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Lesson
1
Negatives and Opposites
Problem Solving:
Bar Graphs
Negatives and Opposites
Vocabulary
What are the numbers to the left of zero on
the number line?
negative numbers
integers
opposites
We use a number line to look at many different kinds of numbers.
These include whole numbers, fractions, mixed numbers, decimal
numbers, and percents.
0 0.1
1
4
50%
0.75
1
1 14
1.5
175%
2
So far, we’ve worked with numbers that appear to the right of zero on
the number line. Now we explore numbers that appear to the left of
zero. These numbers are called negative numbers . We write them with
a negative sign in front. Here are some examples of negative numbers.
−217.58
−100
−56 14 −41.7 −30 12
−12.5
0
Fractions and decimal numbers can be negative numbers as well. When
the numbers only include positive and negative whole numbers and zero
(but no fractions or decimal numbers), we call the numbers integers .
Unit 8 • Lesson 1 537
Lesson 1
A set of integers is written like this: { . . . −2, −1, 0, 1, 2, . . . }. The small
dots on each side of −2 and 2 indicate that we count to infinity in each
direction. Let’s look at a number line showing only integers.
Negative
−4
−3
−2
Positive
−1
0
1
2
3
4
What are opposites?
On a number line, positive numbers are to the right of zero, and they
go on infinitely. Negative numbers are to the left of zero and they
go on infinitely as well. Each number has an opposite . For instance,
−3 is the opposite of 3. Likewise, 5 is the opposite of −5. A number
and its opposite are an equal distance from zero on the number line.
For example, −2 is two units away from zero, and so is 2.
Example 1
Using a number line, show that there is symmetry between opposites.
Line of Symmetry
−5
−4
−3
−2
−1
0
1
2
3
4
5
The distances of two opposite numbers from zero are the same.
There are all types of negative numbers. Remember, fractions and
decimal numbers are the numbers in between the whole numbers on the
number line. Fractions and decimal numbers can be positive or negative.
−1.25 −1 −0.75 −0.5 −0.25
0
0.25 0.5 0.75
1
−1 14
−1
− 34
Apply Skills
Turn to Interactive Text,
page 279.
538 Unit 8 • Lesson 1
− 12
− 14
0
1
4
1
2
3
4
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
1
1.25
1 14
Lesson 1
Problem Solving: Bar Graphs
Vocabulary
What are the important parts of bar graphs?
Bar graphs are one way to display information so that it’s easy to
make comparisons between numbers. In Example 1, the bar graph
shows the number of hours that Cecelia practices soccer every
month. We start to read the graph by looking at the labels. Hours are
on the vertical line (up and down), or vertical axis . Months are on the
horizontal line (across), or horizontal axis .
vertical axis
horizontal axis
Example 1
How do we use bar graphs to display data?
Cecelia’s Soccer Practice
25
Hours
20
15
10
5
0
May
June
July
Aug.
Sept.
Oct.
Month
The number of practice hours changes from month to month.
For example, Cecelia and her family went on vacation in July, so
she practiced only five hours. The amount of practice per month
goes up gradually from August through October.
We will look at bar graphs again later and will see how negative
numbers are shown on a bar graph. For now, let’s familiarize
ourselves with reading the bar graph—pulling out information and
making comparisons. We should know what the important parts
of the graph mean—horizontal axis, vertical axis, and bars—and
how to read the graph.
Problem-Solving Activity
Turn to Interactive Text,
page 280.
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Unit 8 • Lesson 1 539
Lesson 1
Homework
Activity 1
Name the missing numbers on the number lines.
ModelIn Problem 1, the answer for (a) is −5.
1.
2.
(a)
(b)
(c)
(d)
(e)
0
1
2
3
4
5
−25
−20
−15
−10
−5
0
(a)
(b)
(c)
(d)
(e)
−0.25
0
3.
4.
−1
(a)
−0.75
1
4 (b)
−0.5
(c)
3
4
(d)
1
2
(e)
1
4
(a)
1
4
0
(b)
1
2
(c)
3
4
(d)
1
1 14
Activity 2
On your paper, write the opposite of each number. Use the number lines
in Activity 1 and your knowledge of symmetry.
1. −5 ​
1
540 1
4
2. 10 ​
3. − 4 ​
4. 3.75 ​
5. −100 ​
6. 3 ​
Unit 8 • Lesson 1
2
2
3
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.
Lesson 1
Homework
Activity 3
Use the information in the bar graph to answer the questions.
Cecelia’s Soccer Practice
25
Hours
20
15
10
5
0
May
June
July
Aug.
Sept.
Oct.
Month
1. About how many hours did Cecelia practice soccer in May? ​
2. What is the label of the horizontal axis? ​
3. About how many more hours did Cecelia practice in June than in May? ​
4. In which month did she practice soccer most? ​
5. In which month did Cecelia practice soccer least? ​
6. What is the label of the vertical axis? ​
Activity 4 • Distributed Practice
Solve.
1. Write 7% as a decimal number. ​
1
4
2. Rewrite 0.25 as a fraction. ​
3
3. The fraction 4 is the same as what decimal number? ​
4. 1.25 + 8.7 + 2.9 ​
5.
6
7
8
21
· 49 = ​
6. 21.78 − 2.99
7. 6.88 ÷ 0.8 ​
3
1
8. 3 4 + 1 2 1
4
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.
Unit 8 • Lesson 1 541