Vocabulary

Choose the best term from the box.

• Associative • fraction
• decimal • inverse

1. Two operations that undo each other

are called ? operations.
2. A number that contains a point to
separate the ones place from the tenths
place is a ? .
3. A number used to name a part of a
3 The Colorado Potato Beetle
whole is a ? .
feeds on the leaves of eggplant 4. The ? Property of Addition states that
and potato plants. Do you think addends can be regrouped and the sum
the adult Colorado Potato
Beetle or the larva of this beetle
remains the same.
is smaller? You will find out in
Lesson 15-3. Number Sense

Compare the numbers. Use ⬍ or ⬎ for each 䊊.

5. 512 䊊 521 䊊 0.38
6. 0.379

4䊊3 5䊊 4
7. 3
__ 1
__ 1
8. 2__ 1
2__

Number Lines
4 The date a state quarter is Write the number for each point.
released is determined by the
order in which each state was
admitted into the Union. How 0 A B C D 10
can you evaluate an expression
to find the year California 9. C 10. A 11. B 12. D
became a state? You will find
out in Lesson 15-6.
Fractions and Decimals

Writing to Explain Write an answer to

the question.
13. What can you do to make comparing
3 and 0.6 easier?
__
4

Topic 15 341

27291_340-341_FSD.indd 341 12/18/06 5:41:26 PM

 Lesson

15-1 Understanding Integers

What are integers and what situations can 535 feet above
sea level
integers represent?
NS 1.5 Identify The highest point in Louisiana is Driskill
and represent on a 9g^h`^aa
number line decimals, Mountain at five hundred thirty-five feet
fractions, mixed numbers,
BdjciV^c
and positive and negative
above sea level. The lowest point is New
integers. Orleans at eight feet below sea level.
How can you write those highest
and lowest points with integers? CZlDgaZVch

Sea level 8 feet below

sea level

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 4, write an integer for each 5. In the example above, what is the
word description. opposite elevation of Driskill Mountain,
written as an integer?
1. Ten degrees below zero
2. Seventy degrees above zero 6. How far away from sea level is 512 feet
3. Two hundred thirty feet above below sea level?
sea level
7. How would you show sea level
4. Fifty-two feet below sea level represented as an integer?

Independent Practice
In 8 through 12, use the number line to identify the integer at each point.

W Z Y X T
−7 −6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6 +7

8. T 9. W 10. X 11. Y 12. Z

In 13 through 20, write an integer for each word description.

13. A withdrawal of $20 14. A deposit of one hundred dollars

15. A gain of three inches 16. A loss of six yards
17. A loss of 7 pounds 18. A temperature drop of 2 degrees
19. 6 steps forward 20. 10 seconds before blastoff

Animated Glossary
www.pearsonsuccessnet.com

342 *For another example, see Set A on page 360.

27291_342-343_001 342 12/15/06 4:06:40 PM

 Distance above sea level is greater than zero. Numbers that are opposites of each
It is represented by a positive integer. 535 other have the same magnitude
Distance below sea level is less than zero. (distance from 0).
5 and 5 are the same distance from 0.
It is represented by the negative integer. 8
Integers name magnitude (distance) and
direction from 0.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
0
Negative integers Positive integers
–8 + 535 are less than zero. are greater than zero.
The magnitude of 8 is 8. The magnitude of 535 is 535. ⫺
5 is read ⫹
5 is read
The direction is negative. The direction is positive. “negative five.” “positive five.”
Integers are the whole numbers and their 2 is the opposite of 2.
opposites; 0 is its own opposite. 4 is the opposite of 4.

Problem Solving

21. A football team started at the 20-yard 22. Adam has 1_12_ feet of aluminum wire,
line. In the first two plays, the team lost 1.29 feet of copper wire, and 1_58_ feet of
4 yards and gained 4 yards. Where did steel wire. Adam has the most of which
they end up?
kind of wire?

23. A movie company announced that one 24. At midnight, the temperature was
of its releases lost two million, eight 2 degrees. It went down 5 degrees,
hundred fifty-seven thousand, nine then it went up 3 degrees, and then
hundred dollars. Write that number in dropped 2 degrees. What was the final
integer form. temperature? Show your answer on a
number line.

25. Number Sense Julie needs to select an 26. The Mariana Trench is located in the
integer that is two less than 11. What floor of the western North Pacific
number should she pick? How did you Ocean. It is 35,798 feet below sea level.
find the number? Express this depth as an integer.

27. Pam made 28. Writing to Explain Describe how to

$168.75 at a craft fair. She sold 75 of find the surface area of the rectangular
the 125 book covers she made. Which prism shown below. Then find the
expression can you use to find the price surface area.
she charged for each book cover?
A $168.75  75 3.5 ft
B $168.75  (125  75)
C $168.75  125 4.5 ft
5 ft
D $168.75  (125  75)

Lesson 15-1 343

27291_342-343_343 343 10/11/07 5:28:56 PM

 Lesson

15-2 Comparing and Ordering Integers

How do you compare and order
This week’s temperatures (°F)
integers?
Low High
NS 1.5 Identify Alan’s family spent a week at a resort
and represent on a Monday 6º 30º
number line decimals, in Utah. The resort newspaper listed the
fractions, mixed number, low temperature for each night that week. Tuesday 0º 28º
and positive and negative
integers. Wednesday 4º 21º
Which night had a lower temperature,
Thursday or Friday? What is the order from Thursday ⫺
7º 17º
least to greatest of the low temperatures? Friday ⫺
3º 19º

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 4, compare. Use ⬎, ⬍ or ⫽ for 䊊. 7. Compare ⫺7 and ⫺3. Use the ⬎ sign.

1. ⫹4
䊊 3 ⫹ 2. ⫺2
䊊 2 ⫹
8. In the example above, if the
temperature on Wednesday night
3. ⫺ 1䊊 4 ⫺ 4. ⫺10 䊊 11 ⫺
was ⫺9°F, which night would have
been colder, Wednesday or Thursday?
In 5 and 6, order from least to greatest.

5. ⫹8, ⫺5, ⫺2 6. ⫺10, ⫹2, ⫺3 9. In the example above, order the week’s
high temperatures from least to greatest.

Independent Practice
In 10 through 17, compare using ⬎, ⬍ or ⫽ for 䊊.
10. ⫹1
䊊 ⫹3 11. ⫺4
䊊 9 ⫹ 12. ⫹5
䊊 ⫺2 13. ⫺11
䊊 ⫺10

14. ⫺ 8䊊 ⫺15 15. ⫹10 䊊 11 ⫹ 16. ⫺ 7䊊 ⫺6 17. ⫺ 1䊊0

In 18 through 33, order from least to greatest.

18. ⫹1, ⫺7, ⫺5 19. 0, ⫺3, ⫹6 20. ⫺5, ⫹10, ⫺1 21. ⫺4, ⫹11, ⫺6

22. 0, ⫹8, ⫺8 23. ⫹3, ⫹1, ⫹5 24. ⫺2, ⫺8, ⫺1 25. ⫺23, ⫺50, ⫺42

26. ⫹15, ⫺5, ⫹6, ⫺2 27. ⫺20, ⫺1, ⫺9, ⫺13 28. 0, ⫺19, ⫹5, ⫺4 29. ⫺5, ⫺20, ⫺10, ⫺15

30. ⫹6, ⫺3, ⫺2, ⫹7 31. ⫺18, ⫺3, ⫹3, ⫺8 32. ⫺5, 0, ⫹1, ⫺20 33. ⫺6, ⫺7, ⫺8, ⫺9

344 *For another example, see Set B on page 360.

27291_344-345_001 344 12/15/06 4:06:57 PM

 Compare the integers ⫺7 and ⫺3. Order the integers 6, 0, 4, ⫺7, and ⫺3.
Locate ⫺7 and ⫺3 on a number line. Locate the numbers on a number line.
Integers, just like whole numbers,
fractions, and decimals, increase in
– 7 – 6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6
value as you move from left to right.
Thursday Friday
night night From least to greatest, the week’s low
temperatures are ⫺7, ⫺3, 0, 4, and 6.
– 8 – 7 – 6 –5 –4 –3 –2 –1 0 +1 +2
The integer ⫺7 is farther to the left on a Positive numbers are often written
without the ⫹ sign.
number line than ⫺3. So, ⫺7 is less.
⫺7 ⬍ ⫺3
Thursday night was colder than Friday night.

Problem Solving

34. Sam’s class played a history game. 35. Algebra Write an integer for x to make
Team A had a score of 200, Team B each statement true.
had a score of ⫺300, Team C had a score
a x ⬎ ⫺3
of ⫺100 and Team D had a score of 500.
Order the scores, from greatest to least. b x ⬍ ⫹1
c ⫺13 ⬍x

36. Estimation What is a good estimate 37. Reasoning A number, x, is four units to
for the value of point C on the the left of ⫺5 on the number line. What
number line? is the value of x? Is x greater than or less
C
than ⫺5?
– 100 0 +100

38. Writing to Explain Is the explanation 39. Number Sense Which integer is
below correct? If not, tell why not and neither positive nor negative?
write a correct response.
− 13is less than − 12 because it is farther
LVh]^c\idc
to the left on the number line. 6aVh`V ")-$&&-
"-%$&%% BdciVcV
",%$&&,
DgZ\dc
>YV]d
In 40 through 42, use the map at the right. "*)$&&.
"+%$&&- Lndb^c\
"++$&&,
40. Of the states shown, which had the lowest CZkVYV
"*%$&'* JiV]
record temperature? =VlV^^ 8Va^[dgc^V "+.$&&,
&'$&%% ")*$&()

41. Which state had the warmest record low 6g^odcV

temperature? ")%$&'-

42. List the record low temperatures in order from

least to greatest.

Lesson 15-2 345

27291_344-345_345 345 10/11/07 5:29:11 PM

 Lesson

15-3 Integers and the Number Line

How can you name and plot integers, fractions,
and decimals on the same number line?
NS 1.5 Identify Some points are plotted on the number line below. Write a
and represent on a
number line decimals, number to name each of the points A, Z, D, and W. Use integers,
fractions, mixed number, fractions, mixed numbers, or decimals. Name the points that
and positive and negative
integers. represent 2.38, and 1__14, and ⫺1.
A B C D E W X Y Z

–2 0 +1 +2

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 4, use the number line below. 7. In the example above, how could you
Write the number for each point. find the points for fractions?
B D A C
8. In the example above, describe where
–2 –1 0 +1 +2
you would plot ⫹1.35.
1. A 2. B 3. C 4. D
9. What number is represented by
point B above?
In 5 and 6, show each set of numbers on a
number line.
10. Which point above represents 2_34_?
5. 1_14_, 0.75, ⫺1, ⫺2 6. ⫺2, 1_34_, 0.5, 0

Independent Practice
In 11 through 20, use the number line below. Write the number for
each point.
Y A W B X C Z E D V

0 +1 +3

11. W 12. B 13. E 14. D 15. A

16. Z 17. Y 18. C 19. V 20. X

In 21 through 23, order the numbers in each set from least to greatest.
Show each set of numbers on a number line.

21. ⫺2, 1 , 1__

1, __ 2 3
22. 3, ⫺3, 1.5, 2__ 3 , 2.25, ⫺1
23. 0.5, 1__
3 3 4 8

346 *For another example, see Set C on page 360.

27291_346-347_001 346 12/15/06 4:07:13 PM

 Write a number to name a point. Find which point represents a given
The distance between the integers 0 and 1 number.
is divided into 10 equal parts. Since 2.38 is between 2.3 and 2.4 and is closer
Point A is to the left of zero. Point A is at ⫺3. to 2.4, point X represents 2.38.
Point Z is to the right of zero. Point Z is at ⫹3. Since 1_14_ = 1.25 and 1.25 is halfway between
4.
Point D is at 0.4 or ___ 1.2 and 1.3, point E represents 1.25.
10
Since ⫺1 is opposite of ⫹1, point C
Point W is halfway between 1.7 and 1.8.
represents ⫺1.
So, point W is at 1.75 or 1_34_.

Problem Solving

24. Writing to Explain What might you do 25. Number Sense Kathy owes Marty _12_ of
to order the numbers ⫺8, ⫺10, ⫹8, and a dollar. Mina owes Marty 0.6 of a dollar.
⫹5 without using a number line?
Who owes Marty more money?

26. Geometry Find the perimeter of the 27. The larva of the Colorado Potato Beetle
equilateral triangle below. is 0.5 inches long. The adult Colorado
Potato Beetle is _38_ inch long. Which one
3.5 cm is smaller, the adult or the larva?

28. Which number in the following choices 29. Joey got 19 out of 23 questions correct
makes the statement true? on his Social Studies test. If each question
⫺6 was worth 3 points, how many points did
⬎n
Joey earn?
A 0 B ⫺5 C ⫺7 D ⫺4

30. The table below names the lowest 31. Which number line shows a point that
points on each continent. Order the best represents ⫹1.6?
points from lowest to highest. A –3 –2 –1 0 +1 +2 +3
Continent Height (feet)
B –3 –2 –1
Africa ⫺
512 0 +1 +2 +3
⫺
Antarctica 8,327
C –3 –2 –1
Asia ⫺
1,348 0 +1 +2 +3
⫺
Europe 92
⫺
D –3 –2 –1 +1 +2 +3
North America 282 0
⫺
Oceania 52
⫺
South America 131

Lesson 15-3 347

27291_346-347_002 347 1/18/07 12:49:12 PM

 Lesson

15-4 Adding Integers 7 yard gain

How can you add two integers?
A football coach keeps a record of the
NS 2.1 Add, yards gained or lost on each play. What
subtract, multiply, and was the total number of yards gained
divide with decimals;
add with negative or lost after the two plays shown?
integers; subtract positive
integers from negative Choose an Operation Add to find
integers; and verify the 10 yard loss
reasonableness of the the result of joining the two plays.
results.

Another Example How can you add two negative integers?

You know how to represent integers on a number line. In this lesson you
are using a number line to add integers.

On the next play, the football team lost 4 yards and received a
5-yard penalty. How many yards did the team lose on that play?

Find ⫺4 ⫹ ⫺5.

Start at 0. Face the positive integers. Then move backward 5 steps for ⫺5.
Move backward 4 steps for ⫺4.

−9 −4 0
−4 0

You stop at ⫺9. So, ⫺4 ⫹ ⫺5 ⫽ ⫺9

The football team lost a total of 9 yards on that play.

When adding two integers with the same sign, you move in the
same direction on the number line, so you add the magnitude of
the numbers. You move in the same direction, so the sign of the
sum will be the same as the sign of the addends.

Explain It
1. In the example above, why did you move backwards twice?
2. How would the sum be different if the team had gained 4 yards
and the 5-yard penalty was against the other team? How would
that sum look on the number line?

348

27291_348-350_348 348 10/11/07 5:29:23 PM

 Rules for adding Find ⫹7 ⫹ ⫺10.
integers on a Walk forward 7 steps for ⫹7. Then walk backward 10 steps for ⫺10.
number line:
Start at 0; face
the positive
0 +7 –3 0 +7
integers.
Walk forward for To add two integers with different signs you move in different
positive integers. directions on the number line, so subtract the magnitudes of
Walk backward for the numbers. You move farther in the direction of the number
negative integers. with the greater magnitude, so the sign of the sum will be the
sign of that number.
⫹7 ⫹ ⫺10 ⫽ ⫺3 After the two plays, the team lost 3 yards.

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 8, use a number line to find 9. Number Sense The integers ⫹4 and
each sum. ⫺4 are opposites. What statement can

⫹2
you make about the sum of any integer
1. ⫹ ⫺7 2. ⫹4 ⫹ ⫹3
and its opposite?
3. ⫺3 ⫹ ⫺1 4. ⫹5 ⫹ ⫺9
10. The football team gained 12 yards on
5. ⫹5 ⫹ ⫺3 6. ⫺7 ⫹ ⫹9 the first play and then lost 9 yards on
7. ⫺4 ⫹ ⫹7 8. ⫺6 ⫹ ⫺3
the next play. How many yards were
gained or lost after the two plays?

Independent Practice
In 11 through 31, use a number line to find each sum.

11. ⫹2 ⫹ ⫹3 12. ⫺5 ⫹ ⫹4 13. ⫹6 ⫹ ⫺4 14. ⫺8 ⫹ ⫺5

15. 0 ⫹ ⫺4 16. ⫹7 ⫹ ⫺5 17. ⫺9 ⫹ ⫹2 18. ⫹7 ⫹ ⫺6

19. ⫺8 ⫹ ⫺3 20. ⫹7 ⫹ ⫹9 21. ⫺4 ⫹ ⫹8 22. ⫺9 ⫹ ⫺5

23. ⫺6 ⫹ ⫺5 24. ⫺8 ⫹ ⫹7 25. ⫺7 ⫹ ⫺7 26. ⫺6 ⫹ ⫹8

27. ⫺9 ⫹ ⫺8 28. ⫹9 ⫹ ⫺12 29. ⫺11 ⫹ ⫺6 30. ⫹14 ⫹ ⫺5

31. ⫹1 ⫹ ⫺7 ⫹ ⫹12 ⫹ ⫺1 ⫹ ⫹7

32. In 31, how can you find the sum without using a number line?

*For another example, see Set D on page 361. Lesson 15-4 349

27291_348-350_002 349 12/15/06 4:07:36 PM

 Problem Solving

33. Number Sense Mrs. Gomez finds her

checking account balance after each
transaction. A copy of her check register
is shown at the right.
a Would a check be represented by a
positive or a negative integer?
b Would a deposit be represented by a
positive or a negative integer?
c Find the checking account balance 34. Alina is serving juice to her friends. She
after each transaction. On which day 1 cups of juice. If she gives each
has 5__
2
was the balance the greatest? the 1 cup of juice, how many friends
friend __
least? 2
can she serve?
d What is the ending balance?

35. Xavier did math homework for 36. Which expression does not have a value
3 hour, reading homework for __
__ 2 hour, of ⫺5?
4 3
1 hour.
and science homework for __ A ⫺2 ⫹ ⫺3 C ⫺8 ⫹ ⫹3
3
How much time did Xavier spend doing B ⫺6 ⫹ ⫹1 D ⫺8 ⫹ ⫺2
homework?

37. Algebra Use a number line to find each 38. How would you use
missing value. a number line to find ⫺26 ⫹ ⫺2?
a ⫺4 ⫹■⫽9 A Walk forward 26 and backward 2.
b 9 ⫹ ■ ⫽ ⫺1 B Walk backward 26 and forward 2.
c ■⫹3⫽0 C Walk backward 26 and backward 2.
d ⫺7 ⫹
■ ⫽ ⫺13 D Walk forward 26 and backward 24.

39. The sum of Eric’s and Gus’s heights is 40. Writing to Explain How would you
135_34 inches. Eric’s height is 68.5 inches. find ⫺19 ⫹ ⫹32 without using a number
line? What is the sum?
How tall is Gus?

41. What is the prime factorization of 18?

42. Geometry Marcy is making a quilt

and cut a triangle from a piece of fabric.
What is the measure of the third angle
of Marcy’s triangle?

350

27291_348-350_350 350 10/30/07 10:15:08 AM

 Find each sum.

1. ⫺14 ⫹ ⫹10 2. ⫹8 ⫹ (⫺6) 3. ⫺8 ⫹ (⫺12)

4. ⫹9 ⫹ (⫺3) 5. ⫺5 ⫹ (⫺5) 6. ⫺7 ⫹ ⫹8

Find each difference. Simplify if possible.

7. 1
3__ 8. 5
4___ 9. 3
5__ 10. 4
2 12 8
⫺
3
1__ ⫺
1
4 __ ⫺
3
__ ⫺
3
2___
8 4 4 10

11. 5
5 __ 12. 4
1__ 13. 2
3__ 14. 7
6__ 15. 1
4__
6 5 3 8 3
⫺
11
4___ ⫺
1
__ ⫺
1
1__ ⫺
5
4__ ⫺
2
2__
12 2 6 8 3

Error Search Find each sum or difference that is not correct. Write it
correctly and explain the error.

16. ⫺6 ⫹ ⫹4 ⫽ 2 17. ⫺3 ⫹ (⫺8) ⫽ ⫺11 18. 4

5__ 19. 5
9
⫺
2
3__ ⫺
3
1__
3 5
7
2__ 2
3__
9 5

Number Sense
Estimating and Reasoning Write whether each statement is true or false.
Explain your reasoning.
20. If a ⬍ 0 and b ⬍ 0, then a ⫹ b is negative.

21. The difference of 29.13 and 17.95 is greater than 11 and less than 13.

3 and ___
22. The sum of ___ 5 can be simplified.
17 17

23. The quotient of 3,746 ⫼ 50 has a remainder less than 50.

24. The expression w ⫺ 2.5 equals 5.5 when w ⫽ 3.

25. The product of 5 and 2.5 is less than the product of 2 and 5.5.

Lesson 15-4 351

27291_351-351_001 351 12/21/06 6:02:00 PM

 Lesson

15-5 Subtracting Integers (/%%E#B#

•; •8
&%/%%E#B#
•; •8
How can you subtract integers? &%% &%%

On a winter day, George checked the .% .%

temperature during the afternoon and -% -%

NS 2.1 Add,
subtract, multiply, and again at night. How many degrees did
divide with decimals;
,% ,%
add with negative the temperature drop? +% +%
integers; subtract positive
integers from negative Choose an Operation Subtract to find *% *%
integers; and verify the
reasonableness of the the temperature change. )% )%
results.
'% '%

+• &% &%

% (• %

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 6, rewrite each subtraction 7. Reasoning Why is subtracting
problem using addition. Then find the an integer the same as adding its
answer. Use a number line to check. opposite?

1. ⫺1 ⫺ ⫹3 2. ⫺9 ⫺ ⫹4
8. In the example above, suppose the
⫹8
afternoon temperature had been
3. ⫺ ⫹5 4. ⫹4 ⫺ ⫹10 ⫹10°F and the night temperature had

been ⫺4°F. How many degrees did the

5. ⫺6 ⫺ ⫹3 6. ⫺10 ⫺ ⫹1 temperature drop?

Independent Practice
In 9 through 32, rewrite each subtraction problem using addition.
Then find each answer. Use a number line to check.

9. ⫺1 ⫺ ⫹2 10. ⫺8 ⫺ ⫹3 11. ⫺6 ⫺ ⫹8 12. ⫹3 ⫺ ⫹12

13. 0 ⫺ ⫹5 14. ⫹5 ⫺ ⫹6 15. ⫺3 ⫺ ⫹10 16. ⫺7 ⫺ ⫹11

17. ⫺10 ⫺ ⫹5 18. ⫺12 ⫺ ⫹1 19. ⫺7 ⫺ ⫹3 20. ⫹6 ⫺ ⫹3

21. ⫹8 ⫺ ⫹12 22. ⫺4 ⫺ ⫹8 23. ⫺5 ⫺ ⫹2 24. ⫺3 ⫺ ⫹6

25. ⫺5 ⫺ ⫹1 25. ⫺2 ⫺ ⫹2 27. ⫹5 ⫺ ⫹7 28. ⫺9 ⫺ ⫹8

29. ⫹7 ⫺ ⫹9 30. ⫺5 ⫺ ⫹8 31. ⫺9 ⫺ ⫹6 32. ⫹8 ⫺ ⫹11

352 *For another example, see Set D on page 361.

27291_352-353_001 352 12/15/06 4:07:50 PM

 Step 1 Step 2 Step 3
Find ⫺3 ⫺ ⫹6. The subtraction Then walk forward 6 steps for ⫹6.
Start at 0. Face the sign (⫺) means
positive integers. Walk turn around.
backward 3 steps –10 –8 –6 –4 –2 0 2 4 6 8
for ⫺3.
You stop at ⫺9.
So, ⫺3 ⫺ ⫹6 ⫽ ⫺9
–6 –4 –2 0 ⫺3 ⫺ ⫹6 can be thought of as
2 4
⫺3 plus the opposite of ⫹6.
⫺3 ⫺ ⫹6 ⫽ ⫺3 ⫹ ⫺6 ⫽ ⫺9
The temperature dropped 9°F.

Problem Solving

33. The temperature rose from ⫺10°F to 34. Susie cut a blueberry pie into slices.
⫹15°F. How much did the temperature
Each slice was _18_ of the pie. How many
rise? slices of pie were there?

35. Lincoln Elementary students sold school 36. The highest elevation in North America
mugs for a fundraiser. They bought the is Mt. McKinley, Alaska, at 20,320 feet.
mugs for $3 each and sold them for The lowest elevation is ⫺282 feet in
$5 each. If the students sold 248 mugs, Death Valley, California. What is the
how much money did they raise? difference between the two elevations?

37. Owen has a charge account at the

Transaction Amount
school store. His mom deposited $10 in
the account at the beginning of the Pencils $1.50
school year. What is the balance after he Notebook paper $0.89
purchased the items listed at the right? Protractor $1.29
Pocket dictionary $4.49
Trail Mix $1.19

38. Algebra Write the integers that 39. Number Sense The area of California is
could replace x to make each about 163,696 square miles. What is that
statement true. measurement rounded to ten thousand
a ⫺6 ⬍ x ⬍ ⫹2
square miles?

b ⫹6 ⬎ x ⬎ ⫺2
A 200,000 C 164,000
B 163,000 D 160,000

40. Writing to Explain Describe the steps 41. Finish this sentence ”Subtracting ⫹10 is
to find ⫹4 ⫺ ⫹7 on a number line. the same as adding ? .”

Lesson 15-5 353

27291_352-353_002 353 1/18/07 12:49:31 PM

 Lesson

15-6 Simplifying Expressions

How can you evaluate an algebraic expression
with integers?
AF 1.2 Use a The rules of the game Number Signs penalize a player
letter to represent an
unknown number; write for taking too much time. Greg has a score of
and evaluate simple 12 points down. What will be his score if he
algebraic expressions
in one variable by takes too much time on his next turn?
substitution.
Also NS 2.1, AF 1.0 Choose an Operation Subtract to
find a score after losing points.

Another Example How do you evaluate an expression with

more than one variable?
When more than one variable is involved, replace each variable with
a number. If only addition and subtraction are involved, proceed from
left to right, grouping two numbers at a time.

Evaluate a  b  c for a  8, b  6 and c  12.

a  b  c  8  6  12 Replace a with 8, b with 6 and c with 12

 14  12 Add 8  6.
 14  12 Rewrite the subtraction as an addition of the opposite.
 26 Add 14  12.

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
In 1 through 4, evaluate each expression 7. a In the example above, what was the
for n  5 and n  7. opposite operation used to evaluate
the expression?
1. n  5 2. n  9
b Use that operation to rewrite the
3. n  5 4. 8 n expression in the example.
8. a Write an expression to show what
In 5 and 6, evaluate each expression Greg’s score would have been if the
for r  9, s  6, and t  8. penalty for too much time was to
lose 10 points?
5. r  s  t 6. s  t  r
b What would his score have been?

354 *For another example, see Set E on page 361.

27291_354-355_001 354 1/18/07 4:35:17 PM

 Step 1 Step 2
Since 5 points are subtracted from the Evaluate s  5 for s  12.
player’s score, s, the expression s  5 shows s  5  12  5 Replace s with 12.
how to find a player’s score after getting
 12  5 Subtracting an integer is the
the time penalty. same as adding its opposite.
 17 Add 12 and 5.
Greg’s score of 12 points down can be Greg’s score will be 17, or 17 points down if
expressed with the integer 12. he takes too much time on his next turn.

Independent Practice
In 9 through 12, evaluate each expression for x  6 and x  8
9. x  3 10. 4 x 11. x  9 12. 12 x

In 13 through 20, evaluate each expression for a  4, b  2 and c  10

13. a  13 14. c  15 15. b  2 16. 18 a

17. c  b 18. a  b  c 19. a  b  c 20. c  b  a

Problem Solving

21. Which expression names the location 22. Cory uses red paint and blue paint
of a scuba diver who started at 12 feet to make purple paint. How much red
and then moved down 3 feet? paint did he use if he made 9 quarts
A 12  3
of purple?

B 3  12

C 12  3 1 1 1 makes 1 1 1
quart quart quart quart quart quart
D 3  12
red red blue purple purple purple

23. Reasoning The temperature was 9°F at 24. Barry’s CD shelf is 36 inches long. If each
6:00 A.M. Write an expression to name the CD is _38_ of an inch wide, how many CDs
temperature at 3:00 P.M. after it rose 15°F. can Barry put on one shelf?

25. California was the 31st state admitted 26. Writing to Explain Jerome says that
to the Union. The expression 1787  c if you use the LCD when subtracting
represents the year California obtained fractions, you never have to simplify the
statehood. If c  63, what year did answer. Do you agree? Why or why not?
California become a state?
Lesson 15-6 355

27291_354-355.indd 355 1/4/07 10:20:19 PM

 Lesson Problem Solving

15-7 Work Backward

Arnie, Brad, Caren, and Danica sold
nature photographs to raise money
MR 2.3 Use a variety of for their hiking club. Brad raised twice
methods, such as words,
numbers, symbols, charts, as much money as Arnie. Caren raised
graphs, tables, diagrams,
and models, to explain $100 more than Brad, and Danica
mathematical reasoning. raised half as much as Caren. How
Also MR 2.0, NS 2.0
much money did each person raise?

Guided Practice*
Do you know HOW? Do you UNDERSTAND?
You can solve this problem by working 2. Writing to Explain Describe what
backward. Check your work. you did to check your solution to
Problem 1.
1. The Penguins’ hockey practice ended
at 7:00 P.M. The team began practice 3. In the problem above, why is Danica’s
by stretching for _14_ hour. Then they $110 multiplied by 2 to find the
practiced skating and shooting for amount that Caren raised?
_1_ hour. During the last _3_ hour, the team
2 4 4. Write a Problem Write a real-world
played a scrimmage game. What time problem that you can solve by working
did practice start? backward.

Independent Practice
Solve.

5. On a winter night, the temperature dropped

15°F between midnight and 6:00 A.M. By 11:00 A.M.,
the temperature had gone up 7°F. By 3:00 P.M. the
temperature went up another 9°F, making the • What do I know?
temperature 25°F. What was the temperature
• What am I asked to find?
at midnight?
• What diagram can I use to help
understand the problem?
6. Mel spent $9 at the movies, earned $24 mowing • Can I use addition, subtraction,
lawns, and bought a magazine for $5. He had multiplication, or division?
$21 left. How much money did he have at the start? • Is all of my work correct?
• Did I answer the right question?
• Is my answer reasonable?

356 *For another example, see Set F on page 361.

27291_356-357_001 356 12/15/06 4:08:29 PM

 Read and Understand Plan Solve
I know: I know how much each I can start with the amount
Danica raised $110. person made compared Danica raised and work
to someone else. backward.
Danica raised half
as much as Caren. Arnie Brad Caren Danica Arnie Brad Caren Danica
$ 110 $ 110
Caren raised $100 ×2 + 100 ÷2 ÷2 − 100 ×2
more than Brad.
Caren raised 2 ⫻ $110 ⫽ $220.
Brad raised twice as Brad raised $220 ⫺ 100 ⫽ $120.
much money as Arnie. Arnie raised $120 ⫼ 2 ⫽ $60.

7. The numbers show how 8. Mary is knitting a scarf that will be

many shells in each 36 inches long. She knitted 5 inches on
drawer. Meg has a the second day, 8 inches on the third
total of 156 shells. day, and 10 inches on the fourth day.
She organizes them She needs to knit 3 inches more to
by size. How many finish the scarf. How much did she knit
shells are in drawer 1? on the first day?

9. A baby gains about 2_15_ pounds each 10. Briana has 1_14_ cups of sesame seeds left
month for the first three months after in the bag she bought for baking. She
birth. When he was 3 months old, Tyler used the sesame seeds to make muffins,
1 pounds. About how
weighed 14___ bread, and bagels to sell at a bake sale.
10
much did Tyler weigh at birth? How many cups of sesame seeds were
in the bag she bought?

11. Geometry What is the area of a square

garden with a side that measures
18 feet?
2
__ 1
__ 1
cups used cups used 2__ cups used
3 2 3

12. Reasoning Donna, Pam, and Mike 13. Workers need 6 weeks to resurface
worked at a school car wash. Donna 15 miles of road. They resurfaced
washed half as many cars as Mike did. 2_12_ miles the fourth week, 3 miles the
Pam washed 9 more than Donna. Mike
fifth week, and 4_12_ miles the sixth week.
washed 5 fewer than Pam. If Mike
washed 8 cars, how many cars did Pam How many miles did they resurface
and Donna wash? What was the total during the first three weeks?
number of cars washed? A 5 miles B 10 miles
C 6 miles D 11 miles

Lesson 15-7 357

27291_356-357_357 357 10/11/07 5:29:51 PM

 1. The lowest temperature ever recorded 4. Which of the following can be used to
in the United States was in Alaska in represent a deposit of $132? (15-1)
1971. It was about ⫺80° Fahrenheit. ⫹132
A
What is the opposite of ⫺80? (15-1)
B ⫹1
A ⫺80

⫺79
C 0
B
D ⫺132
C ⫹80

D ⫹81 5. Which comparison is true? (15-2)

2. The elevations of some points of ⫺15 ⫺10 ⫺5 0 ⫹5 ⫹10 ⫹15

interest are given in the table. These ⫺12

A ⬍ ⫺6
elevations are above, at, or below sea
level. Which of the following lists the B ⫺12 ⬎ ⫺6
elevations from least to greatest? (15-2) ⫺8
C ⬎ ⫺4

D ⫹3 ⬍ ⫺3
Location Elevation (ft)
Potomac River 1
ⴚ8 6. Which number line shows
New Orleans
4 ? (15-3)
Point F at 1__
Delaware River 0 5
Lake Champlain 95 A F

⫺2 ⫺1 0 ⫹1 ⫹2 ⫹3 ⫹4 ⫹5 ⫹6
A ⫺8, 0, ⫹1, ⫹95
B F
B 0, ⫹1, ⫺8, ⫹95
⫺2 ⫺1 0 ⫹1 ⫹2 ⫹3 ⫹4 ⫹5 ⫹6
C ⫺8, ⫹1, 0, ⫹95
⫹95, ⫺8, ⫹1, C F
D 0
⫺2 ⫺1 0 ⫹1 ⫹2 ⫹3 ⫹4 ⫹5 ⫹6
3. What is the value of 5 ⫺ n
D F
when n ⫽ ⫹7? (15-6)
⫺12 ⫺2 ⫺1 0 ⫹1 ⫹2 ⫹3 ⫹4 ⫹5 ⫹6
A
⫺2
B 7. What is ⫹11 + (⫺18)? (15-4)
C ⫹2 ⫹29
A
D ⫹12 ⫹7
B
C ⫺7

D ⫺29

358

27291_358-359_358 358 11/28/07 5:15:42 PM

 8. What is the integer at Point L? (15-1) 12. Which of the following is equal
M L N
to ⫺6 ⫺ (⫹3) (15-5)
A ⫺6 ⫹ (⫹3)
⫺5 0
B ⫹6 ⫹ (⫹3)
A ⫺6
C ⫹6 ⫹ (⫺3)
B ⫺5
D ⫺6 ⫹ (⫺3)
C ⫺4

D ⫺3 13. What numbers are represented

by Points R and T on the number
9. What is ⫹2 ⫹ (⫺8)? (15-4) line? (15-3)
A ⫺10 R T
⫺6 ⫺15 ⫺10 ⫺5 0 ⫹5 ⫹10 ⫹15
B
C ⫹6 ⫺7
A and 8.5
D ⫹10 ⫺8
B and 8.5
C ⫺7 and 9.5
10. Danny had a ⫺$30 balance for an
amount borrowed from his parents. D ⫺8 and 9.5
Then he borrowed $20 more. What
was his balance then? (15-5) 14. On a winter morning, the outside
temperature was ⫺6°. By noon, it
A $50
had risen 15 degrees. What was the
B $10 temperature in the afternoon? (15-4)
C ⫺$10 ⫺9°
A
D ⫺$50
B 9°

11. After a fundraising dinner, a charity has C 15°

a balance of $2,530. They spent $700 to D 21°
host the dinner. If they made $1,400 on
the event and another $300 afterwards 15. What is the value of a ⫹ b ⫺ c when
from a private donation, how much a ⫽ ⫺3, b ⫽ ⫹7 and c ⫽ ⫹1? (15-6)
money did the charity have before A ⫺11
hosting the dinner? (15-7)
B ⫺10
A $130
C ⫹3
B $1,530
D ⫹5
C $2,130
D $3,530

Topic 15 Test Prep 359

27291_358-359_359 359 11/28/07 5:15:47 PM

 Set A, pages 342–343

Write an integer for each point. Remember that the ⫹ and ⫺ signs
name a direction from zero.
A B
Write an integer for each description.
–5 0 +5
1. Two degrees below zero.
Point A is three units from zero and to the left of
zero. Point A is at ⫺3. 2. Fifty-seven feet above sea level.
3. A loss of three yards.
Point B is two units from zero and is to the right of
zero. Point B is at ⫹2. Write an integer for each point.
A B C D E H
–5 0 +5

4. C 5. E 6. B
7. H 8. D 9. A

Set B, pages 344–345

Compare ⫺4 and ⫹3. Use ⬎, ⬍ or ⫽ . Remember that numbers increase

in value as you move to the right on
Plot the numbers on a number line. a number line, and decrease as you
–5 –4 –3 –2 –1 move to the left.
0 +1 +2 +3 +4 +5

As you move to the right from any point on a

Compare. Use ⬍, ⬎, or ⫽ for each 䊊.
number line, the numbers increase in value. 1. ⫹3
䊊 ⫺3 2. ⫺8
䊊 2 ⫹

⫹3 ⬎ ⫺4 ⫹3 is to the right of ⫺4. 3. ⫺ 9䊊 ⫺7 4. ⫹ 3䊊 2 ⫹

⫺4
5. ⫹ 1䊊 ⫺16 6. ⫺ 14 䊊 10 ⫺
⬍ ⫹3 ⫺4 is to the left of ⫹3.

Set C, pages 346–347

1 , ⫺2, and 0.5 from least to greatest.

Order 4, __ Remember to convert fractions to
4
1 is the same as 0.25.
__ 1 is the same as 0.5.
__ decimals or decimals to fractions to
4 2 make the comparing easier.
Plot the numbers on a number line. Order the numbers from least to greatest.
1
4 0.5 1. ⫺1, 1 , ⫺4, 1.25
__
2
–3 –2 –1 0 +1 +2 +3 +4 1
2. 4, 2.5, ⫺2, __
4
1 , 0.5, 4 3 , 3, ⫺1
3. 3.25, __
The order from least to greatest is ⫺2, __
4 4
1 , ⫺3, 0.68, ⫺2
4. 2__
4
360

27291_360-361.indd 360 1/4/07 10:20:41 PM

 Set D, pages 348–350, 352–353

Use a number line to find ⫹2 ⫹ ⫺5. Remember to move to the right on

a number line when adding positive
integers and move to the left on a
–5 0 +5 number line when adding negative
integers.
Start at 0 and face the positive integers. ⫹6
1. ⫹ ⫺4 2. ⫺7 ⫹ ⫺2
Walk forward 2 steps for ⫹2.
3. ⫺8 ⫹ ⫹2 4. ⫹10 ⫹ ⫺5
Then walk backward 5 steps for ⫺5.
You stop at ⫺3. 5. ⫺12 ⫹ ⫹3 6. ⫺3 ⫹ ⫺9
So, ⫹2 ⫹ ⫺5 ⫽ ⫺3 Remember when using a number line
to subtract integers, the subtraction
Find ⫺2 ⫺ ⫹10. sign means to turn around.
To subtract an integer, add the opposite of the 7. ⫹5 ⫺ ⫹8 8. ⫺3 ⫺ ⫹2
integer being subtracted. ⫺5
9. ⫺ ⫹8 10. ⫺2 ⫺ ⫹4
⫺2 ⫺ ⫹10 ⫺2 ⫹ ⫺10
11. ⫹7 ⫺ ⫹9 12. 0 ⫺ ⫹6
⫺2 ⫹ ⫺10 ⫽ ⫺12

Set E, pages 354–355

Find a ⫹ b ⫺ c for a ⫽ ⫺4, b ⫽ ⫹3 and c ⫽ ⫹5. Remember to replace the variable

with the given values.
a ⫹ b ⫺ c ⫽ ⫺4 ⫹ ⫹3 ⫺ ⫹5 Replace the variables.
Evaluate each expression for x ⫽ ⫹3
⫽ ⫺1 ⫺ ⫹5 Add ⫺4 ⫹ ⫹3.
and x ⫽ ⫺2
⫽ ⫺1 ⫹ ⫺5 Rewrite subtraction ⫹3
1. ⫹x 2. x ⫺ ⫹5
as addition of the
3. x ⫹ ⫹10 4. x ⫺ ⫹1
opposite.
⫺6
5. x ⫺ ⫹3 6. x ⫹ ⫹4
⫽ Add ⫺1 ⫹ ⫺5

Set F, pages 356–357

To work backward, follow these steps. Remember addition and subtraction

undo each other.
Step 1 Identify the unknown initial amount.
1. Jean spent __23 hour on math homework
Step 2 List each change, starting with the and __12 hour on English. Then she spent
initial amount.
3 hour baking. If she finished baking
__
4
Step 3 Start at the end result. Work backward at 8:00, what time did she start doing
using the inverse of each change.
her homework?

Topic 15 Reteaching 361

27291_360-361_002 361 12/15/06 4:09:00 PM