math est 1 - texas state university

MATH QUEST 1TABLE OF CONTENTS

i

CHAPTER 1: EXPLORING INTEGERS ON THE NUMBER LINE

1.1 Building Number Lines 1

1.2 Modeling elevation, temperature, and Time with Number Lines 15

1.3 Modeling Addition on the Number Line 28

1.4 Modeling Subtraction on the Number Line 45

1.5 Adding and Subtracting Large Numbers 68

CHAPTER 2: MODELING PROBLEMS ALGEBRAICALLY

2.1 Variables and Expressions 89

2.2 The Chip Model 110

2.3 Solving Equations 121

2.4 Solving Equations on the Number Line 139

2.5 Graphing on a Coordinate Plane 147

INDEX

Check ups and Activities 156

Additional Resources 192

MATH QUEST 1ACKNOWLEDGMENTS

ii

AUTHORS

Max Warshauer

Hiroko Warshauer

Terry McCabe

TECHNICAL EDITOR

Genesis Dibrell

TECHNICAL AND CONTENT INPUT

Sam Baethge

Sammi Yarto

STAFF SUPPORT

Patty Amende

Copyright © 2018 Texas State University – Mathworks. All rights reserved.

For information on obtaining permission for use of material in this work, please submit written requests to Mathworks, 601 University Drive, San Marcos, TX 78666, fax your request to 512-245-1469, or email to [email protected].

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of Mathworks. Printed in the United States of America.

MATH QUEST 1PREFACE

iii

Welcome to Math Quest Level 1. As you work through the book, you will be introduced to basic ideas about numbers and the use of variables. This will provide a foundation for algebraic thinking and reasoning that is used in your future math courses. In Chapter I, you will learn about the number line, and how to use the number line to model addition and subtraction of positive integers. In Chapter 2, you will learn about variables and use variables to model word problems and solve equations. This will be done both with a number line as well as graphing on a coordinate plane.

This 6th edition of Level 1 revises earlier editions with the contents updated and revised. A special thanks to Genesis Dibrell for her help in editing, and to all of our Math Camp teachers for their help and suggestions. Particularly, we would like to thank Melissa Freese and Molly Bending for teaching in the camp, piloting earlier editions, and suggestions for changes.

1

CHAPTER 1EXPLORING INTEGERS ON THE NUMBER LINE

SECTION 1.1 BUILDING NUMBER LINES

OBJECTIVES

• Graph positive and negative integers on a number line

• Order numbers

Welcome to the world of mathematics! We are going to explore many new ideas and learn the language of mathematics. You will learn how to create and use activities to understand math problems. You will practice explaining what a problem is about by telling and acting out the story of how to explain your solution. This will help you to learn how to visualize a problem and express yourself precisely in developing a solution.

Let’s begin by thinking carefully about numbers. Numbers are part of the mathematical alphabet, just like letters are used in the English language to form words. You have used numbers for counting and representing quantities. When we think of the number one, we have in mind a picture, such as one dot.

Similarly, the number two describes a different quantity. We could, in fact, use a picture with dots to describe the number 2.

We call this way of thinking of numbers the “set model.” While this is one way of using numbers, in fact, numbers are a basic idea in mathematics that can be used to describe many different ideas. Another way of using numbers is to describe locations. We can do this by drawing a picture of the numbers you use on a number line, much like a thermometer.

Chapter 1 Exp lor ing Integers on the Number L ine

2

EXPLORATION: CONSTRUCTING A NUMBER LINE

1. Draw a straight line.

2. Pick a point on the line and call this point the origin. Label the origin with the number 0.

3. Locate 2 on the number line.



6. What could we call the point one unit to the left of 1?

On your number line, pick a point to be the origin. Notice that the line continues in both directions without ending. We show this with the arrows at the ends. Label the origin with 0. We can think of 0 as the address of a certain location on the number line.

0

Next, mark off some distance to the right of the origin and label the second point 1.

0 1

Continue marking off points to the right at the same distance as above and label these points 2, 3, 4, and so on.

0 1 2 3 4

Sect ion 1 .1 Bui ld ing Number L ines

3

What do we mean by distance? The distance between two points on a number line is the number of unit spaces between the points.

The numbers 1, 2, 3, 4, and so on, which we can also write as +1, +2, +3, +4, ..., are called positive integers. In order to label points to the left of the origin, we use negative integers (–1, –2, –3, –4, …). The sign in front of the numeral tells us which side of zero we are on. Zero is not considered a positive or negative number. The positive integers, the negative integers, and zero combined is called the set of integers.

We could call this way of thinking of numbers the “number line model.” So, we can see that numbers can be used in different ways to help us describe the quantity of something using the set model, or a location using the number line model. Notice that the location is also a quantity, namely the number of spaces from the origin to the point labeled with a given number.

4 3 2 1 0 1 2 3 4

Negatives PositivesZero

– – – –

Here is something to think about: How could we use the set model to describe negative integers?


4

ACTIVITY: MAIL MIX–UP

BRAINSTORM: What did you learn from the Mail Mix–up activity?

ACTIVITY: UNITED WE STAND

CLASS PROBLEMS A:

1. The post office is located on the origin of Max Street. We label its address as 0. The laboratory has address 6 and the zoo has address 9. Going in the other direction from the origin, we find a candy shop with address –4 and a space observatory with address –7. Draw a number line representing Max Street. Label each of the above locations on the number line. Watch your spacing.


5

2. We will now make some number lines.

a. Use the line below to mark off and label the integers from 0 to 10 and from 0 to –10. Use a pencil to experiment because you might need to erase. Watch your spacing.

0 1010–

b. Make a number line from –20 to 20. Be careful labeling, or you may not have enough room.

0 2020–


6

GREATER THAN AND LESS THAN

We say 5 is greater than 2 because 5 is to the right of 2 on the number line. “Greater than” means “to the right of” when comparing numbers on the number line. We use the symbol “>” to mean greater than. We write “5 is greater than 2” as“5 > 2.” We also say that 2 is less than 5 because 2 is to the left of 5. “Less than” means “to the left of” when comparing numbers on the number line and can be written as “2 < 5.” Some people like the “less than” symbol because it keeps the numbers in the same order as they appear on the number line.

CLASS PROBLEM B:

3. Rewrite each of the following statements using the symbols < or >. Check to see if your answer is correct by locating the numbers being compared on a number line.

Example: –3 is less than 8 is written as –3 < 8.

This means that –3 is to the left of 8 on the number line.

a. 9 is greater than 6 is written as .

This means that

.

b. 4 is less than 7 is written as .

This means that

.


7

c. –3 is greater than –5 is written as .

This means that

.

d. –4 is less than 1 is written as .

This means that

.

e. 8 is greater than 0 is written as .

This means that

.


8

EXERCISE A:

4. Compare each pair of the numbers below and decide which symbol to use, > or < between the numbers.

Example: 4 > 2

a 2 8 g. 4 –1

b. –2 –5 h. –4 –8

c. –3 5 i. –4 –3

d. 3 –5 j. –5 2

e. –2 –8 k. –7 –3

f. 8 2 l. 4 –3


9

EXPLORATION: SCALING UP

5. Use the line below to mark off the numbers from 0 to 100 and from 0 to –100 with equal distances by tens. Use a pencil to experiment.

0 100100–

a. Is the distance from 0 to 50 the same as the distance from 50 to 100? How can you tell?

b. Measure the distance from the following pairs of numbers: from 10 to 20, 30 to 40, and 70 to 80. Are the distances the same? How can you tell?

c. Find the distance between the following pairs of numbers: from –10 to –20, –30 to –40, and –70 to –80. Are they the same? How can you tell?

d. Do you need to rework your markings? Why or why not?

e. Estimate the location of the following numbers on your number line:

15, 25, 55, –15, –25, –75, 22, 21, –22, –21, 87

Be able to explain how you decided on their placement.


10

VERTICAL NUMBER LINES - THERMOMETERS

Notice that we can move the number line from a horizontal to a vertical position.

We would then have a number line that looks like a thermometer:

1

0

2

1

2

3

3

–

–

–

The numbers become greater as you go up and less as you go down. We can see that 3 is greater than –2, written 3 > –2, since 3 is higher than –2.

Draw a thermometer on the side of the page to help you answer questions #6 –12.

6. It is –7° C when Penelope visits Anchorage, Alaska. It is –9° C when Victoria visits Poughkeepsie, New York. Which temperature is colder? Explain.


11

7. The temperature in Toronto one cold day is –10° C. The next day the temperature is –11° C. Which temperature is colder?

8. One cold day, the temperature in Fargo, North Dakota, is –10° C and –14° C in St. Paul, Minnesota. Which temperature is colder? How much colder?


12

9. At 7:00 a.m., Chicago’s O’Hare Airport has a temperature of –9° C. At 11:00 a.m. that same day, the temperature reads –4° C. Did the temperature rise or fall? By how much did the temperature rise or fall?

10. In Madison, Wisconsin, the morning temperature is –3° C. In the evening the temperature reads –6° C. Did the temperature rise or fall? By how much did it rise or fall?


13

11. Albert is on a flight of stairs 87 steps above the ground floor. Elaine has gone into the basement 78 steps down from ground level (let’s call it the –78th step). Who is farther from ground level?

12. The temperature in McAllen, Texas, on a hot summer day, is 98°F. The temperature in neighboring Mission, Texas, is 103°F. Which city has the hotter temperature? How many degrees hotter is that city?


14

SUMMARY:

What did you learn in this lesson?

What is a number line?

Sect ion 1 .2 Model ing E levat ion, Temperature, and Time with Number L ines

15

SECTION 1.2 MODELING ELEVATION, TEMPERATURE, AND TIME WITH NUMBER LINES

OBJECTIVES:

• Work with distance between points

• Define and use absolute values

• Relate distance with absolute values

• Use number lines in application problems

As you do the following exploration, you can think of locations along a street as points on a number line.

EXPLORATION: FINDING DISTANCE

1. Plot the following locations, using the number line below to represent Max Street. The school has address 7, the dentist has address 3, the grocery store has address –2, and the ice cream shop has address –5.

2. What is the distance between each location and the post office if the post office has address 0?


16

3. What is the distance

a) between the school and the dentist?

b) between the ice cream shop and the grocery store?

c) between the dentist and the grocery store?

d) between the ice cream shop and the school?

e) between the dentist and the ice cream shop?

f) between the school and the grocery store?


17

PRACTICE PROBLEMS A:

Draw a number line below to answer the following questions.

1. Measure the distance from 0 to 5 and from 5 to 10. Are they the same?

2. Measure the distances from 1 to 2, 3 to 4, and 8 to 9. Are they the same?

3. Do you need to rework your markings?


18

4. What is the distance from

a. 0 to 5

b. 5 to 0

c. 0 to –5

d. –5 to 0

e. 3 to 0

f. 0 to 3

g. –3 to 0

h. 0 to –3

i. 2 to 2

j. –2 to –2

k. 0 to 0

l. – 2 to 2

Write down any pattern that you observe about these distances.


19

DISTANCE FROM THE ORIGIN:

Notice that the numbers 10 and –10 are both 10 units from 0. We have a special name for the distance of a number from 0, namely the absolute value of that number.

10 0 10

ten units left of zero ten units right of zero

–

We write:

Absolute value of 10 equals 10 as: | 10 | = 10

Absolute value of –10 equals 10 as: | –10 | = 10

The vertical bars on the left and right of the number are the absolute value symbols. Absolute value gives us a measure of the size or the magnitude of a number, whereas positive and negative tell us the location of the number relative to zero. Since both 10 and –10 are the same distance from 0, they have the same absolute value.

PRACTICE PROBLEMS B:

1. Find the following absolute values.

a. | –3 | =

b. | 5 | =

c. | –7 | =

d. | 0 | =


20

2. Find the distance each number is from zero.

a) –7

b) 9

c) 0

d) –9

3. Find the absolute values of the following numbers.

a) 8

b) –8

c) 10

d) –10

e) 19

f) –21

g) –42

h) 33


21

4. In each problem below, place the correct symbol < , > or = between the pair of numbers.

a) | –7 | | 5 |

b) | 7 | | –3 |

c) | –9 | | 9 |

d) | 32 | | –47 |

e) | –8 | 6

f) 4 | –3 |

g) 62 | 71 |

h) | –28 | 25


22

PROBLEMS:

Draw a number line for each problem and show how it helps explain your answers.

1. In the year 540 BC, Pythagoras, a Greek philosopher and mathematician, formulated the Pythagorean Theorem, which is still used today. Archimedes, another famous Greek mathematician, worked on a variety of problems. In 240 BC, he formulated the area and volume of the sphere. Which formulation occurred earlier in history?

2. If Cave Travis is located 325 feet below the surface and Cave Crockett is located 413 feet below the surface, which cave is farther from the surface of the ground?


23

3. August is atop a hill that is 128 feet above sea level. Hugo is in a cave that is 512 feet below sea level. Who is farther away from sea level?

4. A pilot in a helicopter hovers 463 feet above the surface. There is a spelunker exploring in a cave 436 feet below the surface. Who is closer to the surface?


24

EXERCISES:

1. Find the distances between the following pairs of numbers.

a. 7 and 3

b. 3 and 7

c. –4 and –2

d. –2 and –4

e. 6 and –1

f. –1 and 6

g. –3 and –7

h. –7 and –3

2. Use the number line to explain any patterns you observe.


25

3. Jasper noticed the temperature on the thermometer read 7°C. When he looked at the thermometer 5 hours later, the temperature was 3°C. Did the temperature rise or fall? By how many degrees did the temperature change?

4. The temperature in Boston was –2°C at 7 pm. Later that night, the temperature reached –5°C. What was the change in temperature? Did it get colder or warmer?

5. When Penelope read the thermometer, the temperature read –1°C. When Victoria read the thermometer a few hours later, the temperature read 8°C. What was the change in temperature?


26

CHALLENGE PROBLEMS:

1. If Pythagoras proved the Pythagorean Theorem in the year 540 BC, how long ago did he prove his famous theorem? (Use the current year.)

2. Archimedes discovered the volume of a sphere in the year 240 BC. Whose discovery is more recent, Archimedes’ or Pythagoras’? Explain.


27

SUMMARY:

What did you learn in this lesson?

What is a number line? What is absolute value?


28

SECTION 1.3 MODELING ADDITION ON THE NUMBER LINE

OBJECTIVES:

• Addition using the number line

• The Car Model

Many of you already know what it means to add two numbers such as 3 and 2. You write 3 + 2 and get 5 as the sum. We have a way to show what addition looks like on the number line. We introduce the Car Model here and show that we can add not only positive integers like 2 and 3, but also negative integers.

FOUR-STEP CAR MODEL FOR ADDITION:

Step 1: Draw a number line and place your car at the origin.

Step 2: If the first of the two numbers that you are adding is positive, the car faces right or the positive direction. If the first of the two numbers is negative, the car faces left or the negative direction. Drive to the location given by the first number and park the car.

Step 3: Examine the second of the two numbers you are adding. If this number is positive, point the car to the right or the positive direction. If the second number is negative, point the car to the left or the negative direction.

Step 4: Because you are adding, the car will move forward in the direction that the car is facing. Move the car the distance equal to the absolute value of the second number. You reached the sum of the two numbers.

Here is an example of the Four-Step Car Model for Addition with each step carefully explained. Use your number line and car to follow along.

Consider the problem 3 + 4. The two numbers we're adding are 3 and 4 (which we also know as +3 and +4). We begin with the car at 0, facing to the right. We travel 3 units and stop. Because we are adding 4 to 3, we face towards the right because 4 is positive, and travel 4 more units, going forward for addition.

Here is a picture of 3 + 4. Since the car ends up at location 7, we write 3 + 4 = 7.

7 0 7

3

6541 2 312346 5

4

– – – – – – –

Sect ion 1 .3 Model ing Addi t ion on the Number L ine

29

EXPLORATION 1: CAR MODEL EXAMPLE

Consider the problem –3 + 4. How do we start the process? In which direction do we move first and how far?

We start at 0. We face left because –3 is negative, then move 3 units to the left (drive forward). We then turn the car around to face right because the next number, 4, is positive. We move 4 units to the right (forward), ending up at location 1. The result –3 + 4 = 1 is demonstrated below:

0 7

3

6541 2 3

4

7 12346 5– – – – – – –

–

For the next example, consider the problem –3 + (–4) or simply –3 + –4. Use the Four-step Car Model to find the sum.

Compare your work to the picture and explanation below:

0 7

3

6541 2 3

4

7 12346 5– – – – – – –

–

–

We began by facing left, then moving 3 units to the left. The car then continues to face left because the next number is negative. When the car moves 4 more units to the left it ends up at location –7. We have: –3 + (–4) = –7.


30

PRACTICE PROBLEMS A:

Find the following sums. Use the number line to illustrate your answers.

a. 4 + –5

b. –3 + 9

c. 7 + –4

d. –2 + 6

e. –7 + –5

f. 8 + –9

g. –9 +–2

h. –6 + –6


31

EXPLORATION 2:

Find the sums for each. Use the number line to show your work.

1. a. 5 + 4 =

b. –5 + 4 =

c. 5 + –4 =

d. –5 + –4 =

2. a. 2 + 6 =

b. 2 + –6 =

c. –2 + 6 =

d. –2 + –6 =


32

3. a. 7 + 3 =

b. –7 + 3 =

c. 7 + –3 =

d. –7 + –3 =

4. a) 8 + 5 =

b. –8 + 5 =

c. 8 + –5 =

d. –8 + –5 =

What patterns do you notice?


33

4. Do the following word problems. Show your work on a number line.

a. Camila observes that the temperature is –3°C. If it rises 7°C in the next two hours, what will the new temperature be?

b. Lisa observes that the temperature is 2°C. It falls 9°C that night. What was the low temperature that night?


34

c. Jeremy takes 8 steps forward from his starting point then takes 6 steps back. How far is Jeremy from where he started?

d. David takes nine steps backward from his starting point and then three steps forward. How far is he from where he started?


35

EXPLORATION 3: RELATED SUMS

Find the sum:

a. –5 + –4

b. –4 + –5

c. –2 + 1

d. 1+ –2

Discuss any observations that you notice.

EXERCISE A:

Find the sums for each. Use the number line to show your work.

1. a. –9 + 4 b. 4 + –9

2. a. 7 + 2 b. 2 + 7


36

3. a. –8 + –1 b. –1 + –8

4. a. 0 + –3 b. –3 + 0

What patterns do you observe? Describe several.

EXERCISE B:

Do the following word problems. Use the number line to show your work.

5. Frankie checks the temperature and it is –9°C. If the temperature warms up by 8°C, what is the new temperature?


37

6. Madeline is on the “home” square of a long hopscotch board which extends in both directions. We think of hopping backwards as going in the negative direction.

a. If she hops backwards seven squares and then forward four squares, on which square would she land? Notice that we can write this as a mathematical sentence or equation –7 + 4 = –3.

b. If Madeline hops backwards five squares and then forward eight squares, on which square would she land? Write this as an equation.


38

c. If Madeline hops backwards six squares and then hops backward eight squares, on which square would she land? Write this as an equation.

d. If Madeline hops forwards four squares and then backwards nine squares, on what square would she land? Write this as an equation.


39

7. It was –5 °C in the morning. The temperature rose 8°C. What is the temperature now?

8. Oliver checks the temperature and it is 5°C at 5 p.m. By 10 p.m., the temperature has dropped 8°C. What is the new temperature?


40

9. Yumiko is snorkeling in the San Marcos River 5 feet below the surface. She dives 3 feet deeper. How many feet below the surface is Yumiko now?

10. If Josephine borrows $5, we say her balance is –5 dollars. If Josephine has $14 in her account now but borrows $16, then what is Josephine’s new balance in her account?


41

11. Isaac has –$20 in his account. He needs to borrow $10 more. What will his balance be now?

12. If Vincent has $17 in his account but withdraws $20, what will be his new balance?


42

13. One cold wintry day, the temperature in Wilmington, North Carolina, is a record breaking –4°C. If the temperature falls another 7°C, what will the new temperature be?

14. If a football player has lost 6 yards in one play, 2 yards in another play, and then gains 4 yards in the final play, what is the net gain or loss?


43

You have been using many number lines such as the sentence strip and the number line on the floor. Now consider using a number line that you can imagine in your head.

1. Using a number line in your head, see if you can predict whether the following sums are positive or negative:

a) –30 + 40

b) 30 + –40

c) 10 + 30

d) –10 + –30

e) –10 + –10

f) –10 + 10

2. Explain how you determined whether the sum would be negative or positive.

3. Check your work on a physical number line and find the sum for each of the problems in part 1 above. Check to see if your prediction matches your work.


44

EXPLORATION:

With our car model, the car moves forward when we add. What do you think we should do when we want to subtract a number from another number? Write your best guess of how to do subtraction:

SUMMARY:

Make a list of what you learned in this lesson.

Sect ion 1 .4 Model ing Subtract ion on the number l ine

45

SECTION 1.4 MODELING SUBTRACTION ON THE NUMBER LINE

OBJECTIVES

• Subtraction on the number line

• The Car Model

With our car model, addition involves moving the car forward. What do you think we do when we want to subtract a number from another number? If you said the action of subtraction is backing up, you are correct.

FOUR-STEP CAR MODEL FOR SUBTRACTION:

Step 1: Draw a number line and place your car at the origin.

Step 2: If the first of the two numbers in the subtraction problem is positive, the car faces right, the positive direction. If the first number is negative, the car faces left, the negative direction. Drive to the location given by the first number and park the car.

Step 3: Examine the second of the two numbers you are subtracting. If this number is positive, point the car to the right, the positive direction. If the second number is negative, point the car to the left, the negative direction.

Step 4: Because you are subtracting, the car will move backward from the direction that the car is facing. Move the car the distance equal to the absolute value of the second number. You reached the difference of the two numbers.


46

EXAMPLE:

Use the Four-Step Car Model for Subtraction to find the difference: 5 – 2. Use your number line and car model to follow along.

We first move the car 5 units to the right. Because the second number is also positive, the car remains facing in the positive direction. Because the operation is subtraction, the car must back up 2 units. The car ends at location 3, the difference of 5 and 2. We write 5 – 2 = 3.

¯7 0 7

5

2

6541 2 3¯1¯2¯3¯4¯6 ¯5

Here are the four steps broken down:

Step 1: Start at 0.

Step 2: Notice that the first number is 5 so face the positive direction and go to 5.

Step 3: The second number 2 is positive so continue to face in the positive direction.

Step 4: The operation is subtraction, so go backwards from 5 a distance of 2 units. You should now be at 3.

We write the subtraction sentence as 5 – 2 = 3.

Notice that the only difference between the Four-Step car model for addition and for subtraction is in Step 4. In addition the car moves forward, and in subtraction the car moves backwards.


47

Now try 2 – 5 using the Four-Step Car Model and explain each step to find the difference.

CLASS DISCUSSION:

Use the Four-Step Car Model to show how to find the difference –3 – 5.


48

EXERCISE A:

Use the car model to calculate each of the following problems. Drive carefully.

1. a. –4 + 3 = ________

2. a. –1 – 5 = ________

3. a. 8 – 3 = ________

4. a. 5 – 1 = ________

5. a. 3 – 5 = ________

6. a. –4 – 5 = ________

b. –4 – 3 = ________

b. –1 + 5 = ________

b. 3 – 8 = ________

b. 1 – 5 = ________

b. –3 – 5 = ________

b. 4 – 5 = ________


49

OPPOSITES:

Notice numbers on the number line with the same absolute values. For example, 1 and –1 or 5 and –5. These pairs of numbers are the same distance away from 0. Two numbers whose distance from 0 is the same are called opposites of one another.

We say –1 is the opposite of 1. We also say that 1 is the opposite of –1.

0 1 2 3 4 55 4 3 2 1– – – – –

What is the opposite of 2 on the number line? That is, what number on the other side of 0 is the same distance from 0 as 2? Mark the points on the number line above.

EXPLORATION: NUMBER LINE OPPOSITES.

1. Locate the opposites of 3, –5, and –7.

2. How do you write the opposite of 3?

3. How do you write the opposite of –5?

4. How do you write the opposite of –7?

5. What is the opposite of x if x represents a number?


50

The opposite of the number 6 is –6. The opposite of the number 9 is –9. For a number x, we write –x as the opposite of x. Notice that –x is equidistant from 0 but on the other side of where x is situated. What is the opposite of the number –5? Locate –5 on the number line and use our previous observation above to determine what the opposite of –5, which we write as –(–5), should be. Did you notice that 5 is the number equidistant from 0 and the other side of –5? So, –(–5) must be the same number as 5.

1. What is the opposite of –8? (Write the number in two ways.)

2. What is the opposite of –10? (Write the number in two ways.)

How interesting that the opposite of the opposite of the number x is x itself. We state this as a property.

PROPERTY 1:For a number x, the opposite of the opposite of a number is the original number. –(–x) = x

EXPLORATION: (WHOLE CLASS)

Suppose we have numbers n and m as shown on the number line. Label where the numbers –n and –m would be located. Explain how you decided on the location.


51

EXERCISE B

Find the opposites of the following numbers. Use the number line to illustrate your answers.

1. 6

2. –10

3. –8

4. 7

5. –7

6. –12

7. 24

8. –30

9. x (if x is a positive number)

0 x

10. –a (if a is a negative number)

0a

For example, if x is 3 then we have its opposite –3. The opposite of –3 is –(–3) but notice it is also exactly 3.

03 ( 3)=3– ––


52

11. Find the following sums

a. 4 + (–4) =

b. 6 + (–6) =

c. –9 + 9 =

d. –1 + 1 =

e. Write a rule or pattern that you notice.

PROPERTY 2:The sum of the number and its opposite is zero. We write x + –x = 0 for any number x.

For example, we have 1 + (–1) = 0 and 2 + (–2) = 0. Also, –3 + 3=0 and –4 + 4= 0.

Since 1 – 1 = 0 and 1 + (–1) = 0, we can say that when we subtract 1, it’s the same as adding –1, the opposite of 1. If, in another case we subtract –1, we add the opposite of ¯1 which is 1. In other words, 1 – (–1) = 1 + 1.

You may have noticed this property already when we have subtraction with a negative number.


53

EXPLORATION: (SMALL GROUP)

Calculate the following sums or differences using the car model.

1. a. 7 – 3 = ________

2. a. 3 – 5 = ________

3. a. –2 – 4 = ________

4. a. 0 – 5 = ________

b. 7 + –3 = ________

b. 3 + –5 = ________

b. –2 + –4 =________

b. 0 + –5 =________

What pattern do you see in these problems? Write a rule for this pattern.

Some of the problems above involve subtracting a negative number. Write down the process you used (in your own words) to act out this subtraction. Share with your group what you have written.


54

What you probably noticed was that n – (–m) is the same as n + m. Subtracting –3 from 7 is the same as adding 3 to 7.

PROPERTY 3:For numbers n and m, subtracting a number is the same as adding its opposite. n – m = n + (–m).

PRACTICE PROBLEMS

Verify property 3 by calculating the following sums or differences.

1. a) 3 – 7 = _____

2. a) 1 – 7 = _____

3. a) –5 – 1 = _____

b) 3 + –7 = _____

b) 1 + –7 = _____

b) –5 + –1 = _____

SUBTRACTING A NEGATIVE INTEGER

We next look at 5 – –2 and model this on the number line.

0 7

5¯2

6541 2 3

You found that 5 – –2 = 7.

Is there another way that you can use a Car Model to go from 5 to 7?

Many of you probably noticed that 5 + 2 = 7.

0 7

52

6541 2 3


55

PRACTICE PROBLEMS

1. Check to see if the relationship that you observed works for the subtraction problem 6 – –5 and the addition problem 6 + 5. Show your work using the Car Model on the number line.

2. a) 8 – –2 = _____ b) 8 + 2 = _____

PROPERTY 4:For numbers n and m, n – (–m) = n + m.

You probably noticed that there is a special case of Property 3 when you subtract a negative number. This is Property 4.

We know that –(–m) = m from Property 1, so Property 4 is a special case of Property 3.

Verify Property 3 by calculating the following sums or differences.

3. a) 2 – –6 = _____

4. a) –5 – –4 = _____

b) 2 + 6 = _____

b) –5 + 4 = _____


56

PRACTICE PROBLEMS

Calculate the following sums and differences. Use the car models as needed.

1. a. 3 – 9 = __________

b. 3 + –9 = __________

c. 3 – –9 = __________

2. a. 8 – 5 = __________

b. 8 – –5 = __________

c. 8 + –5 = __________

3. a. –2 + –5 = __________

b. –2 − 5 = __________

c. –2 – –5 = __________


57

4. a. –1 – –5 = __________

b. –1 + 5 = __________

c. –1 + –5 = __________

5. a. –8 – –4 = __________

b. –8 + –4 = ___________

c. –8 – 4 = __________

6. a. –7 – 6 = __________

b. 3 – –3 = __________

c. 0 – –9 = __________


58

EXERCISE C:

Perform the following subtraction problem by rewriting it as an addition problem:

1. 15 – 8 = __________

2. 3 – 9 =__________

3. –4 – 2 = __________

4. –6 – 7 =__________

5. –14 – (–3) =__________

6. 4 – –9 =__________


59

EXERCISE D:

Answer the following problems in two ways: Using a number line to find the temperatures, and also writing down an addition or subtraction problem that answers the question. Which approach do you like to do first?

1. It was 7°C in the morning. The temperature has dropped 8°C. What is the temperature now?

2. It was –4°C in the morning. The temperature rose 2°C by noon. What was the temperature at noon?


60

3. It was –4°C in the morning. The temperature rose 7°C by noon. What was the temperature at noon?

4. It was –4°C at noon. By early evening, the temperature had dropped 7°C. What was the temperature then?


61

5. Make up two problems, one using temperature and one demonstrating another use of negative numbers.

a. Temperature Problem:

b. Negative Number Problem:


62

RANDOM WALK:

This is a fun game that uses addition on the number line.

Scientists use the process of a random walk to study problems. We now describe a simple game based on the idea of a random walk on a number line.

Equipment: You will need a number line with the numbers from –15 to 15 labeled. You will also need a coin for each player, a die (one of a pair of dice), and two small markers.

Set–up: Each player places a marker by the zero on the number line.

Playing Rules: Take turns moving. Flip the coin to decide who goes first. During a turn, a player rolls the die and flips the coin. If the coin lands on heads, the outcome is the value of the die roll. If the coin lands on tails, the outcome is the negative of the value of the die roll. For example, a tails and a roll of 4 gives the outcome of –4. Heads and a roll of 6 gives the outcome of +6. The player then adds this outcome to the position he or she is on to compute the new position. Players should also check that their opponents are making the correct moves!

Object of the Game: To escape past 15 to the right or –15 to the left. The player that lands on a number greater than 15 or less than –15 wins, ending the game.

Example 1: The player is on the +9 position and rolls a 4, together with the coin landing on tails. We then say that the outcome is –4. The new position is computed by 9 + (–4) = +5.

Example 2: The player is on the –7 position and rolls a 5 together with the coin landing on heads. We then say that the outcome is +5. The new position is computed by –7 + (+5) = –2.

Can you compute your next position without using the number line? For example, suppose you are on +3 and you roll a 5 and the coin lands on tails. What is your new position? As you might expect, the moves of each player are unpredictable. They are random. Look up the word random in the dictionary for fun. Do you see why this is called a random walk?

Understand the game: Playing alone or with an opponent, keep track of the number of moves it takes a player to escape. How does the average number of moves change if you restrict the number line to –10 and 10? Try –20 to 20.


63

EXERCISE E:

1. On a day at the race track, Yosh won $5 on the first race, lost $7 on the second race, lost $5 on the daily double, then won $2 on each of the next two races. How much money did Yosh win or lose?

2. The Dallas Cowboys are on their own 20-yard line. On four consecutive plays, running back Emmit Smith rushes for +7, –2, –8, and +15 yards. What yard line are the Cowboys on at the end of this sequence of plays?


64

3. Suppose time is measured in days and 0 stands for today. What number would represent:

a. Yesterday?

b. Tomorrow?

c. The day after tomorrow?

d. A week from today?

e. A week ago?


65

4. At 5 a.m. the temperature in Juneau, Alaska, was –20ºF. If the temperature rises 30ºF, what is the new temperature?

5. Joe is playing the fun game “Mother May I” with Sam. Sam instructs Joe to first take 3 steps forward, then 2 steps backward. He then tells Joe to take 5 steps forward and then 2 more steps forward. The final instruction is to take 3 steps back. How many steps is Joe from his starting point?


66

6. Max went on a diet. The first week he lost 5 pounds and the second week he lost 8 pounds. What was the total amount of weight Max lost?

7. The average temperature in Steven’s Point, Wisconsin, is –4 ºF one cold day and the average temperature on that very same day in Milwaukee, Wisconsin, is –7 ºF. Which city is colder and by how may degrees?


67

SUMMARY:

What properties did you learn?


68

SECTION 1.5 ADDING AND SUBTRACTING LARGE NUMBERS

OBJECTIVES

• Adding and subtracting large numbers

• Review

We continue to explore addition by looking at larger numbers. Let’s do some problems with two digit numbers.

PRACTICE PROBLEMS:

1. Find the following sums.

a. 12 + 17 = _________

b. 19 + 28 = _________

c. –12 + –17 = _________

d. –19 + –28 = _________

CLASS DISCUSSION:

As a class, come up with a set of rules for adding two positive numbers or two negative numbers. Record those rules below:

Sect ion 1 .5 Adding and Subtract ing Large Numbers

69

2. Do the following four problems using the rules you came up with. Find the following sums.

a) 13 + 19 = _________

b) –13 + – 19 = _________

c) 16 + 13 = _________

d) –16 + –13 = _________

Here are our rules. Compare them to rules that you and the rest of the class have discovered.

Rule 1: If you are finding the sum of two positive numbers, add their absolute values and the answer is positive.

Rule 2: If you are finding the sum of two negative numbers, add their absolute values and the answer is negative.

Do these rules agree with what you suggested?


70

Let’s now explore the following problems and try to see how we can modify our rules to handle them:

3. Find the following sums.

a) –13 + 19 = _________

b) 13 + –19 = _________

c) 26 + –33 = _________

d) –26 + 33 = _________

GROUP WORK:

Do you see a pattern? Try writing a rule for these problems. Can you use absolute values to describe what you did?

Does your rule use the number line? Do you have a picture in your mind that makes you believe your rule is correct?


71

4. Do the following four problems using this rule. Find the following sums.

a. 26 + 33 = _________

b. –26 + 33 = _________

c. –45 + 32 = _________

d. 45 + –32 = _________

Below is the picture for the sum –26 + 33. Does your rule give you the answer of 7?

26 0 7

¯ 26

33

–

Here is our rule. Compare it to your rule.

Rule 3: To add two numbers with opposite signs we use the following steps.

Step 1: Take the absolute value of each of the numbers.

Step 2: Subtract the smaller absolute value from the larger absolute value.

Step 3: The answer is the difference found in Step 2 with the sign of the number with the larger absolute value.


72

EXERCISE A:

Practice doing the following problems in class.

Example: –103 + 94 = –9 because

Step 1 Find the absolute values, |¯103 | = 103 and | 94 | = 94

Step 2 Take the difference of the absolute values, 103 – 94 = 9

Step 3 The answer is –9 since the number with the larger absolute value is –103, a negative number.

Notice driving along the number line, we will still be on the negative side!

103 9 0

103

94

–

– –

1 a. –12 + 19 = _________ b. 19 + –12 = _________

2. a. 43 + –27 = _________ b. –43 + 27 = _________

3. a. ¯20 + 30 = _________ b. 20 + ¯30 = _________


73

4. a. –31 + 39 = _________ b. 31 + 39 = _________

5. a) –40 + 29 = _________

b) –65 + 49 = _________

c) 54 + –67 = _________

d) –140 + 182 = _________

e) 235 + –594 = _________

f) –584 + 367 = _________

g) 450 + –750 = _________

h) –640 + 480 = _________

i) –600 + 600 = _________

j) 125 + –125 = _________


74

During a football game, Vincent gained 13 yards on one play and 22 yards on the next play. His total or net yardage gained is calculated by using addition: 13 + 22 = 35 yards. If he gains 16 yards and then loses 9 yards, his net gain can be calculated by 16 – 9 = 7. We can also think of this as addition by writing the sum: 16 + (–9) = 7. The first play he gained 16 yards and the second play he “gained” –9 yards. Thinking of the problem in this way will help in working the following problems. You can draw pictures or diagrams to help you visualize what is going on.

6. During a football game, Francisco lost 17 yards on one play. On the next play, he gained 25 yards. Use addition to find the total yards Francisco gained on these two plays.


75

7. During a football game, Fineas gained 9 yards on the first play but lost 17 yards on the next play. What was the net yardage gained or lost on these two plays? Did Fineas gain or lose yards after the two plays?

8. In Fairbanks, Alaska, the temperature on January 3 was –12ºF. The next day a cold front moved in and the temperature dropped 23ºF. How cold was it then?


76

9. In Toronto, the temperature on January 10 was –23ºF. The next day the temperature rose 16ºF. How cold was it then?

10. Hiroshi borrowed money from his bank account and currently his balance is $45. He needs to borrow $32 more. What will Hiroshi’s balance be after he borrows the $32?


77

Let's remember Property 3 from Section 1.4. To subtract the number b from the number a, a – b, we can rewrite this difference as a sum: a + –b. Simplify the second term –b as needed (for example: –(–5) = 5). Then use the appropriate rule: 1, 2, or 3.

EXERCISE B:

Calculate the following sums or differences. Do you need a number line? Use a calculator to check your answer.

Example: 37– –24 = 61

1. a. 12 – 19 = b. 19 – 12 = c. –20 – 30 =

2. a. –14 – 24 = b. 19 – –12 = c. –20 – –30 =

3. a. –13 + –29 = b. 23 – 37 = c. –19 – 18 =

4. a. 40 – 29 = b. –65 + 49 = c. 54 – –67 =


78

5. a. –43 + –27 = b. –43 – –67 = c. 31 + 48 =

6. a. –39 + 138 = b. 34 + –18 = c. –35 + 19 =

7. a. –35 – – 19 = b. 27 – 52 = c. 31 – –24 =

8. a. –140 –182 = b. 235 – –549 = c. –484 –367 =

9. a. 142 + –150= b. –130 + 180 = c. 160 – –120=


79

In Section 1.2, we learned about the distance between two numbers. In that section, we counted the spaces between two points on the number line to determine distance. You may have noticed that when you are finding the distance between two positive numbers, you can use subtraction. For example the distance between 2 and 5 is 5 – 2 = 3. You subtracted the lesser number from the greater to find the distance. This process also works when one or both of the numbers are negative.

EXAMPLE:

Find the distance between 3 and –2.

Since 3 is greater than –2, we find the distance by subtracting –2 from 3: 3 – –2 = 5. We can check this by counting on the number line.

0 1 2 3 41234 ––––

To find the distance between two points with coordinates m and n on the number line,

• for m > n, find the distance from n to m as m – n.

• for m < n, find the distance from n to m as n – m.

RULE: FINDING THE DISTANCE BETWEEN TWO POINTS ON THE NUMBER LINE

The distance between two numbers, n and m, is given by |n – m|.


80

Calculate the distance between the following pairs of integers:

a. 8 and –2

b. –5 and 4

c. –3 and –8

d. 6 and –6

EXERCISE C:

1. Alex had a sick pig. During one week the pig lost 18 pounds. The next week, the pig lost 15 pounds. How many pounds did the pig lose during these two weeks?


81

2. Do problem 1 again using pounds lost, such as 18 pounds to be a gain of –18.

3. In a town in Canada it was 15°F. The temperature dropped 27°F because of a cold front. What was the temperature after this drop?


82

4. In a town in North Dakota, it was –17°F on February 3. During a warm spell, the temperature rose 23°F. What was the temperature after the warm front moved in?

5. Our family’s pet pig, Jordan, lost 18 pounds during May and gained 13 pounds during June. What was the net gain or loss during these two months?


83

6. The water level at Lake Travis was up 4 feet and then dropped 7 feet. What is the net gain or loss in water level?

PRACTICE PROBLEMS:

1. Compare the pairs of numbers below and place the appropriate symbols between them. Use < or > .

a. 457 _________ 481

b. –23 _________ –32

c. 191 _________ –3


84

2. Find the distance between the following pairs of numbers using the rule for finding distance.

Example: The distance between –2 and 9 can be found using the rule, |–2 – 9| = |–11| = 11

a. 0 and 18

b. 0 and –14

c. –8 and 0

d. 0 and –1

e. 4 and 7

f. –12 and 16

g. –7 and –19

h. –20 and 10

i. 25 and –15

j. –20 and –40

k. –25 and 25

l. –44 and –12


85

3. Find the following sums or differences. Show all your steps and use the number line if you want to.

a. 54 – 63= _________

b. –31 – 24= _________

c. 35 – –22= _________

d. –26 + 21= _________

e. –45 + –75= _________

f. –36 – –44= _________


86

4. A football team is on their 25-yard line. In four consecutive plays a player rushes for +8, –4, –6 and +16 yards. What yard line will this team be on?

5. The temperature in Anchorage, Alaska, is –15°F. The temperature rises 12°F during the day. What is the new temperature?


87

6. Kennedie’s bank account reads –$56. If Kennedie withdraws $23 from her account, what would the balance be?

7. The temperature in Seattle, Washington, is 3ºC. The temperature dips down another 5º. What is the temperature now?


88

89

CHAPTER 2MODELING PROBLEMS ALGEBRAICALLY

SECTION 2.1 VARIABLES AND EXPRESSIONS

OBJECTIVES

• Define variables

• Create expressions

• Model word problems with equations.

Numbers represent definite things in our mind. The number 5 might be 5 marbles, or it might be 5 units on a number line. But it is “five” of something. Often we will, however, want to represent an unknown quantity. To do this we use variables. For example, Amy may have some marbles, but we don’t know how many she has. We could then write:

A = the number of marbles Amy has.

A variable is a letter that can represent a quantity that is not known. Expressions are mathematical phrases that are formed by using numbers, variables, and mathematical operations.

EXPLORATION: BOYS AND GIRLS

A class is known to have 24 students. If we let the letter B be the variable that represents the number of boys in this class and if we let the letter G be the variable that represents the number of girls in this class, then what expression can be used to represent the number of students in this class?

What are some values of B and G that are possible for this class?

Chapter 2 Model ing Problems Algebra ica l ly

90

ACTIVITY 1: NAMING NUMBERS

1. Discuss how most of us have our given name and then often are called by a nickname or term of endearment. There are a variety of names which one might be called and yet which refer to the same person.

2. Suppose we have a known number 6. Notice that we can “disguise” or describe the number 6 as 2 more than 4 or as 3 less than 9 or even twice as big as 3. We can therefore express our number 6 in a variety of ways while still maintaining equality to 6. 6 = 4 + 2 or 6 = 9 – 3 or 6 = 2 · 3.

In this activity, teams try to guess the different ways the teacher has written a given number. Each team gets 3 guesses. For each correct guess, they get 1 point. You can make a list of as many different ways your team can write the chosen number from this list.

a. The number is x = 8

b. The number is A = 7

c. The number is K = 10

Sect ion 2 .1 Var iab les and Express ions

91

CLASS PROBLEMS:

Draw a number line with points and integer coordinates from –10 to 10. Plot 2 points on it and choose one of them to be our variable. Make a story about how our variable can be found using the other number as a starting point. For example, in the picture below, let r be 2 and s be 5. Suppose we start at 2 and want to get to 5. We can say that we start at 2 and add 3 to get to s = 5. We write this as s = 2 + 3. On the other hand, if we start at 5 and want to get to 2, we can say that we start at 5 and subtract 3 to get to r= 2. We can write this as r = 5 – 3.

0 34– 1 2 43– 2– 1–

sr

55–

As a class, create two stories for the integers plotted below. One group of students can create a story that relates g going to h and the other group from h to g.

0 34– 1 2 43– 2– 1–

hg


92

ACTIVITY 2: RELATIONS AND OPERATIONS

Difference in relation (< and >) versus operation (– and +)

1. Two students will demonstrate “2 is less than 5” on the number line.

2. Another pair of students will demonstrate “2 less than 5”

3. Take a minute to articulate what the difference is between these two statements.

4. A different pair of students will situate themselves on the number line. Determine how their situation can demonstrate a relation and then an operation situation. Try writing this mathematically.

5. Do a sketch with “more than” and “addition.”

6. Write what you have learned about variables, expressions, and inequalities. Give examples of each.


93

EXERCISE A:

1. Translate each of the following into a mathematical expression or an inequality. Draw a number line to illustrate each. (We will assume the coordinates are integers and the scaling is 1 unit length.)

Example: 6 less than x can be rewritten as x – 6

xx - 6

a. 2 more than 5

b. 2 is less than 5

c. 2 less than 3

d. 2 is less than 3

e. 5 is more than 2


94

f. 5 is more than –5

g. 2 more than –5

h. 2 more than A

i. 2 less than A

j. 2 is less than A

k. A is more than 2


95

2. Translate the following expressions into word phrases using “less than” or “more than” rather than “plus” or “minus.”

Example: –2 –7 7 less than –2 not –2 minus 7

a. 8 – 5 is written as

b. 4 + 2 is written as

c. –3 – 2 is written as

d. –4 + 5 is written as

e. x + 2 is written as

f. R – 5 is written as

g. 2 < x is written as

h. 4 > y + 3 is written as


96

EQUATIONS:

Let’s begin with a sentence such as, “A number is 3 more than 7.” Rather than simply writing 10, let’s mathematically write what the sentence says. We use numbers, variables, and operations to form an expression. We combine these expressions to form a mathematical sentence called an equation. An equation is simply a math sentence with an equality sign, = , between two expressions.

PROBLEM 1:

Translate the sentence “A number is 3 more than 7” into an equation.

Step 1: We give the unknown number a name and write

N = the number

N is a variable. It stands for the number we are trying to find.

Step 2: We translate the sentence into an equation.

N

A number is 3 more than 7

= 7 + 3

N = 7 + 3

Step 3: We say that we have solved our problem when we write N = 10.

Step 4: Check to see if the answer makes sense. Is ten three more than 7? Yes!

PROBLEM 2:

Set up an equation to find what number is twice as large as six.

Step 1: We define a variable. Let T = a number that is twice as large as six.

Step 2: Then the statement “A number is twice as large as six” becomes

T = (2)(6) = 2 · 6

Notice that when we put the symbols 2 and 6 next to each other with parenthesis around each, it is understood that we mean to multiply them. So (2)(6) = 2 · 6 = 2 × 6. We do not want to use the symbol x, however, since it could be confused with a variable x.

Step 3: T = 12.

Step 4: Check. Is 12 a number that is twice as large as 6? Yes.


97

EXERCISE B:

Make up a variable for the number. Translate each sentence into an equation and solve for the unknown variable. Does your answer make sense?

Example: A number is 2 less than 4 times 10.

Step 1: Let’s call our unknown number N.

Step 2: N = 4(10) – 2 is our equation for the statement above.

Step 3: The right side is equal to 38. So N = 38.

We have solved the problem! Now let’s check.

Step 4: Is 38 equal to 2 less than four times 10? Yes.

1. A number is 4 less than 11.

2. A number is 3 more than twice 15.


98



5. A number is 3 more than a number that is half of 40.


99

CHARTING THE PROCESS:

We have seen how numbers and variables can be used to translate problems into equations. Consider the problem: “Jeremy is 9 years old. In how many years will Jeremy be 15?” We don’t know the answer immediately, even though, looking at the number line, you probably have a very good idea. What steps will we take to arrive at the answer? How might we begin? Use some of the ideas that we have talked about, including using a variable.

Here is a step-by-step approach. Do your steps resemble the following?

Step 1: Define your variable.

Y = the number of years it takes for Jeremy to reach 15.

Step 2: Translate the problem into an equation.

9 + Y = 15 (an equation with one variable Y)

Step 3: Solve for the unknown variable.

If you look on the number line, you’ll notice you have to move right 6 units from 9 to 15. Y = 6.

Step 4: Check.

Substitute Y = 6 into the original equation to see that 9 + 6 = 15.


100

Try the following problems using the four–step process from the previous page. When you “solve” your problem, you should not only find the answer, but also show the way you got to your answer, which is just as important.

EXERCISE C:

Write the following statements in equation form using a variable.

Example: We use the variable N to represent a number that is 3 less than 20. N = 20 – 3

Let N = the number.

1. A number is 2 more than 43.


3. Twice a number is 60.

4. 4 less than a number is 15.

5. A number is 3 more than twice the number 5.

6. 8 less than a number is 50.

7. A number is 6 less than twice the number 10.


101

EXERCISE D:

Do steps 1 and 2 of our four–step process for the following problems.

1. Alejandra is 10. In how many years will Alejandra be 24?

2. If Daniel will be 16 in 7 years, how old is Daniel now?

3. Becky has $73. How much more does she need if she wants to have $98?

4. Jacob has $85 and lends Jeremy $58. How much money does Jacob have left?


102

EXERCISE E:

Write a story problem for the equations below.

1. X + 8 = 25

2. 64 – X = 12


103

EXERCISE F:

In each of the following problems, define a variable, set up an equation, solve, and check. Remember to include your unit of measurement such as degrees C or F, feet, inches, years, etc.

1. In the morning it was a cool 65°F. We use F for Fahrenheit units. By the afternoon the temperature had reached 87°F. What was the increase in temperature from morning to afternoon?

Let T = the increase in temperature, in degrees Fahrenheit.


104

2. On a cold day in Canada the temperature was −8° Celsius at 6:00 A.M. How many degrees must it warm up to reach 5°C? We use C for Celsius units, a part of the metric system used throughout the world.

Let R = the number of degrees (in Celsius) the temperature must rise.


105

3. The tallest mountain in the world is Mount Everest, a part of the Himalayan Mountains near Nepal and Tibet on the continent of Asia. Mount Everest has an elevation of 8848 meters above sea level. The next highest peak is K2 and is also a part of the Himalayas. K2 has an elevation of 8611 meters above sea level. How much taller is Mount Everest than K2?

Let A = the additional altitude, in meters, from K2 to Mount Everest.


106

4. Death Valley in California is the lowest point in North America. It is 282 feet below sea level! Let’s use the notation −282 to mean 282 feet below sea level. The highest point in North America is Mount McKinley or Denali which means The High One in Athabascan. Denali is 20,320 feet above sea level and located in the state of Alaska. There is a big difference in the two elevations. If we let D represent the difference in the elevations, we can write the equation below. Solve for D. (Use a number line.)

Let D = _________________________

We have the equation ¯282 + D = 20,320


107

Another setting where we see a “number line” is in a “time line.”

5. The great Renaissance artist Michelangelo was born in 1475. If Michelangelo were still alive, how old would he be this year? Write a mathematical equation, solve, and check.

Let M = age, in years, Michelangelo would be today.


108

6. The city of Rome, in Italy, is said to have been founded in 753 BC. How many years passed before Christopher Columbus landed in North America in 1492? Represent 753 BC by −753. Write an equation which models this problem. Solve for x and check.

Let x = the number of years that had passed from 753 BC to 1492 AD when Columbus landed in North America.


109

SUMMARY:

Discuss key ideas for the section.


110

SECTION 2.2 THE CHIP MODEL

OBJECTIVES

• Add using chips

• Subtract using chips

What number can we add to 10 so that the sum is 0? If we start with –3, then what number could you add so that –3 and that number equals 0? If you said add the opposite of the number, you would be correct. In other words, –3 + 3 = 0. We introduce the chip model to illustrate how two numbers such as –3 and 3 are “zero pairs” because the sum of the two numbers equals zero.

ACTIVITY 1:

Zero pairs (pair students)

1. Give each pair of students at least 10 of the two-sided chips.

2. Have one student drop the 10 chips gently on the table.

3. What number do you think the arrangement represents? Have the other student count the number of zero pairs and then what “charge” remains.

Notice that if we have 555K after the toss, then we can write 3 + (–1) = 2 because we have one zero pair which is zero and 2 positive chips remaining.

If, however, we have KKK5 after the toss, then we can write –3 + 1 = –2 because we have the one zero pair which is zero and 2 negative chips remaining.

Using six chips, what kind of toss would give the sum of 4? What kind of toss would give the sum of –4? What kind of toss would give the sum of 0? What are all the possible sums?

Sect ion 2 .2 The Chip Model

111

CHIP MODEL FOR ADDITION

1. Represent the two integers in the addition problem using either positive or negative chips as appropriate for the problem.

2. Put together any zero sums, that is any + and – pairs.

3. The sum equals the number of any positive or negative chips left after all the zero pairs have been accounted for as zeros.

For example, we show 5 + –4 using the Chip Model for addition.

Step 1:

Step 2:

Step 3:

+

–++ + + +

– – –

+

–++ + + +

– – –

+

We conclude that 5 + –4 = 1.


112

ACTIVITY 2: CHIP ADDITION

Notice that if we have 5556 we can think of this as an addition problem 3 + –1. Because we have one zero pair and two positive chips left we can conclude that 3 + –1 = 2.

1. Take a pile of chips and carefully toss them on the table. Have your partner write an addition problem that would correspond to the resulting toss. Check to see if you both agree. Now switch to let your partner do the chip toss and you determine the addition problem that corresponds. Check with your partner to see if you agree. Repeat.

2. Using 6 chips, what kinds of toss would give the sum of 4? What kind of toss would give the sum of –4? What kind of toss would give the sum of 0? Are there are sums you cannot get using 6? Explain why you think so.


113

EXERCISE A:

1. Use the chips to demonstrate and find the sum. Draw pictures of chips with the first 4 problems to explain how you found the sum.

a. –3 + 7

b. –6 + 1

c. 6 + –8

d. –3 + –4

e. 3 + –7

f. 5 + –5


114

2. Suppose the following are the results of a toss of chips. Write an addition problem to model each of the following chip results. Find the resulting sum by forming and removing the zero pairs.

a. KKKK5

b. 5KK

c. KK55

d. 55KK55K


115

DOUBLE-SIDED CHIP SUBTRACTION

Just as we can use chips to model addition, we can also model subtraction using chips and the take-away approach.

CHIP MODEL FOR SUBTRACTION

1. Place the number of charged chips that represent the first number in the subtraction problem.

2. Take away the number of chips that correspond to the second number in the subtraction problem.

3. The remaining chips after taking away the appropriate chips in step 2 is the difference.

For example, to find the difference –5 – –3 using the chip model, we have

Step 1: – − – –

Step 2: – − – –

Step 3:

–

–

– –

Start with ¯5

Take away ¯3

Left with ¯2

We say that –5 – –3 = –2.

In the subtraction above, we have enough negative chips to take away since there are five negatives and we want to take away three negatives. However, think about –5 – –6. What can you do when you do not have enough chips to take away? How can we show taking away 6 negative chips when we only have 5 negative chips?


116

One way to do this subtraction is to bring back the idea about the fact that if n is any number then n + 0 = n. Recall that zero added to any number is equal to the original number. Remember that 0 can be represented using zero pairs. For example, 1 + –1 = 0. We can also say 2 + –2 = 0.

In this case, –5 – –6 looks like the following in chips:

Step 1: – − – –

Step 3: – − – –

Step 4:

–

–

Step 2: – − – – – – +

– +

+

Start with ¯5

Still have ¯5

Take away ¯6

Left with 1

KKKKK is the –5 but if we include a zero pair, then we will have –5 represented as KKKKKK5 . We now have six negative chips so we can take away six negative chips. This leaves us with one positive chip.

We can now write this as –5 – –6 = 1.

Find the difference 4 – –2 by using the chip model. Write down the process you used to determine the difference. Remember that you can always add zero pairs if needed.


117

EXERCISE B:

Perform the following operations and use the chip model to justify your answers.

1. Find the sums

a. –2 + 1

b. –4 + 6

c. –3 + –3

d. –3 + 3

e. 7 + –9


118

2. Find the differences

a. –2 – –1

b. –3 – –3

c. –4 – –6

d. 7 – 9

e. –3 – 3


119

EXPLORATION: CHIPPING AWAY

Suppose you have 4 positive chips. What can you add to get the following? Write the mathematical statement and explain.

1. a sum of 5?

2. a sum of 3?

3. a sum of 0?

4. a sum of –1?

Discuss how you decided on your answers.


120

SUMMARY:

Discuss the main ideas with the chip model for addition and subtraction. How does it compare with the number line model of addition and subtraction? Do you have a preference or see a connection between the two models?

Sect ion 2 .3 So lv ing Equat ions

121

SECTION 2.3 SOLVING EQUATIONS

OBJECTIVES

• Model equations with a balance scale

• Use the subtraction property

• Use the addition property

ACTIVITY: KEEPING OUR BALANCE

An equation is a statement that two expressions are the same or equal.

EXAMPLE 1:

Consider the following problem:

If Jeremy were three years older, he would be the same age as his twelve-year-old sister. What is Jeremy’s age?

We let J = Jeremy’s age. Then we can translate this sentence into an equation as follows:

J

If Jeremy were three years older he would be

+ 3 =

twelve years old

12

We have the equation J + 3 = 12. The unknown, J, is Jeremy’s age. Pictorially, this sentence says that J + 3 is the same as 12, and we can draw this on a balance scale:

+

+

+

J+

+

+

+

+

+

+

+

+

+

+

+


122

In order to solve this equation for J, we must find J. To do this, we remove three positive chips from each side of the balance scale.

+

+

+

J+

+

+

+

+

+

+

+

+

+

+

+

This is what we have left.

J+

+

+

+

+

+

+

+

+

We can express this algebraically as follows:

J + 3 = 12

(J + 3) – 3 = 12 – 3

J = 9


123

EXAMPLE 2: FOUR STEP PROCESS WITH BALANCE:

If Wesley finds 5 more marbles, he will have the same number of marbles as John. John has 11 marbles. How many marbles does Wesley have?

Step 1: Define a variable

Let W = the number of marbles that Wesley has.


If Wesley’s marbles were increased by 5 he would have eleven marbles

Step 3: Draw a balance scale which represents the problem. Solve the equation with the balance scale, and then solve it algebraically.

Step 4: Check that your answer makes sense.


124

EXAMPLE 3:

Let’s now work a temperature problem.

If the temperature rises 12º more, it will be 14ºF. What is the temperature now?

Step 1: Define your variable.


Step 3: Solve the problem algebraically. Did you use a balance scale?

Step 4: Check.


125

Solve each equation algebraically for the unknown using the four-step process. Illustrate how the equation would appear on the balance scale. Check that your answer works.

1. B + 4 = 12

2. 7 = A + 2


126

3. 14 = N + 8

4. B + 3 = 10


127

5. 8 + C = 13

6. –5 = T + 2


128

MAKE A RULE:

Write a rule that explains how to solve for the variable X in the equation X + A = B

EXPLORATION:

Note that some of the equations use negative numbers and would be difficult to solve using our balance scale. We need a way to represent negative numbers using a balance scale. Can you think how this might be done on Problem 6 of the previous page?

As long as we subtract the same number from both sides of an equation, the balance scale which represents the equation will remain in balance. The new equation we obtain is said to be “equivalent” to the original equation. We may now state this as the subtraction property for equations:

SUBTRACTION PROPERTYIf A = B then A – C = B – C is an equivalent equation.

Notice that by using the subtraction property,

x + 7 = 11 is equivalent to

x + 7 – 7 = 11 – 7 or better yet,

x = 4


129

EXAMPLE 4:

Suppose that Alice will be 9 three years from now. How old is she now?

Solution:

Step 1: Let A = Alice’s age now

A + 3 = Alice’s age 3 years from now

9 = how old Alice will be in 3 years

Do you see the two descriptions that are equivalent?

Step 2: We write the equation, A + 3 = 9

+

+

+

A+

+

+

+

+

+

+

+

+

Step 3: Solve for the unknown, A.

(A + 3) – 3 = 9 – 3

+

+

+

A+

+

+

+

+

+

+

+

+

A = 6

A+

+

+

+

+

+

Step 4: Check: 6 + 3 = 9


130

ACTIVITY: BALANCE SCALE

EXAMPLE 5:

Whitney was 10 years old 5 years ago. What is Whitney’s age now?

Step 1: Let W = __________________________________________________

W – 5 = __________________________________________________

Step 2: Represent the problem with an equation.

Whitney’s age 5 years ago was ten years old

Finish this problem by solving the equation for W and checking.


131

The property that we now need is the addition property.

Pictorially we see this using a balance scale:

If A = B

A B

Then A + C = B + C.

A BC C

We can write this algebraically like this:

ADDITION PROPERTYIf A = B then A + C = B + C is an equivalent equation.

Let’s finish Whitney’s problem using this property.

Step 3: W – 5 = 10

W + ¯5 = 10

W + ¯5 + 5 = 10 + 5

W = 15

Step 4: If Whitney’s age five years ago was 10, then 10 + 5 = 15 represents Whitney’s current age.


132

EXAMPLE 6:

If Jack were three pounds lighter he would weigh 72 pounds. How much does Jack weigh?

We represent Jack’s weight with an unknown: Let J = Jack’s weight.

We begin by translating the problem into an equation:

If Jack were three pounds lighter he would be 72 pounds

J – 3 72=

Pictorially, we draw J = Jack’s weight

We represent each negative found with negative chips added to Jack’s weight. Is this how you represented negative numbers on a balance scale?

In order to subtract three pounds from J, we simply place three negative chips along with J. This should equal 72.

–

−

–

J 72

To solve for J, we must add three positive chips to both sides of the balance scale. This is because we have three negative chips on the left side of the balance and we want to solve for just Jack's weight.


133

We obtain the picture below.

–

−

–

J 72+

+

+

+

+

+

We add the three positive chips because we know that –3 +3 = 0. From our chip model we know that

= 0+– +

Simply put, the three positive chips and the three negative chips make three pairs of zero sums, and the left side of the balance scale now represents J exactly.

J 75

Algebraically, we have:

J – 3 = 72J + ¯3 = 72

J + ¯3 + 3 = 72 + 3J = 75

As long as we add or subtract the same number to both sides of an equation, the balance scale which represents the equation, will remain in balance and we obtain an “equivalent equation.” When we say that two equations are equivalent, we mean that the solution set for one equation is exactly the same as the solution set for the equivalent equation. So if the balance scale is in balance for one equation, it will be in balance for the other.


134

Exercise A: Solve the following equations

Example: Solve x + 14 = 23

x + 14 – 14 = 23 – 14 (subtract 14 from both sides)

x + 0 = 9 or

x = 9

1. a. x + 3 = 8 b. x – 3 = 8

2. a. 4 + y = 12 b. y – 4 = 12

3. a. 9 = A – 5 b. 9 = A + 5


135

4. a. S – 15 = 34 b. S + 15 = 34

5. a. x + 9 = 7 b. x – 9 = 7

6. a. –2 = T + 2 b. –2 = T – 2


136

7. p – 16 = –3 8. –8 = m – 7

9. x + 23 = 17 10. 50 = H + 50

11. 3 + i = 2


137

12. 9 = a + 8 13. –4 = D – 6

14. –2 = M + 8 15. –10 = 9 + x

16. t – 4 = –3


138

SUMMARY

Write word problems that “translate” to the following equations.

1. x + 4 = 8

2. x – 14 = –3

Now make up a word problem and see if you can translate it to an equation.

Can you solve it? Share it with your class.

Sect ion 2 .4 So lv ing Equat ions on the Number L ine

139

SECTION 2.4 SOLVING EQUATIONS ON THE NUMBER LINE

OBJECTIVES

• Locating general points given the origin and a scale

• Solving equations using the number line

Now that you have a way to solve equations using the balance scale, let’s return to the number line. We will see that provides another way to help us solve some of our equations. In mathematics, it is often useful to have several different ways to solve problems where some methods can be more useful than others, depending on the situation.

EXPLORATION 1:

Suppose a and x are numbers located on the number line as seen below. Locate and label the points that represent the indicated numbers. Use string to act out how you determine your answer.

1. Plot points that represents each of the following: 2a, 3a, −a, −2a, −3a. Explain how you found the points.

0

a


140

2. Plot points that represent each of the following: 2x, 3x, −x, −2x, −3x. Explain how you found the points.

0

x

3. Compare the results from parts 1 and 2. What do you notice?


141

EXPLORATION 2:

Part A: Suppose x is a number that is located on the number line as seen below. Locate and label the points that represent the indicated expressions. The numbers 0 and 1 are also labeled with the distance between them considered one unit.

Plot a point that represents each of the following expressions:

x + 1, x + 2, x – 1, x – 2, x – 3

0

1 x

Explain how you found the points.


142

Part B: Suppose we know the location of each of the expressions as indicated on the number line below. Find the locations for x, y, and z. Explain how you locate each of these points on the number line.

4 3 2 1 0 1 2 3 4

x + 3

– – – – 5 6 7 8 9 108 7 6 5– – – –9–10–

4 3 2 1 0 1 2 3 4

y – 2

– – – – 5 6 7 8 9 108 7 6 5– – – –9–10–


143

4 3 2 1 0 1 2 3 4

z + 2

– – – – 5 6 7 8 9 108 7 6 5– – – –9–10–

Use the information on the previous number line to write the corresponding equations. Use the balance scale method from the previous chapter to solve your equations. You also solved these equations using the number line. Do you see how the two methods are related?


144

EXERCISE A:

Draw a number line for each and use it to model and to solve each of the following equations.

a. x + 3 = 5

b. y + 5 = 2


145

c. z – 4 = 2

d. Discuss how solving these equations on the number line compares with the balance scale method. Include any notes below.


146

SUMMARY

Write what you like about the balance scale way of solving.

What do you like about the number line way of solving equations?

Do you prefer the scale or the number line? Why?

Sect ion 2 .5 Graphing on a Coord inate P lane

147

SECTION 2.5 GRAPHING ON A COORDINATE PLANE

OBJECTIVES

• Create coordinate planes

• Graph ordered pairs

• Set up equations with two variables.

Recall that we use the number line to represent numbers. To each point on the number line we associate a number or coordinate. For example, to graph or plot a point A with the number or coordinate 3 on the number line, we go 3 units to the right of 0. If point B has coordinate −2, then we go 2 units to the left of 0 on the number line. The graph below shows the number line with the points A and B plotted.

4 3 2 1 0 1 2 3 4

A

– – – –

B

Notice that we can move left or right along the number line if the number line is horizontal. We move to the right of 0 if the number is positive and to the left of 0 if the number is negative.

Suppose we now put the number line in a vertical position with the positive side up, like a thermometer:

The point R with coordinate 3 moves up 3 units from 0. The point S with coordinate −2 moves down 2 units from 0.

We can plot points on the number line that have numbers for their coordinates.

Pairs of numbers are used to plot points on a number plane, which we call a coordinate plane.

1

0

2

1

2

3

3

–

–

–

4–

4

R

S


148

A coordinate plane is constructed as follows:

We begin by drawing a horizontal number line and call it the horizontal axis or the x–axis.

4 3 2 1 0 1 2 3 4origin

– – – –

Locate the zero point, which is called the origin. Next, draw a vertical number line through the origin, so that the two zero points overlap at the origin.

P

Quadrant IQuadrant II

Quadrant III Quadrant IV

The vertical number line is called the vertical axis or the y–axis.

These axes divide the plane into four regions. We call these regions Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. The coordinates for the points in the coordinate plane are pairs of numbers.

For example, the point P is identified as (4, 3), where 4 represents the x-coordinate and 3 represents the y-coordinate. The points in the plane are written in the form (x, y) and are called ordered pairs.


149

ACTIVITY: MAKING A COORDINATE PLANE ON THE FLOOR

BRAINSTORM

How might we depict the points in space (i.e. in third dimension)?

What notation might be used?

How might we describe the location of points in the fourth dimension?


150

EXERCISE A:

1. Plot the following points on the coordinate plane. Notice we have already plotted point A on the coordinate plane.

A (2,5) B (5, 2) C (–2,5) D (5,–2) E (–2,–5) F (–5,–2)

G (1,1) H (–1,1) I (0,3) J (4,0) K (–3,0) L (0, –3)

x-axis

y-axis

1 2 3 4 5 6 7

1234567

¯7 ¯6

¯6

¯5

¯5

¯7

¯4

¯4

¯3

¯3

¯2

¯2

¯1 ¯1

A


151

2. Locate the points labeled from A through L and write the coordinates for each of them.

F

B

H

A

K

I

D

J

C

L

G E

A is given by E is given by I is given by

( ) ( ) ( )

B is given by F is given by J is given by

( ) ( ) ( )

C is given by G is given by K is given by

( ) ( ) ( )

D is given by H is given by L is given by

( ) ( ) ( )


152

3. Answer the following questions using the coordinate plane provided.

a. Plot the points (4, 2), (–4, 2), (4, –2), and (–4, –2).

b. Describe any relationships that you see among the points.

c. Plot the point (−3, 5) and then plot three other points that have the same relationship to (−3, 5) as the points plotted in part a do to (4, 2).


153

4. Find 4 closest neighbors to each of the points below. Choose only neighbors with integer coordinates. Use the coordinate plane below to explain.

(3, 5)

(0, 4)

(−2, 4)

(−3, 0)

(3, −2)

(x, y) with x and y integers


154

5. Shade the area of the coordinate plane whose points have positive first coordinates and negative second coordinates.

6. Shade the area of the coordinate plane whose points have negative first coordinates and negative second coordinates.

7. Describe a common characteristic of all the points in the second quadrant.


155

SUMMARY

Discuss what you enjoyed learning in this section. Why?

Check ups and Act iv i t ies

156

CHAPTER 1 SECTION 1 CHECK UP

1. Rewrite 6 is greater than 3 using either the < or > symbol.

2. Rewrite –4 is less than –1 using either the < or > symbol.

3. Plot each pair of points on the number line then compare the numbers using < or >.

a. 2 –8

b. 4 2

c. –5 4


157

4. Where is the “point of origin” on a number line?

5. Write three positive integers: __________________________________

6. Write three negative integers: __________________________________


158


1. On Monday, Mike climbed a hill that is 135 feet above sea level. On Thursday, he went to explore Wonder Cave that is 215 feet below sea level. At which location is Mike further away from sea level? (Use the number line to show this.)

2. An eagle flies 123 meters above the surface and an insect crawls in a cave 97 meters below the surface. Which animal is closer to the surface? (Use the number line to show this.)


159

3. Plot the points on the number line and find the distance between the following pairs of numbers.

a. 8 and 3

distance = ______________

b. 4 and 1

distance = ______________

4. Find the absolute value of the following numbers.

a. 2

b. ¯5

c. 0

d. –43


160


1. Jessica takes 5 steps backward and then 7 steps forward. How far is she from where she started? (Use a number line to show your work.)

2. The temperature is −7 degrees Celsius. If the temperature rises by 3 degrees, what is the temperature now? (Use the number line to show your work.)


161

3. Use the car model to find the sums.

a. 7 + 3 = _____________

b. −6 + 2 =_____________

c. −3 + −9 =_____________

d. 8 + −6 =_____________


162


1. It was 8 degrees at 4:00 pm. By 10:00 pm the temperature had fallen by 12 degrees. What is the temperature at 10:00 pm? (You can use the number line to find your answer.)

2. Use the car model to find the difference.

a. 7 – 3 = _______

b. −6 – 2 = _______


163

c. −3 – −9 = _______

d. 8 – −6 = _______

3. Find the following opposites.

a. opposite of 8 = _____________________

b. opposite of −9 =_____________________

c. opposite of −24 =____________________


164


1. Compare the pairs of numbers using < or >.

a. 457 481

b. 97 57

c. −32 −23

d. −5 −15

2. Find the distance between the following pairs of numbers. Also plot the points on the number line. Watch your scaling.

a) distance between 0 and 18 = ________

b) distance between −25 and 25 = ________

c) distance between 4 and −14 = ________


165

4. The temperature in Anchorage, Alaska, is −15 degrees Celsius. The temperature rises 12 degrees during the day. What is the new temperature?

5. Lisa’s bank account reads −$56. If Lisa withdraws $23 from her account, what will her new account balance be?


166


1. Translate the following into mathematical expressions.

For example: two more than five can be written as 2 + 5

two is less than five can be written as 2 < 5

a. A is greater than 2 can be written as

b. 5 more than 7 can be written as

c. −9 is less than 4 can be written as

d. x less than y can be written as

2. Translate the following expressions into word phrases.

For example: 3 – 5 translates to 5 less than 3.

3 < 5 translates to 3 is less than 5

a. −3 – 2 translates to

b. x + 2 translates to


167

c. 5 > 2 translates to

d. 6 < y translates to

3. Write the following statements in equation form using N for the unknown number.

For example: A number is 3 more than 7. Equation: N = 7 + 3

a. Twice a number is 16. Equation:

b. A number is 7 less than 21. Equation:

c. 6 less than a number is 20. Equation:


168


1. Use the chip model to illustrate the following addition problems, and find the sums.

a. −4 + 3

b. 8 + −5

c. −2 + 2

2. Use the chip model to illustrate the following subtraction problems, and find the differences.

a. −4 – −3

b. 6 – −4

c. −5 – 2


169

3. Using 7 chips, what kind of toss of double-sided chips would give the sum of 3? Use chips to illustrate.


170


Solve each equation algebraically for the unknown using the four-step process.

For example: J + 3 = 12

J + 3 – 3 = 12

J = 9

Check: 9 + 3 = 12

1. B + 2 = 8

2. −7 = T + 5


171

3. x – 19 = −10

4. C – 19 = −10


172


1. Use the number line to model and solve each of the following equations.

a. x – 4 = 10

b. y – 7 = −2

c. z + 5 = 3


173

2. Plot points that represent each of the following: 2m, −m, −2m

0

m


174


Use the coordinate plane below to plot the following ordered pairs. Label each point with the appropriate letter.

A (3, 6)

B (0, 2)

C (7, –3)

D (–3, –3)

E (–2, 0)

F (–4, 1)

G (–9, –1)

H (4, –1)

I (0, –4)

J (–4, 3)


175

SECTION 1.1 MAIL MIX UP

Materials:

Number line, index cards

Instructions:

1. Each of the participants will draw an index card with a number and type of establishment printed on in. The numbers will range from –12 to +12. A facilitator will draw the 0 (zero) card. This number represents the address of a shop, museum, laboratory, garden or observatory on Max Street.

2. The participants will become pieces of mail that would be appropriate to be sent to their stated locations. Participants will be given a few minutes to create what their piece of mail contains and how to act it out. Facilitators will assist participants and make suggestions only if called upon by the participants.

3. When all are ready, the participants will take turns acting out what is in their piece of mail while the rest of the class watches.

4. When all have acted out their mail pieces, the postal workers (facilitators) will load them in order into an imaginary mail truck and set out for Max Street.

5. Meanwhile, the facilitators will take on the roles of an elephant, a hippopotamus, and a rhinoceros, three animals in a zoological garden on Max Street. The three animals have a tremendous argument which will result in the rhinoceros charging off blindly into the street, not looking right or left, and crashing into the mail truck.

6. IN SLOW MOTION the pieces of mail will be jarred loose from their nicely ordered positions and will fly about the truck coming to rest in random order.

7. The postal workers will grab a piece of mail and holler out the number as printed on the card.

8. The mail truck will move up and down Max Street from −12 to +12 until all the pieces are delivered.

9. Discussion.


176

SECTION 1.1 MAIL MIX UP INDIVIDUAL ACTIVITY

Materials:

Number line and triangle cards

Instructions:

Make a 3” × 2” rectangle, drawing a diagonal line to create two triangles. Label each triangle with various places for the students to find on their number line, such as Pizza Hut, McDonald’s, the Library, etc. As you call out where these places are located on the number line, the students label on their own.

SECTION 1.1 GREATER THAN, LESS THAN, OR EQUAL ACTIVITY

Materials:

Folder game board, one die, pawn for each student, tiles

Instructions:

Make a folder game board with problems in each square. The first student rolls the die and moves to that space. The student answers the problem; if correct, the student stays, if incorrect, the student must go back the number of spaces on the die. The students can write the problem on the dry erase board or use tiles with positive and negative numbers on them, along with the greater than, less than, and equal signs on them.

SECTION 1.1 THERMOMETER ACTIVITY

Materials:

Indoor/outdoor thermometer (with numbers both above and below zero)

Instructions:

Use the thermometer as a real life model of a number line. Place it horizontally as well as vertically.


177

SECTION 1.1 UNITED WE STAND

Materials:

One index card per student, numbered with random integers (be sure to include positives, negatives, and zero when you are numbering your cards), colored tape

Instructions:

Before the students get to class, use the colored tape to make an unmarked number line along the floor in the front of your room. Do not mark intervals on the number line.

After shuffling the cards to make sure they are in random order, pass one card to each student. Once all of the cards have been handed out, the students will come up one at a time and position themselves on the number line that you made on the floor. Allow the important issue of spacing to arise naturally as the students arrange themselves.

When all students are standing in the correct order at the front of the room, take this opportunity to ask them some extension questions to check for understanding of the lesson. Some examples of questions you could ask are:

Who is standing in the middle of the line, and why?

If we had a card with the number one million, where would it go?

If we had a card with the number ______, it would be between which two students?

Who represents the biggest number on our line?

Who represents the smallest number on our line?

Is the spacing between integers important?


178

SECTION 1.2 GREATER THAN, LESS THAN, OR EQUAL BOARD GAME

Materials:

Folder game board, one die, a pawn for each student.

Instructions:

You can create a folder game board just as in Section 1.1, but this time incorporate absolute value numbers.


179

SECTION 1.2 TIME MACHINE ACTIVITY

Materials:

Butcher paper 4 meters long, masking tape

Instructions:

1. Divide the participants into four groups.

2. Have each group choose a significant moment in world history from 1000 BC to 3000 AD. The facilitators will make sure that at least one group selects a historical moment in the B. C. range.

3. Give the groups 3 to 5 minutes to practice acting out their selected moment. Facilitators will go from group to group making certain that all participants are involved and answering any questions that may arise. Suggestions are to be made only as needed.

4. Have each group perform their dramatization for the rest of the class. This will be done by having each group enter a time machine that is made up of facilitators. When the time machine opens its doors and the group inside will come tumbling out and flow into their historical dramatization. End each dramatization by having someone in the group tell what the historical event was and the year.

5. After all the groups have completed their dramatizations, have them go to the time line and plot their moment in the appropriate place. A simple stick figure drawing may be used as the plotting point. The time line will be a piece of butcher paper 4 meters long taped to the wall of the classroom. The paper will be divided into centimeters with each centimeter equaling 10 years.

6. Repeat the activity two more times with different historical moments. Questions regarding the number of years from one point to another can then be asked.

7. Discussion.

NOTE: Because the time line goes to 3000 A. D. some of the events selected will be futuristic. These moments may include things like, “the first person landing on Mars,” “the discovery of a new planet,” “the first baby born on the USA moon colony,” etc.


180

SECTION 1.3 CAR MODEL ACTIVITY

Materials:

Butcher paper, masking tape, plastic model cars for a number line on the floor.

Instructions:

Make a number line on the floor, and have students act out the car model.

SECTION 1.3 ADDITION BOARD GAME

Materials:

Blank game board, cards, pawn for each student

Instructions:

Have a blank game board. Have a bag of cards with an addition problem as well as the number of spaces to move. The student then draws a card and states an answer. If the answer is correct, student moves the pawn the specified number of spaces on the card. If the answer is incorrect, the student does not move the pawn. Either place the correct answer on the back of the card or have an answer sheet for the other student to check.


181

SECTION 1.4 SUBTRACTION BOARD GAME

Materials:

Blank game board, cards, a pawn for each student.

Instructions:

This is the same game as in Section 1.3, but this time we have cards with addition or subtraction problems on them.


182

SECTION 1.5 ADDITION AND SUBTRACTION TEAM GAME

Materials:

Cards, dry erase boards, pens, or paper and pencil for each student, a bell

Instructions:

Divide the class into two teams. Have the first person from each team come and draw a card (use the same cards from the subtraction Board Game). The student takes the card back to his/her team. All team members must work the problem and agree on the same answer. When everyone on the team has the same answer, the first person comes up and rings the bell. If the team is correct, they receive a point. If they are incorrect, the other team gets a chance at answering the question and gaining a point.

SECTION 1.5 JEOPARDY

Materials:

Pocket holder, index cards with questions written on them, cards with the five categories, a timer

Instructions:

Divide the team into two groups. Have five categories for students to choose from: Compare the Number, Absolute Value, Addition, Subtraction, Addition and Subtraction. Have 5 cards with questions to go under each category. The questions should range from easy for 10 points to hard for 50 points. If the student gets the question correct, the team gets the number of point value for the question. The students can work together to come up with the answer. If the team says the wrong answer, the other team has a chance to answer the question. Continue to play until all the cards are used. The team with the highest amount of points wins. Use a timer to have the students answer the question in that amount of time - 30 seconds to 1 minute.


183

SECTION 2.1 VARIABLES AND EXPRESSION BOARD GAME

Materials:

One die, folder game board, a pawn for each student, paper and pencil or tiles with number on them and signs <, >, =, +, –

Instructions:

Create a game board with problems on each space such as: 2 more than 5, etc. The student rolls the die and moves to that space. Using tiles, the student lays the answer out on the table or shows the work with paper and pencil. If correct, the student stays there; if incorrect, the student moves back the number of spaces rolled. Be sure to have an answer key. Another way to do the game is to make cards with the problem on them and the answer on the back. The card would also tell the student how many spaces to move.


184

SECTION 2.2 DOUBLE SIDED CHIP ADDITION

Materials:

Double sided chips (coins or checker pieces can be used as long as the two sides are not the same)

Instructions:

We establish one side of the chip to be positive 5 and the other side to be negative K . We also consider any pair of 5 and K to be equivalent to zero and we’ll say:

5K = 0

The Chip Addition is based on a set union concept.

For example:

3 + 2 = 5 is demonstrated with 555 union 55 gives us 55555 all together.

3 + –2 = 1 is demonstrated by 555 union KK gives us 5K5K5 or 5 .


185

SECTION 2.2 DOUBLE SIDED CHIP SUBTRACTION

Materials:

Double sided chips (coins or checker pieces can be used as long as the two sides are not the same)

Instructions:

We use the Take Away Model for subtraction.

Examples:

3 – 2 = 1 is demonstrated with 555 take away 55 leaves only 5 .

−3 – −2 = 1 is demonstrated with KKK take away KK leaves only K .

3 – −2 = 5 is demonstrated with 555 , but unable to take away KK , we introduce into our holdings two zero sums so that we have 5555K5K . When we take away KK , we are left with 55555 .

−3 – 2 = 5 is demonstrated with KKK , but unable to take away 55 , we introduce into our holdings two zero sums so that we have KKK5K5K . When we now take away 55 , we are left with KKKKK .


186

SECTION 2.3 BALANCE SCALE

Materials:

A balance scale made from sentence strips, chips, and a tile for the unknown

Instructions:

Have the students use the Balance Scale to physically do the work on page 103.

SECTION 2.3 THE TIGHTROPE ACTIVITY

Materials:

Yard stick, masking tape

Instructions:

Mark off a tightrope on the floor using masking tape. Let one of the students be the tightrope walker with the yard stick. The other students must help the tightrope walker keep his/her balance. Birds keep landing on the balance bar. Suppose 2 robins weigh the same as 1 blue jay. If the girls are the robins and the boys are the blue jays, begin with 2 girls on one side of the balance bar and 1 boy on the other side. Students can go to one side or the other while the tightrope walker is walking along, but the other students must keep the sides “balanced” by removing themselves or adding themselves. The “birds” keep moving.


187

SECTION 2.4 PINOCCHIO’S NOSE

Materials:

A set of true/false questions each written on an index card

Instructions:

A facilitator represents Pinocchio’s nose. A question is asked of the facilitator. If he answers truthfully, no change occurs. If he answers it untruthfully (he lies), then two people stand by the facilitator to indicate the nose grew that much. If he answers it untruthfully on the next questions two more people come stand next to the first two; the nose keeps growing. No change occurs if Pinocchio answers truthfully. What is the relationship between the number of lies and the length of Pinocchio’s nose? Have the students write this in equation form.

What if three people came and stood by Pinocchio every time he lied? What is the relationship then? Can students write this down in equation form?


188

SECTION 2.5 MYSTERY GRAPH

Materials:

Graph paper, pencil, paper

Instructions:

Have each student create a design on graph paper. The student writes down the coordinates in the order in which the points should be connected. The students share their coordinates with other students. The students try to recreate the designs of the original designer by using coordinates only. Check to see how they match the original design.

SECTION 2.5 DOG BONE GAME

Materials:

Graph paper, pencil

Instructions:

Students pair up. A coordinate plane is drawn on the graph paper with a range of at least −10 to 10 on both the x and y-axes. One student draws a “bone” on the coordinate system that has length 5 and the ends of length 2. The other student, who has not seen the location of the bone must guess the coordinates of the bone by guessing one coordinate at a time. Keep track of the number of incorrect guesses. The object is to guess the location of the bone in the fewest guesses.

Look at the example of a bone below.

You can make this into a “hangman” type of game.


189

SECTION 2.5 MAKING A FLOOR COORDINATE PLANE

Materials:

Masking tape, clear shower curtain

Instructions:

Make a coordinate system on the floor or on the shower curtain with the students. Mark off the axes and determine a reasonable scale that will fit at least 12 or more coordinates on each side of the axes. Have the students stand on different “points” of the coordinate system and discuss the coordinate labeling.

SECTION 2.5 ORIGAMI MATH

Materials:

Origami paper or square sheets of paper and graph paper

Instructions:

Determine the perimeter of the sheet by placing it on the graph paper. Determine the area of the sheet by tracing the outline and then counting the square units. Fold the paper into other rectangular shapes and determine the perimeter and area of each. Fold the paper into triangular shapes and determine the perimeter and area of each.


190

DIAGNOSTIC TEST LEVEL 1 NAME:

1. Add: −3 + 7 =

2. Subtract: −4 – 9 =

3. Solve the equation: −4 = x – 5

Answer:

Answer:

Answer:

4. Draw a number line. Plot the points corresponding to 0, 2, −4, and −1

What is the distance between −4 and 2? Answer:

5. Madeline has x coins. She has two fewer coins than Sam who has y coins. Express the relation between x and y with an equation.

Answer: .


191

6. Plot the points (2, 5), (0, 3), (−5, 0) and (−3, 5) on the coordinate plane below:

7. Judy sells candy bars for $2 each.

a. If she sells 12 candy bars, how much money does she collect?

Answer: .

b. If she collects $18, how many candy bars did she sell?

Answer: .

Addi t ional Resources

192


193

BLANK NUMBER LINES


194

GRIDDED NUMBER LINES


195

BLANK COORDINATE GRID


196

CENTIMETER GRID


197

CENTIMETER GRID

RACE

FINISHto the FINISH

START

Flat Tire!Lose a Turn

Out of Gas!Lose a Turn

Nitrous Boost!Move an extra

2 spaces

Car Placeholders for Game Board

What is the distance from

0 to 6 ?


0 to −4 ?


−3 to 3 ?


1 to −1 ?


0 to −7 ?


0 to 29 ?

What is |13| ?

What is |−3| ?

What is |8| ?

What is |−30| ?

Fill in the box with <, >, or =

0 −7


2 −5


1 −1


3 0


4 |−4|


|−2| |−6|


1 −2


21 |−21|

−3 is to the LEFT or RIGHT

of 0?(choose one)

|−5| is to the LEFT or RIGHT

of −5?(choose one)

8 is to the LEFT or RIGHT

of 1?(choose one)

−9 is to the LEFT or RIGHT

of −6?(choose one)


19 6


−7 10

2 + |−2| |−1| + 6 5 + |−5| |−4| + |−2|

2 + |−8| |−1| + |−3| 1 + −5 4 + −3

2 + −1 10 + −4 3 + −7 7 + −8

−9 + 1 −3 + 3 −2 + 5 −4 + 7

−5 + 6 −8 + 4 −4 + −4 −2 + −9

−10 + −6−1 + −1−7 + −13−3 + −1

2 – |−2| |−1| – 6 5 – |−5| |−4| – |−2|

2 – |−8| |−1| – |−3| 1 – −5 4 – −3

2 – −1 10 – −4 3 – −7 7 – −8

−9 – 1 −3 – 3 −2 – 5 −4 – 7

−5 – 6 −8 – 4 −4 – −4 −2 – −9

−10 – −6−1 – −1−7 – −13−3 – −1

math est 1 - texas state university

Documents