Today for our Family Math project we looked at a problem about rates. Half motivated because I was looking for a fairly light project and half motivated by a rate problem that my older son struggled with last week. See here for the more complicated problem from an old AMC 10:

The 2002 AMC 10A Problem 12

The specific problem we looked at today was a “mowing the lawn” problem and we used some of our lego pieces as props:

The first thing I did was introduce the situation. We have three people who mow a lawn. One person mows at a rate of 4 units per minute, a second person mows at a rate of 6 units per minute, and the third person mows at a rate of 8 units per minute. If the lawn is 288 units, how long does it take each person to mow the lawn?

One reason for introducing the problem this way is to get a little bit of arithmetic practice. The second reason was to show that rate problems aren’t that hard when you know the rates. This second point comes into play later in the project.

The next part of the project was to see how long it would take to mow the lawn if all three people were working together. Since the kids knew the rates for each individual, it wasn’t too difficult for them to add those rates together and calculate the required time.

After we finished that calculation, I changed the question a little bit:

Person A takes 72 minutes to mow the lawn, person B takes 48 minutes, and person C takes 36 minutes. How long will it take the three of them to mow the lawn if they all work together?

This version of the question caused a bit of difficulty.

Since we were stuck on the second problem from the last video, I decided to start a new movie to try to get a new approach. We talked through why this problem was similar to the first problem to see if the similarity gave us any clues as to how to solve this problem. The kids recognized that in the first version of the problem we knew how much work each person did in a minute. Figuring out how to translate the new problem into the old framework was the key idea:

One last challenge problem to end the project – if Unikitty and Buzz Lightyear work together, will they be able to mow the lawn faster than Sensei Wu working alone? I asked this question because I was curious how they would approach it – the way from the start, or the way from the last video, which is more complicated. They chose the way from the second video first and then noticed the easier approach after they solved the problem:

So, a fun project showing two different approaches to the same problem. One approach leads to a relatively straightforward solution, the other is a little more difficult, but knowing that there is an easier approach helps. Understanding how to transform difficult problems into slightly easier problems is an important step in learning about problem solving. It is fun for me to watch the kids learn some of these problem solving techniques.