In the Math Fundamentals problem *Frog Farming*, the goal is to make at least four different rectangular pens, each of which uses 36 meters of fence. Many students thought of this the same way I did, which was to consider half the necessary perimeter as the sum of two adjacent sides.

Rachel B, Seven Bridges Middle SchoolI know that the problem was finding perimeter. I know first you divide 36 by 2 and get 18. Then you find addends of 18 and they are the length and width. I added 12 plus 6 which equals 18 and 12 times 2 plus 6 times 2 equals 36.

Sarah G, Laurel SchoolFirst I decided come up with a length and width for a rectangle that would equal 18 because 18 is half of 36 and you have to multiply that number by two to get the perimeter. I decided on 2 and 16. I checked it by doing 16+16+2+2= 36. One could by length=16, width=2.

Rachel and Sarah and I were thinking about perimeter, in the context of this problem, like this:

Another way I thought of this was as 2(L + W). Hmm…..

Then I was mentoring a few students in this problem and noticed that they were thinking about the problem differently.

Ethan Z, Lorne Park Public SchoolI thought of a rectange which has 4 sides and 2 sides are equal and the other 2 sides are equal because Farmer Mead wants a rectangular pen that uses 36m of fencing. First I got the answer by thinking of 2 equal numbers that add up to less than 36. Then, the last 2 equal numbers are the difference of 36 to the first 2 equal numbers. That’s how I got all the numbers of the first question.

Emily G, Laurel SchoolI used 2 numbers, and doubled 1 number by two (ex. 6×2=12). 36-12 is 24. 24 is an even number that can be split into 2, which is 12 (ex. 24÷2=12.) 24+12 is 36!

Maybe because I had “seen” the problem differently, it took me a few minutes to figure out what these other kids were doing. Then I realized they were “seeing” the problem like this:

This seems more like 2L + 2W! These students tended to use more of a Guess and Check strategy to find solutions, whereas kids who used the first method were a little more systematic from the start. But it was fun to me to see these two different methods to what is a pretty simple idea. I like when simple things are done different ways.

I wonder how you “saw” the problem when I first described it. And how did your kids tend to see it?

Some ** Frog Farming** links in case you are interested:

- The problem [requires a Math Forum PoW Membership].
- Information about accessing
*Frog Farming*(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!