This document contains notes from lessons on rational numbers from a mathematics module. It includes examples and exercises on locating rational numbers on number lines, including integers and fractions. Key points covered are that rational numbers can be written as fractions, integers and some decimals are rational, and the opposite of a rational number has the same magnitude but opposite sign when graphed on a number line. Sample problems have students graphing rational numbers and their opposites on number lines segmented into units of equal length.

1. Module 3 Lesson 5 & 6 Merged.notebook
1
January 22, 2015
Jan 29­4:20 PM
Problem Set Lesson 3

2. Module 3 Lesson 5 & 6 Merged.notebook
2
January 22, 2015
Jan 29­4:21 PM

3. Module 3 Lesson 5 & 6 Merged.notebook
3
January 22, 2015
Jan 29­4:20 PM
Problem Set Lesson 4

4. Module 3 Lesson 5 & 6 Merged.notebook
4
January 22, 2015
Jan 29­4:24 PM

5. Module 3 Lesson 5 & 6 Merged.notebook
5
January 22, 2015
Jan 29­4:24 PM

6. Module 3 Lesson 5 & 6 Merged.notebook
6
January 22, 2015
Aug 26­5:45 PM
MODULE 3 Rational Numbers
Topic A: Understanding Positive and Negative
Lesson 5: The Opposite of a Number’s Opposite
Student Outcomes
§ Students understand that, for instance, the opposite of ‐5 is denoted –(‐5) and is equal to 5. In
general, they know that the opposite of the opposite is the original number; e.g., ‐(‐a) = a
§ Students locate and position opposite numbers on a number line.

7. Module 3 Lesson 5 & 6 Merged.notebook
7
January 22, 2015
Nov 17­2:29 PM
Example 1: The Opposite of an Opposite of a Number
What is the opposite of the opposite of 8? How can we illustrate this number on a number
line?
a. What number is 8 units to the right of 0?
b. How can you illustrate locating the opposite of 8 on this number line?
What is the opposite of 8?
c. Use the same process to locate the opposite of ‐8. What is the opposite of ‐8?
d. The opposite of an opposite of a number is .

8. Module 3 Lesson 5 & 6 Merged.notebook
8
January 22, 2015
Nov 17­2:29 PM
A “” symbol means “the opposite of a number.”
 (5) = 5
Since the opposite of 5 is negative 5, and the opposite of negative 5 is positive 5, then ‐(‐5) = 5.
§ What is the opposite of negative six?
§ What is the opposite of the opposite of 10?
§ How would you write the opposite of the opposite of 12?
§ What does a “” symbol mean?
§ What is the opposite of the opposite of a debit of $12?
§ In general, the opposite of the opposite of a number is the original number; e.g., ‐(‐a) = a.

9. Module 3 Lesson 5 & 6 Merged.notebook
9
January 22, 2015
Nov 17­2:29 PM
Exercise
Complete the table using the cards in your group.

10. Module 3 Lesson 5 & 6 Merged.notebook
10
January 22, 2015
Nov 17­2:29 PM
1. Write the opposite of the opposite of ‐10 as an equation.
2. In general, the opposite of the opposite of a number is the ___________
3. Provide a real‐world example of this rule. Show your work.

11. Module 3 Lesson 5 & 6 Merged.notebook
11
January 22, 2015
Aug 26­8:21 PM
Closing
Please take out your exit ticket for Lesson 5, close your
binder, and complete the exit ticket. This will be collected.
§ What is the opposite of an opposite of a number? Support your answer with
an example.
ú The opposite of an opposite of a number is the original number. The
opposite of the opposite of negative 6 is negative 6 because the opposite of ‐6
is 6. The opposite of 6 is ‐6.
§ What is the relationship between the location of a nonzero number on the
number line and the location of its opposite on the number line?
ú A number and its opposite are located the same distance from 0 on a
number line but on opposite sides of 0.

12. Module 3 Lesson 5 & 6 Merged.notebook
12
January 22, 2015
Jan 22­7:26 AM
MODULE 3 Rational Numbers
Topic A: Understanding Positive and Negative
Lesson 6: Rational Numbers on the Number Line
Student Outcomes
§ Students use number lines that extend in both directions and use 0 and 1 to locate integers and rational numbers on the number line.
Students know that the sign of a nonzero rational number is positive or negative, depending on whether the number is greater than zero
(positive) or less than zero (negative), and use an appropriate scale when graphing rational numbers on the number line.
§ Students know that the opposites of rational numbers are similar to the opposites of integers. Students know that two rational numbers
have opposite signs if they are on different sides of zero, and that they have the same sign if they are on the same side of zero on the number
line.

13. Module 3 Lesson 5 & 6 Merged.notebook
13
January 22, 2015
Jan 22­7:26 AM
What is a rational number?
A rational number is a number that can be
written as a fraction. (The denominator cannot
be zero.)
Integers, Whole Numbers, Some Decimals
Decimals are rational when:
~it terminates (ends)
~it repeats with a pattern

14. Module 3 Lesson 5 & 6 Merged.notebook
14
January 22, 2015
Jan 22­7:28 AM
So what about a number like Pi?
Pi does not terminate and
it never repeats or
develops a pattern.
What do we call it?
Irrational
number and
are written
with a symbol

15. Module 3 Lesson 5 & 6 Merged.notebook
15
January 22, 2015
Jan 22­7:29 AM
Locate and graph the number and its opposite on a number line.
Example 1: Graphing Rational Numbers

16. Module 3 Lesson 5 & 6 Merged.notebook
16
January 22, 2015
Jan 22­7:29 AM
Use what you know about the points, and its opposite, to graph both points on the number line below. The
fraction, , is located between which two consecutive integers? Explain your reasoning.
Exercise 1:
On the number line, each segment will have an equal length of . In the fraction ,
the numerator is and the denominator is . The fraction is located
between and .
0
Explanation:

17. Module 3 Lesson 5 & 6 Merged.notebook
17
January 22, 2015
Jan 22­7:29 AM
Example 2: Rational Numbers and the Real World
The water level of a lake rose 1.25 feet after it rained. Answer the questions below using the
diagram below.
a. Write a rational number to represent the situation.
b. What two integers is 1.25 between on a number line?
c. Write the length of each segment on the number line as a decimal and a fraction.
d. What will be the water level after it rained? Graph the point on the number line.
e. After two weeks of rain, the water level of the lake is the opposite of the water level before
it rained. What will be the new water level? Graph the point on the number line. Explain how
you got your answer.
f. State a rational number that is not an integer whose value is less than 1.25, and describe its
location between two consecutive integers on the number line.

18. Module 3 Lesson 5 & 6 Merged.notebook
18
January 22, 2015
Jan 22­7:30 AM
Closing
Please take out your exit ticket for Lesson 6, close your
binder, and complete the exit ticket. This will be collected.