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# The Constant Area Model, Day 1 of 3

Lesson 10 of 15

## Objective: SWBAT model a geometric scenario with a simple rational function and interpret the model in context.

#### What is "Invariant"?

*5 min*

Discuss the meaning of "invariant" with example of changing area but *invariant* perimeter.

A loop of string can make different shapes all having the same perimeter but having very different areas. How do you know the area is different? Two loops of equal length can prove it: make a large circle with one string, then make a wiggly shape with the other one *inside the circle*. Since the second shape is completely contained by the first, it must have a smaller area but an equal perimeter! (I like this example because it exactly reverses the invariance the students will examine in the coming problem.)

*Coming Soon: Classroom video of this activity!*

A document camera is useful for this quick demonstration.

Once the notion of invariance is understood, hand out the "Tiger, Tiger, Burning Bright..." problem.

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#### Individual Time

*10 min*

Expectation: Students will complete the graph & table on page 1.

Teacher: Some students will need encouragement to see that in this case a 4x9 rectangle may be considered *different* from a 9x4 rectangle. This is reasonable because we're interested in the full range of values that the length of the rectangle may take on, and both 4 and 9 should be included.

It may be worth asking students to explain why it is reasonable to "connect the dots" on their graph. What does that imply about the number of different rectangles that may be formed?

(By the way, if you or your students are into poetry, you might check out this website for a reading of Blake's Tyger poem. In fact, it might be fun to have this recording playing as students enter the room!)

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#### Group Time

*20 min*

Student Expectations:

1) Compare and correct graph & table (parts 1 and 2)

2) Create an equation to model the relationship of length to width.

3) Answer questions 3a and 3b, regarding limitations on the dimensions of the pen and the rate of change of the dimensions with respect to one another.

Teacher:

1) Ensure that students have correct equation.

2) Ask students about *mathematical* vs. *situational* limitations relative to question 3a. What does the mathematical model permit that the actual situation does not? Negative lengths? Impossibly small or large lengths? (**MP 2 & MP 4**)

3) Push students to explain in an intuitive way *why* the length and width do not change at the same rate or by the same amount? (**MP 3**) Encourage the best students to explore this question algebraically as you wait for others to complete the three expectations.

*Please see this video for reflections: Constant Area Day 1, Video Narrative, Group Time*

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#### Sharing Solutions

*10 min*

At this point, different groups or individuals will be asked to come to the front of the room to share and explain their work and then to answer any questions their peers may have. I find it's best to use a **document camera** to make this go smoothly.

After briefly looking at the graph, the table, and the equation, I'll ask students to volunteer to verbally share their answers to questions 3a and 3b. I'll use these initial explanations as a launchpad for the following discussion. *Please see the solutions document with included content standards for details.*

Discuss the importance of paying attention to the limitations of mathematical model, as well as its strengths. These three questions will frame the conversation:

1) What does the mathematical model (equation & graph) reveal to us about the situation that we might not have seen otherwise?

2) What is the domain of the function mathematically? What is the domain in the context of the problem?

3) Is the function continuous mathematically? Is it reasonably continuous in context? (Would you ever choose to make one dimension an irrational length?)

Discuss, if time permits, *why* the area must change if both length and width are changed by the same amount. The sophistication of this conversation will depend on how much time you have and on the interest or ability of the students.

#### Resources

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: What is Algebra?
- LESSON 2: The Music Shop Model, Day 1 of 2
- LESSON 3: The Music Shop Model, Day 2 of 2
- LESSON 4: Letters & Postcards, Day 1 of 2
- LESSON 5: Letters & Postcards, Day 2 of 2
- LESSON 6: Choose Your Own Adventure
- LESSON 7: What Goes Up, Day 1 of 3
- LESSON 8: What Goes Up, Day 2 of 3
- LESSON 9: What Goes Up, Day 3 of 3
- LESSON 10: The Constant Area Model, Day 1 of 3
- LESSON 11: The Constant Area Model, Day 2 of 3
- LESSON 12: The Constant Area Model, Day 3 of 3
- LESSON 13: Practice & Review, Day 1 of 2
- LESSON 14: Practice & Review, Day 2 of 2
- LESSON 15: Unit Test