The Constant Area Model, Day 3 of 3

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SWBAT create general equations to model the relationship between length, width, and perimeter. SWBAT recognize and exploit structural similarities to solve a variety of problem types.

Big Idea

Geometric relationships in the real-world can be modeled with quadratic functions whose solutions can be found through tables, graphs, and quadratics in factored form.

The General Solution

10 minutes

This section consists of a discussion of the first six slides in the Power Point Constant Area, Day 3.  For each slide, I like to use a Think-Pair-Share strategy, with about a minute of individual time, two minutes of pair time, and then sharing out.

Have students create general equations for...

1) ...area in terms of length alone.

2) ...perimeter in terms of length and area.

Then pose the question, "In general, how do we minimize the perimeter of a rectangle without changing its area?"  The answer, of course, is to make it a square!  Students should be able to easily see that the length will be the square root of the given area.

The final question in this section is to create a general equation for the relationship between the three quantities area, perimeter, and length.  The students have already found one such equation in (2) above, but I'd challenge them to find three equations, one solved for P, one solved for A, and one solved for L. (HSA-CED.A.4) These are given on slide 6, along with an equation that shows the explicit quadratic relationship between L and the other two variables.

The important thing to stress here is that all of these are equivalent equations that capture the very same relationship!  In different forms, we can "see" different aspects of the relationship at a glance. (MP 7)

(The solutions to the slideshow problems can be found in the Solutions resource.)

Related Problems

30 minutes

During this section of the lesson, I challenge the students to solve one problem after another from slides 7 through 14.

The first problem (slide 7) is similar to the geometric "constant area" problem with the exception that the notion of perimeter has been replaced by a simple sum. (MP 7)  So, while the same general methods can be used to solve it, the relationships are slightly different.

The second problem (slides 8 - 10) is simply the graphical solution to the previous problem.  Some students will recognize it right away, but others will not.  It's important to give the class a couple of minutes to complete their graphs, and it will speed up the process if you provide them with some graph paper with axes already drawn and labeled.

The next problem (slide 11) is different only in that the solution will not be integers.  I like this change because the students will not simply be able to guess the solution by examining the graph.  They graph will give them a reasonable estimate, but an analytic method is necessary to get a precise solution. (MP 5 & 6)

The next two problems (slides 12 & 13) ask students to reason carefully about the range of values that either the product or sum or two numbers can take on.  The question on slide 12 is solved when students recognize that it is equivalent to the minimum perimeter problem.  (MP 7)  The question on slide 13 is easily represented as a quadratic equation, yeilding a relatively easy-to-find maximum.

The final problem (slide 14) asks students to factor a quadratic expression.  This one is very important because it requires students to explicitly connect factoring to the problem of finding two numbers with a specified sum & product. (MP 7)  I will help them to see that this problem is not any different from problem on slide 11.

The purpose of these problems is to help students see the similarities between the geometric problem they just solved (area vs. length vs. perimeter) and these numeric problems.  If you are pressed for time, I would skip slide 12 to ensure that there is time for the final section of the lesson.

Wrapping Up

5 minutes

The final slide is meant to be a conversation starter about the connections that can be made between a wide variety of concepts, skills, and problem types.  I would encourage students to brainstorm other topics that arose over the course of the last three lessons and to describe how they are connected to the ones shown.  There is no definitive list of connections to be made; the important thing is that students recognize how one mathematical problem may be similar to another one, even if they arise in very different contexts.

As a summary discussion, be sure to bring up the modeling process and the key role it plays in forging connections between the different areas of mathematics.