In this lesson, students see that the difference between two numbers can be positive or negative, but the distance between two numbers is always positive. Using the geometry of the number line (MP7), they see that if you switch the order in which you subtract two numbers, the difference becomes its opposite.
For example, to find the difference in temperature between +70\(^∘\mathrm C\) and +32\(^∘\mathrm C\) we calculate \(70−32=38\), so the difference is 38\(^∘\mathrm C\). The distance between these two is also 38\(^∘\mathrm C\). On the other hand, to find the difference in temperature between +32\(^∘\mathrm C\) and +70\(^∘\mathrm C\) we calculate \(32−70=−38\), so the difference is −38\(^∘\mathrm C\). The distance is still 38\(^∘\mathrm C\). In general, if \(a−b=x\), then \(b−a=−x\). By observing the outcome of several examples, students may conjecture that this is always true (MP8).
Lesson overview
 6.1 Warmup: Number Talk: Missing Addend (5 minutes)

6.2 Activity: Expressions with Altitude (10 minutes)
 Includes "Are you Ready for More?" extension problem
 6.3 Activity: Does the Order Matter? (10 minutes)
 Lesson Synthesis
 6.4 Cooldown: A Subtraction Expression (5 minutes)
Learning goals:
 Compare and contrast (orally) subtraction expressions that have the same numbers in the opposite order.
 Recognize that the “difference” of two numbers can be positive or negative, depending on the order they are listed, while the “distance” between two numbers is always positive.
 Subtract signed numbers, and explain (orally) the reasoning.
Learning goals (student facing):
 Let's bring addition and subtraction together.
Learning targets (student facing):
 I can find the difference between two rational numbers.
 I understand how to subtract positive and negative numbers in general.
Required materials:
 Fourfunction calculators
Required preparation:
 Use of calculators is optional.
 In this lesson, the important insights come from observing the outcome of evaluating expressions.
 Practice evaluating the expressions is of secondary importance.
Glossary:
 Access the complete Grade 7 glossary.
Standards:
 This lesson builds on the standard:CCSS.6.EE.B
 This lesson builds towards the standard:CCSS.7.NS.A.1MS.7.NS.1MO.7.NS.A
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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