units for eighth grade thinking with mathematical...

ISBN-13:ISBN-10:

978-1-269-69461-21-269-69461-8

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Units for Eighth Grade

• Thinking With Mathematical Models Linear and Inverse Variation

• Looking for Pythagoras The Pythagorean Theorem

• Growing, Growing, Growing Exponential Functions

• Butterflies, Pinwheels, and Wallpaper Symmetry and Transformations

• Say It With Symbols Making Sense of Symbols

• It’s In the System Systems of Linear Equations and Inequalities

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Sample only - not for classroom use

PEARSON CUSTOM LIBRARY

MATHEMATICS

© Pearson Education, Inc. Not for distribution.


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Printed in the United States of America.

Printed in the United States of America.

ISBN 10: 1-269-69461-8ISBN 13: 978-1-269-69461-2

ISBN 10: 1-269-69461-8ISBN 13: 978-1-269-69461-2



Table of Contents

P E A R S O N C U S T O M L I B R A R Y

I

1. Thinking With Mathematical Models: Linear and Inverse Variation

1

1Glenda Lappan/Elizabeth Difanis Phillips/James T. Fey/Susan N. Friel

2. Looking for Pythagoras: The Pythagorean Theorem

57


3. Growing, Growing, Growing: Exponential Functions

107


4. Butterflies, Pinwheels and Wallpaper: Symmetry and Transformations

161


5. Say It With Symbols: Making Sense of Symbols

215


6. It’s In the System: Systems of Linear Equations and Inequalities

245




Thinking With Mathematical ModelsLinear and Inverse Variation

Parent Letter

Parent Letter in English

Labsheets

1.3A Truss 1.3B Steps 1ACE Exercises 14–17 2.1A Models 2.1B Models2.2A Graphs 2.2B Tables 2.3A Prices 2.3B Snacks2.4 Rentals 2ACE Exercise 3 2ACE Exercise 6 2ACE Exercise 8 2ACE Exercise 13 2ACE Exercise 20 3.1A 3.1B 3.2 Trips 3ACE Exercises 4–7 4.1 Residuals 4.2A Negative Correlation 4.2B GPA 4.3A Correlation Coefficients 4.3B Top Speed and Maximum Drop 4.3C Top Speed and Track Length 4.3D Duration of Ride and Top Speed 4.3E Number and Age of Riders 4.4A Line Plot and Summary Statistics 4.4B Differences and Squares of Differences 4ACE Exercise 1 4ACE Exercise 3 4ACE Exercise 5 4ACE Exercise 24, parts (c)–(g)

2

Rectangles With Area

From Unit 1 of Connected Mathematics®3: Grade . Copyright © 2014 by Michigan State University, Glenda Lappan, Elizabeth Difanis Phillips, James T. Fey, and Susan N. Friel. Published by Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.

8

2 4Rectangles With Area 2 3

1© Pearson Education, Inc. Not for distribution.


4ACE Exercise 26 5.2A Political Party Choices 5.2B Political Party Choices 5ACE Exercise 15 5ACE Exercise 18, parts (a) and (b) 5ACE Exercise 19 5ACE Exercise 33

Assessments

Check Up 1 Check Up 2 Partner Quiz

Self Assessment Notebook Checklist

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.



Querida Familia:La primera unidad en la clase de matematicas de su hijo(a) es La hora los primos: Factores y multiplos. Esta es la primera unidad sobre el tema de numeros de Connected Mathematics.

Querida Familia:La primera unidad en la clase de matematicas de su hijo(a) es La hora los primos: Factores y multiplos. Esta es la primera unidad sobre el tema de numeros de Connected Mathematics.

Dear Family, This Unit is Thinking With Mathematical Models: Linear and Inverse Variation.

Unit, we will explore situations that can be represented with various mathematical models, including graphs and equations. We will also examine variability and association between two numerical or categorical variables.

Unit Goals

Students will review, extend their understanding of, and improve their skills in working with linear functions and equations. This Unit also introduces concepts associated with nonlinear functions.

Algebraic functions that represent patterns in experimental data are called mathematical models. Students will use these functions to estimate answers to questions about relationships in the data.

This Unit also introduces inverse variation. Students work with inverse variations in several real-world contexts. The Unit also develops student understanding of associations between variables using basic ideas of correlation and two-way tables.

Helping with Homework

You can help with homework and encourage sound mathematical habits as your child studies this Unit by asking questions such as:

What are the key variables in this situation?

What is the pattern relating these variables? Is it linear?

What kind of equation will express the relationship among the variables?

How can you use this equation to answer questions about the relationship?

How can you decide if two categorical or numerical variables are associated?

In your child’s notebook, you can find worked-out examples, notes on the mathematics of the Unit, and descriptions of the vocabulary words.

Having Conversations About the Mathematics in Thinking With Mathematical Models

You can help your child with his or her work for this Unit in several ways:

Have your child share his or her mathematics notebook with you, showing you what he or she has recorded. Ask your child to explain why these ideas are important.

Talk about situations in which a person might collect data and represent relationships with mathematical models, such as tables and graphs.

Review your child’s homework; make sure he or she has answered all the questions and provided clear explanations.

Common Core State Standards

Students develop and use all of the Standards of Mathematical Practice throughout the curriculum. In Thinking With Mathematical Models, students model with mathematics as they use functions to describe the relationship between two variables. This Unit focuses on using algebra to represent data using tables, graphs, equations or inequalities, and rules.

A few important mathematical ideas that your child will learn in Thinking With Mathematical Models are given on the next page. As always, if you have any questions or concerns about this Unit or your child’s progress in the class, please feel free to call.

In this




Important Concepts Examples

Mathematical ModelAn equation or a graph that describes the relationship between two variables. A mathematical model is made by graphing data and finding an equation or a curve to approximate it. A model lets you estimate values between and beyond the data points.

Students model bridge thickness and strength data by:

1. simulating the strength of bridges that have various layers of thickness and collecting data,

2. plotting the data and drawing a line of best fit,

3. finding an equation to model the data (e.g., y = 8x),

4. and using the equation to predict the breaking weights for other bridges. For example, using y = 8x, a bridge of thickness 3.5 layers can hold a load of 28 pennies.

Linear Relationships and FunctionsStudents have learned how to recognize, represent, and analyze linear relationships. They have learned how to solve linear equations. Students will deepen these understandings in this Unit.

In the equation y = mx + b, m indicates the constant ratio change in ychange in x , which is the

slope of the graph. The variable b indicates the y intercept (0, b) of the graph.

Students solve linear equations by

x, y) values in tables and graphs,

properties of equality, and

Direct VariationModels that can be written in the form y = kx.

Students are familiar with direct variation as a special case of a linear function (that is, those with a y-intercept of zero).

Inverse VariationModels that can be written in the form y = k

x .The key learning goals for students are first, that an indirect variation gives a non-linear pattern of change and second, that its equation can be written in the form y = k

x .

The contrasting graphs of

y = 10 - x (line) and y = 10x (curve)

demonstrate that dividing by an increasing variable has a different effect than subtracting an increasing variable does.

Students are familiar with the formula A = /w for finding the area of a rectangle with given length and width. Now, students are asked to look for combinations of length and width that give a fixed area. This leads to the formula / = A

w .

Patterns of Association in Numerical Data

Scatter plots can be used to model association between two quantities. Students describe patterns such as clustering, outliers, positive/negative association, and linear/nonlinear association. For linear data, students write a linear model and assess the fit of the model by judging the closeness of the data points to the line.

Patterns of Association in Categorical Data

Students construct and interpret two-way tables of categorical data and use relative frequencies calculated for rows or columns to describe the association between the two variables. In the two-way table below, students look for an association between gender and political party affiliation.

Independent RepublicanDemocrat

Boys

Girls

8

8

4

2

12

6

10

y

4

2

6

8

10

xO 42 6 8

y = 10x

y = 10 – x




Name Date Class

Number of Rods

Length of Truss 6 7 85432

Labsheet 1.3A Truss

Investigation 1Thinking With Mathematical Models




Name Date Class

CSP Staircase Frames

Number of Rods

Number of Steps 6 7 854321

Labsheet 1.3B Steps





Name Date Class

For Exercises 14–17, tell which graph matches the equation or the set of criteria.

14. y = 3x + 1 15. y = -2x + 2

16. y = x - 3 17. y-intercept = 1; slope = 12

−4 −2 2 4

−4

−2

2

4y

x

Graph A

O

Graph B

O−4 −2 2 4

−4

−2

2

4y

x

O−4 −2 2 4

−4

−2

2

4

x

Graph Cy

O−4 −2 2 4

−4

−2

2

4

x

Graph Dy

Labsheet 1ACE Exercises 14–17





Name Date Class

70

1

10

20

30

40

50

60

2 3 4 5 6 7

x

y

00

Bre

akin

g W

eig

ht

(pen

nie

s)

Layers

Breaking Weight (pennies)

Number of Layers 654321

Predicted by Modely = 10.4x + 1.6

Actual 644840321812

Residual (actual – predicted)

Labsheet 2.1A Models





Name Date Class

70

1

10

20

30

40

50

60

2 3 4 5 6 7

x

y

00

Bre

akin

g W

eig

ht

(pen

nie

s)

Layers

Breaking Weight (pennies)

Number of Layers 654321

Predicted by Modely = 10x

Actual 644840321812

Residual (actual – predicted)

Labsheet 2.1B Models





units for eighth grade thinking with mathematical...

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