##
* *Reflection: Modeling
Reducing Fields - Section 1: Warmup

When students are asked to represent real world contexts mathematically they are engaging in the modeling practice (MP4). For me the most powerful thing about this is that it simultaneously helps students understand the world better and understand the math better. Questions about the context ("how big a gap does a farmer have to have?") along with the constraints of the context (the size of the field) helped them go deeper into the math.

I may also ask them what is the greatest number x could be and discuss why they think that. Some students may say that x can represent any number, but I would encourage them to go back to the context so they can see that in this case there is a limit, it can't be more than the length of the field. Since the context is a farmer growing crops for profit some students may also argue that even the maximum of x=20 is not realistic either. I want to engage them in this type of argument, because it helps not only with sense making, but also with the practice of argumentation (mp3).

Choosing a context that is familiar to your students is crucial. In math class the context needs to help make sense of the math. My school is in a farming community, so this is a very familiar context for my students.

*Using context*

*Modeling: Using context*

# 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