For today's math circle, I adapted a couple of activities from James Tanton's book Solve This: Math Activities for Students and Clubs.
The above book, by the way, is a great book that describes activities that Tanton used while running a math club for college-aged students. Unfortunately, my students are aged 5 through 8, so not everything in his book will work in our class. Some of the activities are a little too mathematically sophisticated for children who can't yet multiply. And other activities require a bit more attention span than that of your typical kindergartener. But all that said, in fact a surprising number of the activities in the book work even for this age group, especially with some minor modification. Today's activity was one of those.
Specifically, we were looking at problems adapted from Chapter 13 of the book. If you're following along with your own copy of Tanton's book, you'll see that to the activities he gave, I added a couple of easier cases to work through first (to warm up the younger children), and I simplified one activity slightly (reduced from a 5x5 grid to a 3x3 grid). Also, I stuck with just two activities rather than try to do all three or more activities listed in there. But otherwise, my activities were pretty similar to his.
Preparation:
1. Prepare one copy of "easy" grid puzzles below for each child, and at least three copies of the "hard" grid puzzles for each child.
2. Bring enough pencils for each child.
3. Just before class, I used masking tape to mark nine x's on the floor for the people-shuffling activity (2nd activity below).
Grid Puzzles.
I learned from last week that any activity I start right at 9:00am (when our math circle starts) will have to have its rules repeated as kids come a little late. So today, I started with an activity with paper and pencil that was easy to explain to late arrivals.
As the children arrived, I handed out pencils and a paper with the following squares printed on them:
The rules of the game are the following. Start with your pencil in the square marked with the X. Draw a path through the grid that meets each square in the grid exactly one time. The path may leave a square through any of its sides, but it can't run diagonally out of a square. (Examples are below if that explanation doesn't make sense.)
Most of the children figured out paths that worked for the above four puzzles quickly, especially the older ones. As the children finished, I handed them a harder puzzle. Here is what the harder puzzle looked like.
Puzzle #5 was difficult, but the students were having particular trouble with puzzle #6. I brought several extra copies of the puzzle so that when they had erased too many times, they could get a fresh puzzle and try again.
I let them work on this for about 15 minutes. Then I polled them to see who had finished their puzzles.
Everyone had finished puzzles #1 through 4. I let the children raise their hands and describe different solutions to puzzle #1. Here are three that they came up with.
I asked for a show of hands on who had finished puzzle #6. No one had. Same for #7. ("We didn't get to #7 because we couldn't finish #6!" explained one child.)
Rather than talk more about these puzzles, I told the children we were going to take a break from the puzzles and do the next activity.
People shuffling.
I asked for four volunteers to come play the next game. They stood on X's marked with masking tape on the floor in a 2x2 grid.
The rules of the game are the following. Every student must move exactly one space. They can move side to side, but not diagonally. The goal is to get every person to move to a different space in the 2x2 grid.
I had my four volunteers run through an example. They switched places in pairs, which worked! I then asked all the students to get into groups of four and see if they could figure out different ways of solving the problem.
We took about five minutes, then put some solutions on the board.
I then asked for nine new volunteers, and asked the children to stand on a 3x3 grid. The rules were the same, the objective the same: Everyone has to move exactly once, with no diagonal moving allowed. Ready set go.
Their first attempt didn't work -- someone on the corner got stuck.
One eager and clever little girl had an idea then. Have the middle person move first, then everyone else switch around them. But unfortunately, that didn't work either.
Another equally clever girl suggested a new alternative. But hers didn't work either!
I then suggested we try moving one person at a time, counting how many moves were made before someone stepped back into an empty space. A sequence of legal moves of people that ended with someone taking the empty space was called a cycle. We worked through a few cycles, and found that they all had to have an even number of steps.
Why was the number even?
This is where things got a little tricky for the younger kids. I showed them that every time someone moved left in a cycle, someone else had to move right. Every time someone moved up, someone else had to move down. That meant moves in the cycle happened in pairs -- so there were an even number of moves!
A couple of the older children seemed to get it now. Because there were nine children, but cycles had an even length, the only way to get everyone to move would involve an even number of children. So one of the nine would be left out.
(I don't know if they really got it, but at that point, the littler ones were getting restless, so I had them all sit down again.)
I had a couple other moving puzzles prepared, but the students voted to go back to the grid puzzles.
Grid Puzzle Solutions.
One of the girls who had been helping with the 3x3 people-shuffling game raised her hand and said she thought that solving puzzle #6 (grid puzzle above) was impossible, just like moving nine people around in a 3x3 grid was impossible.
I announced to the class that she was right! Puzzle #6 was impossible. And our new goal was to figure out why.
Someone suggested that maybe it was because there were 25 squares -- an odd number. But another child pointed out that puzzles #4 and #5 also had an odd number of squares, but we were able to solve them.
Then a boy noticed that in puzzle #6, there were only 3 ways to begin, but in puzzle #5 there were 4 choices for how to begin. That was a good idea. But then someone realized that in puzzle #4, there were only 2 choices for how to begin, but everyone had solved puzzle #4.
I told them I would give them a hint. I drew the 5x5 grid on the board, and started coloring the squares in a checkerboard pattern.
I then let them think for a while and talk about the problem with the others at their table.
They noticed that #4 and #5, which were solvable, started on shaded squares. But #6 and #7, which were not, started on white squares.
Was starting on white squares the problem? Maybe, but puzzle #3 also started on a white square in the 4x4 grid.
By then the children realized the problem was with the 5x5 grid -- something different was happening with that grid than with the 4x4 grid. But what?
After another minute or two, a girl raised her hand and told me she had counted white and shaded squares. (This was the right idea!)
With all the children, we counted 13 shaded squares, but only 12 white squares.
After another minute, I asked the children to tell me what colored squares my path stepped through. If I started on a shaded square, where would I go next?
To a white square.
Why not a shaded square?
Because you can't move diagonally.
So then we realized that if your path started on a shaded square, it would proceed as follows:
Shaded - white - shaded - white -shaded - white - ... through 25 squares (in the 5x5 grid case).
If you started on a shaded square, where would it end?
After a minute or so, they figured out that it would end on a shaded square if there were 25 squares.
Then we counted. That meant it would run through 13 shaded squares, and 12 white ones. Hey! That's how many we had!
Then I talked about a path that started on a white square. Where would it end?
We stepped through the path, and it looked like this:
White - shaded - white - shaded - ... - shaded - White!
It ended on a white square.
"But that's impossible!" shouted one little boy. "You would have to have 13 white squares!"
Exactly.
We all counted together. A path that started on a white square would have to go through 13 white squares and 12 black. But we didn't have 13 white squares, we only had 12 white squares! That meant that the puzzle I gave the students was impossible!
It was time for cookies then. But before I let them get a cookie, I told them conspiratorially that they ought to take a copy of the puzzle home and give it to their parents to try.
They thought that was a hilarious idea, and every one of them came up to get an extra copy of the puzzle.
And to get cookies.
Summary.
I think this activity worked very well for kids this age. Because there were different puzzles of different difficulty level, it engaged all the students regardless of age. Although not all the children seemed to understand all the explanations (especially cycles of even length), they seemed to be having fun and learning something. I would do this activity again for this age group.
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