09 10

26 January 2016

Number Squares

So, I'm thinking about switching to MEP for our math (Hero's close to outgrowing Miquan) seriously enough that I printed out their lesson plans and we're taking them on a bit of a test drive this week. This was the first activity we did (level 4), and I am having so much fun with it that, after Hero(9) was finished, I did some more to see what the pattern was going to do. It starts like this, with the blue set:


So, that was pretty tricky for him (and me, too, when I was looking it over last night). But you're writing the letters, and then in the final grid, we sort of put all the different variations on a single grid, and counted up how many times the final letter hit each square across the diagonal, where the Es are in that 5th blue grid. Next, we did the same thing with a four letter word - MORE. 


This time, he understood what he was supposed to do with the letters, and filled in the chart completely on his own. I helped with the chart where all the words are "stacked" into the same grid, and we counted them up. At first, he was thinking that there was 2 places where those center diagonal Es are used, so we added that up (the equations are at the bottom of the picture), and it was only six - but we'd clearly filled 8 charts, so that couldn't be the number of times they were used. He went back and found the missing numbers and got the correct equation.

We did the same thing for a 5-letter word, but we used Hero's real name, so no pictures of that on the blog. It was more challenging to keep track of - we had a couple of missed solutions and a couple of duplicates, so it took us some time to sort it out and get our equation to work out the way it was supposed to do. Tomorrow, I want to go back and look at that again. He didn't do them in any particular order, so they were easy to mix up. But when I did the next chart, for a 6-letter word, PUZZLE, I did them systematically, and kept things straight no problem. I didn't want to draw the whole grid anymore, so I printed some graph paper for this one. As I was counting things up to see how many times the final E fell in each square, I used some little squares in the middle to help me keep my place.


By now, I'm noticing a whole bunch of interesting patterns. For instance, the letters always fall in the same places. No matter how I follow the rules - turn right or go down - I never need to put two letters in the same block. That's true all the way through. And the equations that this makes are pretty interesting, too. I think that you could make some kind of equation or generalization for how they go:

1+1
1+2+1
1+3+3+1
1+4+6+4+1
1+5+10+10+5+1

I'm not sure how to generalize how that's going into an equation, but there's clearly a pattern, and you could figure out what it is for any iteration, if you could generalize it into an equation. Which would be cool. I'm tempted to do a 7-letter word, which would be 64 squares, just so that I can make a prediction about what the equation would be and then check to see if  I'm right. This is kind of a fun game.

Also, the numbers are squaring. We haven't done the squared numbers unit in Miquan just yet, but after doing this, I think we're going to in the next few days. This is a pretty cool introduction to squares. (Update: Yes, I eventually realized these are exponents, not squares. There's a follow-up post in the works.)

MEP's lesson plan suggested that this activity would take 9 minutes with a whole class of kids. I don't know who thought that was going to be possible; we took more like 20 to do the 3-, 4-, and 5-letter words, and then I spent a few more minutes doing the 6-letter word on my own. So their time estimate was way off. But I really like the way they are showing kids how to look for patterns and encouraging number play. This is a fantastic activity that we had a good time doing.


Part two is here.

1 comment:

Joshua Greene said...

There are lots of wonderful patterns here and I hope you will have an opportunity to continue exploring. Two ideas that might be interesting:
(1) try building up a 4 letter sequence from a 3 letter sequence. In other words, build the tableau for 'abc' and then see how that relates to 'abcd'
(2) think about the patterns within the equations. Even if you can't immediately predict every term of the equation, are there some parts that you recognize?

What you are doing is real mathematics. You've noticed something interesting, are building data, making conjectures, trying to find connections to things you already know, testing, refining your understanding. Even the blind alleys are good; thanks for sharing with us and letting us peak into your experience.

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