C7 interpolate and extrapolate number values from a given graph


Main Focus

Newspapers, magazines, and books often publish information in the form ofa graph. The ability to read, interpret, recognize when data are beingmisrepresented, and construct the appropriate graph to display the data is animportant life skill.



By the end of this unit students will be able to

• evaluate questions used in a survey• collect, group, and display numerical data• determine the measures of central tendency• examine how gaps, clusters, and outliers impact the measures of central

tendency• make decisions about the most appropriate graph to use to display data

Assessment

The following project can be used as the assessment for the unit. It should beintroduced approximately 10 days before the end of the unit and be revisitedeach day for a short period of time to make sure everyone is on task and isworking on his or her projects. Class time should be allocated forpresentations.

Grade 7 Statistics Report

Your report must include the following:

1. Introduction: In this section explain the background to yourinvestigation, why you chose this topic, and how it will benefit yourcommunity.

2. Procedure:• Describe your target population.• Describe your sample and how you selected your sample.• Tell how you collected your data. Include any questions used.

3. Data: Include your data in a table or tally chart. Your report must containat least two different kinds of graphs. One must be a histogram. Thesemay be done by hand or on the computer.

4. Conclusion: Summarize your results. Show how the results of yourinvestigation can be of use to groups or individuals in the community.

5. List what each member of your group did in completing your report.



Project Evaluation

Introduction Your Score

Topic clearly stated ___ / 1

Background information ___ / 1

Population and Sample

Target population identified ___ / 1

Sample described ___ / 1

Sample representative ___ / 1

Procedure

Question phrased well ___ / 2

Collection of data described ___ / 2

Data

Tables ___ / 2

Graph ___ / 1

Graph ___ / 2

Conclusion

Results summarized ___ / 2

Use of survey discussed ___ / 2

Source of error discussed ___ / 2

Group Mark ___ / 5

Total ____ / 25

Communication

Students will share their findings with the other students in their group,reinforcing the use of mathematical terms such as data, samples, estimation,and actual. Class discussions should focus on the type of graph chosen andthe meaning of the data on the graphs.

When the groups present their graphs on the overhead, they will defend whytheir choice of graph is appropriate or not appropriate for their particulartype of data. Oral communication is also reinforced as students makeinferences about the grouped and ungrouped data displays.

Technology Strategies

Students could also be given the opportunity to explore the use ofspreadsheets to sort data, quickly determine the mean and median of largedata sets, and draw different types of graphs quickly. These strategies allowstudents to compare how graphs display data, helping them to understandwhy some types are appropriate and others are not.



Materials

• 10 × 10 grid• activity sheets for each activity• Atlantic Canada Mathematics Curriculum: Grade 7• calculator• graph paper• Mathematics 7: Focus on Understanding (Calabrese, Fournier, and Speijer 2005)• Mental Math in Junior High (Hope, B. Reys, and R. Reys 1998) Lesson 46• newspapers/magazines• novels• number line• percent circle• simulation materials (e.g., Cube-a-Links, pieces of paper, base-10 blocks)• strip graph (strips of paper, approximately 2 cm wide by 12 cm long)

Description of Lessons

Lesson 1: Do you have any questions? (approximately 3–4 days)

In this lesson, students explore questioning in a survey and the part it plays in theresults of the survey.

GCO Grade 7 SCOs


F: Data Management F1 communicate through example the distinction between biassedand unbiassed sampling, and first-and second-hand data



Lesson 2: What are the chances? (5–6 days)

In this lesson, students will collect actual numerical data, group the data, anddisplay it using a histogram. Some teachers may prefer to complete Lesson 4prior to Lesson 2.

GCO Grade 7 SCOs


F: Data Management F2 formulate questions for investigation from relevant contexts





GCO Grade 7 SCOs


F: Data Management F5 construct appropriate data displays, grouping data whereappropriate and taking into consideration the nature of data



C: Patterns and Relations C7 interpolate and extrapolate number values from a given graph


Lesson 3: Using Measures of Central Tendency and Dispersion to DescribeData (approximately 4–5 days)

In this lesson students calculate mean, median, and mode for a data set andmake decisions on which measure of central tendency is most appropriate fora set of data. The impact of gaps, clusters, and outliers on the measures ofcentral tendency is also explored.

GCO Grade 7 SCOs


F: Data Management F8 determine measures of central tendency and how they areaffected by data presentations and fluctuations


Lesson 4: Constructing Appropriate Data Displays (3–4 days)

In the lesson, student will construct various data displays and determinewhich display is most suitable for a situation. The use of technology toconstruct the display is recommended.

GCO Grade 7 SCOs


F: Data Management F5 construct appropriate data displays, grouping data whereappropriate and taking into consideration the nature of data



Lesson Plan 1: Do You Have Any Questions?

Lesson Summary

This lesson looks at how questions in a survey are formulated and the impactthey have on the results of the survey.

Students will be able to determine• if a survey question is biassed• if the sample being surveyed is biassed• from what other sources, besides surveys, data can be found


F1, F2, F3

Assessment

Observe students for• understanding of biassed and unbiassed sampling• understanding of first-hand and second-hand data• choosing an appropriate method of collecting data• ability to rewrite a question that is unclear or misleading

Communication

Students will be discussing survey questions in their small groups and as aclass. They will also be conducting interviews to complete a physicaleducation survey.

Technology

Students will be encouraged to use the Internet as one of their sources whenthey fill out the Comparing Data activity sheets.

Materials

Activity sheets:• Comparing Data• Overhead of Lesson 48, Mental Math in Junior High (Hope, B. Reys, and

R. Reys 1998)• Student Survey on Future Plans• Survey on Calculators in Junior High• That’s a Good Question• What’s Your Bias?



Mental Math

Mental Math Lesson 46 is a good lead-in to this unit where students will beworking with fractions and their decimal equivalents.

Development

Warm-up

Should a traffic light be installed at a certain intersection in our town?

How could this question be answered?• Collect data about pedestrians.• Collect data about vehicular traffic.• What kinds of data should be collected?

– number of pedestrians– number of vehicles– speed of the vehicles– time of day– day and month

• Is it practical and possible to collect all the data that you would like to have?

Do You Have Any Questions?

This activity was taken from Dealing with Data and Chance, Addenda SeriesGrades 5–8 (NCTM 1991).

Give each student a copy of the activity sheet (page 20) entitled StudentSurvey on Future Plans, and ask them to complete it. Pose the followingquestions for discussion after they have finished:• Were all the questions clear to you? If not, tell why.• Did you always know what to answer?• Do you think everyone else answering the question would know what to

answer? Why or why not?• What types of responses were required in the survey? (Multiple choice,

numerical value, short answer, yes/no, fill in the blank?)• Why do you think those conducting the survey asked for the type of

responses they did?• What do you think the data will look like when combined and organized?• What questions do you think the writers of the survey are going to try to

answer with the information they obtain?• How do you think the data will be used?



Activity Sheet: Student Survey on Future Plans

For each question, select one choice. Your answers will be completely confidential;only summary data will be reported. THANK YOU for taking the time to complete thissurvey. Getting your answers and those from others will help assemble accurateinformation on this important topic.

1. How old are you? under 12 12 13 14 15

16 17 18 19 over 19

2. What is your sex? Male Female

3. Do you plan to get married? Yes No

4. Do you plan to have children? Yes No (If no, skip to Question 6.)

5. How many children would you like to have? 1 2 3 4 5 6 7 or more

6. After high school, which of the following do you plan to do?Attend a 4-year programAttend a 2-year programGo to a trade or vocational collegeJoin the Armed ForcesGet a full-time jobNone of these

7. Of the following occupations, which one would you most like to pursue afterschool?

CarpenterComputer programmerDoctorFarmerFirefighter

Forest rangerHairdresserLawyerMechanicSocial worker

StockbrokerTeacherTruck driverNone of these

Forest rangerHairdresserLawyerMechanicSocial worker

StockbrokerTeacherTruck driverNone of these

8. Of the following occupations, which one would you least like to pursue afterschool?

CarpenterComputer programmerDoctorFarmerFirefighter



Activity Sheet: Bias in Questioning

Tell the students that a Thunder Bay radio station took a poll to find outwhether the people in the city preferred the Toronto Maple Leafs or theOttawa Senators hockey team. The people polled were asked these twoquestions:a) On a scale of 1 (low) to 10 (high), how well do you like the great Ottawa

Senators hockey team now that they made it to the semi-finals?b) On a scale of 1 (low) to 10 (high), how well do you like the Toronto

Maple Leafs hockey team?

Ask the students to discuss the following questions in their group and latershare with the whole class.• Will the resulting data be easily tallied and organized?• What kind of results would you expect?• Will the results accurately reflect the comparative feeling people have for

the Maple Leafs and the Senators?• Can you make a conjecture about which team the question writer prefers?

Why do you think so?• What are some guidelines that should be followed in writing effective

survey questions?

Hand out the That’s a Good Question activity sheet (page 22), and ask thestudents to fill it out.

Ask the students to formulate questions for a physical education survey. As aclass, decide on what to find out from students about what they would like todo in physical education class. In groups of four they should come up withsix questions to ask.

Each group should present their questions to the class. As a group, the classdecides on the six questions they think would be best for their survey.

Write these questions in biassed form, and then write the same six questionsagain in non-biassed form.

Each student should interview two people. Ask one the questions in biassedform, the other in non-biassed form.

The following day the results from each set of questions should be combined,and the differences in the results of the two sets should be discussed.



Activity Sheet: That’s a Good Question

Good questions for questionnaires do not show bias or try to influence theperson answering. Choose the better question in each pair below.

1 a) Should inexperienced youth of 16 be allowed to vote?b) Should the voting age be lowered to 16?

2 a) Do you think girls are less athletic than boys?b) Do you agree with most people that girls are less athletic than boys?

3 a) Should kids be allowed to ride bikes on sidewalks?b) Should kids be allowed to endanger others by riding their bikes on the

sidewalk?

4 a) Do you like the wonderfully talented singer Celine Dion?b) Do you think Celine Dion is a good singer?

5 a) Are you a member of a large family?b) How many brothers and sisters do you have?

6 a) Because of the increasing amount of tooth decay, do you thinkteenagers consume too much sugar?

b) Do you think teenagers get too much sugar in their diet?

7 a) Should your city’s water supply be fluoridated?b) Should your city’s water be supplied with the dangerous chemical

fluoride?

8 a) Do you think, as the Minister of Health does, that smoking ishazardous to your health?

b) Do you think smoking can be hazardous to your health?



Activity: Bias in Sample Selection

Discuss with students how your source of information, or where you gatheryour information, could also make your data biassed. See SCO F1 forgeneral examples of places where data could be biassed.

The activity sheet called What’s your Bias? (page 24) could also be used togenerate discussion about how some data could be biassed. Have studentsdiscuss the possible biasses of each pair of the sources. Encourage them tothink of additional sources that might give very biassed or very unbiasseddata about the given topics.

Activity: Sources of Data

Explain to students that some data simply cannot be found by conductingsurveys and must be found from other sources. Discuss what are some othersources of information that are accessible to them.• Is information available from more than one source?• Will all sources give you exactly the same answer? Why not?• Could the source be biassed?• Do you know any organizations that do surveys on a regular basis?

(e.g., Gallup, Neilson, Associated Press, magazines, Statistics Canada)• How do you think these organizations conduct their surveys?• What are the advantages and disadvantages of

– interviews– telephone surveys– mail-in surveys

Activity: Comparing Data

Discuss resources that provide information and data.• almanacs• dictionaries• encyclopedias• Guinness Books of World Records• Internet• periodicals—EBSCO

Have students complete the Comparing Data activity sheet (page 25), thendiscuss any differences in information they might have found and why.



Activity Sheet: What’s Your Bias?

Discuss the possible biasses for each source of information.

1. The health aspect of chewing gum• your dentist• a gum commercial

2. The most popular subject in your school• a poll of the students enrolled in physical education or band• a poll of the whole school

3. The effect bottle deposits have on litter• pop bottling company• Nova Scotia Department of Environment

4. The unemployment in your town/county• a labour union• the unemployment office

5. The amount of pollution caused by cars• Environment Canada• auto makers

6. The effects of burning fields on air quality• a poll of people with asthma• an organization of farmers

7. The health aspects of cigarette smoking• the Minister of Health• a tobacco company

8. The number of accidents caused by drivers under 18• accident reports compiled by a government agency• opinions expressed by teenagers



Activity Sheet: Comparing Data

Different sources often give different data. Look up each of these topics intwo different sources. Explain why the data might be different and point outany bias that a source may have.

1. Estimated World Population

Source #1 Source #2

Population #1 Population #2

Explain difference

2. Height of Mount McKinley

Source #1 Source #2

Height #1 Height #2

Explain difference

3. Number of People Killed in 1964 Alaska Earthquake

Source #1 Source #2

Victims #1 Victims #2

Explain difference

4. Area of the Islands of New Guinea

Source #1 Source #2

Area #1 Area #2

Explain difference



Assessment Questions

The following questions could be posed for further student discussion:

Bilmar Battery Company is conducting a survey to determine which brandsof batteries to carry in their new store. They plan to conduct a survey using aquestionnaire or interviews.

1. When deciding on the type of survey to conduct, which of the advantageswould be the most important to consider?

2. Which disadvantage would be the most important to consider?

3. When conducting an interview, would it be better to read the preparedquestions or to memorize the questions?

4. Some people want to know that what they have to say is importantenough to write down, while others get upset when they see aninterviewer taking notes. Which is better: remembering the answers orwriting them down? Do you think a tape recorder would help?

5. What percentage of the written questionnaires do you think should bereturned in order for the survey to be successful?

6. Which method, the interview or the written questionnaire, do you thinkwould provide the more honest and frank answer?

7. Is the time of the day an interview is conducted important?

Follow-up Activity

Give a copy of the Survey on Calculators in Junior High activity sheet(page 27) to each student. Each survey question is in draft form and hasroom for improvement. Students should write an improved version of eachitem and tell why they made the change.

The changes could then be discussed in small groups, and the group couldturn in one set of agreed-upon changes and explanations for the changes.



Activity Sheet: Survey on Calculators in Junior High School

These items are the first draft of a survey prepared by a grade 7 class. Readeach question carefully. Rewrite text the way you think it should appear onthe final draft. Explain any changes you make.

1. How old are you?

2. Do you consider yourself a good mathematician? Yes or no?

3. Which type of calculator do you prefer to use?a) solar poweredb) battery poweredc) scientificd) other:

4. How often do you use a calculator and for what purposes?

5. Based on recent research, it has been found that technologicaladvancement in today’s society will eventually affect the pedagogicalideology relating to the use of electronic devices in schools. Do you agreeor disagree?

6. No student who has not demonstrated success in basic skills should bepermitted to use a calculator in the classroom. Do you agree or disagree?

7. Should immature junior high school students be exposed to the mindlessactivity of using a calculator to solve mathematical problems?

8. When should calculators be used in school?



Lesson Plan 2: What Are the Chances?

Lesson Summary

At the end of the activity, students will be able to• collect data through actual measurement• group data appropriately using a tally chart, frequency table, spreadsheet,

and sorting• choose appropriate intervals and construct a histogram


F2, F3, F4, F5, F6, F9, C7, C9

Assessment

1 a) Draw a histogram for the reach data for the entire class. Use theactivities on pages 29–31.

b) Write 3 statements describing the reaches of your class.

Communication

Students can be asked to express their opinions (orally or in writing) on thefollowing:• why it is useful to group data• their choice of intervals• their interpretation of the histograms

Teachers should introduce the idea of continuous versus discrete data.

Technology

Technology can be used to sort data. Compare the graphs drawn by studentsagainst those drawn using a spreadsheet program.

Materials

• measuring tapes• unsorted data table



Mental Math

This is a good lesson in which to practise rounding to the nearest one-half.Use decimal numbers that will be similar to the data that will be collected(146.7, 152.2, 151.4, etc.) and have students round to the nearest one-half.

Development

1. Previous grades have mainly dealt with qualitative data (sports teams,colours, etc.). In grade 7, the focus is more on numerical data and how tobest display it.

2. Raw data —> Tally Chart —> Extend to Frequency Table —> Histograma) Work in pairs to collect the following data (the Class Data Table

activity sheet is on page 30):• height (in cm)• reach (in cm)• arm span (in cm)• age (in months)(Discussion on uniformity of measurement must take place beforestudents start to measure.)

b) Students can add their own data on chart paper.



Activity Sheet: Class Data Table

Last Name First Name Height Reach Armspan Age Gender

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32



3. Human GrowthShare this work with your partner. It is generally believed that most girlshave taken a growth spurt by 12 years of age, while the growth spurt forboys is 18 months to two years later. Use the data collected in class tomake two histograms on the same grid. You may need to borrow studentsfrom another class to have equal samples. First, decide on an interval andmake a double tally chart for boys’ and girls’ heights.a)

4 a) Describe the data. Is there anything unusual about the information?b) Use your histogram to estimate the height of the

i) boysii) girls

c) How many boys are taller than the boys’ class average?How many girls are taller than the girls’ class average?

d) Do your results support the statement expressed at the beginning ofthis activity?

5. Do you think these data are typical of all grade 7 classes? Why or why not?

Girls Boys

Interval Tally Frequency Tally Frequency

b) Using two different colours, draw both histograms on the same grid.



Extension

Scatterplots

Scatterplots are used to discover if there is a relationship between twodifferent sets of measurements.

For example, the arm span of a person is often about equal to his or herheight. Your task today is to draw a scatterplot of height data against the armspan data for your class to check for a relationship.

Pick a scale for the horizontal axis that will allow you to plot the heights. Youneed not start at 0. Choose a scale for the vertical axis that will let you fit allof the armspans on it. Plot each person as a point (black dot) on the grid.Cross each person off the list as you go.

Eyeball the pattern on the grid, pick up your pencil, and place it on yourpaper so that it forms a straight line that follows the trend of the data points.When you are happy with its position, use your ruler to draw a straight linethat follows the same path as the pencil. This is your line of best fit.

From now on, use the line that you drew to find answers to these questions.

1. Complete the following chart.

Height Reach(cm) (cm)

148

162

190

183

179

176

2. George found that his height increased by 1 cm last year. By how muchdo you think his reach increased?

3. If his reach increased by 6 cm in 3 years, predict how much his heightshould have increased.



Plant Growth

Using the data shown below, relate the height of the seedlings to the numberof days since planting.

1. Make a scatterplot of the data with days on the horizontal axis. Draw aline of best fit through the data.

Days 18 15 12 13 14 10 18 16 16 17 24 11 19 28 25 21 10

cm 9.5 4 2.5 3 3.5 1 13 4.5 10 9 19 2 13.5 19 18 15 21

2. Use your line to predict the height of a seedling.a) 12 days from plantingb) 20 days from plantingc) 24 days from planting

3. Estimate how long it has been since the seedling has been planted if it isa) 20 cm tallb) 5 cm tallc) 12 cm tall

4. How might a scientist use a scatterplot? Give an example.

Follow-up Activity

Visit a grade primary class and measure the arm span and height of 5students. Determine if the relationship of height versus arm span is the samefor grade primary as for grade 7.



Lesson Plan 3: Using Measures of Central Tendency andDispersion to Describe Data

Lesson Summary

At the end of the activity, students will be able to• determine the range, mean, median, and mode for a data set• indicate if there are gaps, outliers, or clusters in the data• choose the most appropriate measure of central tendency for a data set• determine the impact of fluctuation in the data on each of the three measures

of central tendency


F8, F9

Assessment

1. Create a set of data where all of the following factors would be true:a) The mode is greater than the median and the range is 10.b) The median is greater than the mean and the mode is 140.c) The mode is greater than the mean, and the median is the same as the

mean.

2. a) What is true about a set of mathematics marks where the mean is muchlower than the median?

b) What is true about a set of mathematics marks where the mean is muchhigher than the median?

3. Here are the quiz results that Andrew and Natasha received for first term (seegraph below). Recreate the information on a stem-and-leaf plot.

4. Atlantic Canada Mathematics Curriculum: Grade 7, page 117, numberF8.1, and page 119, numbers F8/9.1 and F8/9.2.

100

80

60

40

20

0

Res

ult

s

QuizAndrew Natasha

1 2 3 4 5 6 7 8 9 10



Find the mean of: 75 80 72 81 96

Estimate = 80 down 5 0 down 8 up 1 up 16

– 5 + 0 – 8 + 1 + 16 = 4

So 4 ÷ 5 = 0.8 Mean = 80.8

Focus

1. Give a quick introduction using this set of the masses of five puppies.

Five Puppies for Sale

Flopsy Mopsy Topsy Dropsy Tiny

3.8 kg 3.8 kg 4.4 kg 4.0 kg 7.2 kg

Ask students to find the mean, median, and mode and discuss which isthe best representation of the mass of the puppies.

2. Students then use the sorted height data for the class to find thea) range b) mean c) median d) mode

3. Have students determine if there are any gaps, clusters, or outliers in thedata.

Communication

1. Students develop a vocabulary list or matching exercise for terms.

2. Students justify the choice of mean, median, or mode as the bestrepresentative of a data set.

Technology

Student data could be placed on a spreadsheet. Use the sort function to arrangethe data from smallest to largest. Sorted data can be used to easily find the range,median, outliers, gaps, and clusters. Use @AVG function to find the mean.

Mental Math

Find the mean by using differences. Estimate the actual mean. Find thedifferences between the estimate and the numbers within the set. Divide thetotal of the differences by the amount of numbers in the set and add to theinitial estimate to find the mean. Keep data sets small.



4. Have students decide which of the measures of central tendency bestrepresents the data.

5. In Mathematics 7: Focus on Understanding (Calabrese, Fournier, andSpeijer 2005), complete appropriate exercises

Extension

Atlantic Canada Mathematics Curriculum: Grade 7, page 119, F8/9.3



Lesson Plan 4: Constructing Appropriate Data Displays

Lesson Summary

At the end of this lesson, students will be able to construct various datadisplays, either manually or with technology, and determine which is/aremost appropriate.


F5

Assessment

bar graph broken-line graph circle graph

histogram scatterplot stem-and-leaf graph

1. Which graph would be the best choice to represent the given situation:a) to track Melanie’s height from birth to age 5b) to predict whether there is a relationship between the number of

minutes spent playing sports daily and mathematics gradesc) to display all the heights of the nine grade 7 classes in St. Andrew

Junior Schoold) to compare the number of goals scored by five different members of a

hockey team for one seasone) to show what fraction of a box of Smarties each colour represents

Communications

Ask students to identify the advantages and disadvantages of the variousforms of data display and explain their decisions on the selection of a displaymethod.

Technology

It is recommended that this lesson be completed using technology toconstruct the graphs. The focus of the lesson is on decision making, not onthe construction of the graphs. A software program such as Excel orLotus 1-2-3 may be used.



Mental Math

Use mental math time to review terms such as range, mean, median, mode,and outliers. Present a sample set of data points such as

• 1, 2, 2, 6, 15• 2, 5, 5, 8, 8, 8• 2 , 2 , 3 , 4 , 4 , 9• 88, 97, 100, 103, 112• or other combinations of compatible numbers

Development

1. Review graphs studied prior to grade 7, indicating the purpose of eachone. Discuss graphs and their uses with students and have them set up achart such as the one below in their scribbler. Have them add the newgraphs studied in grade 7 to the chart.

Graph How It Is Used

Bar To compare different members of the same set

Pictograph To compare members of a set when a strong visualimpact is desired

Broken line To track change in a single entity (usually over time)

Histogram To display continuous data that has been grouped intointervals

Scatterplot To predict a relationship between two different sets ofdata

Circle To compare part to whole relationships

Stem and leaf To display continuous data in intervals when it is usefulto preserve the individual data points

Box and whisker plot To display the spread of a set of data using five keypoints that divide the data into four parts of equalfrequency

12

34

12

14

910

910



a) What is the total number of hours represented in the graph?b) What occupies the most time in a typical student’s day?c) Name three activities that together use up about half of a student’s day.

3. Using the same circle graph displayed in the above question, ask students toa) estimate the fraction of the day used for sleepingb) estimate the percentage of the student’s day spent eatingc) determine what two activities together take up about 33 percent of the

student’s day

4. Look at the information in each of the following tables and decide whichgraph should be used to display the data. Defend your choice. Usingtechnology to draw the graphs is recommended.A construction crew is digging a tunnel to connect the new fitness centrein our town to the existing arena. The table below shows their progress.

Table 1

Number of months digging 1 2 3 4 5

Length in metres of tunnel thus far 15 32 48 69 83

When surveyed about sports, boys and girls in our school reported theseas their favourites.

Table 2

Sport Tally (Girls) Tally (Boys) Frequency (Boys & Girls)

Basketball

Badminton

Hockey

2. Hours Spent in aTypical Student’s Day

Otheractivities

Sleeping

At school

Eating

Recreation



The following shows the television viewing habits of Canadians as takenfrom a 2000 survey.

Table 3

Age group 2–12 13–19 20–29 30+

Hours watching TV per week 20 30 19 20

At track practice the other day, the coach measured the stride length incm of each distance runner.

Table 4

Boys Girls

53 51 47 59 48 45 47 51 51 57

53 54 46 57 62 53 58 64 62 66

66 64 70 71 47 66 62 62

The following table shows how the typical grade 7 student spends his orher day.

Table 5

Activity Sleeping At School Eating Recreation Other

Hours 9 6 2 4 3

Extension

Have each student bring in a graph from another source such as a magazineand decide if the graph could be displayed as another type of graph and if itwould still give the same message.



Section 1B

The following section contains a sample of a yearly plan, a unit plan onpatterns and algebra, and five lesson plans.

Sample Yearly Plan 2

September–October

Number Sense (approximate time: three weeks)

Students will demonstrate number sense and apply number theory conceptsat the beginning of the school year. This will help teachers gauge students’strengths and weaknesses to make recommendations for student programs(e.g., resource time, adaptations, independent student program plans).

GCO Grade 7 SCOs



A2 rename numbers among exponential, standard, and expanded forms


A4 solve and create problems involving common factors and greatestcommon factors (GCF)



Last Week of September–First Week of December

Fractions, Decimals, Ratios, and Percentage (approximate time: nine weeks)

The second topic consists of fractions and decimals and will be covered untiltwo weeks before the break. This unit will build upon the students’ numberskills, which can be applied later to data management and probability.

GCO Grade 7 SCOs



A8 rename single-digit and double-digit repeating decimals tofractions through the use of patterns and use these patterns tomake predictions




GCO Grade 7 SCOs


A: Number Sense A10 illustrate, explain, and express ratios, fractions, decimals, andpercentage in alternative forms





B4 determine and use the most appropriate computational method inproblem situations involving whole numbers and/or decimals


B6 estimate the sum or difference of fractions when appropriate


B8 estimate and determine percentage when given the part and thewhole

B9 estimate and determine the percent of a number

B10 create and solve problems that involve the use of percent

December

Probability (approximate time: two weeks)

GCO Grade 7 SCOs





G4 create and solve problems, using the numerical definition of probability


G6 use fractions, decimals, and percentage as numerical expressionsto describe probability

January–February

Integers (approximate time: five weeks)

The new year will commence with integers. The students should be restedafter the holiday and ready to tackle this important grade 7 topic. This topicwill span a five-week period. Students have been only briefly introduced to

14

12

34



integers in the past. This is the first time they will work on operations withintegers. At the end of the five weeks, your five minutes of mental math canbe used to practise operations on integers.

Students will work with algebra tiles for the first time. These will be used againlater on in the year in the algebra unit.

GCO Grade 7 SCOs


A: Number Sense A12 represent integers (including zero) concretely, pictorially, andsymbolically, using a variety of models


B: Operation Sense B1 Use estimation strategies to assess and justify the reasonableness ofcalculation results for integers and decimal numbers


B3 demonstrate understanding of the properties of operations withintegers and decimal numbers



B13 divide integers concretely, pictorially, and symbolically to solveproblems

B14 solve and pose problems that utilize addition, subtraction,multiplication, and division of integers


February–March

Geometry (approximate time: four weeks)

GCO Grade 7 SCOs






E5 construct polyhedra using one type of regular polygonal face, anddescribe and name the resulting Platonic Solids




March–April

Patterns and Algebra (approximate time: five weeks)

Students have previously covered fractions and decimals and will now makeapplications to patterning.

Patterning will then extend to tables, graphs, and algebraic expressions.

GCO Grade 7 SCOs




C2 summarize simple patterns, using constants, variables, algebraicexpressions, and equations and use them in making predictions


C4 solve one- and two-step single-variable linear equations, usingsystematic trial





C9 construct and analyze graphs to show how change in one quantityaffects a related quantity




D: Measurement D4 construct and analyze graphs of rates to show how change in onequantity affects a related quantity

D5 demonstrate an understanding of the relationships amongdiameter, radius, and circumference of circles, and use therelationships to solve problems



April–May

Data Management (approximate time: three weeks)

Students will solve problems involving data management. This topic is to becompleted by March break. Because statistics provide a bridge betweenmathematics and real-life situations, this is a good opportunity to makeconnections between various curricula.

GCO Grade 7 SCOs


B: Operation Sense B10 create and solve problems that involve the use of percentage





F5 construct appropriate data displays, grouping data whereappropriate and taking into consideration the nature of the data

F6 read and make inferences for grouped and ungrouped data displays




May–June

Geometry and Measurement (approximate time: four weeks)

Geometry and measurement can be covered after algebra because there areconnections between finding missing angle measures, finding angle pairrelationships, and solving algebraic expressions.

GCO Grade 7 SCOs


D: Measurement D1 identify, use, and convert among the SI units to measure, estimate,and solve problems that relate to length, area, volume, mass, andcapacity





GCO Grade 7 SCOs


E: Geometry E7 make and apply generalizations about angle relationships

E8 make and apply generalizations about the commutativity oftransformations


E10 make informal deductions about the minimum sufficientconditions to guarantee that a given quadrilateral is of a particulartype, and to understand formal definitions of the variousmembers of the quadrilateral family



Unit 2: Patterns and Algebra

This sample of a unit overview is from the March/April unit of Yearly Plan 2.It covers all the outcomes in the Patterns and Relations strand, plus somefrom Number Sense, Operation Sense, and Measurement. It should takeapproximately five weeks to complete. The two topics were combined,because one leads into the other as students are asked to develop tables,graphs, and eventually algebraic expressions to describe patterns in numbers.It is scheduled in March/April, because students will have already coveredfractions and decimals and will be able to make applications to patterning.

This unit presents a good opportunity for co-operative learning as studentsshare data, discuss various strategies, point out patterns that they see, andexplain the relationship between the data and/or the patterns.

Outcomes

GCO Grade 7 SCOs


















GCO Grade 7 SCOs



D5 demonstrate an understanding of the relationships amongdiameter, radii, and circumference of circles and use therelationships to solve problems

Lesson Purpose

By the end of the unit, students will be able to solve simple algebraicequations to find the value for a variable.

Students will be able to• add like terms• subtract like terms• use concrete materials to solve an algebraic equation• describe the steps used to solve an algebraic equation• solve algebraic equations using systematic trial• recognize the cover-up method as a method of solving an algebraic

equation

Assessment

Several types of assessment will be used throughout the unit:• Observe as students are working in their groups or going from station to

station. (It is important to choose a few different students to observe everyday throughout the unit.)

• Question students, getting them to explain their thought process as theywork through the problem. (This will give the teacher a good sense ofwhether the instructions were clear or if a concept has been understood.)

• The ability to translate between different modes of representationdemonstrates a good understanding of the concept. Ask students totranslate between the following and others that may be appropriate:– graph to verbal– verbal to graph– table to graph– graph to symbolic– symbolic to concrete– concrete to pictorial– pictorial to table



• Have students write portfolio entries explaining how their thinkingdeveloped during or after doing the assignment.

• Have students write journal entries explaining to an absent friend why thelarge green Alge-Tile square represents x2. (Being able to teach someone isa great way of getting facts straight in your mind.)

• Students can present possible solutions to the rest of the class using theoverhead projector. (Assess the communication skills of the students andcheck for use of the appropriate vocabulary.)

• Students can do pencil-and-paper quizzes, tests, and/or assignments.(These should be used to see if the students understand the conceptstaught so far or if it is necessary to go back and reteach certain aspects ofthe unit.)

Levels of Questions

Level 1 question

Circle the algebraic equationa) 3x + 4 + 13b) 3x + 4 = 13

Level 2 question

Illustrate 2x + 3 – x, using algebra tiles.

Level 3 question

Illustrate and explain each step you would take to solve for x in the equation2x + 6 = 12

Communication

Students will share their findings with the other members of their group,using mathematical terms such as variable, coefficient, constant, expression,equation, like terms, and unlike terms. Class and group discussions provideopportunities to examine patterns, similarities, and differences in terms andto predict what the next term might be. They will write stories that might berepresented by a certain graph or defend their choice of interval on a graph.Students will make conjectures and defend them to the class; they will showhow variables are represented in equations and how pan balances are similarto equations. Some of this work will be done in writing.



Technology

Some students will be encouraged to use Understanding Math, a computerprogram that will give them more examples of the concept they are workingon, while others will be using it to enhance and/or extend the concept.

Materials

• activity sheets• algebra tiles• base-10 blocks• calculators• cards with information about material in each pan for each station• chart paper• coloured pencils• different-sized graduated cylinders or other transparent containers and

some irregular shaped flasks and bottles—one for each group (Theseshould be available from the science supply room.)

• measuring cups of different sizes (Match each cup to the size of thetransparent container to make sure it does not fill up the container tooslowly or too quickly. Ideally, it should be able to take eight measuresbefore it is full.)

• Mental Math in Junior High lessons 6, 8, 12, 30, 47, and 48• pattern blocks• ruler• small paper bags (e.g., halloween style treat bags)• small plastic bags (e.g., sealable sandwich bags)• two-coloured counters (optional)• two-pan balances or drawings of pan balances (one per station)• Understanding Math CD• water



Description of Lessons

Lesson 1: Could You Repeat That, Please? (three days)

In this lesson, students learn to recognize and calculate terminating andrepeating decimals, and to convert fractions to decimals and decimals tofractions.

GCO Grade 7 SCOs




Lesson 2: All around the Square (seven days)

In this lesson, students will use multiple representations to describe patternsand their relations.

GCO Grade 7 SCOs
















Lesson 3: How Do You Measure Up? (four days)

In this lesson, students will analyze graphs, identify patterns, recognize therelationship between two different quantities, write stories about graphs, anddraw a graph to illustrate a story.

GCO Grade 7 SCOs


C: Patterns and Relations C9 construct and analyze graphs to show how change in onequantity affects a related quantity


Lesson 4: Adding and Subtracting Like Terms (two days)

In this lesson, students will identify like and unlike terms and add andsubtract like terms.

GCO Grade 7 SCOs





Lesson 5: Balancing Equations (five days)

In this lesson, students will solve one- and two-step single-variable algebraicequations, using balances and pattern blocks, draw the steps to solve theequation when given a diagram and solve simple equations by guess and check.

GCO Grade 7 SCOs


Patterns and Relatioships C4 solve one- and two-step single-variable linear equations, usingsystematic trial




Lesson Plan 1: Could You Repeat That Please?

Lesson Summary

Students will use patterns to• convert fractions to decimals• convert decimals to fractions

Students will also• recognize terminating and repeating decimals• be able to calculate terminating and repeating decimals


A7, A8

Assessment

There will be observation throughout the lesson, as students use calculators,pencil and paper, and manipulatives to uncover patterns for decimals.Because the lesson should take only a few days, determine ahead of timewhich group of students you want to observe during this activity. A preparedset of questions that you could ask as you observe the students might beuseful, or a simple anecdotal comment might suffice. Keep the commentsbrief and objective, recording what you see and hear.

A written assignment without the use of calculators to be passed in for pointscould include the following.

Given the following decimal representation of fractions as produced by acalculator:

• = 0.09090909 …,• = 0.181818 …,• = 0.272727 …,

a) Predict the decimal for 5/11 and 9/11 and explain why.b) Predict the fraction that will have 0.636363 … as a decimal.c) Predict what the decimal for 8/11 would look like on a calculator display

if the calculator was set up to display 8 places after the decimal. Write thedecimal as a repeating decimal.

d) Describe the pattern for the decimals with the denominator 11.e) Predict the fraction that will have 0.909090 … as a decimal.

111211311



Communication

Students will describe patterns, discuss similarities and differences in thepatterns, and predict what would produce a certain predetermined answer,in small groups and in teacher-facilitated classroom sharing.

Technology

Students will use their calculator to find the decimal form of certainfractions. The activity in the Extensions section of the lesson (page 65) alsouses a spreadsheet software application such as Excel.

Materials

• activity sheets• base-10 blocks• calculators• Mental Math in Junior High, Lessons 47 and 48 (already used in Lessons

4 and 6 in the Data Management and Probability unit)

Mental Math

Use Lessons 47 and 48 from Mental Math in Junior High as a basis for yourinstructions, but ask the students to change the fractions to decimals insteadof percentages. Start with basic ones that they already know such as , , , . Then introduce some repeating ones such as , and continue to the lessobvious ones, such as , , , , .

Development

Warm-up

Write the number 0.999 on the board and ask the students if that is thegreatest decimal less than 1. Ask them to explain their answer in writing andgive an example to support their answer. They could also be asked to showtheir work on a number line.

12

14

110

15

13

23

34

18

120

45

58



Activity: Could You Repeat That Please?

Tell students that, for this activity, they will be using base-10 blocks.

The Flat Represents The Rod Represents The Small Cube RepresentsOne (1) One-Tenth (0.1) One-Hundredth (0.01)

Discuss how you can change a fraction to a decimal number. Direct thediscussion to include changing the denominator to a power of 10. Introducethe term “terminating decimal” when the denominator can be changed to apower of 10.

Using the base-10 blocks, ask students to represent the following:• 1• 3••

Ask them how they could represent 2 using base-10 blocks. Which piecewould represent 1?

Distribute the Fractions to Decimals to Pictures activity sheet (page 57) andask students to fill it out.

1. Ask students to find the decimal approximation of and using acalculator, and then pose the following questions:a) What do you notice about the answers?b) If the calculator could show three more numbers, what do you think

those numbers would be for ? For ?c) Why are these decimals called repeating decimals?

2. Find the decimal approximation of using division with pencil and paper.a) What do you notice about the answer?b) What do you notice about the remainder every time you add a zero

and divide again?c) How would you express that remainder as a decimal?d) Which digit in the decimal approximation for repeats?

3425

52

175

38

13

23

13

23

13

13



3. Use base-10 blocks. The large cube will represent 1.a) How could you represent ?b) What would you trade the cube for to be able to divide it into 3 equal

parts?c) How can you divide the 10 flats into 3 equal parts?d) What would one of those equal parts represent?e) What could you trade the leftover rod for to be able to divide it into 3

equal parts?

4. Continue the sequence up to the small cubea) What do the one leftover flat, rod, and small cube represent?b) Why does the decimal for repeat?

Introduce the idea of the line above the repeating decimal.

Distribute Do You See the Pattern? activity sheet (page 58).

13

13



Activity Sheet: Fractions to Decimals to Pictures

The Flat Represents The Rod Represents The Small Cube RepresentsOne (1) One-Tenth (0.1) One-Hundredth (0.01)

Fraction Decimal Picture

1

0.44

2

Use blocks to explain which is greater, 0.4 or 0.35?

12

330



Do You See the Pattern?

1. When fractions are expressed in decimal form, patterns often occur. Useyour calculator to express each fraction as a decimal and record what yousee in the chart.

Fraction Decimal Approximation

Describe the pattern.

Use your pattern to predict the decimal form of the following fractions,then check with your calculator to see if you are correct.

Fraction Decimal Predicted Calculator Answer

Based on the pattern you described, predict the fraction form of thedecimal then use your calculator to check your answer.

Decimal Equivalent Fraction Predicted or

0.555555 ...

0.777777 ...

Therefore, should = = .

2. If you know that = 0.166666 ..., = 0.333333 ..., = 0.5, and = 0.66666 ..., wouldyou predict to be equal to a number between 0.8 and 0.9 or between0.7 and 0.8? Explain your prediction, then check your answer on yourcalculator.Write as a repeating decimal.How many digits are in the repeating sequence?

19

99

29

39

49

59

69

16

26

36

46

56

17



Write each digit from the repeating sequence once in clockwise orderaround a circle.

Find the decimal approximations for the other sevenths fractions.


What do you notice when you compare the repeating sequence of digitsfor the decimal approximation of the sevenths fractions?

3. Use your calculator to express each fraction as a decimal, and record whatyou see in the chart.


Describe the pattern.

4. Use your pattern to predict the decimal form of the following fractions,then check with a calculator to see if you are correct.

Fraction Decimal Predicted Calculator Answer

Was your pattern correct? If not, use what you learned from the last threedecimal equivalents to describe what you think the pattern is now.

113

213

313

413

513

613

27

37

47

57

67



Extensions

1. Using pencil and paper, find the decimal equivalent of to 10 decimalplaces. What do you notice?

2. Some fractions with denominators such as 15 have patterns that changefrom repeating to terminating as the numerators increase. Predict whatother numerators will produce terminating decimals and predict therepeating decimals for other fractions.


3. Complete the Investigating Repeating Decimals activity (page 113 inMinds on Math Grade 7.)

Follow-up Activities (homework)

1. Compare the decimal approximations of fractions with a denominator of14 to the decimal approximation of fractions with a denominator of 7.

2. Do you suppose this is true of all fractions whose denominators aremultiples of 7? Use examples to explain your answer.

3. Chris has a calculator that displayed 2.373737374. Chris concluded thatit was not a repeating decimal. Explain why Chris drew this conclusionand whether you believe it is correct.

115

117

215

315

415

715



Investigating Repeating Decimals

The world around us is full of patterns. You can find patterns in naturalobjects and objects created by humans. Numbers can also involve patterns ofrepeating digits. You may notice that such patterns sometimes occur whenyou express fractions in decimal form. You can use a spreadsheet to explorethese patterns.

1. Start a new spreadsheet file. Enter the numbers and formulas shownbelow.

A B C

1 Numerator Denominator Decimal

2 1 99 = A2/B2

3 = A2+1 99 = A3/B3

Format column C to show numbers as fixed decimals with 10 places. Youmay need to make column C wider so there is room to display all thedigits. Copy the formula in row 3 to row 4.a) What does the formula in cell C2 tell the computer to do?b) What does the formula in cell A3 tell the computer to do?c) What fractions are being expressed as decimals? Write the decimal

form for each of these fractions.

2 a) Based on your results for exercise 1, predict the decimal form for eachfraction:

b) Use your spreadsheet to check your predictions in part a. What aresome different ways to do this?

3 a) Revise your spreadsheet. Use it to express , , and as repeatingdecimals.

b) Based on the results of part a, predict the decimal form for each of thefollowing fractions. Check your predictions using your spreadsheet.

4. Use your spreadsheet to investigate the decimal form of these fractions.a) , , …b) , , …c) , , …

d) , , …e) , , …f) , , …

g) , , …h) , , …

5. Express each fraction in decimal form.a)b)

c)d)

4 5 12 28 53 6699 99 99 99 99 99

4 5 14 37 685999 999 999 999 999

1999

2999

3999

16

26

36

111

211

311

116

216

316

133

233

333

137

237

337

164

264

364

1101

2101

3101

1271

2271

3271

551011032

25743091



Patterns and Algebra

Lesson Plan 2: All Around the Square

Lesson Summary

Students will be able to• describe patterns• describe relations

Students will be able to accomplish this by using multiple representationssuch as• algebraic formulas• concrete representation• graphs• pictures• tables• verbal descriptions


B16, B17, B18, C1, C2, C3, C4, C5, C6, C7, C8, C9

Assessment

Information on the students’ levels of performance will be gathered throughboth informal and formal means. Ask questions, observe problem-solvingstrategies, and note if the student is able to express verbally the method that heor she used to solve the perimeter problem. Evidence of being able to translatefrom one mode of representation to another (e.g., from table to algebraicequation to graph) will also be assessed. Students might choose to include thisactivity in their portfolio and write a paragraph to explain why they chose it andhow their thinking developed during or after doing the assignment.

Communication

There will be lots of opportunity during the lesson for students to shareideas, listen to others share ideas, ask questions, explain their thought processand even justify their solutions. Oral and written communication willprovide students with the opportunity to think about problem solving,develop explanations, and reflect on their own understanding. Teachers willintroduce students to new words such as algebraic expression, algebraicequation, variable, constant, and coefficient. Students will describe in theirjournals the difference between an equation and an expression.



Technology

Technology is not suggested for this lesson, although students might want touse a spreadsheet application to graph the data they collected in their tables.Students could also use data recorded in a spreadsheet to help them develop aformula to show the patterns.

Materials

• activity sheet• algebra tiles• chart paper• coloured pencils• pattern blocks• Mental Math in Junior High (Hope, B. Reys, and R. Reys 1998) Lesson 12

Mental Math

Mental Math in Junior High, Lesson 12 introduces the idea that balancing oradding the same thing to both numbers in a subtraction problem does notchange the answer. This will prepare them for balancing the scale by adding ortaking away the same amount on both sides.

Development

Warm-up

Because the rest of the lessons in this unit focus on equations, expressions, andthe different ways of expressing the same idea, it would be appropriate toexpress the “Number of the Day” in as many ways as possible.

Write the date (number only) on the board. Ask students to find different waysof expressing that number. For example, if today’s date is October 21st, the daycould be expressed as follows:

10 × 2 + 1

XXI

30 – 9

5 groups of 4 plus 1

42 ÷ 233 – 6

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ... + 1

3 × 7

twenty-one

21



Activity: All around the Square

• Give each group of four a set of pattern blocks. Ask each student to selecteight squares and discuss their properties in their group.– How many sides does a square have?– What can you say about the sides of a square?– What can you say about the angles in a square?– Because the angles are equal, what is the measure of each one?– What do we mean by “perimeter”?– If 1 side of the square measures 1 unit, what is the perimeter of the

square?• Give each student a copy of the Writing Algebraic Equations activity

sheet (page 66).• Model on the overhead (or draw on the board) what you mean by a

square train. A train is formed when squares are joined side to side.• Ask the students to make a train of two of these squares and sketch it as

they add each square.– What is the perimeter of a two-square train?– Add another square. What is the perimeter now?– What about when you add a fourth square? A fifth? A sixth?

• Tell the students to record information by filling in the perimeter columnof the table on the activity sheet.– What do you see happening to the perimeter as another square is

added to the train?• Tell students to examine the train and determine how they can calculate

its perimeter without 1-to-1 counting of each side.• Ask the students to write a number sentence to express how they would

find the perimeter of their six-square train without counting each side.• Have chart paper on the wall and ask students to come up and record

their number sentences. Hopefully there will be several different ones.(Leave three lines between each number sentence, as the numbersentences for 8, 20, and (n) squares will be added later, to check if thepattern works).

• Tell the students to add two more squares to their trains.• Ask them if they can determine the perimeter of this train using the same

method as they did for the six-square train.• On the chart paper, write each new number sentence for solving for eight

squares below the same number sentence for six squares.• Tell students to fill in the number sentence column of their chart to show

how they solved for the perimeter.



• Ask students to write how they could find the perimeter for any numberof squares.

• Ask students to write an algebraic equation, replacing the number ofsquares by the variable (n) to describe how to find the perimeter (p) of atrain with any number of squares.

• Return to the chart with all the equations. Using algebra tiles, show howall the equations can be simplified to (p) = 2n + 2.

• Ask students to graph the table of values.– How do the characteristics of the graph relate to the model of the

train?– Should the dots be connected? Why or why not?– What is the perimeter of a train built with 15 squares?– How many squares were needed to produce a perimeter of 18 units?

25 units?– As you increase the number of squares by 1, what happens to the

perimeter of your train?• Predict what the perimeter would be if you had used 21 squares.

Extensions

Construct a train with triangles, one with trapezoids, and another one withhexagons. Graph the relationship between the number of blocks used andthe perimeter on the same graph as that of the squares. Use a different colourpencil to graph the results for each pattern block. Compare the slope of eachline.



Activity Sheet: Writing Algebraic Equations

A square train is formed when you join the squares side to side. Sketch a2-square train and add “cars” to it as you go along.

Fill the perimeter column in the table below for the perimeter of squaretrains that have up to 6 squares.

Number of Squares Number Sentence Perimeter

1

2

3

4

5

6

8

20

n

Write a number sentence to express the method you used to calculate theperimeter of a 6-square train. Remember to use the order of operations.

As soon as someone tells me how many squares there are in the train, I canfind the perimeter by

If (p) represents the perimeter and (n) represents the number of squares inthe train, an algebraic expression to find the perimeter of a train with anynumber of squares could be

In its simplest form, the equation is

Plot the graph of the relationship between the perimeter and the number ofsquares used.



Lesson Plan 3: How Do You Measure Up?

Lesson Summary

Students will be able to• construct graphs based on stories and data they have collected• create stories that describe the activity in a graph• analyze graphs• identify patterns• recognize the relationship between two different quantities


C9, D4

Assessment

Students will be asked to sketch a graph after reading a short story. Circulatearound the room, assessing students for their ability to choose theappropriate axis for each quantity, create an appropriate scale, and sketch agraph that will tell the story.

Students will also be asked to write a story that would describe the activitythat is taking place on a graph.

Communication

Students will be given some mathematical information in order to create astory. They will also read a short text and communicate the story line ingraph form. As in other lessons, they will express their thinking andmathematical ideas orally. Circulate around the class asking questions andlooking for clarification.

Technology

The Graphing module of Understanding Math has some very good examplesof graphs of rate that could be helpful to students who need more practice ininterpreting graphs.



Materials

• activity sheet• different sizes of graduated cylinders or other transparent containers and

some irregular shaped flasks and bottles—one for each group (Theseshould be available from the science supply room.)

• graph paper• measuring cups of different sizes (Match them to the size of the transparent

container, to make sure it does not fill up the container too slowly or tooquickly. Ideally, it should be able to take eight measures and not be full.)

• Mental Math in Junior High (Hope, B. Reys, and R. Reys 1998) Lesson 30• ruler• water

Mental Math

Lesson 30 of Mental Math in Junior High deals with a daily life issue where wecompare unit rates and/or costs of equal amounts to determine best prices.

Development

Refer to the grade 7 curriculum guide SCOs C9 and D4 for more excellentassessment suggestions and instructional strategies. This activity was taken fromthe Patterns and Functions, Addenda Series Grades 5–8 (NCTM 1991).

Warm-up

Write the numbers 12, 16, 24, 33, and 48 on the board. Ask the students todetermine which one they think does not belong. Ask them to explain theirchoice. Encourage students to look for more than one answer.

How Do You Measure Up?

1. Supply each group with a straight-sided transparent container (a differentsize for each group, if possible), some water, and a measuring cup. Studentswill also need a ruler.

2. Hand out the activity sheet entitled How Do You Measure Up? (page 70).Demonstrate adding the same amount of water each time to thetransparent container and measuring the height of the water (in cm) in thecontainer after each addition.



3. Ask students to do the same thing with their containers and to record theheight of the water, after each measure is added, in the table on theactivity sheet. Make sure each group has room in the container to addeight measures of water. If the students measure carefully, this should beenough to convince them that they have a linear relationship. (Encouragethem to use the word “constant” when describing the growth of the line.)

4. Tell the students to graph the results from the table. Check that theywrite the number of measures on the x-axis and the height of the water onthe y-axis. Ask them to explain their decision.

5. Ask the studentsa) What patterns do you notice in both the table and the graph?b) Will the pattern go on forever?c) What would the graph look like if you kept on pouring in measures?d) What does it depend on?e) What would happen if you had used a different measure?f ) What is the slope of the line? What does it represent?

Where’s the Straight Line?

1. Repeat the preceding experiment using irregular shaped bottles or flasksfrom the science lab and then complete the activity sheet Where’s theStraight Line? (page 71).

2. Before beginning, ask the studentsa) What do you think will happen as each measure is added?b) Will this container fill at the same rate as the container you used in the

previous experiment?

3. Tell the students to add the water one measure at a time and to record theheight in the table. This time they should fill the bottle. Tell them tograph the results.

4. Ask them questions that focus on the relationship between the growth inthe height of the water and the shape (cross-section) of the containers:a) How did your data and graph differ from those in the experiment

with the straight-sided containers?b) What is increasing in your experiment?c) When is the height increasing most rapidly?d) When is it increasing most slowly?e) Do the height of the water, the number of measures, and the shape of

the bottle all relate to each other?f ) What does the height depend on?g) What does the graph record?



Activity Sheet: How Do You Measure Up?

Number of Measures Height of the Water (cm)

1

2

3

4

5

6

7

8

Graph the data from the table on the grid below.



Activity Sheet: Where’s the Straight Line?

Number of Measures Height of the Water (cm)

1

2

3

4

5

6

7

8

Graph the data from the table on the grid below.



Activity: Every Graphs Tells a Story

Mr. Jones walks every day. The three graphs below show the distance hetravelled over the last three days. Use the graphs below to create a story thatdescribes his walk on each of the three days. Don’t forget to label the axis.

Activity: Every Story Has a Picture

A tour bus left the station and drove slowly through downtown traffic. Itstopped at a historic site for a short time then travelled quickly along thehighway to its next stop, where it parked and the tourists had lunch. Thetour bus then took a side road back to the garage, slowing down at severalscenic locations. Sketch and label a graph illustrating this particular tour.



Extensions

Students could repeat the experiment using two containers that haveapproximately the same capacity. Container A is long and narrow whileContainer B is short and wide. They should add the same measure to eachcontainer and measure and record the height of the water after eachaddition and then construct a graph for each container.

Ask the students• What happens after you add a measure to container A?• Does the same thing happen to container B?• How are the graphs the same?• How are they different?• What else, besides the number of measures added, does the height of

the water in the container depend on?• What makes the height of the water in the container grow quickly?• What makes the height of the water in the container grow slowly?


Students could do some of the activities in the Graphing module of theUnderstanding Math computer program.



Lesson Plan 4: Adding and Subtracting Like Terms

Lesson Summary

Students will be able to• identify like and unlike terms• add like terms• subtract like terms


B16, B17, B18

Assessment

As a follow-up to the discourse that will take place in the classroom in bothsmall groups and as a whole class when students present their findings on theoverhead, ask the students to explain in their journals why the big greenrectangle represents x and why the big green square represents x2. It is alsoimportant that they understand and are able to explain the zero principleand when it is used.

Communication

Students will communicate their ideas and discoveries to each other and toyou, both verbally and symbolically. Students will make conjectures in theirgroups, and you should ask them to then share their ideas with the class. Theinstant feedback or discussion that follows provides a type of formativeevaluation. Allow students to use the tiles at the overhead to present anddiscuss their findings.

Technology

Using the Algebra module of Understanding Math could help some studentswith some extra examples that they need in order to understand the idea ofadding like and unlike terms.

Materials

• activity sheets– Combining Like Terms– Expressing Equality– Subtracting Like Terms

• algebra tiles



• Mental Math in Junior High (Hope, B. Reys, and R. Reys 1998) Lesson 6• overhead projector (optional)• Understanding Math CD (optional)

Mental Math

Mental Math in Junior High Lesson 6 and a review of the zero principleusing integers will help students as they learn about combining like terms.

Development

Warm-up

Have students look around the room to find objects that are exactly the same(e.g., desks). Have them discuss why they are the same and talk about“congruent.” Discuss how you can add or combine all the individual deskstogether. Ask them to find another object in the room that there are several of(e.g., chairs). Talk about the fact that you can add all the chairs together if youwish, but cannot add chairs and desks together because they are not like things.

Combining (Adding) Like Terms

For the purpose of this activity, we will assume that the students have alreadybeen introduced to algebra tiles and they know the names of all the pieces.

Ask students to display one example of each shape in both colours on theirdesks. You might also want to display them on an overhead projector. Askthem• What shapes are congruent?• Explain.• Does the colour affect whether or not the shapes are congruent?• Explain.• Take the rest of the pieces out of the bag and place congruent tiles

together in the same column (i.e., all the x2 in the same column, all the xin another column, all the y in another, etc.).

• How many large green squares do you have?• How many large white squares do you have?• How many green and white rectangles?• How many small red and white squares?• How many medium-sized orange and white squares?• Notice that x2 and –x2 are congruent or have the same area. The same is

true of y and –y, 1 and –1, etc, …• Each column or group represents terms that are alike—they are called

like terms.



• If I said I had 3x2 what would the “3” in 3x2 tell you? What would the –2in –2x tell you?

• Introduce the words variable, constant, and coefficient.• Write these groups on the board and ask them to select the like terms in

each group.a) 4x, 12, 5, 3x2, 6x, 18b) 4x, 2, –y, 2y, xyc) y, y2, –y, –y2, –5d) 3x2, –x2, –2x2, –y2

• Hand out the Combining (Adding) Like Terms activity sheet (page 78).

Subtracting Like Terms

Students should have used Alge-Tiles or two-coloured counters to model thesubtraction of integers. We can now extend this idea of modelling to includethe subtraction of like terms.• Write this expression on the board: (3x + 5y) – (x + 2y)• Remind students that subtraction means to “take away,” and therefore

(3x + 5y) – (x + 2y) means to take x + 2y away from 3x + 5y.• Ask students to model 3x + 5y with the appropriate Alge-Tile pieces.

• Circulate around the room to ensure that this expression has beenmodelled properly.

• Ask students if they can take away or remove x + 2y• Ask students what will remain.• Have students put the following in their notes:

(3x + 5y) – (x + 2y)

(3x + 5y) – (x + 2y) = 2x + 3y



Greater care has to be taken to effectively model expressions such as5x + 2y – (4x – 3y).• Ask students to model 5x + 2y with the appropriate Alge-Tile pieces and

check to see they have done so correctly.

• Remind students that we want to take away 4x – 3y from thisarrangement and ask how this might be done. (Give sufficient time forstudents to think about this and discuss what could be done.)

• You may need to remind students that just as we used the zero principleto model the subtraction of integers, we can also use the zero principlehere. We can add zeros to the diagram in the form of x and –x or y and–y, etc., until we have the tiles we want to “take away.”

• Ask them what we should add as our zeros.• Circulate around the room until you are convinced that everyone has

added the correct number of zeros. (They should add 3 sets of y and –y).• Now ask them to “take away” (4x – 3y)• Now ask them to record what they did in their scribblers.

5x + 2y – (4x – 3y)

5x + 2y – (4x – 3y) = x + 5y• Hand out the Subtracting Like Terms activity sheet (page 80).



Activity Sheet: Combining (Adding) Like Terms

1. Use algebra tiles to represent the following expressions and draw yourrepresentation in the table.

Expressions Representation

2x2 + 4x + 1

3x2 + 5x + 2

Combine the like terms symbolicallyin this box and pictorially in the boxon the right.

2. Write the algebraic expression that represents each group of tiles.

Expressions Representation

Combine the like terms pictoriallyin this box and symbolically in the boxon the left



3 a) What terms can be combined when algebraic expressions arecombined?

b) What remains the same when algebraic terms are combined?c) What changes when algebraic terms are combined?

4 a) What value would 3p + 1 be equal to if p = 2?b) What if p was equal to 6?c) Fill in the table to make the statment true for the given value of a.

Equation Value of a

2a + 3 = 9 3

2a + 3 = 0

2a + 3 = –10

2a + 3 = 4

2a + 3 = 0.5

2a + 3 = –1

5. In question 1, if x = 2, what would be the value of each expression?



Activity Sheet: Subtracting Like Terms

Illustrate the following subtractions using algebra tiles.

Expressions Illustration

(2x – x – 3) – (2x – x – 3)

Difference:

(3x + 4y) – (2x + y)

Difference:

(2x + x) – (3x – 2x)

Difference:

Extensions

1. Given an equation, the student should be able to express that equation inwords.Example: 60 – y = 48.• Sixty decreased by a number is equal to 48.

2. A natural progression for some students would be to write simpleequations to represent sentences, then solve them. Students could then begiven the following problem.Example: Four times a number is 68.• You can choose any letter to represent the number. Let’s choose n.

Four times a number would be 4 × n or 4n. Therefore the equationwould be 4n = 68.

3. Choose the sentence that describes the equation.Example: b + 3 = 12a) Three times a number is equal to 12.b) Three more than a number is equal to 12.c) A number decreased by Three is equal to 12.

A sample activity called Expressing Equality follows.



Activity Sheet: Expressing Equality

1. Write the algebraic equation for the following sentences.a) Triple a number is thirty-three.b) A number decreased by seven is fifteen.c) A number subtracted from ninety-nine is forty-seven.d) A number increased by five is thirteen.e) The product of eight and a number is fifty-six.

2. Write a sentence for each equation.a) w – 4 = 11b) 2a = 14c) 3s = 3.75d) r + 7 = 10e) = 18

3. In the following problems choose the sentence that describes theequation.a) 11c = 13

• Eleven plus a number is 13.• Eleven less a number is 13.• A number multiplied by 11 is 13.

b) m – 7 = 5• Seven less than a number is 5.• Five more than a number is 7.• A number divided by 7 is 5.

4. In the following problems choose the equation that describes thesentence.a) A number divided by 2 is 12.

• 2n = 12• n + 2 = 12• = 12

b) Five less than a number is 9.• d – 5 = 9• d + 9 = 5• 5d = 9

s3

n6



Lesson Plan 5: Balancing Equations

Lesson Summary

Students will be able to• solve one- and two-step single-variable algebraic equations using balances and

pattern blocks• draw the steps to solve the equation when given a diagram• solve simple equations by guess and check


C4, C5

Assessment

• Students will be asked to describe verbally what they did to solve theequation. For some students you can do this one-on-one at their desk, whileothers can be asked to describe their process to the class using the overhead tomanipulate their material and show how they came up with the answer.

• Students should also be informally assessed on their ability to successfullyphysically move the materials around to solve the equations at each station.

• Pencil-and-paper questions should be solved to verify student understandingand consistency in finding the solution.

• Students should make a portfolio entry on solving for the value of a variableby using systematic trial.

Communication

Students will discuss methods for discovering the contents of the bags, how thevariable is represented in the model, and how the balances are like equations.They explain to others how they solved the equation, using the material to helpthem explain their process. They will also communicate through drawings andwriting, as they show the steps they performed to solve the problem.

Technology

Understanding Math has an Algebra module that could be used for somestudents as an extension to what they have done in class.

The Illuminations section of NCTM website, illuminations.nctm.org, has severalonline activities and resources that would be appropriate alternative methods forbalancing algebraic expressions. Example: Seeilluminations.nctm.org/tools/index.aspx and look for Pan Balance–Expressions.



Materials

• algebra tiles or two-coloured counters• cards with information about material in each pan for each station• Mental Math in Junior High (Hope, B. Reys, R. Reys 1998) Lesson 8• Pattern Blocks• small paper bags (e.g., halloween style treat bags)• small plastic bags (e.g., sealable sandwich bags)• two-pan balances or drawings of pan balances (one per station)

If you have finished Mental Math Lesson 6 on finding compatibles, continueon to lesson 8, which extends the idea of looking for something that whenadded to another number will give you a particular answer. This lesson usesdecimal numbers.

Mental Math

Lesson 8 of Mental Math in Junior High is an excercise about searching forcompatable decimals.

Development

Warm-up

Tell students that a × b = 120. What are some possible values of a and b?Now add that a + b also equals 43. What are the values of a and b?

Balancing Equations

This activity should be set up as stations. Each station would require a two-pan balance, pattern blocks, and small paper bags. If two-pan balances arenot available, this activity can be done on paper with drawn versions of two-pan balances. The students will be asked to determine the contents of each“full” paper bag at each station.

Preparation

1. Photocopy the cards (double-sided) that go at each station on ticket boardor construction paper.

2. For each of the stations you will need a card, a full paper bag(s) (see p. 86)for contents, specified pattern blocks (see p. 85 for quantity), emptypaper bag(s). Put all materials in a plastic bag and label.

3. Divide your class into groups of four.4. Set up as many stations as you have two-pan balances.



5. At each station place one of the prepared plastic bags.6. Instruct students not to peek inside the paper bag. They must place the

materials listed on front of the card in one side of the pan balance and thematerial listed on the back of the card on the other side of the panbalance.

Before the students get started, discuss how the balance is like an equation inthat what is on one side of the scale must be equal to what is on the otherside. Demonstrate how removing one thing from one side will cause the scaleto become unbalanced. Show them how removing the same thing from eachside of the scale, e.g., a full bag from each side, or one triangle from eachside, will keep the scale balanced. You might even want to demonstrate howto solve for the content of the bag by actually going through an example withthem.1. Assign a group of students to each station.2. Students need to determine the contents of the full bag (each full bag at

that station contains the same number and shape of blocks). After theyhave solved the equation, either you or the students need to make sure the“full bags” contain the right number and shape of blocks, and then puteverything in the plastic bag as they found it.

3. Upon your signal, students rotate to the next station and repeat theprocess. It is important to remind students that a different student shouldbe picked at each station to physically perform the task of solving theequation. This is a good time to observe for co-operation andcomprehension.

If two-pan balances are not available, enlarge and photocopy the two-panbalance drawing and place one at each station. Students can proceed in thesame way to solve the equations except they will not see the effect of the scalenot being balanced if they forget to take something off from both sides.

After this activity, it is a natural progression to move on to equations with numbersand variables. In the grade 7 curriculum guide, C4 and C5 (pages 62–65)provides several examples of solving equations by using simple equations and othermore complicated ones that require the guess-and-check method.

Mathematics 7: Focus on Understanding (Calabrese, Fournier, and Speijer2005) is an excellent resource, using algebra tiles to help solve for linearequations in one variable.

Check student understanding by having them enter a page in their portfolio.Ask them to find the value of p in the following equation by using systematictrial.

5p + 8 = 63

Ask them to explain if there is more than one possible answer and justifytheir answers.



Cards for Each Station

Card Front Back

1 1 full bag 5 blue rhombus

1 hexagon 1 hexagon

3 triangles 3 triangles

1 empty bag

2 3 full bags 1 full bag


4 hexagons

2 brown rhombus

2 empty bags

3 2 full bags 4 full bags

2 squares

4 blue rhombus

6 brown rhombus

2 empty bags


1 hexagon 2 hexagons

2 squares 4 squares


1 empty bag

5 2 full bags 4 full bags

5 trapezoids 1 trapezoid

2 empty bags

Card Front Back

6 2 full bags 6 squares

2 squares 4 blue rhombus

2 blue rhombus 2 triangles

2 empty bags

7 1 full bag 5 trapezoids

1 hexagon 1 hexagon


1 empty bag


1 trapezoid 2 trapezoids

2 brown rhombus 2 brown rhombus

3 triangles

1 empty bag

9 2 full bags 6 squares

4 blue rhombus

2 triangles

2 empty bags


2 blue rhombus 4 blue rhombus

1 hexagon 3 hexagons


2 brown rhombus

2 squares

2 trapezoids

2 empty bags



Answer Sheet

What is the content of the “full bag” at each station?

Two-Pan Balance

Station Solution

1 5 blue rhombus

2 2 hexagons

1 brown rhombus

1 triangle

3 1 square

2 blue rhombus

3 brown rhombus

4 1 hexagon

2 squares

3 triangles

5 2 trapezoids

Station Solution

6 2 squares

1 blue rhombus

1 triangle

7 5 trapezoids

8 1 trapezoid

3 triangles

9 3 squares

2 blue rhombus

1 triangle

10 1 hexagon

1 trapezoid

1 square

1 brown rhombus

1 blue rhombus

Extensions

The cover-up method is suggested as an extension in the curriculum guide.This simple method of solving an equation is easy to teach and fun for thestudents to use.



Example:

Encourage students to record the solution in an organized way.4m + 5 = 25• cover the 4m• we know that what we covered up has to be equal to 20 because 20 + 5 is

equal to 25, so4m = 20• cover the m• we know that what we covered up has to be equal to 5 because 4

multiplied by 5 is equal to 20, som = 5

Some students will no doubt realize that what you have done is subtract 5from each side to keep the equation balanced, then divided each side by 4.

There are more examples in the curriculum guides and you could use any ofthe questions in Mathematics 7: Focus on Understanding (Calabrese, Fournier,Speijer 2005).


Balance Logic

Encourage students to talk about what they are doing as they work in pairs tosolve the problems.

With each of the following activities there are two or three balances. Studentsshould put coloured cubes on the squares to match the letter inside thesquare. They must combine the information in the balances to solve theequations at the bottom of each page. Point out to students that cubes of onecolour can be substituted for cubes of another colour when they are found tobe equal. They should also realize that when the same colour cubes are onboth sides of the scale, they can be removed and the scale is still balanced.

Example:



then:

1 Y = B

1 Y = G

1 G = B

Activity: Balance Logic 2

If:

and

then:

1 G = P

1 B = G

1 B = P


If:

and




If:

and

then:

1 P = Y

1 Y = G


If:

and

then:

1 B = G

1 G = O




If:

and

and

then:

1 Y = P

1 P = B

1 B = O

1 Y = O




If:

and

and

then:

1 P = O

1 P = Y

1 P = G

1 O = Y

1 Y = G

1 O = G


SECTION 2: LANGUAGE AND MATHEMATICS

LANGUAGE AND MATHEMATICS

The ability to read, write, listen, speak, think, and communicate visuallyabout concepts will develop and deepen students’ mathematicalunderstanding. It is through communicating about mathematical ideas thatstudents articulate, clarify, organize, and consolidate their thinking. Writingand talking about mathematics makes mathematical thinking visible.Language is as necessary to learning mathematics as it is to learning to read.

As teachers, you must assist students as they learn how to expressmathematical ideas, explain their answers, and describe their strategies. Youcan encourage students to reflect on class conversations and to “talk abouttalking about mathematics” (Principles and Standards for School Mathematics,NCTM, p. 128). You should ask questions that direct students’ thinking andpresent tasks that help students think about how ideas are related.

Principles and Standards states that instructional programs should enable allstudents to• organize and consolidate their mathematical thinking through

communication• communicate their mathematical thinking coherently and clearly to peers,

teachers, and others• analyze and evaluate the mathematical thinking and strategies of others• use the language of mathematics to express mathematical ideas precisely

As students progress through the grades they should be developing morefluency in their ability to talk about mathematics and their learning. Theirdevelopment of language skills is not the primary goal, but students requirefrequent opportunities to use language to make sense of mathematical ideasthrough sharing of problem-solving strategies and modelling. Students learnthrough discourse. Communication should include sharing ideas, askingquestions, and explaining and justifying ideas. As well, students needexperience in writing about their mathematical conjectures, questions, andsolutions. “Teachers need to explicitly discuss students’ effective andineffective communication strategies” (Principles and Standards for SchoolMathematics, NCTM, p. 197).

As teachers, you need to refine your own listening, questioning, andparaphrasing techniques, both to direct the flow of learning and to providemodels for student dialogue. By asking questions that require students toarticulate and extend their thinking, you are able to assess students’understanding and make appropriate instructional decisions.

SECTION 2:



As students progress into junior high grades they should be expected notonly to present and explain a strategy used to solve a problem but also tocompare, analyze, and contrast the usefulness, efficiency, and elegance of avariety of strategies. Students must be encouraged to provide explanations ofmathematical arguments and rationales, as well as procedural descriptions orsummaries. The use of oral and written communication provides studentswith opportunities to• think about solving problems• develop explanations• acquire and use new vocabulary and notation• explore various ways of presenting arguments• critique and justify conjectures• reflect on their own understanding as well as that of others

“Teachers should build a sense of community ... so students feel free to expresstheir ideas honestly and openly, without fear of ridicule” (Principles andStandards for School Mathematics, NCTM, p. 269).

Classroom Strategies

During the junior high school years, students continue to develop thediscourse of mathematics as they begin to think in increasingly abstract ways.From the solid language-based approach of the early school years, teachers ingrades 7, 8, and 9 provide further opportunities for students to extend anddeepen their abilities to reason and communicate. The curriculum requiresthat students relate to and respond to a wide variety of experiences usingmany communicative forms of representation to express their work. Teachersneed to utilize strategies that will allow students to process what they arelearning, to organize their ideas in ways that are meaningful to them, and toreflect on and extend their learning.

The integration of the five mathematics strands from the Atlantic CanadaMathematics Curriculum (Number Concepts/Number and RelationshipOperations, Patterns and Relations, Shape and Space, Data Management, andProbability) with the three language strands from the Atlantic Canada EnglishLanguage Arts Curriculum (Speaking and Listening, Reading and Viewing, andWriting and Other Ways of Representing) provides the richness and diversityin curriculum and instruction that will enable students to confidently exploremathematical concepts and ideas. The essence of the three language strandscan be found in the Atlantic Canada English Language Arts Curriculum: Grades7–9 on pages 119–152. Appendix I: Outcomes Framework to InformProfessional Development for Teachers of Reading in the Content Areasidentifies the expectations of all content area teachers with respect to literacy.



To help teachers deliver the prescribed curriculum, this section identifiesclassroom strategies that support teachers as they develop the vocabulary ofmathematics, initiate effective ways to manoeuvre informational text, andencourage students to reflect on what they have learned. Along with asuggestion of how to use the strategy is a mathematical application. Whenattempting to use these strategies, teachers are reminded of the suggestionsfound in Secondary Science: A Teaching Resource (p. ix):• start small—try something that feels comfortable and build competence

with time• network with others• take advantage of professional development opportunities

When teachers use these strategies in the instructional process or asassessment tasks, the expectations for students must be made explicit. Thestudent’s understanding of the mathematics involved and maintaining theintegrity of the curriculum are still the foremost concerns.

Name of Strategy Before During After Assessment

1. Concept Circle X X X X

2. Frayer Model X X X X

3. Concept Definition Map X X X X

4. Word Wall X X X

5. Three Read X X

6. Graphic Organizer X X X X

8. K-W-L X X X

7. Think-Pair-Share X X

9. Think-Aloud X X

10. Academic Journals—Mathematics X X

11. Exit Cards X X



1. Concept Circle

A concept circle is a way for students to conceptually relate words, terms,expressions, etc. As a before activity, it allows students to predict or discoverrelationships. As a during or after activity, students can determine the missingconcept or attribute or identify an attribute that does not belong.

The following steps illustrate how the organizer can be used:• Draw a circle with the number of sections needed.• Choose the common attributes and place them in the sections of the circle.• Have students identify the concept common to the attributes.

This activity can be approached in other ways.• Supply the concept and some of the attributes and have students supply

the missing attributes.• Insert an attribute that is not an example of the concept and have students

find the one that does not belong and justify their reasoning.

Concept Circle

Concept: Student Answer: Sum of opposites is always zero

Duringthe day, the

temperaturerose to 10°C; at

night it droppedto –10°C

Have $4 inpiggy bank;

owe John $4

(+3) + (–3)

two sets of (–3)plus three setsof (2)

2×(–3) + 3×(2)



2. Frayer Model

The Frayer Model is a graphic organizer used to categorize a word and buildvocabulary. It prompts students to think about and describe the meaning of aword by• giving a definition• describing the main characteristics• providing examples and non-examples of the word or concept

It is especially helpful to use with a concept that might be confusing becauseof its close connection to another concept. The following is a rendition of thetemplate found in Appendix C.

Definition (learners)

Examples

The following steps illustrate how the organizer can be used.• Display the template for the Frayer Model and discuss the various headings

and what is being sought.• Model how to use this map by using a common word or concept. Give

students explicit instructions on the quality of work that is expected.• Establish the groupings (e.g., pairs) to be used and assign the concept(s) or

word(s).• Have students share their work with the entire class.

This is an excellent activity to do in poster form to display in class. Eachgroup might do the same word or concept, or different words or conceptscould be assigned.

Facts/characteristics

Non-examples

Word



3. Concept Definition Map

The purpose of a concept definition map is to prompt students to identify themain components of a concept, show the interrelatedness, and buildvocabulary. Information is placed into logical categories, allowing students toidentify properties, characteristics, and examples of the concept. Thefollowing is a rendition of the template found in Appendix D.

Essential Characteristics

• a measure of central tendency• can be used to represent a set

of data• sum of the data divided by the

number of items in the set• can be used to find the sum of

an entire data set

Mean

Examples

• average incomes ($50,000)• average age• average height of grade 7

students (160 cm)• average time per night

spent on homework

The Frayer Model

Non-essential Characteristics

• most often called “average”• the most common “average” or

measure of central tendency in use• easily computed on a spreadsheet• most affected by gaps or outliers in

data• is not the opposite of nice

Non-examples

• John’s “average” speed in travellingfrom the airport to Halifax is 80 km/hour (rate).

• average age at retirement (mode)

Comparisons What is it? Properties

IllustrationsWhat are some examples?



The following steps illustrate how the organizer can be used.• Display the template for the concept definition map.• Discuss the different headings, what is being sought, and the quality of

work that is expected.• Model how to use this map by using a common concept.• Establish the concept(s) to be developed.• Establish the grouping (e.g., pairs) and materials to be used to complete

the task.• Complete the activity by having the student write a complete definition of

the concept.

Encourage students to refine their map as more information becomes available.

Concept Definition Mapping

Comparison Category

What is it?

Multiplication

Properties

What is it like?

base

exponent

power

The base ismultiplied by itselfand the exponentdirects how manytimes the base ismultiplied.

What are someexamples?

103 = 10 × 10 × 10

= 1000

Powers

24 = 2 × 2 × 2 × 2

= 16

0.12 = 0.1 × 0.1

= 0.01

2 × 3

2 (3 + 4)

3a

2•3

05 = 0 × 0 × 0 × 0 × 0

= 0

1100 = 1

42

42

42



4. Word Wall

A mathematics word wall is based upon the same principle as a reading wordwall found in many classrooms. It is an organized collection of words that isprominently displayed in the classroom and helps students to learn thelanguage of mathematics. A word wall could be dedicated to a concept, bigidea, or unit in the mathematics curriculum. Words are printed in bold blockletters on a card and then posted on the wall or bulletin board.

Illustrations placed next to the word on the wall would add to the students’understanding. Students may also elaborate on the word in their journals byillustrating, showing an example, and using the word in a meaningfulsentence or short paragraph. Students can each be assigned a word and itsillustration to display on the word wall. Room should be left to add morewords and diagrams as the unit or term progresses.

As a new mathematical term is introduced to the class, students can defineand categorize the word in their mathematics journals under an appropriateunit of study. Then the word can be added to the mathematics word wall sostudents may refer back to it as needed. Students will be surprised at howmany words fall under each category and how many new words they learn touse in mathematics.

Note: The word wall is developed one word at a time as the new terminologyis encountered.

To set up a word wall• Determine the key words that students need to know or will encounter in

the topic or unit.• Print the word in large block letters and add the appropriate illustrations.• Display the card when appropriate.• Regularly review the words as a warm-up or refresher activity.

5. Three Read

Using this strategy, found in Toward a Coherent Mathematics Program (NovaScotia Department of Education 2002), the teacher encourages students toread the problem three times before they attempt to solve it. There arespecific purposes of each reading.

First Read

The students try to visualize the problem in order to get an impression of theoverall context of the problem. They do not need specific details at this stage,only a general idea so they can describe the problem in broad terms.



Second Read

The students begin to gather facts about the problem to make a morecomplete mental image of the problem. As they listen for more detail, theyfocus on the information to determine and clarify the question.

Third Read

The students check each fact and detail in the problem to verify the accuracyof their mental image and to complete their understanding of the question.

During the three read strategy, the students discuss the problem includingany information needed to solve the problem. Reading becomes an activeprocess that involves oral communication among students and teachers andalso written communication as teachers encourage students to recordinformation and details from their reading, as well as to represent what theyread in other ways with pictures, symbols, or charts. The teacher facilitates theprocess by posing questions that ask students to justify their reasoning, tosupport their thinking, and to clarify their conclusions.

6. Graphic Organizer

A graphic organizer can be of many forms: web, chart, diagram, etc. Graphicorganizers use visual representations as an effective tool to do such things as• activate prior knowledge• make connections• organize• analyze• compare and contrast• summarize

The following steps illustrate how the organizer can be used.• Present a template of the organizer and explain its features.• Model how to use the organizer, being explicit about the quality of work

that is expected.• Present various opportunities for students to use graphic organizers in the

classroom and as assessment tools.

Students should be encouraged to used graphic organizers on their own as away of organizing their ideas and work.



( )

Venn Diagram

Classifying Numbers by Their Factors

Students were asked to place the following numbers in the appropriatesections of the Venn diagram to satisfy the classification.

{2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 25, 30}

7. K-W-L (Know/Want to Know/Learned)

K-W-L (Know/Want to Know/Learned) is an instructional strategy that guidesstudents through a text and uses a three-column graphic organizer toconsolidate the important ideas. Students brainstorm what they know about thetopic and record it in the K column. They then record what they want to knowin the W column. During and after the reading, students record what they havelearned in the L column. The K-W-L strategy has several purposes; to• illustrate a student’s prior knowledge of a topic• give a purpose to the reading• help a student monitor his/her comprehension

Know Want to Know Learned

Numbers divisible by 2 N2

( )Numbers divisible by 3 N3

( )Numbers divisible by 5 N5

2

3

4

5

6

8

9

10

12

14

15

16

18

20

21

25

30



The following steps illustrate how K-W-L can be used.• Present a template of the organizer to the students, explain its features, and

be explicit about the quality of work that is expected.• Ask them to fill out the first two sections: what they know and what they

want to know before proceeding.• Check the first section for any misconceptions in thinking or weaknesses

in vocabulary.• Have the students read the text, taking notes as they look for answers to

the questions they posed.• Have students complete the last column to include the answers to their

questions and other pertinent information.• Discuss this new information with the class and address any questions that

were not answered.

Variations on this strategy are explained in Secondary Science: A TeachingResource, pages 2.9–2.15 (Nova Scotia Department of Education 1999).

K-W-L Concept Sum of the Interior Angles of a Quadrilateral

Know Want to Know Learned

The sum of the interiorangles of a triangle is180°.

a + b + c = 180°

What is the sum of theinterior angles of aquadrilateral?

Investigate variousquadrilaterals

Student Answer

The sum of the interiorangles of a quadrilateral is360°

a + b + c = ?d + e + f = ?

a + b + c = ?d + e + f = ?

Soa + b + c + d + e + f =

Further Investigation

What is the sum of the interior angles of a pentagon?



8. Think-Pair-Share

Think-Pair-Share is a learning strategy designed to encourage students toparticipate in class and keep them on task. It focuses students’ thinking onspecific topics and provides the students with an opportunity to collaborateand engage in meaningful discussion (Van de Walle 2006).• First, teachers ask students to think individually about a newly introduced

topic, concept, or problem. This provides essential time for each student tocollect their thoughts and focus their thinking.

• Second, each student pairs with another student, and together the partnersdiscuss each other’s ideas and points of view. Students are more willing toparticipate because they do not feel the peer pressure that is involved whenresponding in front of the class. Teachers ensure that sufficient time allowseach student to voice his or her views and opinions. Students use this timeto talk about personal strategies, compare solutions, or test ideas with theirpartners. This helps students to make sense of the problem in terms ofprior knowledge.

• Third, each pair of students shares with the other pairs of students in alarge-group discussion. In this way, each student has the opportunity tolisten to all the ideas and concerns discussed by the other pairs of students.Teachers point out similarities, overlapping ideas, or discrepancies amongthe pairs of students and facilitate an open discussion to expand upon anykey points or arguments they wish to pursue.

9. Think-Aloud

Think-Aloud is a self-analysis strategy that allows students to gain an insightinto the thinking process of a skilled reader as he or she works through a pieceof text. Thoughts are verbalized and meaning is constructed aroundvocabulary and comprehension. It is a useful tool for such things asbrainstorming, exploring text features, and constructing meaning whensolving problems. When used in mathematics, it can reveal to teachers thestrategies that are part of a student’s experience and those that are not.

The Think-Aloud process will encourage students to use the followingstrategies as they approach a piece of text:• connect new information to prior knowledge• develop a mental image• make predictions and analogies• self-question• revise and fix up as comprehension increases



The following steps illustrate how to use a think-aloud.• Explain that reading in mathematics is important and requires students to

be thinking and trying to make sense of what they are reading.• Identify a comprehension problem or piece of text that may be challenging

to students; then read it aloud and have students read it quietly.• While reading, model the process by verbalizing what you are thinking,

what questions you have, and how you would approach a problem.• Then model this process a second time, but have a student read the

problem and do the verbalizing.• Once students are comfortable with this process, a student should take a

leadership role.

10. Academic Journals Portfolio—Mathematics

An academic journal in mathematics is an excellent way for students to keeppersonal work and other materials that they have identified as beingimportant for their personal achievement in mathematics. The types ofmaterials that students would put in their journals would be• strategic lessons—lessons that they would identify as being pivotal as they

attempt to understand mathematics• examples of problem-solving strategies• important vocabulary

Teachers are encouraged to allow students to use these journals in anassessment. This way it will emphasize to the student that the material that isto be placed into their journal has a purpose. Mark these journals only on thebasis of how students are using them and whether or not they have theappropriate entries.

Math Journals

The goal of writing in mathematics is to provide students with opportunitiesto explain their thinking about mathematical ideas and then to re-examinetheir thoughts by reviewing their writing. Writing will enhance students’understanding of mathematics as they learn to articulate their thoughtprocesses in solving math problems and learning mathematical concepts.



11. Exit Cards

Exit cards are a quick tool for teachers to become better aware of a student’sunderstanding. They are written student responses to questions that have beenposed in class or solutions to a problem-solving situation. They can be used atthe end of a day, week, lesson, or unit. An index card is given to each studentwith a question that promotes understanding on it and the student mustcomplete the assignment before he or she is allowed to “exit” the classroom.The time limit should not exceed 5–10 minutes and the student drops thecard into some sort of container on the way out. The teacher now has a quickassessment of a concept that will help in planning instruction.



Grade 7 Mathematics Language

• integers• LCM (least common

multiple)• multiples• percentage• pictorial• power• ratio

Operations

• associative• commutative• distributive• like terms

• repeatedmultiplication

• repeating decimals• scientific notation• standard form• terminating decimal• expanded form• symbolic

Number Sense

• base• common factors• common multiples• divisibility rules• exponent• exponential form• GCF (greatest

common factor)

• numerical expressions• order of operations• properties of operations• properties of zero

• unlike terms• variable• whole numbers• zero principle

Patterns & Relationships

• affected quantity• algebraic equations• algebraic expressions• constants• equations• extrapolate

• interpolate• linear equations• one-step single-variable

linear equation• predictions• related quantity

• systematic trial• table of values• two-step single-

variable linearequation

Measurement

• accuracy• capacity• circumference• diameter

• graphs of rate• length• mass• precision

• radius• rate• SI units• volume

Geometry

• adjacent angles• complimentary

angles• coordinates• deduce• equiangular• Euclidean• image

• inferences• line symmetry• non-parallel• obtuse• platonic solids• pre-image• quadrants• reflection

• rotation• translation• transversal• triangle

classifications• vertiically opposite

angles



Data Management

• analyze• biassed data• bi-modal• circle graph• clusters• conjecture• distorted• ethical• extreme values• frequency distribution

• frequency table• gaps• grouped data• histogram• hypothetical

situations• inferences• interval• leading question• lower bound of

interval

• measure of centraltendency

• non-numerical data• outliers• percentage circle• range of the scale• raw data• scatterplot• unbiassed data

Probability

• area models• independent events• likely events• one in four

• one in two• possible outcomes• simulation

• tree diagrams• uncertain events• zero probability



SECTION 3:ASSESSMENT AND EVALUATION

Introduction

Assessment and evaluation are essential to student success in mathematics.The purpose of assessment is manifold: assessment yields rich data to evaluatestudent learning, the effectiveness of teaching, and the achievement of theprescribed curriculum outcomes. However, assessment without evaluation isinsufficient, as the collection and reporting of data alone are not entirelyuseful unless the quality of the data is evaluated in relation to the outcomes.To this end, teachers use rubrics, criteria, marking keys, and other objectiveguides to evaluate the work of their students.

Assessment

Assessment is the process of collecting information about student learning (forexample, through observation, portfolios, pencil and paper tests, performance).Assessment is the gathering of pertinent information.

Evaluation

Evaluation follows assessment by using the information gathered to determine astudent’s strengths, needs, and progress in meeting the learning outcomes.Evaluation is the process of making judgements or decisions based on theinformation collected in assessment.

Assessments for Learning

Assessments are designed with a purpose. Some assessments are designed byteachers as assessments “for” learning. The purpose of these assessments is, inpart, to assist students in their progress towards the achievement of prescribedcurriculum outcomes. In such assessments, the tasks used by teachers shouldinform students about what kinds of mathematical knowledge andperformances are important.

As well, assessments for learning help teachers to know where their studentsare on the learning continuum, track each student’s progress, and plan what“next steps” are required for student success. Following assessments forlearning, teachers help students toward the achievement of a mathematicsoutcome by providing them with further opportunities to learn. In this way,such assessments take a developmental perspective, and track students’ growththrough the year.



Assessment as Learning

Some assessments for learning designed specifically to encourage studentinvolvement provide students with a continuous flow of informationconcerning their achievement. When students become involved in the processof assessment, it becomes assessment “as” learning. Assessment techniquessuch as conversation, interviews, interactive journals, and self-assessment helpstudents to articulate their ideas and understandings and to identify wherethey might need more assistance. Such techniques also provide students withinsight into their thinking processes and their understandings. This kind ofassessment is used, not only to allow students to check on their progress, butto advance their understandings, to encourage them to take risks, to allowthem to make mistakes, and to enhance their learnings. This kind ofassessment also helps students to monitor and evaluate their own learning, totake responsibility for their own record keeping and to reflect on how theylearn.

Teachers should keep in mind that such assessment practices may beunfamiliar to students at first, and that the emphasis on their being activelyinvolved and thinking for themselves will be a challenge for some students.Such practices, however, enable teachers and students, together, to form a planthat ensures students are clear about what they have to do to achieveparticular learning outcomes

Assessments of Learning

Assessments “of ” learning provide an overview of a student’s achievement inrelation to the outcomes documented in the Atlantic Canada mathematicscurriculum that form the basis for the student’s learning requirements. Whenan assessment of learning achieves its purpose, it provides information to theteacher for the grading of student work in relation to the outcomes.

Final assessments of learning should be administered after the student has hadthe fullest opportunity to learn the intended outcomes in the mathematicsprogram. Assessments of learning check for a student’s achievement againstthe outcomes. It should be noted that any assessment for learning that hasreveals whether a student has met the intended outcome can also beconsidered assessment of learning and the evaluation of that assessment maybe used to report on the student’s achievement of the outcome.


SECTION 3: ASSESSMENT AND EVALUATION

Assessment “as”/”for”/”of” Learning

* Note: A teacher may discover, in the process of assessment for learning, that the student hasmet the mathematics outcome. At this point the “assessment for learning” becomesan “assessment of learning.” The evaluation of this assessment may be used to reporton the student’s achievement in relation to the mathematics outcome. The student isready to move on.

Figure 1: Assessments “as,” “for,” and “of ” learning are what teachers do in abalanced classroom assessment process.

Alignment

Assessments serve teaching and learning best when teachers integrate themclosely with the ongoing instructional/learning process, when assessments areplanned in advance, and when both formative and summative assessments areused appropriately. The nature of the assessments used by the teacher must beappropriate to and aligned with curriculum, so that students’ progress ismeasured by what is taught and what is expected. When learning is the focus,curriculum and assessment become opposite sides of the same coin, eachserving the other in the interest of student learning and achievement.Assessments, therefore, should inform classroom decisions and motivatestudents by maximizing their confidence in themselves as learners. For thisreason, teachers need to be prepared to understand the fundamental conceptsof assessments and evaluation.

Choosing and using the right kinds of assessments are critical, and teachersneed to be aware of the strengths and weaknesses of their assessment choices.As well, employing a variety of appropriate assessments improves thereliability of their evaluation, and can help to improve both teaching andlearning.



Planning Process: Assessment and Instruction

What would the classroom assessment process look like?1. The teacher needs to have a clear understanding of the outcomes that are to

be achieved and the multiple ways that students can be involved in thelearning process. The teacher must• be able to describe what the student needs to achieve• collect and create samples that model what the learning looks like• decide what kinds of evidence the student can produce to show that

he/she has achieved the outcome(s) (i.e., design or select assessment tasksthat allow for multiple approaches)

• select or create learning activities that will ensure student success

2. To introducing the learning, the teacher• discusses the outcomes, and what is to be achieved• shows samples and discusses what the product of the learning should look

like• plans with students by setting criteria for success and developing time lines

– activates prior knowledge– provides mini-lessons if required to teach/review prerequisite skills

3. After assigning the learning activity, the teacher.• Provides feedback and reminds students to monitor their own learning

during the activity.– feedback to any student should be about the particular qualities of the

work, with advice on how to improve it, and should avoid comparisonswith other students

– the feedback has three elements: recognition of the desired performance;evidence about the student’s current understanding; and someunderstanding of a way to close the gap between the first two

Figure 3: Assessment as/for learning are the foundation of classroomassessment activities leading to assessment of learning.



• Encourages students to reflect on the learning activity and revisit thecriteria:– the discourse in the classroom is imperative-the dialogue should be

thoughtful, reflective, and focused to evoke and exploreunderstanding

– all students should have an opportunity to think and express their ideas– the teacher must ask questions (See pages 115–116 for more

discussion on “questioning.”)– the questions must require thought– the wait time must be long enough for thinking to take place

• Uses classroom assessments to build student confidence– Tests given in class and exercises given for homework are important

means for providing feedback.– It is better to have frequent short tests than infrequent long ones.– New learning should first be tested within about a week.– Quality test items, relevance to the outcomes, and clarity to the

students are very important.– Good questions are worth sharing, collaboration between and among

teachers is encouraged.– Feedback from tests must be more than just marks .– Quality feedback from tests has been shown to improve learning when

it gives each student specific guidance on strengths and weaknesses.– Instruction should be continuously adjusted, based on classroom

assessments.

• Encourages students to self-assess, review criteria, and set new goals tohelp them take responsibility for their own learning:– Students should be taught self-assessment so that they can understand

the purposes of their learning and understand what they need to do toimprove. (See pages 138–139 for examples of “self-assessment”sheets.)

– Students reflect on their learning journals. (See page 138 for examplesof prompts to promote writing in journals and logs.)

4. Use cumulative assessment• Eventually, there is a time when students must be able to demonstrate

what they have learned, what they understand, and how well they haveachieved the outcomes.

• Assessment should reflect the outcomes and should focus on the students’understanding, as well as their procedural skills. (See page 114 forexamples of the kinds of questions students should be answering.)



Assessing Students’ Understanding

What does it mean to assess students’ understanding?

Students should be asked to provide evidence that they can• identify and generate examples and non-examples of concepts in any

representation (concrete, context, verbal, pictorial, and symbolic)• translate from one representation of a concept to another• recognize various meanings and interpretations of concepts• identify properties and common misconceptions• connect, compare, and contrast with other concepts• apply concepts in new/novel/complex situations

What does it mean to assess students’ procedural skills?

Students should be asked to provide evidence that they can• recognize when a procedure is appropriate• give reasons for steps in a procedure• reliably and efficiently execute procedures• verify results of procedures analytically or by using models• recognize correct and incorrect procedures

What follows are five examples from grade 7 that address the five bulletsabove. Each question is connected to the following stem problem:

To win the contest Marla had to answer the skill testing question given here:

–5 – 2(6÷ 3 + 32)2 –4

Help her find the answer.

1. Marla begins her attempt to get the answer by taking 2 away from –5. Isthis correct?

2. Ralph begins to help Marla by dividing 6 by 3, and adding 9. Why does hebegin there?

3. Bobby uses his scientific calculator, entering the whole question as written,left to right, then pushes enter, and the calculator shows the correct answer55.5. Is this the most efficient way to get the answer? Explain.

4. Ralph begins to help Marla by dividing 6 by 3, and adding 9. That giveshim11, which he then squares and multiplies by two. He then divides thatanswer by negative 4 and gets –60.5. Explain using pictures of two-colourtiles how he then gets the correct answer of 55.5.



5. Provide three possible student responses to the above problem, and askstudents to identify any correct or incorrect procedures, and explain theirdecisions.a) a correct response using symbolsb) a response not using BEDMAS correctlyc) a response where the students gets the 242, but then subtracts that

from 5 before dividing by –4.

Questioning

When we teach we ask questions. Are we planning the kinds of questions thatwe want to ask? The effective use of questions may result in more studentlearning than any other single technique used by educators.

When designing questions, avoid those that can be answered with “yes/no”answers. Remember the levels. Questions written at the knowledge levelrequire students to recall specific information. Recall questions are a necessarybeginning since critical thinking begins with data or facts. There are timeswhen we want students to remember factual information and to repeat whatthey have learned. At the comprehension level the emphasis is onunderstanding rather than mere recall, so we need to ask open-endedquestions that are thought provoking and require more mental activity.Students need to show that concepts have been understood, to explainsimilarities and differences, and to infer cause-and-effect relationships.

Comprehension-level questions require students to think, so we mustestablish ways to allow this to happen after asking the question.• Ask students to discuss their thinking in pairs or in small groups—the

respondent speaks on behalf of the others• Give students a choice of different answers—let the students vote on the

answer they like best—or ask the question to the class, then after the firststudent responds, without indicating whether that was correct or not,redirect the question to solicit more answers from other students.

• Ask all students to write down an answer, then select some students to reada few responses.

When redirecting questions (asking several students consecutively withoutevaluating any) and after hearing several answers, summarize or have the classsummarize what has been heard. Or, have students individually create (inwriting) a final answer to the question now that they have heard severalattempts. Lead the class to the correct or best answer to the question (if thereis one). Inattentive students who at first have nothing to offer when asked,should be re-asked during the redirection of the question to give them asecond chance to offer their opinions.



Discourse with the Whole Group

Sometimes it is best to begin discussion by asking more divergent levelquestions, to probe all related thoughts and bring students to the awareness ofbig ideas, then to move towards more convergent questions as the learninggoal is being approached. Word the questions well and explain them in otherwords if students are having trouble understanding. Direct questions to theentire class. Handle incomplete or unclear responses by reinforcing what iscorrect and then asking follow-up questions. There are times when you wantto restate correct responses in your own words, then ask for alternativeresponses. Sometimes it is important to ask for additional details, seekclarifications, or ask the student to justify a response. Redirect the question tothe whole group if the desired response is not obtained. Randomize selectionwhen many hands are waving. Ask a student who is always the first to wave ahand to ask another student for an answer, then to comment on thatresponse. As the discussion moves along, interrelate previous students’comments in an effort to draw a conclusion. It is particularly important to askquestions near the end of the discussion that help make the learning goalclear.

Questioning is a way of getting to assessing student progress, and animportant way to increase student learning. As well, it is a way to get studentsto think, and to formulate and express opinions.

Examples of Effective Questions

Critical Thinking

• Why did ... ?• Give reasons for ...• Describe the steps ...• Show how this ...• Explain why ...• What steps were taken to ... ?• Why do you agree (disagree) with ... ?• Evaluate the result of ...• How do you know that ... ?

Cause/Effect Relationship

• What are the causes of ... ?• What connection exists between ... ?• What are the results of ... ?• If we change this, then ... ?• If these statements are true, then

what do you think is most likely tohappen ... ?

Problems

• What else could you try ... ?• The diagrams in the problem suggest

the following relationship:x 1 2 3 4 5y 1 3 6 10 15

• What would you say is the y-valuewhen ... ?

Comparison

• What is the difference ... ?• Compare the ...• How similar are ... ?• Contrast the ...



Personalized

• Which would you rather be like ... ?• What would you conjecture ... ?• Which do you think is correct ... ?• How would your answer compare ...?• What did you try ... ?• How do you feel about ... ?• What would you do if ... ?• If you don’t know ... how could you

find out?

Levels of Questioning

In recent years the Nova Scotia Department of Education has administeredelementary and junior high mathematics assessements. In an attempt to setstandards for these assessements, committees of teachers prepared tables ofspecifications. These committees also decided that there would be a blend ofdifferent complexity levels of questions and determined what percentage ofthe whole assessment should be given for each level of question. They alsoagreed that the assessments would use a combination of selected responsequestions and constructed response questions, some of which might requireextended responses, others short responses.

Level 1: Knowledge and Procedure (Low Complexity)

Key Words

• compute • find • measure • recognize• evaluate • identify • recall • use

• Level 1 questions rely heavily on recall and recognition.• Items typically specify what the student is to do.• The student must carry out some procedure that can be performed

mechanically.• The student does not need an original method of solution.

The following are some, but not all, of the demands that items of“low complexity” might make:• recall or recognize a fact, term, or property• recognize an example of a concept• compute a sum, difference, product, or quotient• recognize an equivalent representation• perform a specified procedure

Descriptive

• Describe ...• Tell ...• State ...• Illustrate ...• Draw (sketch) ...• Define ...• Analyze ...



• evaluate an expression in an equation or formula for a given variable• solve a one-step word problem• draw or measure simple geometric figures• retrieve information from a graph, table, or figure

(See page 120 for more examples of Level 1 questions.)

Level 2: Comprehension of Concepts and Procedures (Moderate Complexity)

Key Words

• classify • estimate • interpret • organize• compare • explain

• Items involve more flexibility of thinking and choice.• Questions require a response that goes beyond the habitual.• Method of solution is not specified.• Questions ordinarily have more than a single step.• The student is expected to decide what to do using informal methods of

reasoning and problem-solving strategies.• The student is expected to bring together skills and knowledge from

various domains.

The following illustrate some of the demands that items of “moderatecomplexity” might make:• make connections between facts, terms, properties, or operations• represent a situation mathematically in more than one way• select and use different representations, depending on situation and

purpose• solve a word problem involving multiple steps• compare figures or statements• explain and provide justification for steps in a solution process• interpret a visual representation• extend a pattern• retrieve information from a graph, table, or figure and use it to solve a

problem requiring multiple steps• formulate a routine problem, given data and conditions• interpret a simple argument(See pages 121 for more examples of Level 2 questions.)



Level 3: Applications and Problem Solving (High Complexity)

Key Words

• analyze • explain • investigate • prove• describe • formulate

• Questions make heavy demands on students.• The students engage in reasoning, planning, analysis, judgment, and

creative thought.• The students must think in an abstract and sophisticated way.

The following illustrate some of the demands that items of “high complexity”might make:• explain relations among facts, terms, properties, or operations• describe how different representations can be used for different purposes• perform a procedure having multiple steps and multiple decision points• analyze similarities and differences between procedures and concepts• generalize a pattern• formulate an original problem• solve a novel problem• solve a problem in more than one way• explain and justify a solution to a problem• describe, compare, and contrast solution methods• formulate a mathematical model for a complex situation• analyze the assumptions made in a mathematical model• analyze or produce a deductive argument• provide a mathematical justification

(See pages 122 for more examples of Level 3 questions.)

Recommended Percentages for Testing

Level 1: 25–30%

Level 2: 40–50%

Level 3: 25–30%



Sample Questions

Level 1

a) reflection through the y-axisb) a translation [2R, 2U]c) a rotation 90° clockwised) reflection through the x-axis

3. Select the answer that best describes the measures of these angles.

a) a = 65°, b = 65°b) a = 65°, b = any #c) a = 65°, b = 135°d) a + b = 185, b = a

4. What value placed in the box makes the following statement true 3 × 2 = 24

2. Select the transformation that best describes the diagram below.

1. Classify each type of triangle in two ways.a)

b)

c)



Level 2

2. Fred has $6.00 in his pocket. He wants to buy 2 bottles of juice at $1.69each and 3 snack bars at $0.94 each. Estimate to determine whether ornot he has enough money to purchase these items. (Show the series ofrough calculations you made to justify your answer.)

3. a) Solve the following equation pictorially and symbolically. Shaded piecesare positive. Unshaded pieces are negative.

b) Create a story problem that represents the picture.

b) Place the fractional value that represents the number of blocks shadedin the appropriate place on the number line below.

c) Provide an alternative representation using fraction tiles or circle piecesto represent the fraction.

1 a) What fraction of the entire diagram does the shaded portion represent?

4. a) Place each of the following on a number line: 1 , – , 1.4, 0.7525

14

b) Explain how you determined the placement of 1 as compared with 1.4.14



Level 3

1. Sally is trying to find the measure of angle D in the diagram. Examine hersolution and explain whether you think it is correct or incorrect. If youfind an error, tell what is wrong, then fix the error and complete thesolution to find the measure of D.

2. The cost of the production on opening night at the musical theatre was$600. Every seat in the theatre costs $12. What percentage of the seatsneed to be sold to break even?

3. The diagram below shows a kitchen floor with an island counter.Determine the cost of installing ceramic tile if a 30 cm × 30 cm tile costs$2.98.

4. Marla has two garden plots in her yard that she wants to convert to grass.A bag of grass seed covers 3 square metres at the cost of $4.50 per bag.How much will it cost to cover the two plots?

80° + 35° = 115°180° – 115° = 75° = AEB AEB = DEC DEC = 75° DEC = DCE = 75° ( )75° + 75° = 150°360° + 150° = 210° D = 210°



Scoring Open-Ended Questions Using Rubrics

Students should have opportunities to develop responses to open-endedquestions that are designed to address one or more specific curriculumoutcomes. Open-ended questions allow students to demonstrate theirunderstandings of mathematical ideas and show what they know with respectto the curriculum that should lead to a solution.

Often, responses to open-ended questions are marked according to a rubricthat allows the teacher to assign a level of achievement towards an outcomebased on the solution written by the student.

For each individual open-ended question, a rubric should be created to reflectthe specific important elements of that problem. Often these individualrubrics are based on more generic rubrics that give examples of the kinds offactors that should be considered. Details will vary for different grade levelsbut the basic ideas apply for all levels.

How do you begin thinking about a rubric? Consider this. Let us say that youasked your students to write a paper on a particular mathematician. Theyhanded in their one-page reports, and you began to read them. As you read,you were able to say, Hey, that one is pretty good, this one is fair, and thisthird one needs a lot of work. So you decide to read them all and put theminto the three piles: good, fair, and weak. A second reading allows you toseparate each of the three piles into two more piles: top of the level andbottom of the level. When done you have six levels, you could describe thecriteria for each of those six levels in order to focus more on the learningoutcome. You have created a rubric.

Some rubrics have criteria described for six levels, some five levels, and somefour levels. Some people like four levels because it forces the teacher todistinguish between acceptable performance, and below (not acceptable),there is no middle—you either achieve a level 2 (not acceptable work) or level3 (acceptable).

The first example that follows includes the criteria for a generic four-levelrubric, found in the NCTM booklet (listed in Authorized LearningResources) Mathematics Assessment: Myths, Models, Good Questions, andPractical Suggestions, edited by Jean Stenmark, University of California. Inchoosing to use this rubric a teacher would change the generic wording to fitthe given problem or open-ended situation. Some teachers like to assign aname to the different achievement levels. For example, they may call the “toplevel” responses, exceptional; the “second level,” good; the “third level,” notquite; and the “fourth level,” needs more work. Many schools simply assign anumber rating, usually the top level receiving a 4, then going down, 3, then 2,then 1. Students strive for a level 3 or 4. Some schools assign letters to thecategories, giving A to the top level, then B, C, and D.



Achievement Level Criteria

Top Level • contains complete response with clear, coherent,unambiguous, and elegant explanation

• includes clear and simple diagram• communicates effectively to an identified audience• shows understanding of the question’s mathematical

ideas and processes• identifies all the important elements of the question• includes examples and counter-examples• gives strong supporting arguments• goes beyond the requirements of the problem

Second Level • contains a good solid response, with some of thecharacteristics above, but not all

• explains less elegantly, less completely• does not go beyond requirements of the problem

Third Level • contains a complete response but the explanationmay be muddled

• presents arguments, but they are incomplete• includes diagrams, but they are inappropriate or

unclear• indicates understanding of the mathematical ideas,

but they are not expressed clearly

Fourth Level • omits significant parts or all of the question andresponse

• has major errors• uses inappropriate strategies

A second example of a rubric follows that allows for six achievement levels.This example can be found in the booklet Assessment Alternatives inMathematics, prepared by the Equals staff and the Assessment Committee ofthe California Mathematics Council Campaign for Mathematics, under theleadership of Jean Stenmark.

The top two levels of the above rubric have been titled “DemonstratedCompetence” and will get a 6 and a 5. The next two levels called“Satisfactory” will get a 4 and a 3, while the bottom three levels, “InadequateResponse,” receive a 2, 1, or 0.



(minor flaws)

(serious flaws)

Achievement Level Criteria

6. Exemplary Response • contains complete response, with clear, coherent,unambiguous, and elegant explanation

• includes clear and simplified diagrams• communicates effectively• shows understanding of the mathematical ideas and

processes• identifies all the important elements• may include examples and counter-examples• presents strong supporting arguments

5. Competent Response • contains fairly complete response, with reasonablyclear explanations

• may include an appropriate diagram• communicates effectively• shows understanding of the mathematical ideas and

processes• identifies the most important elements of the problem• presents solid supporting arguments

4. Satisfactory • completes the problem satisfactorily but explanationmight be muddled

• presents incomplete arguments• includes diagram but it is inappropriate or unclear• understands the underlying mathematical ideas• uses mathematical ideas effectively

3. Nearly Satisfactory • begins problem appropriately, but fails to completeor may omit parts of the problem

• fails to show full understanding of the mathematicalideas and processes

• may make major computational errors• may misuse or fail to use mathematical terms• contains response that may reflect inappropriate strategy

2. Incomplete • contains explanation that is not understandable• includes diagram that may be unclear• shows no understanding of the problem situation• may make major computational errors

1. Ineffective Beginning • contains words that do not reflect the problem• includes drawings that misrepresent the problem situation• copies part of the problem but without attempting a

solution• fails to indicate which information is appropriate to

the problem

0. No Attempt • has no evidence of anything meaningful



Understandingconcepts

• demonstrates nounderstanding ofthe mathematicalconceptsrequired in theproblemsituation

• demonstrateslittleunderstanding ofthe mathematicalconceptsrequired in theproblemsituation

• demonstratessomeunderstandingof themathematicalconceptsrequired in theproblemsituation

• demonstrates athoroughunderstandingof themathematicalconceptsrequired in theproblemsituation

Application ofprocedures

• includes a fewcalculations and/or use of skillsand proceduresthat may becorrect, butinappropriate forthe problemsituation

• makes severalerrors incalculations and/or use of someappropriate skillsand procedures

• may make a fewerrors incalculationsand/or use ofskills andprocedures

• uses accuratecalculations andappropriateskills andprocedures

Communication • includes noappropriatearguements

• includes noappropriatediagrams

• improperly useswords and terms

• incompletearguments

• diagraminappropriate orunclear

• may misuse offail to usemathematicalwords and terms

• makes a fairlycompleteresponse, withreasonably clearexplanations andappropriate useof words andterms

• may include anappropriatediagram

• communicateseffectively

• presents somesupportingarguments whenappropriate

Expectation Level 1 Level 2 Level 3 Level 4

Problem solving • shows noor very littleunderstandingof themathematicalideas andprocessesrequired

• uses nostrategies

• shows littleunderstanding ofthemathematicalideas andprocessesrequired

• attemptsinappropriatestrategy orstrategies

• showsunderstandingof themathematicalideas andprocessesrequired

• uses someappropriatestrategies

• shows thoroughunderstandingof themathematicalideas andprocessesrequired

• usesappropriatestrategies

A third kind of rubric that is becoming more popular these days includesmore than one domain when assigning the levels. For example, in the nextgeneric rubric the solution attempt will follow the criteria for four levels ofachievement, but in four domains: problem solving, understanding concepts,application of procedures, and communication. The teacher assigns levels ofachievement for each domain.

• makes completeresponse, withclear, coherent,unambiguous,and elegantexplanation and/or use of wordsand terms

• includes clearand simplifieddiagrams

• communicateseffectively

• presents strongsupportingarguments whenappropriate



The use of rubrics for assessing open-ended problem situations allows theteacher to indicate how well the student has demonstrated achievement of theoutcomes for which the assessment item has been designed in each of theseimportant domains. In reporting to the student or the parent, the teacher canmake very clear what it is that the student has not accomplished with respectto full achievement of the outcome(s). Over time, as the outcome(s) isassessed again, progress or lack thereof becomes very clear when the criteriaare clearly indicated.

Achievement levels can be changed into percentage marks (if that is the desire)by adding together the achievement levels obtained, dividing by themaximum levels obtainable, and changing that ratio to a percentage. Forexample, Freddie received a level 3, 4, 3, 2, 3, 4, 3, 3, 3, and 2 during theterm when open-ended problem-solving opportunities were assigned. Hecould have obtained a grade of 4 each time, so his ratio is 30 out of a possible40, giving him a 75 percent mark.

Example of a Rubric for a Level 3 Question

Next, let us examine an example of an open-ended problem situation forgrade 7. We will develop a specific rubric from this last generic rubric, thenmark four students’ actual work pages. We would like to thank the studentsfrom Scotia Middle School in Antigonish for sending us their solutions. Inparticular, a thank you to four students whose work we will evaluate.

The problem: Scotia Middle School is holding a competition to design a newflag for their school. All students in the school are eligible to submit a design;however, all designs must meet the following conditions: (1) all designs mustbe submitted on grid paper, using 10 rows of 14 squares; (2) two-fifths of theflag overall has to be gold; (3) one-sixth of the remaining part of the flag hasto be white; and (4) all the rest of the flag has to be black. Include anexplanation in words of how you designed the flag.

Design an entry for this flag competition being sure that your entry satisfiesall four conditions.



Understandingconcepts

• demonstrates nounderstanding ofarea and/orfractions andhow they areused todetermine thenumber ofsquares

• demonstrateslittleunderstanding ofarea andfractions andhow they areused todetermine thenumber ofsquares

• demonstrates agoodunderstanding ofarea and fractionsand how they areused (maybe a fewmisunderstandings)to determine thenumber of squares

• demonstrates athoroughunderstandingof area andfractions andhow they areused todetermine thenumber ofsquares

Application ofprocedures

• determines areausing an incorrectprocedure andsubdividesincorrectly bymisusing fractionprocedures

• makes inaccuratecalculations todetermine areaand/or tosubdivide usingfractions

• uses almostaccuratecalculations(small errors) todetermine areaand to subdivideusing fractions

• uses accuratecalculation todetermine areaand to subdivideusing fractions

Communication • uses terms andwords incorrectlyor not at all andmakes errorssubdividing thediagram, ordiagram poorlydone

• uses terms, andwords incorrectlyin explanationsand makes someerrors insubdividing theareas on thediagram

• uses terms andwords correctlyin explanationsand subdividesthe diagrambased on areasdetermined

Expectation Level 1 Level 2 Level 3 Level 4

Problem solving • demonstrates nounderstanding ofthe use offractions tosubdivide thearea according tothe instructions

• demonstrateslittleunderstanding ofthe use offractions tosubdivide thearea according tothe instructionsand uses onlysome strategies

• demonstrates agoodunderstandingof the use offractions (maybea fewmisinterpretations)to subdivide thearea accordingto theinstructions anduses appropriatestrategies

Demonstrates athoroughunderstandingof the use offractions tosubdivide thearea accordingto theinstructions anduses appropriatestrategies

We will use this rubric:

• uses correctterms, words,andexplanationsclearly andsubdivides thediagramappropriately



Problem Solving: [3]: Correct use of fractions to determine areas for colours,but misinterpreted condition #3, “one-sixth of the remaining part.”

Understanding Concepts: [4]: The concepts of area and fraction computationare correct, even though one condition was not correct.

Application of Procedures: [3]: The calculations of the number of squares foreach colour have been performed correctly except for black, and the indicatedcorrect number of squares for each colour has been allocated on the diagram.Black should be calculated for “checking” purposes.

Communication: [3]: Explanation not as clear as it could be. For example,“drew out 4/10 and counted the squares ...”, and “... the rest has to be black ...”

Student One



Student Two

Problem Solving: [4]: Correct use of fractions to determine areas for colours,and conditions all met.

Understanding Concepts: [4]: The concepts of area and fraction computationare correct. There is a recording error (140 ÷ 56). The divide sign should be asubtract sign, but it was clear that she knew to subtract.

Application of Procedures: [4]: The calculations of the number of squares foreach colour have been performed correctly, and the indicated correct numberof squares for each colour has been allocated on the diagram.

Communication: [1]: No explanation given.



Student Three

Problem Solving: [3]: Correct use of fractions to determine areas for colours,but misinterpreted condition #3, “one-sixth of the remaining part.”

Understanding Concepts: [4]: The concepts of area and fraction computationare correct, even though one condition was not correct.

Application of Procedures: [3]: The calculations of the number of squares foreach colour have been performed correctly except for black, and the indicatedcorrect number of squares for each colour has been allocated on the diagram.Black should be calculated for “checking” purposes.

Communication: [1]: Serious flaws in the written explanation. Reference tothe area 10 by 14 as the perimeter. Saying in one spot that two-fifths of theflag had to be gold, then saying that to determine this she took one-sixth ofthe perimeter 140 to determine that 56 will be gold. She also refers todividing one-sixth “into” 140, instead of “one-sixth of 140.”



Student Four



Problem Solving: [4]: Correct use of fractions to determine areas for colours,and conditions all met.

Understanding Concepts: [4]: The concepts of area and fraction computationare correct and well presented.

Application of Procedures: [4]: The calculations of the number of squares foreach colour have been performed correctly, and the indicated correct numberof squares for each colour has been allocated on the diagram and in a veryinteresting design, which she points out is a “Bunny on unicycle with trainingwheels.”

Communication : [4]: Very well explained process.



Assessment Techniques

Observations

All teachers learn valuable information about their students every day. Ifteachers have a systematic way of gathering and recording this information, itwill allow the teacher to provide valuable feedback about student progress toboth students and parents. First, teachers must decide what they are lookingfor and which students will be observed today. Then the recording methodcan be decided. John A. Van de Walle’s Elementary and Middle SchoolMathematics; Teaching Developmentally, 4th Edition, provides ideas on pages72–74. For example:

Super

• clear understanding• communicates concepts in

multiple representations• shows evidence of using ideas

without prompting

On Target

• understands, or is developing well• uses designated models

Not Yet

• shows some confusion• misunderstands• models ideas only with help

Use the checklist for several days during the development of a topic or duringan activity and record the names of observed students and their achievement.

A generic observation checklist specifically noting what outcomes are beingachieved, and to what extent, as the learning develops may be useful toaccompany the above checklist.

Name: Grade:

Not Yet OK Super Comments

Topic: Concept

specifics from associated outcomeson the conceptual developmentrelated to the topic to be observed

Topic: Procedures

specifics from associated outcomeson the procedural aspects of thetopic to be observed



Specifically, the following checklist shows what this may look like at thegrade 7 level, when developing the topic of geometry.

Name: Grade:

Geometry: angles and triangles Not Yet OK Super Comments

• classifying triangles

• understands relationshipsbetween angle measure and sidelength

• construct angle bisector using avariety of methods

• construct perpendicularbisectors using a variety ofmethods

• find missing angle measuresusing angle pair relationships

• understand angle relationshipsfor two parallel lines intersectedby a transversal

• understand angle pairrelationships for two non-parallel lines intersected by atransversal

• with a model understands thatthe sum of the angle measuresof a triangle is 180°

Name: Grade:

Geometry (3-D) Not Yet OK Super Comments

• sketch 3-D objects using avariety of materials andinformation

• sketch various views of 3-Dobjects

• build 3-D models using a varietyof materials and information



Name: Grade:

Transformational Geometry Not Yet OK Super Comments

• Draw, describe, and applytranslations

• Draw, describe, and applyreflections

• Draw, describe, and applyrotations

• Uses a combination of thesetransformations

• Create and describe designsusing translations, rotationsand reflections

A focused checklist can be kept for each student during the development ofparticular topics.

Other examples can be seen in the NCTM booklet, Mathematics Assessment:Myths, Models, Good Questions, and Practical Suggestions, page 33.

Daily Observation Sheet

Name:

Date Activity Observed Behaviour Progress Suggestions

Conversations (Student Interviews)

Conversations and student interviews encompass all the formal and informaldialogue with a student that may provide more detail about themathematical development of the student. Information yielded by studentinterviews provides the teacher with a clearer direction for appropriate andtimely support or extensions with individual students. It also informsplanning for differentiated instruction and assessment in the classroom.Teachers should prepare and have a collection of student interview tasks.



With these, the teacher can conduct an interview with one student or a fewstudents at a time, while the class is working on a common task, on self-directed activities, or in centres. Teachers could collaborate in creating thesetasks within a family of schools. Creating these tasks is an excellent activity fora Professional Learning Community (PLC) or a Network LearningCommunity (NLC).

How is a formal interview created?

• Identify the key concept.• Identify aspects of the concept which may be problematic for the student.• Review the expected outcomes of the previous grade level to determine

the prerequisite skills needed to work with the current concept.• Create a task that will allow the student to demonstrate understanding of

each of the prerequisite skills for the concept.• Gather a variety of manipulative materials, including pencil and paper for

drawing, for use by the student.• Prepare an overall plan for the interview including key questions to ask

during the interview.• Document answers, observations and comments.

How is an Interview Conducted?

• Have all materials ready.• Arrange a comfortable working space.• Sit beside the student.• Explain the purpose of the assessment to the student.• Present one task, one item, or one question at a time.• Stick to the plan but be prepared to be flexible based on student

response.• Take brief notes of student responses.• Conclude the interview with a supportive comment.• Elaborate on notes immediately after the interview.

During the Interview

• Use neutral language (use responses that encourage but do not confirmthat something is right or wrong).

• Avoid cueing or leading the student to the right answer, withoutconsciously knowing it.

• Use imperatives rather than questions, using expressions such as “showme, tell me, tell me more, explain why you did this,” to elicit moreelaboration from the student.



• Ask the student about drawings, use of manipulatives, and explanations,to make connections between thinking and use of representations.

• Use “wait time” after presenting a task or a question, to allow for studentreflection and response. Wait again after the response. (This provides thestudent with thinking time and the teacher with time to think about thedirection of the interview.)

• Don't interrupt the student. Let thoughts flow freely and encourage themto use their own words and their own ways of writing things down.

• Thank the student at the end of the interview, but do not give excessivepraise.

After the Interview:

• Look for patterns (correct answers or errors) rather than isolated answers.• Make note of evidence of correct or incorrect learning.• Track terminology used correctly or incorrectly.• Record rules and generalizations correctly or incorrectly applied.• Note areas for a follow-up interview.

Examples of checklists can be seen in the NCTM booklet MathematicsAssessment: Myths, Models, Good Questions, and Practical Suggestions, page 33.

Student: Week:

Observation and/or Interview Notes

Tell me how you solved the problem.

Is there another way to solve the problem?

Where did you get stuck?



Journal/Log Entries

Writing in mathematics is a very important way to help students clarify theirthinking. Have students write explanations, justifications, and descriptions.

Here are some writing prompts for journals or logs:

• I think the answer is ...• I think this is so, because ...• Write an explanation for a student in a lower grade, or for a student who

was absent when this was taught.• What do I understand? What don’t I understand?• I got stuck today because ...• How do you know you are right?• Summarize concepts by drawing pictures of things that represent the

concept, and of things that do not represent the concept.

The best assessment of these writings would be to respond to the students’writing with the intention to develop a written conversation that might leadto clarifying misconceptions and to giving better explanations about conceptsor mathematical ideas being developed in the classroom, about how to studyor prepare for tests, and so on.

The teacher need only keep a checklist of who has submitted (and when)journal or log entries and to whom he or she has responded.

Self-Assessment

The NCTM booklet, Mathematics Assessment: Myths, Models, Good Questions,and Practical Suggestions says that “self-evaluation promotes metacognitionskills, ownership of learning, and independence of thought.”

When students assess their own learning, the teacher and the student will beable to see• signs of change and growth in attitudes, mathematical understandings,

and achievement• alignment of how well students are performing with how well they think

they are performing• a match between what the teacher thinks the student can do, and what the

student thinks the student can do

Some prompts include• How well do you think you understand this concept?• What do you believe ...? How do you feel about ...?



• How well do you think you are doing?• When you work with a group, what are your strengths? What are your

weaknesses?

Self-assessment checklists can be developed for each activity in which theteacher wants the students to perform self-assessment. What follows are somesuggested yes/no/not sure type questions from which more specific checklistscan be developed. These suggestions come from the NCTM bookletMathematics Assessment: Myths, Models, Good Questions, and PracticalSuggestions, put together by Jan Stenmark, University of California. See pages58 and 59 in that resource for other ideas.

1. Sometimes I don’t know what to do when I start a problem.2. I like mathematics because I can figure things out.3. The harder the problems, the better I like to work on them.4. I usually give up when a problem is really hard.5. I like the memorizing part of mathematics the best.6. There is more to mathematics than just getting the right answer.7. I think that mathematics is not really useful in everyday living.8. I would rather work alone than with a group.9. I like to do a lot of problems of the same kind rather than have different

kinds all mixed up.10. I enjoy mathematics.11. There’s always a best way to solve a problem.12. I liked mathematics when I was younger, but now it’s too hard.13. Put an “×” on this scale where you think you belong:

I am not good I am good atat mathematics. mathematics.


SECTION 4: TECHNOLOGY


Integrating Information Technologyand Mathematics

The public school program includes “technological competence” as one of theessential graduation learnings. This means that technological activities andfamiliarity with technology and its impact on society are considered essentialfor inclusion in every curricular area. The Nova Scotia Department ofEducation has developed a document entitled The Integration of Informationand Communication Technology within the Curriculum (2005). Thisdocument provides a foundation for the use of information technologieswithin classrooms in Nova Scotia.

Many other technological tools can be used as part of curricular activities tohelp satisfy the essential graduation learning requirement for “technologicalcompetence.” For the purposes of this section of the resource, however, onlyinformation technologies (computers, software, and Internet) will bediscussed.

When planning lessons for students, it is important to consider the key-stageoutcomes for the integration of technology. Where appropriate, lessons shouldinclude a technological approach or activities, as well as other approaches tofacilitate student learning.

The following information is excerpted from Nova Scotia Department ofEducation’s Public School Program and The Integration of Information andCommunication Technology within the Curriculum (2005).

Technological Competence—What Is It?

Graduates will be expected to use a variety of technologies, demonstrate anunderstanding of technological applications, and apply appropriatetechnologies for solving problems.

Graduates will be able, for example, to• locate, evaluate, adapt, create, and share information, using a variety of

sources and technologies• demonstrate understanding of and use existing and developing

technologies• demonstrate understanding of the impact of technology on society• demonstrate understanding of ethical issues related to the use of

technology in a local and global context



The Role of Technology—What Is It?

Just as computers and other technology play a central role in developing andapplying scientific knowledge, they can also facilitate the learning of science. Itfollows, therefore, that technology should have a major role in the teaching andlearning of mathematics.

Computers and related technology (data projectors, CD and DVD players,analog-digital interfaces, graphing calculators) have become valuable classroomtools for the acquisition, analysis, presentation, and communication of data inways that allow students to become more active participants in research andlearning. In the classroom, such technology offers the teacher more flexibility inpresentation, better management of instructional techniques, and easier recordkeeping. Computers and related technology offer students a very importantresource for learning the concepts and processes of mathematics throughsimulations, graphics, sound, data manipulation, and model building. Thesecapabilities can improve mathematical learning and facilitate communication ofideas and concepts. It should be remembered that, although there is anemphasis on the use of computers as technological tools, there is a wide varietyof other technological tools, which will also be of special benefit for amathematics classroom. The following guidelines are proposed for theimplementation of computers and related technology in the teaching andlearning of mathematics.

Tutorial software should engage students in meaningful interactive dialogueand creatively employ graphs, sound, and simulations to promote acquisition offacts and skills, promote concept learning, and enhance understanding.

Simulation software should provide opportunities to explore concepts andmodels that are not readily accessible in the classroom, e.g., those that require• expensive or unavailable materials or equipment• hazardous materials or procedures• levels of skills not yet achieved by the students• more time than is possible or appropriate in real-time classrooms

Analog-digital interface technology could be used to permit students to collectand analyze data as scientists and statisticians do, enabling real-worldmathematical experiences.

Databases and spreadsheets should be used to facilitate the analysis of data byorganizing and visually displaying information.

Networking among students and teachers should be encouraged to permitstudents to emulate the way mathematicians work and, for teachers, to reduceteacher isolation. Using tools such as the World Wide Web should beencouraged, as it provides instant access to an incredible wealth of informationon any imaginable topic.



Many software applications have a nearly universal use in classrooms asmethods of acquiring and demonstrating knowledge. Students may use word-processing applications to write about their experiences. They might usesoftware to create presentations as teaching tools for other students or as a wayof presenting knowledge they have gained. Concept-mapping software can beused to create flow charts, to visually show information, or to organize andplan other work. Students can take photographs and/or videos thatdemonstrate some aspect of their classroom learning and incorporate theseinto reports or create slide shows and videos of their activities. Specificsoftware applications that address specific outcomes for various grades areavailable through the Nova Scotia School Book Bureau.

In order to effectively implement computers and other technology inmathematics education, teachers should• know how to effectively and efficiently use the hardware, software, and

techniques described above• know how to incorporate computers and other technology into

instructional strategies• become familiar with the use of computer applications such as management

tools for grading and creating reports, inventories, and budgets• exemplify the ethical use of computers and software• seek to provide equitable access for all students

Information Technology—Why Use It?

When used properly, information technology (IT)• facilitates students’ communication, problem solving, decision making,

and expression• helps students to manipulate information so as to discover patterns and

relationships and construct meaning• helps students formulate, in the pursuit of knowledge, both conclusions

and complex questions for further research• permits students and teachers to access and share current learning content

and information more immediately• provides students with special needs fuller access to their learning

environment• allows Nova Scotians to develop and maintain competitive advantages in

the global information economy



Learning Outcomes Framework

There are five components to the learning outcomes framework for theintegration of IT within curriculum programs.

1. Basic Operations and Concepts

Concepts and skills associated with the safe, efficient operation of a range ofinformation technologies

2. Social, Ethical, and Human Issues

The understanding associated with the use of IT that encourages in studentsa commitment to pursue personal and social good, particularly to build andimprove their learning environments and to foster stronger relationships withtheir peers and others who support their learning

For further information, such as the key-stage outcomes for technologicalcompetence, refer to the Nova Scotia Department of Education TheIntegration of Information and Communication Technology within theCurriculum (2005).

3. Productivity

The efficient selection and use of IT to perform tasks such as• the exploration of ideas• data collection• data manipulation, including the discovery of patterns and relationships• problem solving• the representation of learning

4. Communications

The use of specific, interactive technologies that support collaboration andsharing through communication

5. Research, Problem Solving, and Decision Making

The organization, reasoning, and evaluation by which students rationalizetheir use of IT



Using Software Resources in Mathematics Classrooms

A wide variety of software titles is useful in all classroom areas, including themathematics classroom. These include word processors, databases, spreadsheets,graphics programs, presentation and/or multimedia production programs, videoproduction programs, concept-mapping applications, and Internet browsers.

Many aspects of the mathematics curriculum can make use of basic software suchas word-processing applications. Any written work, done either independently orin groups, could be produced on the computer, if this is appropriate. Much of thiswork could be done both at home and in school through the use of memory sticksor e-mailing the document. Data manipulation using tables and calculations andthe collection of information, both qualitative and quantitative, can be made moreefficient and effective through the use of databases and spreadsheets. Data chartscan easily be converted into graphs through spreadsheet applications.

There are several presentation and multimedia programs available to schools,either as part of the Information Economy Initiative (IEI) projects or through theAuthorized Learning Resources (ALR) list. These programs are excellent forallowing students to present information collected on a topic, to augment groupreports on work done and research gathered, or for many other activities withinthe mathematics classroom. Software programs such as PowerPoint, Comic Life,and KidPix Deluxe enable students to present their knowledge in creative andinnovative ways. If students are presenting their knowledge and understandingthrough skits and dramatic activities, the use of video-editing software (iMovie forApple computers or Pinnacle Studio for Windows computers) can greatly enhancetheir productions.

Concept-mapping software is an excellent method for collecting the informationshared by students in brainstorming activities, as well as a quick way for students todisplay information collected on topics done in class. Outlining, thought webbing,and other organizers are easily created and used through these types of software.One such software program that is available in most schools in Nova Scotia isInspiration.

Graphics programs can be used to create illustrations, cartoons, and diagrams.These can then be inserted into computerized reports, presentations, conceptmaps, etc. Very basic images can be created in software programs such as Paint (inthe Accessories on Windows computers). More sophisticated images can be madeusing the drawing tools of graphics programs such as Painter, Adobe Illustrator, orAdobe PhotoShop.

Specific software programs for different grades and aspects of the mathematics curriculum arelisted in Authorized Learning Resources. An annotated list of software titles is included atthe end of this section.



Grades 7–9 Mathematics Software Titles

• ArcView• Fathom• Geometer’s Sketchpad• Hot Dog Stand: The Works• Inspire Data• Let’s Do Math: Tools and Things• Mighty Math Cosmic Geometry• Secondary Mathlab Toolkit• Tessellation Exploration• Tinkerplots• Understanding Math Plus, including the following modules:

– Understanding Algebra Plus– Understanding Equations Plus– Understanding Exponents Plus– Understanding Fractions Plus– Understanding Graphing Plus– Understanding Integers Plus– Understanding Measurement and Geometry Plus– Understanding Percent Plus– Understanding Probability Plus

• Zoombinis Logical Journey• Zoombinis Mountain Rescue• Inspiration• word processor• spreadsheet• database• presentation software

Technological Tools for Mathematics

“Technology” does not equal “computers.” There are many other technologicaltools that will provide students with excellent learning opportunities inmathematics. Some recommendations for technological tools that are availableand can be used as part of curricular activities are listed below.



One per school (minimum)

• photocopier (activities such as congruency and scale)• video camera• still camera• VCR or DVD player and/or TV and videos (for a list of videos available

through the Nova Scotia Department of Education, seelrt.EDnet.ns.ca/media_library/catalogues/math.html

• computer and computer peripherals such as scanner, printer, and CDburner

One per classroom

• overhead projector• audio recorder/player for tape cassettes and CDs (if possible)

• microphone (for recording on the computer if possible)

One class set per classroom

• a basic calculator such as the TI-108 (grades P–3)• TI-15 with fraction capabilities (grades 4–8)• TI-34 Explorer Plus scientific (grades 4–8)• TI-73 graphic calculator with graphics (grades 7–9)• TI-84+ graphing calculator (grades 9–12)• TI-Nspire CAS (grades 9–12)

Several versions of graphing calculator software are also available forcomputers. Both Macintosh and Windows includes a calculator within theirsystem that can be viewed as either a standard or a scientific calculator.

Using Calculators in Mathematics Classrooms

Calculators cannot replace the human brain. Students will always need to beable to read and understand problems, write appropriate equations, chooseoperations to be used, interpret the solution presented, and determine theappropriateness of the answer. Students must provide the important andcomplex thought processes required to do mathematics. Calculators canprovide many varied instructional opportunities for teachers, allowingstudents to explore patterns and relationships. Calculators should not beused solely as an alternative to paper-and-pencil computation.

In early grades, the calculator can assist in helping the development ofstudents’ number sense. They can begin to carry out activities that allowstudents to construct mathematical relationships, especially in such areas asthe understanding of 10 and 100 as abstract units. Problem solving may be



more effective when students are allowed to use calculators to solve thearithmetical portions of a problem. They are more likely to use a calculator toperform a variety of exploratory computations that will help themunderstand the problem asked, while reducing the amount of time spent onpaper-and-pencil computation. Calculators could help ensure that time in amathematics classroom can be spent on concept development, problem-solving, mental arithmetic, and estimation activities.

As students move into the upper elementary grades, fractional calculators canbe very helpful. The use of a fraction-friendly calculator will cause the focus tobecome building meaning for fractions and mathematical relationships. Whenthis type of calculator is combined with traditional learning tools, it helpsstudents explore their world through investigation and experimentation andhelps develop skills in mathematical patterns and reasoning.

Graphing calculators are very useful in junior and senior high school classes.Linear and non-linear relationships can be easily understood when studentscan create visual images of them with a few keystrokes. Whether used inalgebra, trigonometry, or calculus, these tools require student understandingand skill in the use of algebraic logic for the entry of expressions, functionrules, equations, and inequalities. This means that students will requireinstruction in order to use these tools effectively. Students can spend theirtime concentrating on understanding, setting up, and choosing appropriateoperations and equations. This can open the door for students to begindiscovering topics and relationships on their own and to find alternative ideasand solutions for activities presented.

Using the Internet in Classrooms

The Internet is one of the most powerful tools available for the gathering ofcurrent data and research results. Through the Internet, students can accessmany resources that may not be easily available in the typical mathematicsclassroom. They may access online articles and reports of current research.They can find and correspond with mathematicians regarding their work.There are literally millions of mathematics sites on the Internet, but teachersmust be aware of the necessity for safety and critical examination of all sitesused. A good starting point for information about Internet basics is in the“Information for...” section in the left menu of EDnet.ns.ca. Follow theselinks from that menu.Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Search Engines

Sites used in schools should be evaluated, either by the teacher before thestudents use the site, or by the students as part of the learning activity, with



the teacher acting as a supervisor. Websites have been evaluated at a varietyof sites on the Internet, and an overview of many of these can be found atEDnet.ns.ca using “Information for...” in the left menu. Follow these links.Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Evaluating Web Sites

Safety

There are many important issues surrounding the safety and well-being ofstudents using the Internet. Predators “stalk” chat rooms and can also gleanlarge amounts of information about children through school and classroomweb pages. There are many inappropriate sites—pornographic and hate sitesbeing only two types. Many websites are produced by people who have specialinterests and present themselves as credible and reasonable, but may provideinformation that is biassed and/or incorrect. Websites addressing safety issues,as well as critical thinking about the Internet, can be found at EDnet.ns.causing the “Information for...” section in the left menu. Follow these links.Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Internet Safety

Critical Awareness

Much information on the Internet today is of the same character as thatprovided in supermarket tabloids: sensational, biassed, and inaccurate.Students must have the skills and awareness to recognize sites that purport tocontain serious information, but have unacceptable levels of unfoundedinformation in them. The ability to recognize unbiassed information and todistinguish it from material that is unscientific is an important skill forstudents. Information on this topic can be found at EDnet.ns.ca using the“Information for...” section in the left menu. Follow these links.Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Internet Safety

Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Evaluating Web Sites

Search Engines

When students search the Internet, they can spend over half their class timelooking for usable information on a topic. It is much more efficient for theteacher to do a search prior to the class period and provide the students withseveral web addresses (URLs) that will give them information to get started.



This will avoid the difficulty of students searching in such a general way thatmost of the sites returned in the search are not useful or may be completelyinappropriate. EDnet.ns.ca provides many links to information about websearches. Using the “Information for...” section in the left menu, follow theselinks:

Educators —> Classroom and Curriculum Resources —> Curriculum-RelatedWeb Sites —> Integration of Information Technology —> Search Engines

The most common search engines also provide tips on searching, which canassist the user in ensuring that his or her search is more focused. Every searchengine does the searching a little differently, and it is probably best, for thebeginner, to learn to use a common search engine, such as Google.The following example shows how you might search for information aboutriver tank aquariums.

Using just the word tank in the search box could give you over 140 000 000sites about Thomas The Tank Engine, storage tanks, fish tanks, armoredvehicles, septic tanks, stainless steel tanks, water tanks, and polyethylenetanks, to name a few.

A search for aquarium finds links for items such as Long Beach Aquarium,home aquarium, Monterey Bay aquarium, aquarium aish, aquarium supplies,fish tropical, and Baltimore Aquarium on over 45 000 000 sites.

Using the search engine to find the phrase river tank will narrow the search tolinks about topics such as: tanks, snapping turtle, rivers, alligator snappingturtle, river tank system, monitor fishtank, river tank squarium fish, and soon. You will still have over 1 000 000 sites in the list, however.

Search strings can also be used to narrow a search. A search string, typed intothe search engine, with one space after each word or phrase can be tailoredto a particular set of items. If a + is typed directly in front of a word or aphrase, most search engines will find only sites that contain that word andmay contain any words included without the +. Here is a sample searchstring that is likely to find information specifically about setting up, using,and purchasing a river tank aquarium.

+aquarium +“river tank” +fish

Many search engines allow more complex searches. The Advanced Searchfeatures of many of the popular search engines allow for Boolean searches,using AND, OR, AND NOT, or NEAR as instructions. Using parenthesesaround portions of a complex Boolean string can group items.



Many search engines also provide specific searches of a different type.Searches for pages that have certain items, links, images, text, or anchors canbe done using special search commands typed into the regular search box.

link:URLtext will find pages with a link to the page having the URL given.

link:www.EDnet.ns.ca finds pages with a link to any page on theDepartment of Education’s server.

An easy reference guide to the advanced features of Google can be found at:google.ca/advanced_search, and then select the “advanced Search Tips” linkon the page.

Alphabetical Software List for Mathematics, with Annotations

Title: ArcView

Grade levels: 3–12

ArcView permits students to interpret data, create maps, do presentations,and explore many real geographic information system databases. Fromcommunity studies in elementary grades to advanced problem solving usingenvironmental or original data, many courses can make use of this software toprovide experiences previously impossible. This is a professional-levelprogram that is easy to learn and very powerful. It encourages studentanalysis and critical thinking. Several Nova Scotia municipalities use GISdata, and students may have access to real data for their projects.

Title: Fatham


This software and accompanying materials support student understandingand accomplishment in data analysis and statistics. It contains sliders,function plots, and simulation tools. It supports dynamic dragging of datafrom sources such as web documents and allows the user to create multiplerepresentations of the data. Several sets of sample data provided with theprogram are based on U.S. Census information; this information is of limitedvalue for Nova Scotian classrooms. However, the program does allow datafrom other files and/or websites to be imported easily, including filesdownloaded from E-Stat and Statistics Canada. The program comes with asingle user Hybrid CD, quick reference guide, learning guide, referencemanual, three-ring binder, and blackline activity masters included. A book,“Data in Depth: Exploring Mathematics with Fathom” is sold separately for$39.95 (US). This resource appears to fit well with the outcomessurrounding GCO F in the senior high school math curriculum.



Title: The Geometer’s Sketchpad


This program can be used to construct a variety of figures. Once the figure isdrawn, students or teachers can transform it with the mouse while preservingthe geometric relationships of the construction. The geometric constructionslead to generalizations, as you see which aspects of geometry change andwhich stay the same. This program brings geometry to life. The user canconstruct geometric figures and measure all attributes of the figure. As afigure is modified, the measurements reflect changes in the size and shape,which allow sstudents to construct their own learning. It is useful as ademonstration tool as well as a hands-on student activity.

Title: Hot Dog Stand: The Works


Students are challenged with running a simulated small business. Theprogram provides an environment where students practise critical thinking,problem solving, and communication skills. It is an excellent vehicle for co-operative learning. Students take on the role of a vendor and are responsiblefor keeping records, determining prices, planning market strategies, makinginformed decisions based on their data analysis, and handling unexpectedevents. It provides tools for approximations, graphing, word processing, andsign creation. A calculator is also available. This is a good way to introducepractical entrepreneurial skills and to encourage independent learning.

Title: How the West Was 1 + 3 × 4

Grade levels: 4–9

This software provides a fun way for students to practise basic order ofoperations on integers. Students must choose operations to maximize theirresult and should also consider strategy: which answer gets them to their goalthe fastest way. The program allows for individual, competitive, and co-operative approaches. Given three randomly generated numbers, studentsconstruct arithmetic expressions using addition, subtraction, multiplication,division, and parentheses to advance a locomotive and a stagecoach in a racealong a number trail. Students must use problem-solving strategies to makethe biggest move, bump their opponent, take a shortcut, or jump to the nexttown.



Title: Inspiration

Grade levels: P–12

Inspiration is a concept mapping program. Concept mapping allowsstudents, teachers, or anyone planning a project, event, or experience, todraw organizing diagrams. These can be various geometric shapes or objectswith lines connecting as relational, logic, or time paths. Various shapes andstyles make this program adaptable to almost any level. This program is easyto use and well designed. It makes good use of colours, graphics, visualthinking and mapping skills. It encourages open-ended and co-operativelearning activities and could be used in any subject area. It can also be usedfor problem solving, data collection, presentations, review, and assessment.Canadian Symbols are provided for use with this software.

Title: InspireData


InspireData comes from the same company that produces Inspiration. Itprovides a similar visual approach to data management activities. Studentscan collect and explore data in a dynamic environment, displaying theirinformation in tables, Venn diagrams, pie charts, stack plots, and scatterplots. The software provides an opportunity to create questionnaires, with thedata being automatically added to the file. Slide shows to present results andfindings can also be created through this tool.

Title: Let’s Do Math: Tools and Things


This program provides a variety of interactive tools for learning mathematics.It also contains a wide-ranging and well-organized database of approximately900 terms and definitions. Although it is a searchable math dictionary, it usesa hyperlink format to allow individual exploration and has many interactiveexamples and activities. It is very easy to use with a good format and excellentdiagrams.



Title: Mighty Math Cosmic Geometry


This program teaches basic geometry concepts and problem-solving skills. Itcontains sections on length, perimeter and area, surface area and volume,attributes of shapes and solids, constructions, transformations, and two-dimensional and three-dimensional coordinates, with many levels of difficulty.The sections on tessellations and transformations are very good. It is easy tonavigate, especially for those with computer game experience. There are manyactivities to familiarize the students with geometry applications: a three-dimensional maze, assembling robots, creating tessellating patterns, makingmovies, etc.

Title: Secondary Math Lab Toolkit


This program contains sections on algebra, geometry, and advanced algebra.This toolkit-style program is both powerful and flexible. Some of the features ofthe program include a symbolic editor, assignment editor, coordinate graphwindow, algebra tiles, solver, calculator, matrix calculator, graphing calculator,probability editor, and geometric figure generator. The symbol editor is easy touse. The spreadsheet will calculate values from linked equations. The linking tothe graph maker is a very smooth process. Clicking on the graph of a functionand moving it about causes the linked equation to instantaneously update,reflecting the changes in the graphed figure. Most secondary schools are likelyto own copies of this title, as it was a title recommended for purchase duringthe IEI project. This program matches well with all motion geometry outcomesin GCO E at grade 4, 5, 6, and 7 levels. The teacher’s binder is very helpful,but teachers should also refer to curriculum guides as well as other resourcesthatare available, such as blackline masters and reference books.

Title: Tessellation Exploration

Grade levels: 3–8

Students can view, explore, and create tessellations. They can choose the type oftransformation to use, the shape of the tile, and then make modifications to theshape. They can add “stamps” to make their creations into identifiable images.Slide shows can be made; tessellations can be imported as backgrounds orexported as images. Students can also investigate different types oftransformations and test themselves on their understanding as to the changesthey would see in an image if different transformations were carried out on it.This program matches well with all motion geometry outcomes in GCO E atgrade 4, 5, 6, and 7 levels. The teacher’s binder is very helpful, but teachersshould also refer to curriculum guides as well as other resources that areavailable, such as blackline masters and reference books.



Title: Tinkerplots


TinkerPlots is a younger version of Fathom. It is a data-management andstatistical-analysis tool suitable for students in grades 4 to 9. It can be used inseveral discipline areas, including mathematics, science, and social studies.

The program provides sample data and activities for students, but they canalso collect their own data and input it for analysis. It allows varying sorts,displays, and queries for the data. External data can be easily imported intothe program. This software provides students with a tool that allows them toask questions and investigate information in many different ways.

Title: Understanding Math

Grade levels: 4–12 (varies by title)

This series contains sections on concepts with graphic explanations, examples,interactive practice, and cumulative checks. The series is a comprehensiveCanadian program in a tutorial format. It covers most of the math topics inthe junior high curriculum. With good use of colour and animation,concepts are developed and checked as the lessons proceed. This would beespecially useful for students who have missed time or who did not grasp aconcept in class.

• Understanding Algebra 6–12• Understanding Equations 7–12• Understanding Exponents 6–12• Understanding Fractions 4–12• Understanding Graphing 6–12• Understanding Integers 6–12• Understanding Measurement and Geometry 5–12• Understanding Percent 5–12• Understanding Probability 5–12



Title: Zoombinis Logical Journey

Grade levels: 3–8

This program helps students explore and apply fundamental principles oflogic, problem solving, and data analysis. The 12 puzzles, each with fourlevels of difficulty, work together to support growth in mathematicalconcepts and thinking skills. It encourages and strengthens creative problem-solving skills. The puzzles are open-ended, so there is more than one path to asolution. The teacher should be available to give guidance to students if theyneed it. This program does not address specific curriculum outcomes, butdoes provide a great deal of development in the underlying skills of logicalreasoning, patterning, compare/contrast, classifying, predicting/confirming,and cause/effect. A Practice mode is available to enable teachers and studentsto access levels and difficulties they may not yet have reached in the gameformat.

Title: Zoombinis Mountain Rescue

Grade Levels: 3–8

This program is the second CD about the Zoombinis and their activities. Itprovides a new set of challenges for students. This program stands alone, andstudents do not need to be familiar with Logical Journey in order to be ableto use this software. It helps students explore and apply fundamentalprinciples of logic, problem solving, and data analysis. The puzzles, each withthree levels of difficulty, work together to support growth in mathematicalconcepts and thinking skills. It encourages and strengthens creative problem-solving skills. The puzzles are open-ended, so there is more than one path to asolution. The teacher should be available to give guidance to students if theyneed it. In spite of the challenge of some of the problems, the feedback isappropriate, and many keep at the problems until solved.


APPENDICES

APPENDICES


APPENDICES

Appendix A: SCOs by StrandGCO Grade 7 SCOs



A2 rename numbers among exponential, standard, and expandedforms


A4 solve and create problems involving common factors andgreatest common factors (GCF)



A7 apply patterning in renaming numbers from fractions and mixednumbers to decimal numbers



A10 illustrate, explain, and express ratios, fractions, decimals, andpercentages in alternative forms


A12 represent integers (including zero) concretely, pictorially, andsymbolically, using a variety of models





B4 determine and use the most appropriate computational methodin problem situations involving whole numbers and/or decimals


B6 estimate the sum or difference of fractions when appropriate


B8 estimate and determine percentage when given the part and thewhole






APPENDICES

GCO Grade 7 SCOs


B: Operation Sense B13 divide integers concretely, pictorially, and symbolically to solveproblems

B14 solve and pose problems that utilize addition, subtraction,multiplication, and division of integers


B16 create and evaluate simple variable expressions by recognizingthat the four operations apply in the same way as they do fornumerical expressions



C: Patterns and Relations C1 describe a pattern, using written and spoken language andtables and graphs









D: Measurement D1 identify, use, and convert among the SI units to measure,estimate, and solve problems that relate to length, area, volume,mass, and capacity



D4 construct and analyze graphs of rates to show how change in onequantity affects a related quantity

D5 demonstrate an understanding of the relationships amongdiameter, radii, and circumference of circles and use therelationships to solve problems






APPENDICES

GCO Grade 7 SCOs


E: Geometry E5 construct polyhedra using one type of regular polygonal face, anddescribe and name the resulting Platonic Solids


E7 make and apply generalizations about angle relationships

E8 make and apply generalizations about the commutativity oftransformations


E10 make informal deductions about the minimum sufficientconditions to guarantee that a given quadrilateral is of a particulartype, and to understand formal definitions of the various membersof the quadrilateral family













G4 create and solve problems, using the numerical definition ofprobability


G6 use fractions, decimals, and percentages as numericalexpressions to describe probability

1

41

2

3

4


APPENDICES

Appendix B: The Process Standards

Communication Standard

Instructional programs from pre-primary through grade 12 should enable allstudents to organize and consolidate their mathematical thinking throughcommunication; communicate their mathematical thinking coherently andclearly to peers, teachers, and others; analyze and evaluate the mathematicalthinking and strategies of others; and use the language of mathematics toexpress mathematical ideas precisely.

Problem-Solving Standard

Instructional programs from pre-primary through grade 12 should enable allstudents to build new mathematical knowledge through problem solving;solve problems that arise in mathematics and in other contexts; apply andadapt a variety of appropriate strategies to solve problems; and monitor andreflect on the process of mathematical problem solving.

Connections Standard

Instructional programs from pre-primary through grade 12 should enable allstudents to recognize and use connections among mathematical ideas;understand how mathematical ideas interconnect and build on one another toproduce a coherent whole; and recognize and apply mathematics in contextsoutside of mathematics.

Reasoning and Proof Standard

Instructional programs from pre-primary through grade 12 should enable allstudents to recognize reasoning and proof as fundamental aspects ofmathematics; make and investigate mathematics conjectures; develop andevaluate mathematical arguments and proofs; and select and use various typesof reasoning and methods of proof.

Representation Standard

Instructional programs from pre-primary through grade 12 should enable allstudents to create and use representations to organize, record, andcommunicate mathematical ideas; select, apply, and translate amongmathematical representations to solve problems; and use representations tomodel and interpret physical, social, and mathematical phenomena.


APPENDICES

Appendix C: Large Frayer Model


APPENDICES

Appendix D: The Concept Definition Map

ComparComparComparComparComparisonsisonsisonsisonsisons What It Is?What It Is?What It Is?What It Is?What It Is? PrPrPrPrProperoperoperoperopertiestiestiestiesties

IllusIllusIllusIllusIllustrtrtrtrtrationsationsationsationsationsWhat Are Some Examples?


APPENDICES

Appendix E: What Is Mental Math?

Mental math is the mental manipulation of knowledge dealing withnumbers, shapes, and patterns to solve problems. Both estimation and mentalcomputation (often referred to as “mental math”) are done “in one’s head”and have been deemed important skills by the National Council of Teachersof Mathematics (NCTM). Computing mentally to find approximate or exactanswers is frequently the most appropriate and useful of all methods.

Mental computation, estimation, pencil and paper, the use of manipulatives,computers, and calculators are all tools that are used to solve problems.Whenever a problem situation is encountered in mathematics, the firstdecision to make is whether it requires an exact or an approximate answer. Ifan approximate result is sufficient, an estimate can usually be arrived atmentally. If an exact answer is needed, an estimate could be obtained prior toformal calculations.


APPENDICES

Teacher and student attention to mental computational estimation strategieshelps students to develop number and operation sense, check reasonablenessof solutions, recognize errors on calculator displays, and develop confidencewhen dealing with numbers. Students might need to estimate beforecalculating in order to judge the reasonableness of results obtained using othermethods. A thorough understanding of, and facility with, mentalcomputation also allows students to solve complicated multi-step problemswithout spending needless time figuring out calculations and is a valuableprerequisite for proficiency with algebra.

Estimation and mental math are not topics that can be isolated as a unit ofinstruction; they must be integrated throughout the study of mathematics.There are specific strategies and algorithms for mental computation that mustbe taught and practised regularly at the appropriate grade level. It is importantto remember that• skills in mental math take time to develop• mental math is best developed in context• a variety of strategies for one type of computation should be generated and

shared by students• students need to identify why particular procedures work; they should not

be taught computational tricks without understanding• students should be able to transfer, relate, and apply mental strategies to

paper-and-pencil tasks• helping students to make connections between the representations will

help them develop mental images of shapes, patterns, and numbers andwill assist them in solving problems

Mental computation strategies, both for exact and approximate answers, areoften the most useful of all strategies; therefore they should be frequentlyrevisited throughout the year. Every student of mathematics grades 1–9 isexpected to engage in five minutes of mental math each day. These dailyactivities are part of a well-thought-out, coherent plan. As each additionalstrategy is introduced, enough practice must be given to allow for proficiencyin the new strategy as well as all previous strategies. Strategies need to becarefully developed. Attention must be paid to instruction that promoteunderstanding, reinforcements that allow students to internalize theirthinking, and assessments that determine if the skill has been acquired.


APPENDICES

Appendix F: Recommended Resource List—Grade 7

GCO A: Students will demonstrate number sense and apply number-theoryconcepts.

• Complete Book of Cube-a-Link, The, Revised Edition• Developing Number Sense in the Middle Grades: Addenda Series, Grades 5–8• Fraction Factory Binder, The• Fraction Factory Games and Puzzles Binder, The• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illuminations (illuminations.nctm.org/index2.html)• Mental Math in Junior High• Principles and Standards for School Mathematics• Teaching Student-Centered matematics, Grades 5–8, Volume 3• Understanding Mathematics, Mac/Win CD (complete series)• Understanding Rational Numbers and Proportions: Addenda Series, Grades

5–8

GCO B: Students will demonstrate operation sense and apply operationprinciples and procedures in both numeric and algebraic situations.

• Alge-Tiles Resource Binder• Complete Book of Cube-a-Link, The, revised edition• Developing Number Sense in the Middle Grades: Addenda Series, Grades 5–8• Fraction Factory Binder, The• Fraction Factory Games and Puzzles Binder, The• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illuminations (illuminations.nctm.org/index2.html)• Mental Math in Junior High• Principles and Standards for School Mathematics• Teaching Student-Centered matematics, Grades 5–8, Volume 3• Understanding Mathematics, Mac/Win CD (complete series)• Understanding Rational Numbers and Proportions: Addenda Series, Grades

5–8


APPENDICES

GCO C: Students will explore, recognize, represent, and apply patterns andrelationships, both informally and formally.

• Alge-Tiles Resource Binder• Complete Book of Cube-a-Link, The, revised edition• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illuminations (illuminations.nctm.org/index2.html)• Patterns and Functions: Addenda Series, Grades 5–8• Principles and Standards for School Mathematics• Understanding Mathematics, Mac/Win CD (complete series)

GCO D: Students will demonstrate an understanding of, and apply conceptsand skills associated with, measurement.

• Geometry in the Middle Grades: Addenda Series, Grades 5–8• Measurement in the Middle Grades: Addenda Series, Grades 5–8• Principles and Standards for School Mathematics• Geoboard Collection: An Adventure in Mathematics, The, Intermediate Level

7-9• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illumination (illuminations.nctm.org/index2.html)• Understanding Mathematics, Mac/Win CD (complete series)

GCO E: Students will demonstrate spatial sense and apply geometricconcepts, properties, and relationships.

• Complete Book of Cube-a-Link, The, revised edition• Geometry in the Middle Grades: Addenda Series, Grades 5–8• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illumination (illuminations.nctm.org/index2.html)• Principles and Standards for School Mathematics• Understanding Mathematics, Mac/Win CD (complete series)


APPENDICES

GCO F: Students will solve problems involving the collection, display, andanalysis of data.

• Dealing with Data and Chance: Addenda Series, Grades 5–8• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illumination (illuminations.nctm.org/index2.html)• Principles and Standards for School Mathematics• Triple “A” Mathematics Program, Data Management and Probability• Understanding Mathematics, Mac/Win CD (complete series)• Understanding Rational Numbers and Proportions: Addenda Series,

Grades 5–8

GCO G: Students will represent and solve problems involving uncertainty.

• Dealing with Data and Chance: Addenda Series, Grades 5–8• Good Questions: Great Ways to Differentiate Mathematics Instruction• Halifax Regional School Board (www.hrsb.ns.ca/program/)• NCTM: Illumination (illuminations.nctm.org/index2.html)• Principles and Standards for School Mathematics• Triple “A” Mathematics Program, Data Management and Probability• Understanding Mathematics, Mac/Win CD (complete series)


APPENDICES

Appendix G: Mathematics ResourcesMaterials can be ordered through the Nova Scotia School Book Bureau on the Website https://w3apps.ednet.ns.ca/nssbb/default.asp

Every student should have

Mathematics 7: Focus on Understanding—student text

graph paper

protractor

pencil

coloured pencils or markers

Every teacher must have

Atlantic Canada Mathematics Curriculum: Grade 7 (Department of Education)

Mathematics 7: Focus on Understanding—student text

Mathematics 7: Focus on Understanding—teacher resource

Mathematics 7: A Teaching Resource (Department of Education)

Foundation for Atlantic Canada Mathematics Curriculum (Department of Education)

Toward a Coherent Mathematics Program: A Study Document for Educators(Department of Education)

Every classroom should have

It is necessary that teachers have use of the following materials to deliver the grade 7mathematics curriculum:

10 × 10 grids

graph paper

number line

overhead projector

Algebra tiles Class Set

Algebra tiles Overhead Set

Algebra tiles Resource Binder

Base-10 Beginners Set

Base-10 Overhead Blocks

Fraction Factory Pieces: Overhead

The Fraction Factory Binder

The Fraction Factory Pieces (15 sets in 5 containers)

Geometry Drawing Tools (Bulls-Eye Compass)

Mental Math in the Junior High Grades

Overhead Two-Coloured Counters

Pattern Blocks (wooden, set of 250)

Pattern Blocks: Overhead

Polydrons

Two Colour Counters (200) ( 6 class sets)


APPENDICES

Every classroom should have

It is recommended that teachers have use of the following materials to deliver thegrade 7 mathematics curriculum:

The Complete Book of Cube-A-Link, Revised Edition

Cube-A-Link

Fraction Block Binder

Geoblocks

Geoboard: Overhead

Geometric Models

Geo-Strips

measuring tapes

Pattern Blocks Binder for Intermediate Grades (6–8)

Pattern Block Template

Teaching Student-Centered Mathematics, Grade 5–8, Volume 3

Trundle Wheel

Every school should have

Assessment Standards for School Mathematics

class sets of scientific calculators

Decimal Squares: Teacher’s Guide and Manipulatives

Exploring Geoblocks

Geoboard - class set

Geoboard Collection: An Adventure in Mathematics

Exploring Geometry with The Geometer Sketchpad

Exploring Algebra with The Geometer Sketchpad

Mathematics Assessment: Myths, Models, Good Questions and Practical Suggestions

Math-Vu Mirror

NCTM 1998 Yearbook: The Teacher and Learning of Algorithms in School Mathematics

NCTM 1999 Yearbook: Developing Mathematical Reasoning in Grades K–12

NCTM 2000 Yearbook: Learning Mathematics for a New Century

NCTM 2001 Yearbook: The roles of Representations in School Mathematics

NCTM 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions

NCTM 2003 Yearbook: Learning and Teaching Measurement

Number Sense, Grades 6–8

Principles and Standards for School Mathematics

Pythagoras Plugged In, Proofs and Problems for The Geometer Sketchpad

Tangram Pieces (1 set) (class set)

The Geometer’s Sketchpad, 50 Disk Lab Pac

The Geometer’s Sketchpad, site/network package

The Intermediate Geoboard

The Link-A-Board (11 × 11 pin)

Tessellation Exploration

24 Game: (Double Digits) Add, Subtract, Multiply, and Divide


APPENDICES

Other Nova Scotia School Book Bureau Titles

101 Mathematical Projects

20 Thinking Questions for Base Ten Blocks, Grades 6–8

20 Thinking Questions for Fraction Circles, Grades 6–8

20 Thinking Questions for Geoboards, Grades 6–8

20 Thinking Questions for Pattern Blocks, Grades 6–8

20 Thinking Questions for Rainbow Cubes, Grades 6–8

A Collection of Math Lessons, Grades 6 through 8

About Teaching Mathematics

Actions with Fractions

Active Achievements, Grade 7

Addenda, Grades 5–8: Geometry in the Middle Grades

Addenda, Grades 5–8: Measurement in the Middle Grades

Addenda, Grades 5–8: Patterns and Functions

Addenda, Grades 5–8: Dealing with Data and Change

Constructing Ideas About Data Analysis, Grades 6–8

Constructing Ideas About Geometry, Grades 6–8

Constructing Ideas About Fractions, Decimals and Percent Grades 6–8

Constructing Ideas About, Complete Set, Grades 6–8

Elementary and Middle School Mathematics: Teaching Developmentally, 4th Edition

Explain It!: Answering Extended-Response Math problems, Grades 7–8

Exploring With Power Polygons

Gage Mathematics Assessment, Grade 7

Gage Mathematics Assessment, Grade 7, Teacher Edition

Harcourt Math Assessment: Measuring Student Performance, Grade 7

In the Balance, Algebra Logic Puzzles, Grade 7–9

Investigate and Discover Geometry, Shapes, and Measurement Using the Math-vuMirror, Intermediate (Grades 7–9)

Mathematics ... A Way of Thinking

M-CUBED Motivating Mathematics with Manipulatives

Navigating through Algebra in Grades 6–8

Navigating through Data Analysis in Grades 6–8

Navigating through Geometry in Grades 6–8

Niktu: A Game of Algebraic Reasoning Level 5 Kit (Grade 7)

Proportional Reasoning

Relational Geosolids

Remediation with a Difference

Textile Math: Multicultural Explorations through Patterns, Grades 6–8

TI-73 for the Viewscreen, Model 73VSC/CBX

TI-73 (10 pack)

TI-73 and Viewscreen, Model 73VS11/CBX

Writing in Math Class: A Resource for Grades 3–8

Zoombinis Logical Journey

Zoombinis Mountain Rescue EEV


APPENDICES

Appendix H: Outcomes Framework to InformProfessional Development for MathematicsTeachers

To be effective, professional development experiences for teachers must reflectand respond to their individual learning needs and build on their priorknowledge and experiences. It is recognized that mathematics teachers will be atdifferent stages on the continuum of professional development required toexpand their knowledge base and extend their repertoire of effective practices. Itis also recognized that effective professional development is ongoing and requireslong-range planning and long-term commitment. Substantial commitment isneeded to ongoing professional development to ensure that educational practicesreflect the requirements of the Atlantic Canada mathematics curriculum.

The following outcomes framework is intended to assist administrators, schoolstaff, and individual teachers in planning and designing appropriateprofessional growth experiences, for themselves and for others, over time and ina range of contexts. It provides the “big picture” for professional developmentin mathematics education and reference points that may be helpful indetermining the focus of professional development sessions and identifyingindividual teachers’ learning needs for future professional growth opportunities.

Mathematics teachers need to have an understanding of concepts and strategiesand the ways in which they are developed throughout the strands that make upthe mathematics curriculum.

To do so, teachers need to• know the curriculum outcomes prescribed for the grade level(s) they teach• know the progression and development of outcomes before, during, and after

the grade(s) they teach• broaden and clarify their mathematical understandings of number concepts

and operation, patterns and relation, measurement and geometry, and datamanagement and probability

• broaden and clarify their mathematical understandings of the strategiesstudents need to solve problems in number concepts and operations, patternsand relations, measurement and geometry, and data management andprobability

• be aware of current research in mathematics education


APPENDICES

Mathematics teachers need to have an understanding of the unifying ideas inmathematics and the ways they permeate the strands that make up themathematics curriculum.

To do so, teachers need to• model and emphasize aspects of problem solving, including formulating and

posing problems• model and emphasize mathematical communication using written, oral, and

visual forms• demonstrate and emphasize the role of mathematical reasoning• represent mathematics as a network of interconnected concepts and strategies• emphasize connections between mathematics and other disciplines and

between mathematics and real-world situations• model and value the five ways to represent a concept: contextually, concretely,

pictorially, verbally, and symbolically

Mathematics teachers are expected to organize instruction to facilitate students’understanding and growth in mathematics.

To do so, teachers need to• have a complete understanding of the pedagogy that is embedded throughout

the mathematics curriculum• use a repertoire of varied teaching strategies• know how to access and utilize a range of resources to address diverse learning

needs• know how to design and select worthwhile tasks to support the mathematics

curriculum• have knowledge of the role of discourse in the mathematics classroom• know how to group students effectively based on classroom dynamics and the

nature of an activity• know scope and sequencing for concept development, as well as an

appropriate time frame to support this development• be aware of current research regarding the ways students learn mathematics• model and teach literacy skills relevant to mathematics• value all ways of knowing and doing mathematics• be able to work collaboratively to develop, implement, monitor, and evaluate

programming for students with special needs• be able to support teacher assistants working with students• reflect on their own practice


APPENDICES

Mathematics teachers are expected to assess the mathematical development oftheir students in relation to the specific curriculum outcomes (SCOs)

To do so, teachers need to• understand the different purposes of assessment• know when to use different assessment tools to determine a student’s

mathematical achievement in relation to the curriculum• determine the cognitive levels indicated in an outcome and design assessment

tasks that appropriately address that outcome• know how to use assessment information to plan instruction and to challenge

or support students as appropriate• know how to record, organize, track, as well as communicate and report

assessment information• analyze and reflect upon a variety of information to make a professional

judgment about student progress

Mathematics teachers need to have an understanding of how an appropriatelearning environment supports the learning process.

To do so, teachers need to• provide and structure the time necessary for students to explore mathematics

and engage with significant ideas• use physical space and materials in ways that facilitate students’ learning of

mathematics• provide contexts that encourage the development of mathematical skill and

proficiency• respect and value students’ ideas, ways of thinking, and mathematical dispositions• expect and encourage students to work independently and collaboratively to

make sense of mathematics• expect and encourage students to take learning risks• expect and encourage students to display mathematical competency• extend opportunities for mathematics learning beyond the classroom to home

and community

Mathematics teachers are expected to promote a positive mathematicaldisposition in their students.

To do so, teachers need to• model a positive disposition to do mathematics• demonstrate the value of mathematics as a way of thinking• demonstrate the applications of mathematics in other disciplines, in the

home, community, and workplace• promote students’ confidence, flexibility, perseverance, curiosity, and

inventiveness in doing mathematics through the use of appropriate tasks• be lifelong learners of mathematics


APPENDICES

Appendix I: Outcomes Framework to InformProfessional Development for Teachers of Readingin the Content Areas

To be effective, professional development experiences for teachers, like alllearning experiences, must reflect and respond to their individual learningneeds, building on their prior knowledge and experiences. It is recognized thatcontent area teachers will be at different stages on the continuum ofprofessional development required to expand their knowledge base and extendtheir repertoire of effective practices and strategies that support students’development as readers. It is also recognized that effective professionaldevelopment is ongoing and requires long-term commitment and long-rangeplanning.

The following outcomes framework is intended to assist leadership teams,administrators, school staffs, and individual teachers in planning anddesigning appropriate professional growth experiences, for themselves and forothers, over time and in a range of contexts. It provides the “big picture” forprofessional development in reading education and reference points that maybe helpful in determining the focus of in-service sessions and identifyinglearning needs for future professional development opportunities for contentteachers.

Content area teachers are expected to teach students to read for meaning.

To do this content area teachers need to know• the features, structure, and patterns of various information texts and the

corresponding demands on the reader• the integration of various sets of knowledge that contribute to meaning

making including– personal experience– literary knowledge– world knowledge

• the reading strategies of sampling, predicting, and confirming/self-correcting that proficient readers use as they identify words andcomprehend text


APPENDICES

Content area teachers are expected to assess the reading development oftheir students in order to plan effective instruction for them.

To do this content area teachers need to know• the reading behaviours that characterize emergent, early, transitional, and

fluent stages of development so that they can monitor students’ growth asreaders

• effective strategies for gathering information about students’ understandingof concepts and ideas relevant to the subject area curriculum and theinfluence reading has on their level/degree of understanding

• planning strategies for whole-class, small-group, and individual instructionon the basis of the assessment of learners’ reading development

Content area teachers are expected to organize instruction to facilitatestudents’ reading growth.

To do this content area teachers need to• assist readers of informational text throughout the reading process by

supporting comprehension at all stages of reading– before (activating)– during (acquiring)– after (applying)

• model and teach literacy skills relevant to specific curriculum areas• know classroom management and organizational strategies to facilitate

whole-class, small-group, and individual instruction• collaborate effectively with others who share responsibility for supporting

students’ reading development


APPENDICESBIBLIOGRAPHY

BIBLIOGRAPHY

Barton, M. L., and Heidema, C. Teaching Reading in Mathematics 2nd

Edition. Alexandria, VA: McRel, 2002.

Black, Paul, Dylan William. “Inside the Black Box,” Phi Delta Kappan, Oct.98, Vol. 80, Issue 2: 139–147.

Buehl, Doug. Classroom Strategies for Interactive Learning. Newark, DE:International Reading Assoc. Inc., 2001.

Calabrese, Beth, Nancy Fournier, Jacob Speijer. Mathematics 7: Focus onUnderstanding. Whitby, ON. McGraw-Hill Ryerson Ltd., 2005.

Equals and the Assessment Committee of the California MathematicsCouncil, Campaign for Mathematics. Assessment Alternatives inMathematics: An Overview of Assessment Techniques That PromoteLearning. Berkley, CA: University of California, 1989.

Hope, Jack A., Barbara J. Reys, and Robert E. Reys. Mental Math in JuniorHigh. Palo Alto, CA: Dale Seymour Publications, 1998.

National Council of Teachers of Mathematics. Dealing With Data andChance. Addenda Series, Grades 5–8. Reston, VA: NCTM, 1991.

———. Mathematics Assessment: Myths, Models, Good Questions, andPractical Suggestions. Reston, VA: NCTM, 1991.

———. Patterns and Functions. Addenda Series, Grades 5–8. Reston, VA:NCTM, 1991.

———. Principles and Standards for School Mathematics. Reston, VA:NCTM, 2000.

Nova Scotia Department of Education. Atlantic Canada MathematicsCurriculum: Grade 7. Halifax, NS: Province of Nova Scotia, 2000.

———. Atlantic Canada English Language Arts Curriculum Guide: Grades7–9. Halifax, NS: Province of Nova Scotia, 1998.

———. The Integration of Information and Communication Technologywithin the Curriculum. Halifax, NS: Nova Scotia Department ofEducation, 2005.

———. Secondary Science: A Teaching Resource. Halifax, NS: Province ofNova Scotia, 1999.

———. Toward a Coherent Mathematics Program: A Study Document for

Van de Walle, John A. Teaching Student-Centered Mathematics, Grades 5–8,Volume 3. Toronto, ON: Pearson Education Canada Inc., 2006.

———. Elementary and Middle School Mathematics: TeachingDevelopmentally, 4th Edition. New York, NY: Longman, 2000.

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