PreCalculus Modeling

After establishing the parent functions in PreCalculus we take a little side trip to explore modeling.

Students had the Norman Window problem to do this summer as part of their summer prerequisites packet. Everyone (predictably) had a difficult time with this. They are so used to following a particular procedure to solve a problem with all the same type of problem grouped together. They really have very little experience with modeling.

PROBLEM – A Norman window is going to be built with the perimeter of 30 ft. A Norman window has the shape of a rectangle with a semi-circle attached at the top.  The diameter of the circle is equal to the width of the rectangle.  Your job is to determine the dimensions of the Norman window that allows the maximum amount of light to pass through.  

The modeling we do in this class is still pretty structured but we try to do things with a little bit of open ended discussion to start.

So we start off our modeling adventure with this problem:

I ask students - any thoughts on what to do, discuss. Coming back together there are a lot of unformed ideas but they do know how to maximize the volume via the graph in a graphing calculator (we don't do derivatives in this course, that will be next year). Maybe if I gave them a lot more time there would be some progress in solving but they had all summer to do the Norman window problem with very poor result.

So we talk about using our great mathematical toolbox of concepts and formulas. There is so much out there we can use. And in a job there won't be "steps to follow" in solving a problem. You'll have to consider all the possible tools available to you. Some brainstorming will usually happen and then you'll go forward and see what you can do.

So back to those tools available to us. I ask students to pick out some mathematical concepts in this problem and underline them.

Great - now we just have to merge all those concepts together in a way that we can analyze and answer the question.

Students are not good at writing functions or merging functions. So they need to see what that looks like. So we go back and do some easier problems. We highlight what concepts are necessary and write them out with specific information for the problem. Then we merge the concepts together in one function. Here are some examples we do.

2)     A wire of length x is bent into the shape of a square.

i)  Express the perimeter of the square as a function of x.

a) Using your calculator, draw a sketch of the function.  State the window and label the zeros and max/min point(s). 

b)  What is the parent function?

c)  Using interval notation, state Domain  & Range                        

d)  State the domain and range of the problem situation.

     

ii)  Express the area of the square as a function of x.

a) Using your calculator, draw a sketch of the function.  State the window and label the zeros and max/min point(s). 

b)  What is the parent function?

c)  Using interval notation, state Domain  & Range                     

d)  State the domain and range of the problem situation.      

3)     A right triangle has one vertex on the graph of y = x³, x > 0, at (x, y), another at the origin, and the third on the positive y-axis at (0, y). (see the figure) Express the area of the triangle as a function of x.

 

While doing some of these we also consider the graphs of the resulting functions. We compare the "parent" graph and the "problem situation" - how and why they may differ. This is important. Those smart math kids are so good at regurgitating back formulas without truly understanding what they mean. Being able to communicate the requirements of the problem situation shows good understanding of the problem.

Finally we go back to the starting example above and solve it. I get the students to do a lot of the work in their small groups with leading questions. I also have them draw both graphs - of the parent function (a nice cubic) and the problem situation (first quadrant, looks kind of parabola like....). And what's a good window for the problem situation? Is that the domain & range? Nope, so how do we find that? Lots of back and forth and then students get to work on another piece of the problem.

Turns out the figure that has maximum volume is actually a cube. I thought there would be some ahas going around the classroom but surprisingly to me most of them have never worked through a problem where they find what dimensions of a rectangle has a perimeter of ____ (any fixed value) and found that the rectangle closest to a square has the greatest area. I figured they saw that somewhere in their math career - middle school? high school geometry? sigh.

So this was day 1 of modeling (82 minute block) with homework that had some structured 2D modeling problems.

Next class...

We start with a quiz on a previous topic (graphing calculator skills) and then they work on the Norman Window again in small groups. Hopefully with better success after giving some framework to modeling with geometric functions n the previous class.

After that we do a little prep for their box problem project. Each table has a piece of colored grid paper (17 by 25) with a square of a fixed dimension drawn in each corner (squares all the same size, each table has different square sizes than other tables).

They cut out those squares and fold up to create a box that is open on the top.

  Each table compares all their different sized boxes.

 I tell them that with this project they have to find the box with the maximum volume and what size square must be cut out of each corner to create this box. There are a bunch of other details they have to communicate. All found in the description in the folder below.

Everyone gets a personalized box project description. I fill in different dimensions for each of the students. Unfortunately there can be a fair amount of academic dishonesty in our school, especially with stressed out advanced student. So this is a necessary step to keep everyone honest. A bit of a pain to grade, but worth the piece of mind.

All documents found HERE.