The purpose of this lesson is to develop the rules for multiplying two negative numbers. Students use the familiar fact that \(\displaystyle \mbox{distance} = \mbox{velocity}\times\mbox{time}\) to make sense of this rule. They interpret negative time as time before a chosen starting time and then figure out what the position is of an object moving with a negative velocity at a negative time. An object moving with a negative velocity is moving from right to left along the number line. At a negative time it has not yet reached its starting point of zero, so it is to the right of zero, and therefore its position is positive. So a negative velocity times a negative time gives a positive position. When students connect reasoning about quantities with abstract properties of numbers, they engage in MP2.
There is also an optional activity where students can use another approach to understanding why the product of two negative numbers is positive, by examining patterns in an extended multiplication table that includes both positive and negative numbers (MP7).
Lesson overview
 9.1 Warmup: Before and After (5 minutes)
 9.2 Activity: Backwards in Time (15 minutes)

9.3 Optional Activity: Cruising (15 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.

9.4 Optional Activity: Rational Numbers Multiplication Grid (10 minutes)
 There is a digital applet in this activity.
 Lesson Synthesis
 9.5 Cooldown: True Statements (5 minutes)
Learning goals:
 Generalize (orally) that the product of two negative numbers is positive.
 Interpret signed numbers used to represent elapsed time before or after a chosen reference point.
 Use patterns to find the product of signed numbers, and explain (orally and using other representations) the reasoning.
Learning goals (student facing):
 Let's multiply signed numbers.
Learning targets (student facing):
 I can explain what it means when time is represented with a negative number in a situation about speed and direction.
 I can multiply two negative numbers.
Required materials:
 copies of blackline master
Required preparation:
 It is optional to provide 1 copy of the Rational Numbers Multiplication Grid blackline master to each student.
Glossary:
 Access the complete Grade 7 glossary.
Standards:

This lesson builds on the standard: CCSS.6.RP.A.3.bMS.6.RP.3bMO.6.RP.A.3b
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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