Connected Mathematics 2 - Math For Alabamamathforalabama.com/media/pdf/CMP2_Joint_Usage.pdf ·...

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A Winning Combination Connected Mathematics 2 and Prentice Hall Mathematics ©2010 Course 1, 2, 3 Lappan, Fey, Fitzgerald, Friel, Phillips Grade Six Lappan, Fey, Fitzgerald, Friel, Phillips Grade Seven Lappan, Fey, Fitzgerald, Friel, Phillips Grade Eight

Transcript of Connected Mathematics 2 - Math For Alabamamathforalabama.com/media/pdf/CMP2_Joint_Usage.pdf ·...

A Winning Combination

Connected Mathematics 2and

Prentice Hall Mathematics ©2010Course 1, 2, 3

Lappan, Fey, Fitzgerald, Friel, Phillips

Grade Six

GRA

DE SIX

Prime TimeFactors and Multiples

Bits and Pieces IUnderstanding Fractions, Decimals, and Percents

Shapes and DesignsTwo-Dimensional Geometry

Bits and Pieces IIUsing Fraction Operations

Covering and SurroundingTwo-Dimensional Measurement

Bits and Pieces IIIComputing With Decimals and Percents

How Likely Is It?Understanding Probability

Data About UsStatistics

Grade Six Units

Classroom tested, proven effective!Before work began on CMP2, mathematics teachers in morethan 18 school districts—that's over 80 teachers—reviewedConnected Mathematics. More than 100 classroom teachers tried out Connected Mathematics 2 at 49 schools all across thecountry. This classroom testing allowed the authors to carefullystudy and revise the program to make sure the materials helpmath students like you every day, in every classroom.

Lappan, Fey, Fitzgerald, Friel, Phillips

GRA

DE SEV

EN

Grade Seven

Grade Seven Units

Classroom tested, proven effective!Before work began on CMP2, mathematics teachers in more than18 school districts—that's over 80 teachers—reviewed ConnectedMathematics. More than 100 classroom teachers tried outConnected Mathematics 2 at 49 schools all across the country. Thisclassroom testing allowed the authors to carefully study and revisethe program to make sure the materials help math students likeyou every day, in every classroom.

Variables and PatternsIntroducing Algebra

Stretching and ShrinkingUnderstanding Similarity

Comparing and ScalingRatio, Proportion, and Percent

Accentuate the NegativeIntegers and Rational Numbers

Moving Straight AheadLinear Relationships

Filling and WrappingThree-Dimensional Measurement

What Do You Expect?Probability and Expected Value

Data DistributionsDescribing Variability and Comparing Groups

Lappan, Fey, Fitzgerald, Friel, Phillips

GRA

DE EIG

HT

Grade Eight

Thinking With Mathematical ModelsLinear and Inverse Variation

Looking for PythagorasThe Pythagorean Theorem

Growing, Growing,GrowingExponential Relationships

Frogs, Fleas, and Painted CubesQuadratic Relationships

Kaleidoscopes, Hubcaps,and MirrorsSymmetry and Transformations

Say It With SymbolsMaking Sense of Symbols

The Shapes of AlgebraLinear Systems and Inequalities

Samples and PopulationsData and Statistics

Grade Eight Units

Classroom tested, proven effective!Before work began on CMP2, mathematics teachers in more than18 school districts—that's over 80 teachers—reviewed ConnectedMathematics. More than 100 classroom teachers tried outConnected Mathematics 2 at 49 schools all across the country. This classroom testing allowed the authors to carefully study andrevise the program to make sure the materials help math studentslike you every day, in every classroom.

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Grade 6

CMP2 and Prentice Hall Mathematics Course 1, 2, 3 ©2010A Winning Combination Pearson’s Connected Mathematics 2 (CMP2) is a complete curriculum that helps students develop understanding of important mathematical concepts, skills, procedures, and ways of thinking and reasoning in numbers, geometry, measurement, algebra, probability, and statistics. CMP2 embeds important mathematical ideas in the context of interesting problems. As students explore a series of connected problems, they develop understanding of the embedded ideas and abstract powerful ideas, problem-solving strategies, and ways of thinking.

The following pages provide an overview of the mathematical concepts presented in CMP2. In addition, you will find options from Prentice Hall Mathematics Course 1, 2, 3 for additional practice of these rich mathematical ideas and concepts. Exercises from Prentice Hall Mathematics can be used to reinforce what your students are learning.

Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Prime Time

Investigation 1: Factors and Products

Chapter 4: Number Theory and Fractions 4-1–Divisibility and Mental Math 4-3– Prime Numbers and Prime Factorization

The Factor Game engages students in distinguishing between numbers with many factors and numbers with few factors. Students are then guided through an analysis of game strategies and introduced to the definitions of prime and composite numbers. The ACE questions are rich in connections to situations in which factors, multiples, and prime numbers are significant.

Investigation 2: Whole-Number Patterns and Relationships

Chapter 4: Number Theory and Fractions 4-4–Greatest Common Factor 4-5–Equivalent Fractions

Students explore the relationship among factors, factor pairs, and rectangles made to fit the factor pairs for a given number. The geometric interpretation of factor pairs adds to students’ understanding of multiplication as well as factors of a number. By looking at the products and sums of even and odd numbers the students are introduced to the idea of making conjectures and providing arguments to prove or disprove the conjectures. Venn diagrams show the relationship among common multiples, common factors, least common multiples, and greatest common factors.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 3: Common Multiples and Common Factors

Chapter 4: Number Theory and Fractions 4-1–Divisibility and Mental Math 4-4–Greatest Common Factor 4-7–Least Common Multiple

The concepts of least common multiple and greatest common factor, though not formally introduced, are used naturally throughout the problems and in the ACE section. The context of the problems and questions helps make clear whether a solution involves finding a common multiple, a common factor, the least common multiple, or the greatest common factor.

Investigation 4: Factorizations: Searching for Factor Strings

Chapter 4: Number Theory and Fractions 4-3– Prime Numbers and Prime Factorization

Students discover the Fundamental Theorem of Arithmetic: a whole number can be factored into a product of primes in exactly one way. Factor trees are used as a systematic way of finding the prime factorization of a number. The last problem in the Investigation helps students use prime factorizations to find the greatest common factor and least common multiple of two or more numbers. The discussion of why 1 is not a prime occurs in the ACE section.

Investigation 5: Putting It All Together

Chapter 4: Number Theory and Fractions— Culmination of this Chapter

Problem Solving Handbook: Look for a Pattern, Act It Out, Make a Table, Make a Drawing, or Work a Simpler Problem

This problem provides the students with the opportunity to use what they have learned in the unit to solve something more challenging.

Bits and Pieces I

Investigation 1: Fundraising Fractions

Chapter 4: Number Theory and Fractions 4-5–Equivalent Fractions 4-8–Comparing and Ordering Fractions

Students explore three components of understanding fractions: the visual model, word names for fractions, and symbols for fractions. The part-whole interpretation of fractions is developed. Students make fraction strips to study the progress toward a fundraising goal. The aim is to focus on the meaning of such phrases as, “two-thirds of the goal has been reached.”

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 2: Sharing and Comparing With Fractions

Chapter 4: Number Theory and Fractions 4-5–Equivalent Fractions 4-6a– Activity Lab: Exploring Improper

Fractions 4-6– Mixed Numbers and Improper Fractions 4-8– Comparing and Ordering Fractions

Chapter 5: Adding and Subtracting Fractions

5-1– Estimating Sums and Differences Estimation using benchmarks is introduced

The most important concept in understanding and using rational numbers is equivalence of fractions. This concept underlies operations with fractions, changing representations of fractions, and reasoning proportionally. Comparing fraction strips is used to motivate an investigation of equivalence and the creation of a number line that contains all of the information of the individual fraction strips. The idea of using benchmarks to estimate the size of fractions and to make comparisons is introduced.

Investigation 3: Moving Between Fractions and Decimals

Chapter 1: Whole Numbers and Decimals 1-5a–Activity Lab: Exploring Decimal Models 1-5–Understanding Decimals 1-6–Comparing and Ordering Decimals

Chapter 4: Number Theory and Fractions 4-9–Fractions and Decimals

Students are introduced to decimal representations of fractions and explore the place-value interpretation of decimals. They investigate a 100-square grid and explore how it could continue to be subdivided to show 1,000 parts or 10,000 parts. This process of subdividing and naming the new parts is very important mathematically; the underpinnings of the infinite process are met in this problem. The process will continue to help students understand equivalence of fraction and equivalence of decimals as well as to see the connections between fractions and decimals.

Investigation 4: Working With Percents

Chapter 7: Ratios, Proportions, and Percents

7-6a–Activity Lab: Modeling Percents 7-6–Percents, Fractions, and Decimals

Percents are introduced as another form of representation. A database of information about cats is used as a context for understanding percent. Students are engaged in activities requiring them to move among fractions, decimals, and percents.

Shapes and Designs

Investigation 1: Bees and Polygons

Chapter 8: Tools of Geometry 8-5–Exploring and Classifying Polygons 8-7–Line Symmetry

Students sort polygons by common properties and develop rotation and reflection symmetries of a shape. Students also explore which shapes will tile a plane.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 2: Polygons and Angles

Chapter 8: Tools of Geometry 8-2–Angles 8-3–Special Pairs of Angles

Students are introduced to three basic ways of thinking about angles and the ideas behind angle measurement. It gives students practice in estimating angle measurements based on a right angle. Students then explore a problem that looks at the possible consequences of making measurement errors.

Investigation 3: Polygon Properties and Tiling

Chapter 8: Tools of Geometry 8-4–Classifying Triangles 8-5–Exploring and Classifying Polygons

Students focus on some basic properties of familiar polygons, using tiling as a context.

Investigation 4: Building Polygons

Chapter 8: Tools of Geometry 8-4– Classifying Triangles 8-5– Exploring and Classifying Polygons

This investigation is based on the general question, “Is the shape of a polygon determined exactly by the lengths of its sides and the order in which those sides are connected?”

Bits and Pieces II

Investigation 1: Estimating With Fractions

Chapter 4: Number Theory and Fractions 4-5–Equivalent Fractions

Chapter 5: Adding and Subtracting Fractions

5-1–Estimating Sums and Differences

This investigation focuses on estimating sums of fractions. Students play two games in which they use benchmarks to help them estimate how big sums are.

Investigation 2: Adding and Subtracting Fractions

Chapter 5: Adding and Subtracting Fractions

5-1–Estimating Sums and Differences 5-2a– Activity Lab: Modeling Fraction

Operations 5-2–Fractions With Like Denominators 5-3a– Activity Lab: Modeling Unlike

Denominators 5-3–Fractions With Unlike Denominators 5-4–Adding Mixed Numbers 5-5–Subtracting Mixed Numbers

This investigation prepares students to figure out how to add and subtract fractions by emphasizing flexibility in finding equivalent fractions and in changing between fractions and decimals. In the course of struggling with the problems, most students will invent ways of adding fractions that you and the class can help make more explicit and efficient. The last problem gives students an opportunity to summarize what they have developed by asking them to write efficient algorithms for adding and subtracting fractions.

Investigation 3: Multiplying With Fractions

Chapter 6: Multiplying and Dividing Fractions

6-1a– Activity Lab: Modeling Fraction Multiplication

6-1–Multiplying Fractions 6-2–Multiplying Mixed Numbers 6-5– Solving Fraction Equations by

Multiplying

Developing computational skill with and understanding of fraction multiplication is focused on in this investigation. Various contexts and models are introduced to help students make sense of when multiplication is appropriate. Through the process of solving and characterizing how the problems are categorized, students develop a general algorithm for fraction multiplication.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 4: Dividing with Fractions

Chapter 6: Multiplying and Dividing Fractions

6-3a–Activity Lab: Fraction Division 6-3–Dividing Fractions 6-4–Dividing Mixed Numbers

Students develop the meaning of division with fractions and the strategies and algorithms for dividing them. Everyday situations are used to help students make sense of when division is an appropriate operation. Other operations with fractions are reviewed and the relationship between division and multiplication is explored.

Covering and Surrounding

Investigation 1: Designing Bumper Cars

Chapter 9: Geometry and Measurement 9-3–Perimeters and Areas of Rectangles

This investigation builds experience with analyzing what it means to measure area and perimeter and develops efficient strategies for finding area and perimeter of rectangles. Students will be able to write rules for finding area and perimeter of a rectangle and be able to explain why these work.

Investigation 2: Changing Area, Changing Perimeter

Chapter 9: Geometry and Measurement 9-3–Perimeters and Areas of Rectangles

Students explore fixed area and fixed perimeter problems, which are sometimes referred to as maximum and minimum problems. It helps to strengthen students’ understanding of area and perimeter and how they are related.

Investigation 3: Measuring Triangles

Chapter 8: Tools of Geometry 8-4–Classifying Triangles

Chapter 9: Geometry and Measurement 9-4–Areas of Parallelograms and Triangles

Students are introduced to finding areas and perimeters of triangles by using grids, arranging triangles to form parallelograms, and measuring with rulers. Special triangles—such as isosceles and 30-60-90 triangles—are explored.

Investigation 4: Measuring Parallelograms

Chapter 9: Geometry and Measurement 9-4a–Activity Lab: Comparing Areas 9-4–Areas of Parallelograms and Triangles

Students cut and rearrange parallelograms to make rectangles and develop strategies for using what they know about finding the area of a rectangle to find the area of a parallelogram. Students will develop good formulas for finding the area of a rectangle and a parallelogram. The use of grids and other more informal reasoning methods is encouraged.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 5: Measuring Irregular Shapes and Circles

Chapter 9: Geometry and Measurement 9-5a–Activity Lab: Exploring Circles 9-5– Circles and Circumferences 9-6–Area of a Circle

The investigation begins with counting techniques for estimating areas and perimeters of non-regular shapes. Students then discover how diameter and radius of a circle are related to its circumference and area. Students develop an understanding of the number Pi, which helps their understanding of expressions they may have seen for circumference and area of a circle.

Bits and Pieces III

Investigation 1: Decimals—More or Less!

Chapter 1: Whole Numbers and Decimals 1-7–Adding and Subtracting Decimals

This investigation develops addition and subtraction of decimals. The focus is also on the place value interpretation of a number and what that means for adding or subtracting numbers. Students articulate an algorithm for adding and subtracting.

Investigation 2: Decimal Times

Chapter 1: Whole Numbers and Decimals 1-8–Multiplying Decimals 1-8b– Activity Lab: Multiplying and Dividing

by 10, 100, and 1,000

The focus is on developing an algorithm for multiplying decimals. Students look at products, find missing factors, and use estimation as a way to determine where the decimal has to be in the product of decimal numbers.

Investigation 3: The Decimal Divide

Chapter 1: Whole Numbers and Decimals 1-8b– Activity Lab: Multiplying and Dividing

by 10, 100, and 1,000 1-9–Dividing Decimals

An algorithm for division of decimals is developed in this investigation. Students solve a set of contextualized problems that provide a common sense way to think about decimal division based on what they already know about whole-number and fraction division. Patterns in division and in terminating and repeating decimals are included.

Investigation 4: Using Percents

Chapter 7: Ratios, Proportions, and Percents 7-7–Finding the Percent of a Number 7-9–Estimating With Percents

Students look at real situations in which one encounters percents. The situations of discounts, taxes, and tips help students think about taking a percent of a number.

Investigation 5: More About Percents

Chapter 7: Ratios, Proportions, and Percents 7-7–Finding the Percent of a Number 7-8–Circle Graphs 7-9–Estimating With Percents

In this investigation, students are asked to devise a general strategy for finding a percent when they are dealing with totals that are more than or less than 100.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

How Likely Is It?

Investigation 1: A First Look at Chance

Chapter 10: Exploring Probability 10-2–Probability 10-4–Making Predictions From Data

Students flip a coin 30 times and then compute the experimental probability of a head occurring on a toss of a coin—first by using only their own data and then by using the entire class’s data. To get an even larger set of data, a computer (if available) is used to generate data and produce a graph of the fraction of tosses that result in heads. The graph is used to help students begin to understand the long-term regularity in the behavior of coins. The phrase, “equally likely” is introduced, and students are asked to decide whether various events are equally likely.

Investigation 2: Experimental and Theoretical Probabilities

Chapter 10: Exploring Probability 10-2–Probability 10-3–Experimental Probability 10-4–Making Predictions From Data

Probability is formally introduced in a game show setting. This whole-class activity is used to form a working definition of probability and to emphasize specific characteristics of probability: that the sum of the probabilities of all outcomes is 1; that probability is a number from 0 to 1; and that a probability of 0 or 1 has a particular significance. Students are asked to make comparisons between experimental and theoretical probabilities, and they have their first experience with making an organized list to find theoretical probabilities.

Investigation 3: Making Decisions With Probability

Chapter 10: Exploring Probability 10-2–Probability 10-3–Experimental Probability 10-4–Making Predictions From Data

Spinners are introduced as a new context for thinking about probabilities. Students explore that a spinner has a continuous range of possibilities by subdividing into any number of sections. Students also analyze various methods for making fair decisions and devise a simulation to find experimental probabilities.

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Grade 6 CMP2 Unit PH Mathematics Course 1 Investigation Summaries

Investigation 4: Probability, Genetics, and Games

Chapter 11: Exploring Probability 10-2–Probability 10-3–Experimental Probability 10-4–Making Predictions From Data

Students determine how many in their class can curl their tongues and use this data to make predictions about the probability of any one person being able to curl his or her tongue. This experimental method is compared with the way in which geneticists predict the traits of children by examining the genetic makeup of the parents. Students then gain experience with determining whether a person will have the tongue-curling trait based on genetic probabilities.

Data About Us

Investigation 1: Looking at Data

Chapter 2: Data and Graphs 2-2–Median and Mode 2-3–Frequency Tables and Line Plots 2-4–Bar Graphs and Line Graphs

Students investigate the distribution of numbers of letters in names. Students consider what’s typical about a data set of lengths of names and then consider their class’s data. They are introduced to or review the use of tables, line plots, and bar graphs to represent data; ways to describe the shape of a distribution; and the use of two measures of center (the mode and median) and a measure of spread (the range) that can be used to characterize a distribution.

Investigation 2: Using Graphs to Explore Data

Chapter 2: Data and Graphs 2-6–Stem-and-Leaf Plots

Students learn strategies for grouping and displaying data in intervals. The stem-and-leaf plot (or stem plot) is a useful tool for grouping data in intervals of 10, and it helps students see patterns in the data. Students examine two given data sets. These data provide the vehicle for introducing and exploring the stem plot.

Investigation 3: What Do We Mean by Mean?

Chapter 7: Data and Graphs 2-1–Finding the Mean

This investigation focuses on developing the concept of mean. The notion of “evening out” or “balancing” the distribution at a point (the mean) located on the horizontal axis is modeled by using cubes and stick-on notes. These models support the algorithm for finding the mean: adding up all the numbers and dividing by the total number of numbers.

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Grade 7

Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Variables and Patterns

Investigation 1: Variables, Tables, and Coordinate Graphs

Chapter 9: Patterns and Rules 9-1a– Activity Lab: Choosing Scales

and Intervals 9-1– Patterns and Graphs 9-3–Patterns and Tables 9-4a– Activity Lab: Generating Formulas

From a Table 9-4–Function Rules 9-5–Using Tables, Rules, and Graphs 9-5b– Activity Lab: Three Views

of a Function 9-6–Interpreting Graphs

Chapter 10: Graphing in the Coordinate Plane

10-2–Graphing Linear Equations

This investigation introduces the idea of a variable and three different ways to represent relationships between variables: verbal description, tables, and graphs. The context for the problems is the planning of a bike tour by a group of college students. Students interpret and create representations and begin to think about the strength and weaknesses of each type of representation.

Investigation 2: Analyzing Graphs and Tables

Chapter 9: Patterns and Rules 9-1a– Activity Lab: Choosing Scales

and Intervals 9-1– Patterns and Graphs 9-3–Patterns and Tables 9-4a– Activity Lab: Generating Formulas

From a Table 9-4–Function Rules 9-5–Using Tables, Rules, and Graphs 9-5b– Activity Lab: Three Views

of a Function 9-6–Interpreting Graphs

Chapter 10: Graphing in the Coordinate Plane

10-2–Graphing Linear Equations

Students make decisions about costs and profit for the bike tour by making and using tables and graphs. The last problem involves matching verbal descriptions with related graphs.

Investigation 3: Rules and Equations

Chapter 2: Equations and Inequalities 4-1a–Activity Lab: Describing Patterns 4-1– Evaluating and Writing Algebraic

Expressions 4-2– Using Number Sense to Solve Equations

Chapter 9: Patterns and Rules 9-1– Patterns and Graphs 9-1a– Activity Lab: Choosing Scales

and Intervals 9-3–Patterns and Tables 9-4a– Activity Lab: Generating Formulas

From a Table 9-4–Function Rules 9-5–Using Tables, Rules, and Graphs9-5b– Activity Lab: Three Views

of a function

Students develop strategies for writing symbolic equations, or formulas, to represent relationships between quantitative variables. They first write equations involving one operation and then move to two-operation equations. Students write equations for revenue, expenses, and profit for the bike tour.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 4: Calculator Tables and Graphs

Chapter 9: Patterns and Rules 9-5b– Activity Lab: Three Views

of a Function

Chapter 10: Graphing in the Coordinate Plane

10-2–Graphing Linear Equations 10-2b–Activity Lab: Representing Data 10-3b–Activity Lab: Exploring Slope

These two problems help students learn to make and use tables and graphs on a graphing calculator. Students review the strategies and techniques developed and compare their own work with tables and graphs generated on a graphing calculator.

Stretching and Shrinking

Investigation 1: Enlarging and Reducing Shapes

Chapter 5: Ratios, Rates, and Proportions 5-5a– Activity Lab: Exploring

Similar Figures

Similarity is introduced at an informal level. Students use their intuition about enlargements and reductions to answer questions. Students make drawings of similar figures using a pair of rubber bands and then compare side lengths, angle measures, perimeters, and areas of the original and enlarged figures.

Investigation 2: Similar Figures

Chapter 5: Ratios, Rates, and Proportions 5-5–Using Similar Figures 5-5b–Activity Lab: Drawing Similar Figures

Students build a good working definition of “similar” in mathematical terms. They begin to see connections between geometry and algebra. Using a coordinate grid, they draw several geometric figures with some that are similar and others that are not. As students investigate transformations they explore the algebraic rules that cause images to change size and to move about the coordinate plane. Students find that for two figures to be similar, corresponding angles must be congruent and corresponding sides must grow or shrink by the same factor.

Investigation 3: Similar Polygons

Chapter 5: Ratios, Rates, and Proportions 5-5–Using Similar Figures

Students continue to work with similar figures and explore the relationship between the areas of figures. Through experiments with rep-tiles (shapes where copies are put together to make larger, similar figures), students explore the relationship between the areas of two similar figures. Triangles are also discussed. These experiences help them build mental images to support their evolving ideas about the relationship between scale factor and area.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 4: Similarity and Ratios

Chapter 5: Ratios, Rates, and Proportions 5-1–Ratios 5-3–Proportions 5-5–Using Similar Figures

Equivalent ratios are used to test if figures are similar. Students compare ratios of the sides within rectangles. They find that to decide if non-rectangular shapes are similar they need information about angle measures as well. They find that the length of missing sides of two similar figures can be determined using either ratios or scale factors.

Investigation 5: Using Similar Triangles and Rectangles

Chapter 5: Ratios, Rates, and Proportions 5-5–Using Similar Figures

Real-world problems are used for students to apply their knowledge about similarity of triangles. They use both the shadow and mirror methods and compare their data to decide which method gives more consistent results.

Comparing and Scaling

Investigation 1: Making Comparisons

Chapter 5: Ratios, Rates, and Proportions 5-1–Ratios

This investigation focuses on the language of comparisons and ratios in the context of advertisements. Discussion of how to decide whether to use a difference, ratio, fraction, or percent to make a particular comparison.

Investigation 2: Comparing Ratios, Percents, and Fractions

Chapter 5: Ratios, Rates, and Proportions 5-1–Ratios

Students investigate in more depth how ratios can be formed and scaled up or down to find equivalent ratios.

Investigation 3: Comparing and Scaling Rates

Chapter 5: Ratios, Rates, and Proportions 5-2– Unit Rates and Proportional Reasoning 5-3b– Activity Lab: Interpreting Rates

Visually 5-6a– Activity Lab: Scale Drawings

and Models 5-6–Maps and Scale Drawings

This investigation takes a specific focus on rates, scaling rates, and finding and interpreting unit rates as strategies. It looks at scaling in numerical contexts. Students are asked to draw rate tables as well as to write rules or equations.

Investigation 4: Making Sense of Proportions

Chapter 5: Ratios, Rates, and Proportions 5-3–Proportions 5-3b– Activity Lab: Interpreting Rates

Visually 5-4a– Activity Lab: Using Proportions

With Data 5-4–Solving Proportions

All the problems in this investigation can be set up as a proportion in classical form, yet they are solved in a variety of equivalent ways. The strategies used are based on students’ knowledge of equivalent fractions. This investigation does not cover cross multiplication.

Accentuate the Negative

Investigation 1: Extending the Number System

Chapter 1: Decimals and Integers 1-6–Comparing and Ordering Integers

Positive and negative numbers in the form of integers, fractions, and decimals are also represented on a number line. Students are given experiences with positive and negative numbers, ordering, and informal operations in a variety of contexts.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 2: Adding and Subtracting Integers

Chapter 1: Decimals and Integers 1-7a– Activity Lab: Modeling Integer

Addition and Subtraction 1-7–Adding and Subtracting Integers

Students formulate algorithms for addition and subtraction of positive and negative numbers. Different representation models are used (number line and chip board).

Investigation 3: Multiplying and Dividing Integers

Chapter 1: Decimals and Integers 1-8a– Activity Lab: Modeling Integer

Multiplication 1-8–Multiplying and Dividing Integers

The number line model and fact families as well as the contexts of time, distance, and speed are used to develop students’ understanding of multiplication and division of positive and negative numbers. The cases of combinations of “signs” for multiplication are explained by looking at number patterns.

Investigation 4: Properties of Operations

Chapter 1: Decimals and Integers 1-9–Order of Operations and the Distributive Property 1-9b–Activity Lab: Properties and Equality

Chapter 2: Exponents, Factors, and Fractions

2-1–Exponents and Order of Operations

Students compare algebraic properties of the operations on positive and negative numbers to those of the number system of only positive numbers (whole numbers).

Moving Straight Ahead

Investigation 1: Walking Rates

Chapter 10: Graphing in the Coordinate Plane

10-2–Graphing Linear Equations 10-2b–Activity Lab: Representing Data

Chapter 11: Displaying and Analyzing Data 11-7a– Activity Lab: Two-Variable Data

Collection

The contexts for this investigation are the rates at which students walk and the donation per kilometer that sponsors pay for a walkathon. Students look at the change in the rate and its effects on various representations. They recognize that graphs of linear functions are straight lines and they begin to see that as the independent variable changes by a constant amount, there is a corresponding constant change in the dependent variable.

Investigation 2: Exploring Linear Functions With Graphs and Tables

Chapter 10: Graphing in the Coordinate Plane

10-2–Graphing Linear Equations 10-2b–Activity Lab: Representing Data

Chapter 11: Displaying and Analyzing Data 11-7a– Activity Lab: Two-Variable Data

Collection

This investigation helps students deepen their understanding of patterns of change. Constant rate of change and the y-intercept are formalized in this investigation. Students interpret the y-intercept as a special point on a line, an entry in a table, or as the constant b in the equations y = mx + b. They predict constant rate and decide whether relationships are decreasing or increasing.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 3: Solving Equations

Chapter 4:Equations and Inequalities 4-2– Using Number Sense to Solve Equations 4-2b–Activity Lab: Keeping the Balance 4-3a–Activity Lab: Modeling Equations 4-3– Solving Equations by Adding and

Subtracting 4-4– Solving Equations by Multiplying

or Dividing 4-5–Exploring Two-Step Problems 4-6–Solving Two-Step Equations 4-8– Solving Inequalities by Adding

or Subtracting 4-9– Solving Inequalities by Multiplying

or Dividing

Students begin to make connections among points on a line, a pair of data points in a table, and the solution to an equation. They use the properties of equality with equations in pictorial form and transition into solving equations symbolically by adding and subtracting the same variable or multiplying or dividing by the same nonzero number or variable on both sides of an equation.

Investigation 4: Exploring Slope

Chapter 10: Graphing in the Coordinate Plane

10-3–Finding the Slope of a Line 10-3b–Activity Lab: Exploring Slope

In this investigation students find the ratio of vertical change to horizontal change between two points on a line and connect this to the constant rate of change. Students find the slope of a line given two points on the line and then find the

y-intercept using either a table or a graph. The equation of the form y = mx + b is written. Finally, students apply their knowledge of slope to explore parallel and perpendicular lines. Graphing calculators help students explore many lines before making conjectures.

Filling and Wrapping

Investigation 1: Building Boxes

Chapter 8: Measurement 8-8–Three-Dimensional Figures

Students design flat patterns for cubic and rectangular boxes. They find the area of flat patterns and discover the association between this area and the surface area of the related box.

Investigation 2: Designing Rectangular Boxes

Chapter 8: Measurement 8-9–Surface Areas of Prisms and Cylinders 8-10– Volumes of Prisms and Cylinders 8-10b– Activity Lab: Generating Formulas

for Volume

Students continue their exploration of surface area and investigate its relationship to volume. Volume is defined as the number of unit cubes it takes to fill a rectangular box; surface area is defined as the amount of wrapping it takes to enclose a box. Students seek more efficient ways to determine the number of cubes a right prism would hold.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 3: Prisms and Cylinders

Chapter 8: Measurement 8-9– Surface Areas of Prisms and Cylinders 8-10– Volumes of Prisms and Cylinders 8-10b– Activity Lab: Generating Formulas for

Volume

Students use the same process to determine the surface area and volume of a cylinder. Then students design a rectangular box with the same volume as a given cylinder and compare the surface area of the two shapes.

Investigation 4: Cones, Spheres, and Pyramids

Chapter 8: Geometry and Measurement 8-8–Three-Dimensional Figures

Hands-on activities are used to compare the volumes of a cone, a sphere, and a cylinder of equal radius and equal height.

Investigation 5: Scaling Boxes

Chapter 8: Measurement 8-9–Surface Areas of Prisms and Cylinders 8-10– Volumes of Prisms and Cylinders 8-10b– Activity Lab: Generating Formulas

for Volume

The effects of changing the dimensions or the volume of a rectangular prism in the context of designing compost containers.

What Do You Expect?

Investigation 1: Evaluating Games of Chance

Chapter 12: Using Probability 12-1–Probability 12-2a–Activity Lab: Exploring Probability 12-2–Experimental Probability 12-2b–Activity Lab: Random Numbers

This investigation uses a variety of situations that provide students with a chance to review both experimental and theoretical probabilities, equally likely events, fair/unfair games, and strategies for determining theoretical probabilities. These situations also introduce two-stage events.

Investigation 2: Analyzing Situations Using an Area Model

Chapter 12: Using Probability 12-3–Sample Spaces 12-3b–Activity Lab: Using Data To Predict 12-4a– Activity Lab: Exploring

Multiple Events

Theoretical probability of two-stage events is analyzed using the area model. The two-stage events used are spinning spinners, choosing paths in a game, and choosing a marble at random from a container chosen at random.

Investigation 3: Expected Value

Chapter 12: Using Probability 12-3–Sample Spaces

In Investigation 3, the two-stage event is a one-and-one free-throw situation. After determining experimental probabilities that the player will get a score of 0, 1, or 2, students find the theoretical probability by using an area model. Students determine the long-term average (expected value) for the situation and explore expected value in a variety of different probability settings.

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Grade 7 CMP2 Unit PH Mathematics Course 2 Investigation Summaries

Investigation 4: Binomial Outcomes

Chapter 12: Using Probability 12-3–Sample Spaces 12-3b–Activity Lab: Using Data To Predict 12-4a– Activity Lab: Exploring

Multiple Events 12-4–Compound Events

By taking a four-item true-false quiz where each answer is determined by tossing a coin, students are introduced to binomial situations. Students then find the expected value (or average score) for guessing. Students also use lists or trees to determine outcomes. Pascal’s Triangle is explored in the exercises.

Data Distributions

Investigation 1: Making Sense of Variability

Chapter 1: Decimals and Integers 1-10–Mean, Median, Mode, and Range

Chapter 11: Displaying and Analyzing Data 11-1–Reporting Frequency

In this investigation, students explore how data in a distribution vary. The measures of variability are described using categorical data and numerical data. Situations in which reporting frequency as actual counts and percents are explored. Displaying data in tables and graphs is used to help identify patterns and to determine what is typical about a distribution.

Investigation 2: Making Sense of Measures of Center

Chapter 1: Decimals and Integers 1-10–Mean, Median, Mode, and Range

Chapter 11: Displaying and Analyzing Data 11-1–Reporting Frequency 11-2–Spreadsheets and Data Displays

Students explore the three measures of center, mean, median, and mode to describe what is typical about a distribution. The mean is often called the “average” of the data and is thought of the amount each person gets if everyone gets an equal share. The mode is the data value that occurs most frequently and the median is the midpoint in an ordered distribution. The shape of the distribution influences where the median and mean are located and students experiment with making changes to distributions.

Investigation 3: Comparing Distributions: Equal Numbers of Data

Chapter 11: Displaying and Analyzing Data 11-3b–Choosing the Best Display 11-6–Using Data to Persuade 11-7–Exploring Scatter Plots

In this investigation, students compare distributions of data with the same numbers of data values. Using reaction times, data is compared using different representations. Scale is explored using value bar graphs.

Investigation 4: Comparing Distributions: Unequal Numbers of Data Values

Chapter 11: Displaying and Analyzing Data 11-3b–Choosing the Best Display 11-4–Random Samples and Surveys 11-6–Using Data to Persuade

Students develop strategies to compare two or more distributions with unequal amounts of data about roller coasters. Representations of survey data are compared. Students also use a database to make comparisons about speed.

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Grade 8

Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Thinking With Mathematical Models

Investigation 1: Exploring Data Patterns

Chapter 3: Real Numbers and the Coordinate Plane

3-5a–Activity Lab: Tables and Graphs 3-5–Equations, Tables and Graphs 3-5b–Activity Lab: Matching Graphs

The central objectives of this investigation are to refresh student understanding of linear relationships and to contrast linear and nonlinear patterns. Students will work with inverse and quadratic relationships but are not expected to name these specific types of relationships or to represent them symbolically.

Investigation 2: Linear Models and Equations

Chapter 1: Integers and Algebraic Expressions

1-6– Solving Equations by Adding and Subtracting

1-7– Solving Equations by Multiplying and Dividing

Chapter 3: Real Numbers and the Coordinate Plane

3-4–Graphing in the Coordinate Plane 3-5a–Activity Lab: Tables and Graphs 3-5–Equations, Tables and Graphs

Chapter 6: Equations and Inequalities 6-1a– Activity Lab: Modeling

Multi-Step Equations 6-1–Solving Two-Step Equations 6-3–Solving Multi-Step Equations 6-5– Solving Inequalities by Adding

or Subtracting 6-6– Solving Inequalities by Multiplying

or Dividing

Chapter 11: Functions 11-2–Relating Graphs to Events 11-2b–Activity Lab: Line Graphs 11-3–Functions 11-3b–Activity Lab: Rate of Change 11-4–Understanding Slope 11-5a–Activity Lab: Graphing Equations 11-6–Writing Rules for Linear Functions

This investigation introduces the idea of using a mathematical model to approximate patterns in data. Students review methods of writing equations to match given information and methods for solving linear equations. Informal methods to solve inequalities are also used.

Investigation 3: Inverse Variation

The goal of Investigation 3 is to acquaint students with inverse variation, one of the fundamental nonlinear patterns of variation. Students should become comfortable with interpreting numeric and graphic patterns associated with inverse variations and representing such relationships with symbolic equations.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Looking for Pythagoras

Investigation 1: Coordinate Grids

Chapter 3: Real Numbers and the Coordinate Plane

3-4–Graphing in the Coordinate Plane 3-4b–Activity Lab: Finding the Midpoint 3-5a–Activity Lab: Tables and Graphs

Students review coordinate grids as they analyze a map in which streets are laid out on a grid. Students use this information for finding the distance between two points on a grid without measuring. Students investigate geometric figures on coordinate grids. They also calculate areas of several figures drawn on a dot grid.

Investigation 2: Squaring Off

Chapter 3: Real Numbers and the Coordinate Plane

3-1– Exploring Square Roots and Irrational Numbers

Students explore the relationship between the area of a square drawn on a dot grid and the length of its sides. This provides an introduction to the concept of square root.

Investigation 3: The Pythagorean Theorem

Chapter 3: Real Numbers and the Coordinate Plane

3-2a– Activity Lab: Exploring the Pythagorean Theorem

3-2–The Pythagorean Theorem

Students develop and explore the Pythagorean Theorem. They then investigate a geometric puzzle that verifies the theorem, and they use the theorem to find the distance between two points on a grid. In the last problem, they explore and apply the converse of the Pythagorean Theorem.

Investigation 4: Using the Pythagorean Theorem

Chapter 3: Real Numbers and the Coordinate Plane

3-3–Using the Pythagorean Theorem

Students explore an interesting pattern among right triangles, apply the Pythagorean Theorem to find distances on a baseball diamond, investigate properties of 30-60-90 triangles, and find missing lengths and angles measures of a triangle composed of smaller triangles.

Growing, Growing, Growing

Investigation 1: Exponential Growth

Chapter 2: Rational Numbers 2-6–Formulas 2-6b–Activity Lab: Using Formulas 2-7–Powers and Exponents 2-7b–Activity Lab: Evaluating Expressions 2-8a– Activity Lab: Multiplying

by Powers of 10 2-8–Scientific Notation

In Investigation 1, students explore situations that involve repeated doubling, tripling, and quadrupling. Students are introduced to one of the essential features of many exponential patterns: rapid growth. Students make and study tables and graphs for exponential situations, describe the patterns they see, and write equations for them, looking for a general form of an exponential equation. Linear and exponential patterns of growth are compared.

Investigation 2: Examining Growth Patterns

Investigation 2 focuses on exponential relationships with y-intercepts greater than 1. Each problem in the investigation presents information about an exponential pattern in a different form—in a verbal description, in an equation, and as a graph—helping students develop flexibility in moving among representations.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Investigation 3: Growth Factors and Growth Rates

This investigation has students study non-whole-number growth factors other than 1 and relate these factors to growth rates. Students also explore how the growth rate and the initial value affect the growth pattern.

Investigation 4: Exponential Decay

Investigation 4 introduces students to exponential decay—patterns of change rather than increases. These decreasing relationships are generated by repeated multiplication by factors between 0 and 1 called, “decay factors.” Strategies for finding decay factors and initial population, and for representing decay patterns, are similar to those used for exponential growth patterns.

Investigation 5: Patterns With Exponents

Chapter 12: Polynomials and Properties of Exponents

12- 3a–Activity Lab: Exploring Exponents 12-3–Exponents and Multiplication 12-3b–Activity Lab: Scientific Notation 12-4–Multiplying Polynomials 12-5–Exponents and Division

Students develop patterns for operating with exponents in this investigation. Patterns among the ones digits of powers are used to predict patterns of ones digits for powers that would be tedious to find directly. Students look for relationships among numbers written in exponential form. Graphing calculators are used to study the effects of exponents.

Frog, Fleas, and Painted Cubes

Investigation 1: Introduction to Quadratic Relationships

Chapter 11: Functions 11-7– Quadratic and Other Nonlinear Functions 11-7b– Activity Lab: Changing

Representations

Chapter 12: Polynomials and Properties of Exponents

12-1–Exploring Polynomials 12-2–Adding and Subtracting Polynomials 12-3a–Activity Lab: Exploring Exponents 12-3–Exponents and Multiplication 12-3b–Activity Lab: Scientific Notation 12-4–Multiplying Polynomials 12-5–Exponents and Division

Students explore the relationship between length and area of rectangles with a fixed perimeter to learn that this relationship is a quadratic function. Students examine the characteristics of tables and graphs of quadratic functions. They learn to recognize quadratic patterns in graphs and tables and to write equations for these patterns.

Investigation 2: Quadratic Expressions

Chapter 11: Functions 11-7–Quadratic and Other Nonlinear Functions 11-7b– Activity Lab: Changing

Representations

Students investigate how increasing one dimension of a square (sides of length n) by 2 and decreasing the other dimension by 2 affects the area. Students explore a visual representation of the Distributive Property. They discover that they can represent quadratic relationships as the product of two linear expressions, called factored form, or as the sum of one or more terms called expanded form. Students make connections among tables, graphs, and equations to learn which form of a quadratic equation provides specific information about the relationship.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Investigation 3: Quadratic Patterns of Change

Chapter 11: Functions 11-7– Quadratic and Other Nonlinear Functions 11-7b– Activity Lab: Changing

Representations

Students investigate quadratic relationships in sequences of triangular, square, and rectangular numbers. By comparing and analyzing patterns in tables, graphs, and equations, they express these quadratic relationships with equivalent expressions and predict whether variations in the handshake problem are quadratic.

Investigation 4: What Is a Quadratic Function?

Chapter 11: Functions 11-7– Quadratic and Other Nonlinear Functions 11-7b– Activity Lab: Changing

Representations

This investigation uses the classic projectile motion problems to extend students’ understanding of quadratic polynomials and their graphs. Students find significant intercepts, and maximum or minimum points, from quadratic relationships given in standard form.

Kaleidoscopes, Hubcaps, and Mirrors

Investigation 1: Three Types of Symmetry

Chapter 3: Real Numbers and the Coordinate Plane

3-6–Translations 3-7a–Activity Lab: Exploring Reflections 3-7–Reflections and Symmetry 3-8a–Activity Lab: Exploring Rotations 3-8–Rotations

Students identify different kinds of symmetry: reflection, rotation, and translation. Various drawing tools are used to test or draw figures with symmetry.

Investigation 2: Symmetry Transformations

Chapter 3: Real Numbers and the Coordinate Plane

3-6–Translations 3-7a–Activity Lab: Exploring Reflections 3-7–Reflections and Symmetry 3-8a–Activity Lab: Exploring Rotations 3-8–Rotations

Chapter 4: Applications of Properties 4-5–Similarity Transformations 4-5b– Activity Lab: Geometry Software

and Dilations

In Investigation 2, students identify properties of symmetry. How points and their images are related to the line of reflection is discovered. In a rotation, students determine how points, their images, and the center of rotation are related. Points and images are also explored in a translation. Students use these descriptions of symmetry to think about tessellations.

Investigation 3: Exploring Congruence

Chapter 4: Applications of Proportions 4-4–Similar Figures and Proportions 4-4b–Activity Lab: Ratios of Similar Figures 4-5–Similarity Transformations

Students discover connections between symmetry and congruence. Using symmetry transformations, students are able to compare the size and shape of figures to see whether they are congruent. Students also construct figures that are congruent to other figures. Polystrips are used to explore congruent triangles and quadrilaterals.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Investigation 4: Applying Congruence and Symmetry

Chapter 4: Applications of Proportions 4-4–Similar Figures and Proportions 4-4b–Activity Lab: Ratios of Similar Figures 4-5–Similarity Transformations 4-7a–Activity Lab: Using Similar Figures 4-7–Similarity and Indirect Measure

The focus of this investigation is for students to apply what they have learned about congruence and symmetry. Using properties of congruent triangles, students solve problems about shapes and measurement.

Investigation 5: Transforming Coordinates

Chapter 4: Application of Proportions 4-5a–Activity Lab: Exploring Dilations 4-5–Similarity Transformations 4-5b– Activity Lab: Geometry Software

and Dilations

In this investigation, students look at transformation of figures drawn on grids. Students describe symmetry transformations by telling what happens to a general point (x,y) and develop rules for symmetry on a coordinate grid.

Say It With Symbols

Investigation 1: Equivalent Expressions

Chapter 1: Integers and Algebraic Expressions

1-1– Algebraic Expressions and the Order of Operations

1-5–Properties of Numbers

Students generate and justify the equivalence of two or more symbolic expressions for the same situation. They are encouraged to think about problems in a variety of ways, leading to different, yet equivalent, expressions. Equivalency is discussed in terms of graphs and tables and the validity of the reasoning each expression or equation represents. The structures of the problems lead to important properties of numbers, especially the Distributive Property.

Investigation 2: Combining Expressions

Chapter 1: Integers and Algebraic Expressions

1-1– Algebraic Expressions and the Order of Operations

1-5–Properties of Numbers

In this investigation, students combine expressions to write new expressions either by adding or subtracting expressions or by substituting an equivalent expression for a given quantity in an expression or equation. They use the properties of real numbers to write equivalent expressions as they continue to connect symbolic expressions with real-world contexts.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Investigation 3: Solving Equations

Chapter 1: Integers and Algebraic Expressions

1-1– Algebraic Expressions and the Order of Operations

1-5–Properties of Numbers

Chapter 6: Equations and Inequalities 6-1a– Activity Lab: Modeling

Multi-Step Equations 6-1–Solving Two-Step Equations 6-2a–Modeling Expressions 6-2–Simplifying Algebraic Expressions 6-3–Solving Multi-Step Equations 6-4– Solving Equations With Variables

on Both Sides

Chapter 11: Functions 11-6–Writing Rules for Linear Functions

Students continue to use the Distributive and Commutative Properties to write equivalent expressions. They also use properties of equalities to solve linear equations with parentheses and to solve quadratic equations by factoring.

Investigation 4: Looking Back at Functions

Chapter 11: Functions 11-1–Sequences 11-1b–Activity Lab: Exploring Sequences 11-2–Relating Graphs to Events 11-3–Functions 11-4a–Activity Lab: Rate of Change 11-4–Understanding Slope 11-5a–Activity Lab: Graphing Equations 11-5–Graphing Linear Functions 11-6–Writing Rules for Linear Functions

In this investigation, students describe the underlying pattern of change represented by a symbolic statement. They also write symbolic equations to represent specific patterns of change and to find answers to specific questions. Many of the algebraic ideas from the algebraic strand are pulled together.

Investigation 5: Reasoning With Symbols

Chapter 1: Integers and Algebraic Expressions

1-1– Algebraic Expressions and the Order of Operations

1-5–Properties of Numbers 1-6a–Activity Lab: Modeling Equations 1-6– Solving Equations by Adding

and Subtracting 1-6b–Activity Lab: Number Squares 1-7– Solving Equations by Multiplying

and Dividing 1-7b–Activity Lab: The Cover-up Method

In this investigation, students explore why number puzzles work. This focuses on another important aspect of understanding symbols and writing equivalent expressions and their role in confirming or proving a conjecture. Students also explore algebraic expressions that represent even and odd integers and the patterns that emerge from squaring an odd number and then subtracting one.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

The Shapes of Algebra

Investigation 1: Equations for Circles and Polygons

In this investigation, students find patterns in the coordinates of points on line segments and circles. This includes writing and using equations of circles by looking at patterns in their graphs, coordinates, and equations. Coordinates of points that divide line segments in various ratios will be studied.

Investigation 2: Linear Equations and Inequalities

Chapter 6: Equations and Inequalities 6-3–Solving Multi-Step Equations 6-4– Solving Equations With Variables

on Both Sides 6-5a–Activity Lab: Graphing Inequalities 6-5– Solving Inequalities by Adding

or Subtracting 6-6a– Activity Lab: Inequalities

and Negative Numbers 6-6– Solving Inequalities by Multiplying

or Dividing

Investigation 2 develops an understanding of graphic and symbolic methods for analyzing and solving systems of linear equations and linear inequalities.

Investigation 3: Equations With Two or More Variables

Students use coordinate graphs to display solutions of linear equations of the form ax + by = c and to find solutions of systems of linear equations. The ability to change linear equations of the form ax + by = c to y = mx + b is developed.

Investigation 4: Solving Systems of Linear Equations Symbolically

Several strategies for finding solutions of systems of linear equations are developed: graphing, substituting, and combining. Students choose the most efficient solution method for a given system of equations.

Investigation 5: Linear Inequalities

Students explore situations that can be modeled with linear inequalities. By graphing linear inequalities and systems of linear inequalities, students are able to describe points that lie in the determined regions. Students solve systems of linear inequalities to find values that satisfy several conditions. This leads students to solving problems using systems of linear equations and inequalities.

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Grade 8 CMP2 Unit PH Mathematics Course 3 Investigation Summaries

Samples and Populations

Investigation 1: Comparing Data Sets

Chapter 9: Using Graphs to Analyze Data 9-2–Displaying Frequency 9-2b–Activity Lab: Making Histograms 9-6–Box-and-Whisker Plots 9-6b– Activity Lab: Making

Box-and-Whisker Plots

In this investigation, students group data into intervals to make histograms and use the five-number summaries of data to make box plots. These graphs are then used to analyze and compare data distributions.

Investigation 2: Choosing a Sample From a Population

Chapter 10: Probability 10-3–Conducting a Survey 10-3b– Activity Lab: Simulations With

Random Numbers 10-4a– Activity Lab: Comparing Types

of Events

Investigation 2 explores sampling techniques. By exploring data from random samples, students make predictions and draw conclusions about a population.

Investigation 3: Solving Real-World Problems

Chapter 9: Using Graphs to Analyze Data 9-4a– Activity Lab: Reading

Graphical Displays 9-4–Reading Graphs Critically 9-4b– Activity Lab: Making Graphs

to Tell a Story 9-5–Stem-and-Leaf Plots 9-6–Box-and-Whisker Plots 9-6b– Activity Lab: Making

Box-and-Whisker Plots 9-7a–Activity Lab: Scatter Plots 9-7–Making Predictions from Scatter Plots 9-9–Choosing an Appropriate Graph

By applying their knowledge of statistics and data displays, students solve real-world problems. Students compare sample distributions using measures of center (mean, median), measures of variability (range, minimum and maximum data values, percentiles), and displays that group data (histograms, box-and-whisker plots).

Investigation 4: Relating Two Variables

Chapter 9: Using Graphs to Analyze Data 9-7a–Activity Lab: Scatter Plots 9-7–Making Predictions from Scatter Plots

Students make scatter plots to look for relationships between pairs of variables. Where possible, a line to fit the patterns in the points is drawn and will be used to predict the value of one variable given the value of the other.