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# Reducing Fields

Lesson 18 of 23

## Objective: SWBAT use the distributive property and an area model to factor variable expressions.

## Big Idea: Students will work to solve a problem using distributive property in a real world context.

*54 minutes*

#### Warmup

*15 min*

In this warmup Reducing Farmer Fred students are given a diagram of a rectangular field with outer dimensions of 10 and 20 meters. They are told the farmer needs to reduce the width of his field in order to create a gap between his field and his neighbor's field in order to prevent cross-polination of their crops. They are asked what the area would be if it is reduced by 2 meters, then by 5 meters.

When I go over this warmup I set up a table warmup Reducing Farmer Fred complete with table at the bottom and first ask what the area would be if it wasn't reduced at all and I fill in an area for x=0. Students volunteer to share and explain their solutions to the other two questions (x=2 & x=5). I expect them to have solved in one of two ways:

- Using the distributive property to subtract 10x2 and 10x5 from the original area of 200
- Shortening the length and multiplying by the difference

Then I ask them to calculate the area if x = 10 and and I include that in the table and ask them to explain how they did the math. Next I put a variable in the table for "x" and ask how they would represent the area (how they would show what to do with whatever value x is). I expect they will come up with 10(20-x) or 200-10x. Finally, I ask them how they can both be true. If they can't explain I would ask how they would show someone that 10(20-x) was equal to 200-10x, which should prompt them to use the distributive property.

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#### Problem Solving

*25 min*

This is an open ended problem with multiple possible solutions. I give students a diagram warmup Reducing Farmer Fred and John of two neighboring farmer's fields with dimensions. I tell them they need a 10 meter gap between their fields and ask what the areas of the two fields could be. I expect some student push back on this type of problem, because students are used to following directions and finding the one right answer. It often throws them when they have a little more autonomy in the decision making process and in this case they need to see that it depends on how much each farmer reducing the length of his field.

**Prompting questions may help them get started:**

- "what are the original dimensions of Farmer Frank's field? what's the height? the length?"
- "what are the original dimensions of Farmer Joe's field?", etc.
- "If Frank reduces his field by 2 meters, by how many meters does Joe need to reduce his field?"
- "If Frank reduces his by 7 meters...?"

**As I talk and listen to each group I may share some highlights with the class like:**

- "Robert's group is drawing a diagram to help them visualize it better"
- "Johnnie's group has come up with a couple of possibilities"
- "Hallie's group wants to make it as fair as possible", etc.

**I may also highlight some of the ways they are working together.**

- "I like that Jessica just asked what her partner 'what do you mean by that?'"
- "I like what close attention Logan is paying to his partner's work!", etc.

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#### Extension

*14 min*

**I draw a diagram of two fields, one 20 by 20 and one 5 by 10 and ask if it would be fair if both farmers reduced their fields by 5 meters if these were the original sizes**.

Most of them realize intuitively that it would be unfair, but I tell them they should try to use some **mathematical evidence** to try to convince us.

I am hoping students will calculate the original areas of both fields (400 sq. m. and 50 sq. m.) and also the resulting areas after reducing each one by 5 meters. I will look for a student who has started and ask them to share this idea if not everyone has. This way the idea is coming from them and not me.

**Most students will be able to articulate in some way that the farmer with the smaller field loses a larger part of his field.** When I ask what they mean by a larger part I am hoping someone may point out that he lost half of his field while the other farmer lost much less than half or a quarter. **Requiring them to give mathematical evidence is part of developing the practice of argumentation while at the same time making connections to related math.**

If there is time, I may also ask them what they think would be a fair way to reduce these fields. This is a good way to get them to articulate and support an argument and to critique the argument of another. It also helps to put them in touch with their sense of **ratio & proportion**.

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Farmer John and Farmer Fred Day 1 of 2
- LESSON 2: Farmer John and Farmer Fred Day 2 of 2
- LESSON 3: Let's Break It Down
- LESSON 4: Halloween Candy to Zombies
- LESSON 5: Extending Farmer Frank's Field with the Distributive Property
- LESSON 6: Who's Right?
- LESSON 7: To Change or Not to Change
- LESSON 8: Let's Simplify Matters
- LESSON 9: Clarifying Our Terms
- LESSON 10: Breaking Down Barriers
- LESSON 11: Number System Assessment
- LESSON 12: Garden Design
- LESSON 13: Ducks in a Row!
- LESSON 14: The Power of Factors
- LESSON 15: Forgetful Farmer Frank
- LESSON 16: Common Factor the Great!
- LESSON 17: Naughty Zombies
- LESSON 18: Reducing Fields
- LESSON 19: Common Factor the Great Defeats the Candy Zombies!
- LESSON 20: The Story of 1 (Part 1)
- LESSON 21: The Story of 1 (Part 2)
- LESSON 22: Simple Powers
- LESSON 23: Equivalent expression assessment