In this lesson students see how to use the distributive property to write a compact expression for situations where one quantity is described in relation to another quantity in language such as "half as much again" and "one third more than." If \(y\) is half as much again as \(x\), then \(y = x + \frac12 x\). Using the distributive property, this can be written as \(y = (1 \frac12)x\). Students apply this sort of reasoning to various situations. A warmup activity activates their prior knowledge of using the distributive property to write equivalent expressions. When students look for opportunities to use the distributive property to write equations in a simpler way, they are engaging in MP7.
In the next lesson they will consider similar situations involving fractions expressed as decimals. These two lessons prepare them for later study of situations involving percent increase and percent decrease.
Lesson overview
 4.1 Warmup: Notice and Wonder: Tape Diagrams (5 minutes)

4.2 Activity: Walking Half as Much Again (10 minutes)
 Includes "Are you Ready for More?" extension problem
 4.3 Activity: More and Less (10 minutes)
 4.4 Optional Activity: Card Sort: Representations of Proportional Relationships (10 minutes)
 Lesson Synthesis
 4.5 Cooldown: Fruit Snacks and Skating (5 minutes)
Learning goals:
 Apply the distributive property to generate algebraic expressions that represent a situation involving adding or subtracting a fraction of the initial value, and explain (orally) the reasoning.
 Coordinate tables, equations, tape diagrams, and verbal descriptions that represent a relationship involving adding or subtracting a fraction of the initial value.
 Generalize a process for finding the value that is “half as much again,” and justify (orally and in writing) why this can be abstracted as \(\frac32 x\) or equivalent.
Learning goals (student facing):
 Let's use fractions to describe increases and decreases.
Learning targets (student facing):
 I can use the distributive property to rewrite an expression like \(x+\frac12x\) as \((1+\frac12)x\).
 I understand that “half as much again” and “multiply by \(\frac32\)” mean the same thing.
Required preparation:
 If doing optional activity 4, print and cut up slips from the Representations of Proportional Relationships Card Sort blackline master.
 Prepare 1 copy for every 2 students.
 These can be reused if you have more than one class.
 Consider making a few extra copies that are not cut up to serve as an answer key.
Glossary:

tape diagram  A tape diagram is a group of rectangles put together to represent a relationship between quantities. For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint. If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.
 Access the complete Grade 7 glossary.
Standards
 This lesson builds on the standards:CCSS.7.RP.A.1MS.7.RP.1MO.7.RP.A.1CCSS.6.RP.A.3MO.6.RP.A.3bMS.6.RP.3 CCSS.6.EE.A.3MS.6.EE.3MO.6.EEI.A.3
 This lesson builds towards the standard:CCSS.7.RP.A.3MS.7.RP.3MO.7.RP.A.3
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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