The Constant Area Model, Day 2 of 3

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Objective

SWBAT interpret the symmetry of a rational function in terms of the two dimensions of a rectangle. SWBAT create an equation to model the relationship between perimeter and length by combining two other equations.

Big Idea

Symmetric points on a graph reveal a deeper connection between the length, width, and perimeter of a tiger's pen that can be modeled with a new equation.

Individual Time

10 minutes

At the beginning of class, briefly let the class know that today we are continuing our work with the constant area model.  Then hand out the "Constant Area, Day 2" document, and instruct students to work individually for about 10 minutes.  During that time, they should aim to complete the front of the handout.  They will also need to work with the graph they made yesterday, so I hope they still have it! (In case not, it's good to have a copy ready to pass around.)

Expectations:

1) Label perimeter on all points.

2) Join two points and interpret intercepts.

3) Join all equal-perimeter point pairs, observe & describe parallelism.  (See solutions document for details.)

Teacher:

1) Ensure that points are labelled & joined correctly.

2) Ask as many students as possible, "This line appears to cross the x-axis at ____.  What does that number mean in the context of the tiger's cage?" (MP 2 & 4)

3) Ask as many students as possible, "You've written that all of these lines are parallel, and they certainly look that way.  How do you know that they really are or always must be?" If students are going to call two lines parallel, they should be able to argue that the lines meet the definition of parallelism! (MP 3 & 6

4) Ask as many students as possible, "Are there any points on your graph that do not have an equal-perimeter pair?"

Group Time

20 minutes

Expectations:

1) Create equation for P in terms of L.

2) Graph equation, identify minimum, & interpret in context.

3) Explain asymptotic behavior of graph in context.

4) Answer summary question.

Teacher:

1) Expect students to have difficulty creating equation!  The only hint I would give is the following: "So, you're finding it difficult to create an equation for P in terms of L alone.  Do you think you could create an equation for P in terms of both L and W?  Could you use this to solve the problem at hand?"  Once students have P = 2L + 2W, I would want them to consider on their own how they might replace W with some expression in terms of L. (MP 1)

2) Students will tend to identify and label the lowest point of the graph visually, so consider whether or not you want them to provide reasoning.  The lowest point appears to be (6, 24), but is it really?  I will always ask my students about this, but I would accept a variety of arguments since they don't yet have the tools to identify the minimum point analytically.  In this case, they'll have to use the tools they have (MP 5) to obtain a reasonably precise answer (MP 6).

3) As students consider the asymptotic behavior, a good question to ask is this: "Suppose the graph met the vertical axis at some point.  What would that point represent in context? Why couldn't such a thing happen?" (MP 2 & 4)

4) Another excellent question about the graph (for advanced students): "Your graph appears to straighten out for larger values of L.  Why does this happen?  What would it mean in context if the graph were actually straight?"  It's helpful here to write the equation in the form P = 2L + (72/L) and consider the limit of (72/L) as L grows larger and larger. (MP 4 & 7)  This is a useful foreshadowing of some later concepts: end behavior, asymptotes and limits!

Discussing the Model

15 minutes

1a) Let's go back to the original question.  What did you learn about the situation by constructing and studying this mathematical model?

1b) How did the model help you answer the question?  Did it teach you anything besides the answer to the original question?  Hopefully, they recognized that two different models were used.  The first was simple length-versus-width and the equal-perimeter lines could be used to minimize the perimeter.  The second was the explicit length-versus-perimeter model in which the the lowest point on the graph corresponded to the minimum perimeter. (MP 4)

2) This was originally posed as an optimization problem.  Could we have used linear programming to solve it? (MP 7In a way, the answer to this is yes.  The "perimeter lines" the students drew in part 4 are just like the cost lines they drew in the music shop problem, and the feasible region is the portion of the plane above the lenght-versus-width curve. I will use GeoGebra in class to illustrate this explicitly.