Open any calculus text to the chapter on applications of the derivative and you will find a farmer who is trying to maximize a rectangular pasture with a fixed amount of fencing. Or, perhaps, the farmer has a fixed area to enclose and wants to minimize the fencing needed. Often times the farmer wants to construct the pen along side an existing structure so that fencing is only needed on three sides.
Why wait until calculus (which many students may never reach) to mess around with the idea of optimization? If you scaffold your questioning, upper elementary students (grades 3, 4, 5) can get a lot out of this problem. Here’s how I got k-12 teachers thinking about this idea at our annual state math conference yesterday:
I had the teachers put their designs on the board and then the conversation really took off. (Gentle reader, you sketch out your own designs on paper or in your mind before you proceed.)
Me: Are these all of the possible designs?
Teachers: We think so.
Me: How many designs are there?
Teachers: Well, are we counting a 2 ft by 16 ft the same as a 16 ft by 2 ft?
Me: I would, since multiplication is commutative.
Teachers: Then there are nine.
Me: What about a side of 4.5 feet? Would that be possible? Or a side of Pi feet?
Teachers: Well in that case there are infinitely many designs.
Me: Right. So there are nine unique designs if we restrict our sides to the natural numbers.
Me: So, is there a design which gives the greatest area? Do we have it on the board?
Teachers: Yes. The 9 ft by 9 ft rectangle gives the greatest area (81 sq ft).
Me: What shape is it?
Teachers: A square.
Me: Could a fourth grader make a graph of width versus area to show that this is the greatest? What would that graph look like?
Teacher: An exponential function!
Me: No, it wouldn’t be an exponential function. Let’s plot some points and see.
Teachers: The width of nine feet is at the highest point so that is the largest area.
Me: Right. The width of nine feet produces the maximum value for the area. It optimizes the area. High school teachers — what kind of function is that?
Teacher: Quadratic! I said exponential but I am not awake yet!
(in fairness this session started at 7:40…)
Again, I had the teachers work on their designs and then put them on the board. (Again, gentle reader, you sketch out your own designs on paper or in your mind before you proceed.) Which design gives the greatest area? Again, great conversation ensued. After a little while…
Teacher: How could we prove that true?
Me: Well, you could use algebra or calculus.
After another little while, this is kind of what the white board looked like:
To continue with this task, we might need to be middle schoolers (6, 7, 8) or high school geometry students. I leave the remaining designs to you. For fun, you could comment on this post by telling us what shape you chose and what its area is and/or what shape maximizes the area.
At the end of the session I gave each participant a handout which anyone can download for free by clicking this hot link: The Fickle Farmer