Saxon mathc course 3

The Practice Set provides a chance to solve problems involving the new concept.. When we solve a problem, it helps to ask ourselves some questions along the way.Follow the Process Ask Yo

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Course 3 Student Edition

Stephen Hake

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A C K N O W L E D G E M E N T S

Staff Credits

Editorial: Jean Armstrong, Shelley Farrar-Coleman, Marc Connolly, Hirva Raj, Brooke Butner, Robin Adams,

Roxanne Picou, Cecilia Colome, Michael Ota

Design: Alison Klassen, Joan Cunningham, Deborah Diver, Alan Klemp, Andy Hendrix, Rhonda Holcomb

Production: Mychael Ferris-Pacheco, Heather Jernt, Greg Gaspard, Donna Brawley, John-Paxton Gremillion

Manufacturing: Cathy Voltaggio

Marketing: Marilyn Trow, Kimberly Sadler

E-Learning: Layne Hedrick, Karen Stitt

This book was made possible by the signifi cant contributions of many individuals and the dedicated efforts of a talented team at Harcourt Achieve

Special thanks to:

• Melody Simmons and Chris Braun for suggestions and explanations for problem solving

in Courses 1–3,

• Elizabeth Rivas and Bryon Hake for their extensive contributions to lessons and practice

in Course 3,

• Sue Ellen Fealko for suggested application problems in Course 3

The long hours and technical assistance of John and James Hake on Courses 1–3, Robert Hake on Course 3, Tom Curtis on Course 3, and Roger Phan on Course 3 were invaluable in meeting publishing deadlines The saintly patience and unwavering support of Mary is most appreciated

– Stephen Hake

ISBN 1-5914-1884-4

in whole or in part, without permission in writing from the copyright owner Requests for permission should be mailed to: Paralegal Department, 6277 Sea Harbor Drive, Orlando, FL 32887.

Saxon is a trademark of Harcourt Achieve Inc

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iii

ABOUT THE AUTHOR

Stephen Hake has authored five books in the Saxon Math series He writes from

17 years of classroom experience as a teacher in grades 5 through 12 and as a math

specialist in El Monte, California As a math coach, his students won honors and

recognition in local, regional, and statewide competitions

Stephen has been writing math curriculum since 1975 and for Saxon since 1985

He has also authored several math contests including Los Angeles County’s first

Math Field Day contest Stephen contributed to the 1999 National Academy of

Science publication on the Nature and Teaching of Algebra in the Middle Grades

Stephen is a member of the National Council of Teachers of Mathematics and the

California Mathematics Council He earned his BA from United States International

University and his MA from Chapman College

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C O N T E N T S O V E R V I E W

Table of Contents v

Letter from the Author xvii

How to Use Your Textbook xviii

Introduction to Problem Solving 1

Section 1 6

Lessons 1–10, Investigation 1 Section 2 72

Lessons 11–20, Investigation 2 Section 3 139

Lessons 111–120, Investigation 12 Appendix: Additional Topics in Algebra 785

Glossary with Spanish Terms 859

Index 897

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Table of Contents v

T A B L E O F C O N T E N T S

Integrated and Distributed Units of Instruction

Lessons 1–10, Investigation 1Section 1

• Rates and Average

• Measures of Central Tendency

• The Coordinate Plane

Activity Coordinate Plane

Maintaining & Extending

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Activity Pythagorean Puzzle

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Table of Contents vii

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• Multiplying and Dividing Integers

• Multiplying and Dividing Terms

• Drawing Geometric Solids

Activity 1 Sketching Prisms and Cylinders Using Parallel

Projection

Activity 2 Sketching Pyramids and Cones

Activity 3 Create a Multiview Drawing

Activity 4 One-Point Perspective Drawing

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Table of Contents ix

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• Using Unit Multipliers to Convert Measures

• Converting Mixed-Unit to Single-Unit Measures

• Nets of Prisms, Cylinders, Pyramids, and Cones

Activity Net of a Cone

• Collect, Display, and Interpret Data

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Table of Contents xi

Activity 1 Probability Simulation

Activity 2 Design and Conduct a Simulation

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Activity Make a Scatterplot and Graph a Best-fit Line

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Table of Contents xiii

Activity Random Number Generators

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Activity Calculating Interest and Growth

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Table of Contents xv

Activity 1 Modeling Freefall

Activity 2 Using the Graph of a Quadratic Function

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• Using Scatterplots to Make Predictions

Activity Using a Scatterplot to Make Predictions

• Proof of the Pythagorean Theorem

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Letter from the Author xvii

Dear Student,

We study mathematics because of its importance to our lives Our school schedule, our trip to the store, the preparation of our meals, and many of the games we play involve mathematics You will find that the word problems

in this book are often drawn from everyday experiences

As you grow into adulthood, mathematics will become even more important In fact,

your future in the adult world may depend on the mathematics you have learned

This book was written to help you learn mathematics and to learn it well For this to

happen, you must use the book properly As you work through the pages, you will see

that similar problems are presented over and over again

Solving each problem day after day is the secret to success.

Your book is made up of daily lessons and investigations Each lesson has three parts

1 The first part is a Power Up that includes practice of basic facts and mental math

These exercises improve your speed, accuracy, and ability to do math “in your

head.” The Power Up also includes a problem-solving exercise to familiarize you

with strategies for solving complicated problems

2 The second part of the lesson is the New Concept This section introduces a new

mathematical concept and presents examples that use the concept The Practice

Set provides a chance to solve problems involving the new concept The problems

are lettered a, b, c, and so on

3 The final part of the lesson is the Written Practice This problem set reviews

previously taught concepts and prepares you for concepts that will be taught in

later lessons Solving these problems helps you remember skills and concepts for a

long time

Investigations are variations of the daily lesson The investigations in this book often

involve activities that fill an entire class period Investigations contain their own set of

questions instead of a problem set

Remember, solve every problem in every practice set, written practice set, and

investigation Do not skip problems With honest effort, you will experience

success and true learning that will stay with you and serve you well in the future.

Temple City, California

L E T T E R F R O M A U T H O R S T E P H E N H A K E

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have colorful photos to distract you from learning The Saxon approach lets

you see the beauty and structure within math itself You will understand more

mathematics, become more confident in doing math, and will be well prepared

when you take high school math classes

Learn Something New!

Each day brings you a new concept,

but you’ll only have to learn a small

part of it now You’ll be building on

this concept throughout the year so

that you understand and remember it

by test time

Power Yourself Up!

Start off each lesson by practicing

your basic skills and concepts, mental

math, and problem solving Make your

math brain stronger by exercising it

every day Soon you’ll know these

facts by memory!

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How to Use Your Textbook xix

Lesson 55 377

different rectangles that appear in the net Each rectangle appears twice, because the front and back faces are congruent, as are the top and bottom faces and the left and right faces.

Activity

Net of a Cone

Materials needed: unlined paper, compass, scissors, glue or tape, ruler.

Using a compass, draw a circle with a radius of at least two inches Cut out the circle and make one cut from the edge to the center of the circle Form the lateral surface

of a cone by overlapping the two sides of the cut The greater the overlap, the narrower the cone Glue or tape the overlapped paper so that the cone holds its shape.

To make the circular base of the cone, measure the diameter of the open end of the cone and use a compass to draw a circle with the same diameter

(Remember, the radius is half the diameter.) Cut out the circle and tape it in place using two pieces of tape

base lateral surface

area of the lateral surface of a cone is to calculate the area of the portion of a circle represented by the net of the lateral surface Use a protractor to measure the central angle of the lateral surface of the cone you created The measure of that angle is the fraction of a 360° circle represented by the lateral surface Use a ruler

to measure the radius Find the area of a whole circle with that radius Then find the area of the sector by multiplying the area of the whole circle by the fractioncentral angle360 Answers will vary

central angle radius

Top

Front Right Left

Bottom

Back

cut

Lesson 43 297

Practice Set a What is the lateral surface area of a tissue

box with the dimensions shown? 112 in 2

b What is the total surface area of a cube with edges 2 inches long? 24 in 2

c Estimate the surface area of a cube with edges 4.9 cm long 150 cm 2

Written Practice Strengthening Concepts

* 4.

11 inches × 11 inches × 12 inches? 198 in 3

5.

(43)How many square inches is the surface area of the shipping box

described in problem 4? 333 in 2

(39, 40) Find a the area and b circumference of the circle with a radius of 6 in

Express your answer in terms of π a 36π in 2b 12π in

9 in.

5 in.

4 in.

7.

Exercise Your Mind!

When you work the Written Practice

exercises, you will review both today’s

new concept and also math you learned

in earlier lessons Each exercise will

be on a different concept — you never

know what you’re going to get! It’s like

a mystery game — unpredictable and

challenging

As you review concepts from earlier in

the book, you’ll be asked to use

higher-order thinking skills to show what you

know and why the math works

The mixed set of Written Practice is just

like the mixed format of your state test

You’ll be practicing for the “big” test

every day!

Get Active!

Dig into math with a hands-on activity Explore a math concept with your friends as you work together and use manipulatives

to see new connections in mathematics

Check It Out!

The Practice Set lets you check

to see if you understand today’s new concept

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Dive into math concepts and explore

the depths of math connections in

the Investigations

Continue to develop your

mathematical thinking through

applications, activities, and

extensions

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Problem-Solving Overview 1

PROBLEM-SOLVING OVERVIEW

Problem Solving

problems We face mathematical problems in our daily lives, in our careers, and in our efforts to advance our technological society We can become powerful problem solvers by improving our ability to use the tools we store in our minds In this book we will practice solving problems every day

four-step

problem-solving

process

Solving a problem is like arriving at a destination, so the process of solving

a problem is similar to the process of taking a trip Suppose we are on the mainland and want to reach a nearby island

Mainland

Island

Problem-Solving Process Step 1:

Understand Know where you are and where you want to go

Take the journey to the island

Verify that you have reached your desired destination

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When we solve a problem, it helps to ask ourselves some questions along the way.

Follow the Process Ask Yourself Questions Step 1: Understand What information am I given?

What am I asked to find or do?

Step 2: Plan How can I use the given information to

solve the problem?

What strategy can I use to solve the problem?

Step 3: Solve Am I following the plan?

Is my math correct?

Step 4: Check (Look Back) Does my solution answer the question

that was asked?

Is my answer reasonable?

Follow the Process Ask Yourself Questions

Below we show how we follow these steps to solve a word problem

Example

Josh wants to buy a television He has already saved $68.25 He earns $35 each Saturday stocking groceries in his father’s store The sizes and prices of the televisions available are shown at right If Josh works and saves for 5 more weekends, what is the largest television he could buy?

Solution

Step 1: Understand the problem We know that Josh has $68.25 saved

We know that he earns $35.00 every weekend We are asked to decide which television he could buy if he works for 5 more weekends

Step 2: Make a plan We cannot find the answer in one step We make a

plan that will lead us toward the solution One way to solve the problem is to find out how much Josh will earn in 5 weekends, then add that amount to the money he has already saved, then determine the largest television he could buy with the total amount of money

Step 3: Solve the problem (Follow the Plan.) First we multiply $35 by 5

to determine how much Josh will earn in 5 weekends We could also find 5 multiples of $35 by making a table

Amount $35 $70 $105 $140 $175

Televisions

15" $149.99 17" $199.99 20" $248.99

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$35

$175Josh will earn $175 in 5 weekends

Now we add $175 to $68.25 to find the total amount he will have

$175+ 68.25

$243.25

We find that Josh will have $243.25 after working 5 more weekends When

we compare the total to the prices, we can see that Josh can buy the 17″

television

Step 4: Check your answer (Look Back.) We read the problem again

The problem asked which television Josh could buy after working five weekends We found that in five weekends Josh will earn $175.00, which combined with his $68.25 savings gives Josh $243.25 This is enough money to buy the 17″ television

1 List in order the four steps in the problem-solving process

2 What two questions do we answer to help us understand a

problem?

Refer to the text below to answer problems 3–8

Mary wants to put square tiles on the kitchen floor The tile she has selected is 12 inches on each side and comes in boxes of 20 tiles for

$54 per box Her kitchen is 15 feet 8 inches long and 5 feet 9 inches wide How many boxes of tile will she need and what will be the price of the tile, not including tax?

3 What information are we given?

4 What are we asked to find?

5 Which step of the four-step problem-solving process have you

completed when you have answered problems 3 and 4 ?

6 Describe your plan for solving the problem Besides the arithmetic,

is there anything else you can draw or do that will help you solve the problem?

7 Solve the problem by following your plan Show your work and any

diagrams you used Write your solution to the problem in a way

someone else will understand

you used the information correctly Be sure you found what you were asked to find Is your answer reasonable?

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to travel from the mainland to the island Other strategies might not be as effective for the illustrated problem For example, choosing to walk or bike across the water are strategies that are not reasonable for this situation.

Problem-solving strategies are types of plans we can use to solve

problems Listed below are ten strategies we will practice in this book You may refer to these descriptions as you solve problems throughout the year

Act it out or make a model Moving objects or people can help us visualize

the problem and lead us to the solution

Use logical reasoning All problems require reasoning, but for some

problems we use given information to eliminate choices so that we can close in on the solution Usually a chart, diagram, or picture can be used to organize the given information and to make the solution more apparent

Draw a picture or diagram Sketching a picture or a diagram can help us

understand and solve problems, especially problems about graphs or maps

or shapes

Write a number sentence or equation We can solve many word problems

by fitting the given numbers into equations or number sentences and then finding the unknown numbers

Make it simpler We can make some complicated problem easier by using

smaller numbers or fewer items Solving the simpler problem might help us see a pattern or method that can help us solve the complex problem

Find a pattern Identifying a pattern that helps you to predict what will

come next as the pattern continues might lead to the solution

Make an organized list Making a list can help us organize our thinking

about a problem

Guess and check Guessing a possible answer and trying the guess in

the problem might start a process that leads to the answer If the guess

is not correct, use the information from the guess to make a better guess Continue to improve your guesses until you find the answer

Make or use a table, chart, or graph Arranging information in a table,

chart, or graph can help us organize and keep track of data This might reveal patterns or relationships that can help us solve the problem

Work backwards Finding a route through a maze is often easier by

beginning at the end and tracing a path back to the start Likewise, some problems are easier to solve by working back from information that is given toward the end of the problem to information that is unknown near the beginning of the problem

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9 Name some strategies used in this lesson

The chart below shows where each strategy is first introduced in this textbook

or Equation

This is a description of the way we solved the problem about tiling a kitchen

First, we round the measurements of the floor to 6 ft by 16 ft We multiply to find the approximate area: 6 × 16 = 96 sq ft Then we count

by 20s to find a number close to but greater than 96: 20, 40, 60, 80,

100. There are 20 tiles in each box, so Mary needs to buy 5 boxes of tiles We multiply to find the total cost: $54 × 5 = $270.

10 Write a description of how we solved the problem in the example

Other times, we will be asked to write a problem for a given equation

Be sure to include the correct numbers and operations to represent the equation

11 Write a word problem for the equation b = (12 × 10) ÷ 4.

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she drive in total?

A line B line segment C ray

Problem: A robot is programmed to take two steps forward, then one step

back The robot will repeat this until it reaches its charger unit, which is ten steps in front of the robot How many steps back will the robot take before it reaches the charger unit?

Understand We are told a robot takes two steps forward and then one step back We are asked to find how many steps back the robot will take before it reaches its charger unit, which is ten steps away

also select a student to act out the robot’s movements for the class

1

As a shorthand, we will use commas to separate operations to be performed sequentially from left to right This is not a standard mathematical notation.

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Lesson 1 7

Solve We use a number line for our diagram We count two spaces forward from zero and then one space back and write “1” above the tick mark for 1 From that point, we count two spaces forward and one space back and write “2” above the 2 We continue until we reach the tick mark for 10 The numbers we write represent the total number of backward steps by the robot

We reach 10 after having taken eight steps back The robot reaches the charger unit after taking eight steps back

New Concept Increasing Knowledge

The numbers we use in this book can be represented by points on a number line

Number line

Points to the right of zero (the origin ) represent positive numbers Points

to the left of zero represent negative numbers Zero is neither positive nor negative

but not all integers are whole numbers We will study numbers represented

by points between the tick marks later in the book

Classify Give an example of an integer that is not a whole number

(members) of the set within the braces

The set of integers

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The absolute value of a number is the distance between the number and the origin on a number line Absolute value is always a positive number because

it represents a distance Thus, the absolute value of −5 (negative five) is 5, because −5 is 5 units from zero Two vertical bars indicate absolute value

“The absolute value of negative five is five.”

−5 = 5

We can order and compare numbers by noting their position on the number line We use the equal sign and the greater than/less than symbols (> and <, respectively) to indicate a comparison When properly placed between two numbers, the small end of the symbol points to the lesser number

When you move

from left to right

three is less than negative one

follows a rule

The set of even numbers

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dots at −3, −1, 1, and 3 The ellipses show that the sequence continues, so

we darken the arrowheads to show that the pattern continues

e −3 f 3

g Analyze Write two numbers that are ten units from zero

h Write an example of a whole number that is not a counting number

Written Practice Strengthening Concepts

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(1) Conclude What is the sum when you add −2 and its opposite?

For multiple choice problems 28−30, choose all correct answers from

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far does he travel?

a row of 12 push pins be?

problem

solving

A number of problems in this book will ask us to “fill in the missing digits”

in an addition, subtraction, multiplication, or division problem We can work backwards to solve such problems

Problem: Copy this problem and fill in the missing digits:

$_0_8

− $432_

$4_07 Understand We are shown a subtraction problem with missing digits We are asked to fill in the missing digits

digits one-by-one This is a subtraction problem, so we can look at each column and think of “adding up” to find missing digits

Solve We look at the first column and think, “7 plus what number equals 8?” (1) We write a 1 in the blank We move to the tens column and think,

“0 plus 2 is what number?” We write a 2 in the tens place of the minuend (the top number) Then we move to the hundreds column and think, “What number plus 3 equals a number that ends in 0? We write a 7 in the difference and remember to regroup the 1 in the thousands column We move to the thousands column and think, “4 plus 4 plus 1 (from regrouping) equals what number?” We write a 9 in the remaining blank

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$9028

The fundamental operations of arithmetic are addition, subtraction, multiplication, and division Below we review some terms and symbols for these operations

Terminology

addend + addend = sum minuend − subtrahend = difference factor × factor = product dividend ÷ divisor = quotient

18 ÷ 9 = 2

Three numbers that form an addition fact also form a subtraction fact

5 + 3 = 8

8 − 3 = 5 Likewise, three numbers that form a multiplication fact also form a division fact

4 × 6 = 24

24 ÷ 6 = 4

We may use these relationships to help us check arithmetic answers

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b 25

× 18200250

450 ✓

Symbols for Multiplication and Division

Math Language

The term real

numbers refers

to the set of all

numbers that can

be represented

by points on a

number line

The table below lists important properties of addition and multiplication

properties apply to all real numbers A variable can take on different values

a ∙ b = b ∙ a 3 ∙ 4 = 4 ∙ 3

of Multiplication Associative Property

(a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5)

of Addition Associative Property

(a ∙ b) ∙ c = a · (b · c) (3 ∙ 4) ∙ 5 = 3 ∙ (4 ∙ 5)

of Multiplication Identity Property

of Addition Identity Property

of Multiplication Zero Property

of Multiplication

We will use these properties throughout the book to simplify expressions and solve equations

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Instead of first multiplying 25 and 15, we can use properties of multiplication

to rearrange and regroup the factors so that we first multiply 25 and 4 Changing the arrangement of factors can make the multiplication easier to perform

Step: Justification:

(25 ∙ 15) ∙ 4 Given

(15 ∙ 25) ∙ 4 Commutative Property of Multiplication

15 ∙ (25 · 4) Associative Property of Multiplication

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Solution The letters x and y represent numbers and are unknowns in the equations.

from the sum

g Justify Lee simplified the expression 5 · (7 · 8) His work is shown

below What properties of arithmetic did Lee use for steps 1 and 2 of his calculations?

h Explain Explain how to check a subtraction answer

i Explain Explain how to check a division answer

For j and k, find the unknown.

(2) Analyze What is the quotient when the product of 20 and 5 is divided

by the sum of 20 and 5?

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Lesson 2 17

3.

10 + 20 = 30Write two subtraction facts using these three numbers

4.

10 ∙ 20 = 200Write two division facts using the same three numbers

* 5.

(2) a Using the properties of multiplication, how can we rearrange the

factors in this expression so that the multiplication is easier?

(25 ∙ 17) ∙ 4

b What is the product?

c Which properties did you use?

Justify Decide if the statements in 9–10 are true or false, and explain why.

(1) a What is the absolute value of –12?

b What is the absolute value of 11?

Analyze In 13–17, name the property illustrated.

(2) a Name the four operations of arithmetic identified in Lesson 2.

b For which of the four operations of arithmetic do the Commutative and Associative Properties not apply?

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For multiple choice problems 24–26, choose all correct answers.

* 24.

A counting numbers (natural numbers)

B whole numbers

C integers

D none of these

* 25.

B whole numbers

C integers

D none of these

* 26.

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how much does object B weigh?

When we read a novel or watch a movie we are aware of the characters, the setting, and the plot The plot is the storyline Although there are many different stories, plots are often similar

Many word problems we solve with mathematics also have plots

Recognizing the plot helps us solve the problem In this lesson we will consider word problems that are solved using addition and subtraction The problems are like stories that have plots about combining, separating, and comparing

Problems about combining have an addition thought pattern We can

calculate a desired result

some + more = total

s + m = t

Ibs

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