Learn on PengiIllustrative Mathematics, Grade 7Chapter 5: Rational Number Arithmetic

Lesson 1: Interpreting Negative Numbers

In this Grade 7 lesson from Illustrative Mathematics Chapter 5, students review signed numbers by interpreting negative numbers in real-world contexts such as temperature and elevation above or below sea level. Students practice reading thermometers, comparing positive and negative values on a number line, and ordering rational numbers from least to greatest. The lesson also introduces absolute value as a way to describe a number's distance from zero and reinforces the concept of opposites.

Section 1

Negative Numbers in the Real World

Property

Negative numbers appear in real-world measurements that are two-sided, with a value of zero acting as a reference point.
Examples include temperature (degrees below freezing), elevation (below sea level), and finance (debits or debt).

Examples

  • The temperature rose from a low of −8∘-8^\circF to a high of 15∘15^\circF. The total temperature spread is the distance from −8-8 to 00 plus the distance from 00 to 1515, so 8+15=23∘8+15=23^\circF.
  • A submarine at −300-300 feet ascends to −120-120 feet. The submarine traveled a vertical distance of ∣−300−(−120)∣=∣−180∣=180|-300 - (-120)| = |-180| = 180 feet.

Section 2

Defining Integers and Opposites

Property

Negative numbers are numbers less than 0.
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.
The notation −a-a is read as “the opposite of aa.”
The whole numbers and their opposites are called the integers.
The integers are the numbers …,−3,−2,−1,0,1,2,3,…\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots.

Examples

  • The opposite of 1515 is −15-15, as both are 15 units from zero.
  • The opposite of −9-9 is 99. This can be written as −(−9)=9-(-9) = 9.
  • If y=−25y = -25, then −y-y means the opposite of −25-25, which is −(−25)=25-(-25) = 25.

Explanation

Integers expand our number system to include negative values, which are like mirror images of positive numbers across zero. The 'opposite' of a number is simply its reflection on the other side of the number line.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Negative Numbers in the Real World

Property

Negative numbers appear in real-world measurements that are two-sided, with a value of zero acting as a reference point.
Examples include temperature (degrees below freezing), elevation (below sea level), and finance (debits or debt).

Examples

  • The temperature rose from a low of −8∘-8^\circF to a high of 15∘15^\circF. The total temperature spread is the distance from −8-8 to 00 plus the distance from 00 to 1515, so 8+15=23∘8+15=23^\circF.
  • A submarine at −300-300 feet ascends to −120-120 feet. The submarine traveled a vertical distance of ∣−300−(−120)∣=∣−180∣=180|-300 - (-120)| = |-180| = 180 feet.

Section 2

Defining Integers and Opposites

Property

Negative numbers are numbers less than 0.
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.
The notation −a-a is read as “the opposite of aa.”
The whole numbers and their opposites are called the integers.
The integers are the numbers …,−3,−2,−1,0,1,2,3,…\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots.

Examples

  • The opposite of 1515 is −15-15, as both are 15 units from zero.
  • The opposite of −9-9 is 99. This can be written as −(−9)=9-(-9) = 9.
  • If y=−25y = -25, then −y-y means the opposite of −25-25, which is −(−25)=25-(-25) = 25.

Explanation

Integers expand our number system to include negative values, which are like mirror images of positive numbers across zero. The 'opposite' of a number is simply its reflection on the other side of the number line.