The scenario for the Pre-Algebra Problem of the Week Building Fences has this paragraph:
We decided to put the fencing around a rectangular garden that we have been planning. The twenty-foot side of our shed will be along one of the edges of the garden.
We later learn that the author has bought 36 feet of fencing and wants to build a garden with the largest possible area.
Because the description of how the garden is fenced is not described with mathematical precision, just everyday talk, there are a lot of possible assumptions to make.
At first, the ambiguity in how the garden is fenced bothered me. I thought, “this isn’t a good math problem. We might trip students up. We are tricking them. It isn’t fair.”
But this problem comes from real life. Our colleague really did describe wanting to fence her garden this way, and really did want to figure out the maximum possible area she could fence.
I started realizing that in math problems that don’t come neatly packaged in textbooks, but rather arise from conversations about day-to-day life, the language isn’t always perfectly precise. We need to make assumptions, and know when we are making assumptions, to cope with the messy, ill-defined problems that happen outside of math books.
So I was thrilled to see how many students said things like, “at first I thought the shed would be one full side of the garden, but then I realized the fence could stick out from both sides of the shed,” or, “I had an aha! moment when I realized they didn’t have to use fencing along the 20 feet of the shed.” The students realized they had been making assumptions which made the possible maximum area smaller.
Ultimately, the assumptions that lead to the largest maximum area are to assume that the shed is only part of one side of the garden, and that no fencing is used along the shed. Other answers aren’t wrong, but they could be optimized.
Which then makes me wonder, what routines do we have in our classroom for ferreting out and challenging assumptions? There are some math problems (such as this one) that can’t even be solved without challenging assumptions! So how do we learn to do that?
Some Building Fences links in case you are interested
- The problem [requires a Math Forum PoW Membership].
- Information about accessing “Building Fences” (and a selection of all our PoWs) for 21 days with a free Math Forum trial account.
- Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!