Integers and Opposite Numbers

These lessons, with videos, examples and solutions, help Grade 6 students learn about real-world positive and negative numbers and zero.

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New York State Common Core Math Grade 6, Module 3, Lesson 3, Lesson 4

Grade 6, Module 3, Lesson 3 Worksheets (pdf)
Grade 6, Module 3, Lesson 4 Worksheets (pdf)

The following figure shows positive numbers, negative numbers and opposite numbers. Scroll down the page for examples and solutions.

Lesson 3 Student Outcomes

Students use positive and negative numbers to indicate a change (gain or loss) in elevation with a fixed reference point, temperature, and the balance in a bank account.
Students use vocabulary precisely when describing and representing situations involving integers; e.g., an elevation of - 10 feet is the same as 10 feet below the fixed reference point.
Students choose an appropriate scale for the number line when given a set of positive and negative numbers to graph.

Example 1: A Look at Sea Level

The picture below shows three different people participating in activities at three different elevations. What do you think the word elevation means in this situation?

Exercises

Refer back to Example 1. Use the following information to answer the questions.
The diver is 30 feet below sea level.
The sailor is at sea level.
The hiker is 2 miles (10,560 feet) above sea level.

Write an integer to represent each situation.
Use an appropriate scale to graph each of the following situations on the number line to the right.

Also, write an integer to represent both situations.
a. A hiker is 15 feet above sea level.
b. A diver is 20 feet below sea level.

For each statement there are two related statements: i and ii. Determine which related statement is expressed correctly (i and ii), and circle it. Then correct the other related statement so that both parts, i and ii, are stated correctly.
a. A submarine is submerged 800 feet below sea level.
b. The elevation of a coral reef with respect to sea level is given as -250 feet.

Lesson 4 Student Outcomes

Students understand that each nonzero integer, a, has an opposite , denoted -a; and that a and -a are opposites if they are on opposite sides of zero and are the same distance from zero on the number line.
Students recognize the number zero is its own opposite.
Students understand that since all counting numbers are positive, it is not necessary to indicate such with a plus sign.

Example 1: Every Number has an Opposite

Locate the number 8 and its opposite on the number line. Explain how they are related to zero.

Exercises 2–3

Locate the opposites of the numbers on the number line.
9, -2, 4, -7
Write the integer that represents the opposite of each situation. Explain what zero means in each situation.
a. 100 feet above sea level.
b. 32 degrees below zero.
c. A withdrawal of $25

Example 2: A Real World Example

Maria decides to take a walk along Central Avenue to purchase a book at the bookstore. On her way, she passes the Furry Friends Pet Shop and goes in to look for a new leash for her dog. The Furry Friends Pet Shop is seven blocks west of the bookstore. After she leaves the bookstore, she heads east for seven blocks and stops at Ray’s Pet Shop to see if she can find a new leash at a better price. Which locations, if any, are the furthest from Maria while she is at the bookstore?
Determine an appropriate scale and model the situation on the number line below.
Explain your answer. What does zero represent in the situation?

Exercises 4–6

Read each situation carefully and answer the questions. 4. On a number line, locate and label a credit of $15 and a debit for the same amount from a bank account. What does zero represent in this situation? 5. On a number line, locate and label 20 C below zero and 20 C above zero. What does zero represent in this situation? 6. A proton represents a positive charge. Write an integer to represent protons. An electron represents a negative charge. Write an integer to represent electrons.

Closing

What is the relationship between any number and its opposite when plotted on a number line? How would you use this relationship to locate the opposite of a given number on the number line? Will this process work when finding the opposite of zero?

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Lesson 3 Problem Set

Write an integer to match the following descriptions.
a. A debit of $40
b. A deposit of $225
c. 14,000 feet above sea level
d. A temperature increase of 40°F
e. A withdrawal of $225
f. 14,000 feet below sea level

For Problems 2–4, read each statement about a real-world situation and the two related statements in parts (a) and (b) carefully. Circle the correct way to describe each real-world situation; possible answers include either (a), (b), or both (a) and (b).

A whale is 600 feet below the surface of the ocean.
a. The depth of the whale is 600 feet from the ocean’s surface.
b. The whale is -600 feet below the surface of the ocean.
The elevation of the bottom of an iceberg with respect to sea level is given as -125 feet.
a. The iceberg is 125 feet above sea level.
b. The iceberg is 125 feet below sea level.
Alex’s body temperature decreased by 2°F.
a. Alex’s body temperature dropped 2°F.
b. The integer -2 represents the change in Alex’s body temperature in degrees Fahrenheit.
A credit of $35 and a debit of $40 are applied to your bank account.
a. What is an appropriate scale to graph a credit of $35 and a debit of $40? Explain your reasoning.
b. What integer represents “a credit of $35” if zero represents the original balance? Explain.
c. What integer describes “a debit of $40” if zero represents the original balance? Explain.
d. Based on your scale, describe the location of both integers on the number line.
e. What does zero represent in this situation?

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Lesson 4 Problem Set

Find the opposite of each number, and describe its location on the number line.
a. -5
b. 10
c. -3
d. 15
Write the opposite of each number, and label the points on the number line.
a. Point A: the opposite of 9
b. Point B: the opposite of -4
c. Point C: the opposite of -7
d. Point D: the opposite of 0
e. Point E: the opposite of 2
Study the first example. Write the integer that represents the opposite of each real-world situation. In words, write the meaning of the opposite.
a. An atom’s positive charge of 7
b. A deposit of $25
c. 3,500 feet below sea level
d. A rise of 45°C
e. A loss of 13 pounds
On a number line, locate and label a credit of $38 and a debit for the same amount from a bank account. What does zero represent in this situation?
On a number line, locate and label 40℃ below zero and 40℃ above zero. What does zero represent in this situation?

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