Two nice problems shared by David Coffey and Federico Chialvo

I saw two neat problems for kids on twitter this week and went through them with the boys this morning.

First up, the problem shared by David Coffey:

I couldn’t find any multicolored items to use, so we used snap cubes instead of cards. The first thing that we tried to do is see if we could solve the problem at all. The boys had a couple of ideas at the beginning, and it was interesting to hear the discussion. My younger son noticed that a procedure my older son was following was going in a circle, for example. My older son noticed that the last step would have to involve swapping out 3 orange cubes for blue cubes. That idea will help us find the minimum number of steps in the next video.

Eventually we found a way to swap the orange and blue cubes in 5 steps.

Next we tried to see if there was a way to swap the cubes in fewer than 5 steps. My younger son noticed that it would take at least 3 steps for a complete swap since you have to move 7 cubes, 3 at a time. He then noticed a way to make the complete swap in three steps.

At the end we compare both procedures that we found. This was a nice little activity.

The next problem we tackled this morning was from Federico Chialvo

The challenge on this problem is that my older son knows how to calculate each of these areas from studying geometry. I had him do that calculation at the end, but first I wanted to hear ideas that didn’t involve calculation. The “no calculations” requirement turned this into a fairly challenging task.

I split the 9 minute conversation into two pieces just to make it easier to watch. Their geometric instincts are to chop the hexagon into triangles and rectangles. Chopping up the hexagon in this way is interesting, but since you get 30-60-90 triangles my older son’s urge to calculate is hard to suppress 🙂

As we try to move away from calculating, they notice that the triangles they are looking at might actually form a square that is roughly equal in size to the original square. Eventually we see that the remaining rectangle has an area that is a little bit more than half the area of our original square. All of that information comes together to produce an estimate that the area of the hexagon is bit more than 2.5x the area of the square.

We finish up with a quick calculation of the area of the hexagon by my older son. The calculation shows that the hexagon is 6*\sqrt{3} times the area of the square. So, our original estimate of 2.5 times wasn’t that far off – yay!

Three versions of the same problem

My younger son and I have been studying angles for the last week. I decided to run through a few clock problems for a review. The secret about this exercise is that all three problems are pretty much the same problem in a slightly different form.

Question 1: How many degrees does the hour hand move in 10 minutes. I love the way he talks through this problem.

Question 2: What acute angle do the hour and minute hand make at 2:10?

I really didn’t know how he’d approach this problem. He got a little confused in the beginning thinking that the hour hand would move 1/3 of an hour in 10 minutes. Eventually he found that it moved 1/6th of the way through and that observation got him to the answer. I was a little surprised that he didn’t notice a connection between this problem and the first problem.

Question 3: What acute angle do the hour and minute hand make at 9:50?

The interesting thing here is that he does pick up on the fact that this one is similar to the previous problem. That observation allows him to get to the answer quickly. It was nice to see him also notice at the end that all of the answers were the same. Fun little project.