A “Mowing the Lawn” problem with Legos

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.

A neat probability problem from James Tanton

Last month I bought a copy of James Tanton’s “Solve This:  Math Activities for Students and Clubs” on the recommendation of Fawn Nguyen.   As is true with all of Fawn’s recommendations (that aren’t related to college football), it has been a joy to go through.

I picked out a pretty challenging problem to try out for our Family Math today.  My goal was not to give the boys a complete understanding of the solution to the problem, but rather to show them a situation that they could understand and speculate about a little.    Although walking them through the clever solution at the end proved to be a little difficult, I’m really happy with how this problem engaged them and look forward to doing more problems from this book later in the year.

We started with a simple explanation of the problem and used several lego figures to help with the illustration.  Before diving in to the solution, we spent a few minutes just talking about what they thought the answer would be.

Next we tried a few examples.   Almost comically, every time we flipped the coin we got heads, so Unikitty kept falling off the cliff at the first step.  Finally we got a long sequence where we had more tails than heads for a long time – actually a really long time.   This long sequence was a lucky illustration of just how complicated this problem can get if you try to look at it case by case.   Sorry that it is hard to see heads / tails in the flips on the camera – I probably should have used something that had different colored sides rather than a coin.

Next we went to our whiteboard to try to work out the math.   It was interesting to me that both boys thought that the series 1/2 + 1 /4 + 1 / 8 + 1 / 16 + . . . . would show up in the solution.  I wanted to approach the problem with binary trees to show them that the series they were looking for doesn’t show up quite as easily as they thought it would.    I also wanted to illustrate this approach because we’d looked at binary trees last week:  https://mikesmathpage.wordpress.com/2014/09/07/binary-trees-and-pascals-triangle/

Though this part of walking through the problem wasn’t as clear as it could have been, I’m happy for the kids to see that you don’t always march right to the solution of a problem in a straight line.  I’m also happy for them to see that problems that are relatively easy to illustrate can sometimes be a little more complicated than they seemed when you start thinking about them more carefully.

Finally the clever mathematical solution.  The idea used in solving this problem is a little bit over their heads, but it is a great mathematical idea for them to see nonetheless.  The fun, and actually pretty amazing, part of this solution is the idea of finding a clever piece of symmetry in the problem.  That symmetry allows you to write down an equation whose solution is the probability that you are looking for.  Quite a remarkable idea – I don’t know what the probability is, but I know a quadratic equation that it has to solve!

As it took a while to walk through this solution, I chose to talk pretty informally.  At the end, I let my younger son see that his guess at the solution – i.e. that the probability we are looking for is 1 – does indeed solve the equation.  With my older son, I let him play with the quadratic equation a little and see that p = 1 was the only root.

Tough stuff, but hopefully still a fun example:

I’m really happy that Fawn recommend James Tanton’s book to me.   Despite the difficulty of this first project, the boys were engaged all the way through.  Can’t wait to try out a few more.