Nothing special. Part a takes all the fun out of it and immediately abstracts the situation into a math problem. No one got a chance to build a garden.
How I would modify it:
- Give each student a couple of drinking straws (the length of which would replace the 40 ft. noted in the problem)
- Take a page in their notebook and draw a line as the fence
- Give them 10-15 minutes to build a couple gardens/playgrounds/squares, offering no hints as to how they might make the biggest one
- Compare gardens with their table members, determine which person at the table made the biggest based on sight
- Calcuate the actual area of their gardens
- Plot our numbers of width vs. area
From there it could get complex in a number of different ways depending on what you wanted to do.
- Can we get the same area from two different widths?
- Is there a definitive "biggest" garden?
- Can we generalize the shape of this fence with variables?
- What if I added another condition: x fencing and y sod?
- What if the existing fence wasn't there?
And with the idea properly motivated, I might present a scenario like #53 and have them jump right to the function. In fact, a series of problems like that might be the classwork. If you want a technology angle, I might have them generate something like this:
I've had numerous problems trying to jump right in with things like #53 because students don't think in terms of functions.
This was a fun thought exercise and fun to do with Algebra II. Pre-Cal is proving to be a bit more difficult but I will try.