Dots in a Square from Math Without Words

Recently, my third grade class and I have fallen in love with Math Without Words by James Tanton. This is a delightful collection of math puzzles, ranging in difficulty from fairly easy to quite hard, but all mathematically beautiful. Occasionally, I find one that is somehow related to the topic that we’re studying, but usually I pick out ones that I think the students would enjoy and that seem just barely within their reach. I also often have them work in pairs or groups of 3.

True to the book’s title, the puzzles come with no word descriptions; one has to figure out what is being asked based on a diagram or picture. Whenever I give out the puzzles, the reaction is predictable: “Huh? What is this? What are we supposed to do here?” But fairly quickly, the students start getting some ideas about what the pictures might mean, and soon there are heated discussions about how to best solve the problems.

One of my favorite discussions ever happened when I gave the students a puzzle as we were just starting our perfect squares unit. We had at this point spent several weeks on multiplication. Here was the picture “description”



And then came the problems. The first one was 1 + 2 + 3 + 2 + 1. Well who needs a diagram to explain to them how to do that? The next few were slightly larger, but still the students could do the additions fairly easily, which they proceeded to do. But then came… 1 + 2 + 3 + …+ 99 + 100 + 99 + … + 3 + 2 + 1. And that’s when they started analyzing the diagram more closely. A few fearless students started on the additions, but gave up pretty quickly.

Here is a conversation that I witnessed among a group of three students. This isn’t an exact transcript (partially because I don’t remember who said what exactly), but it is very close to it.

S1: Hey, 25 is the number of dots in the square!
S2: (after counting them carefully) Oh yeah, you’re right.
S1: And 5 is the number of dots in that longest line in the middle.
S3: So we just need to draw a square with 100 dots in the middle and count the total number of dots.
S2: Somehow I don’t think that’s going to be any easier than adding up all those numbers.
S3: You have a better idea?
S2: Wait, 5 is also the number of dots in each row of the square.
S1: And there are 5 rows.
S3: Well this problem has a 5 in it and the one we’re trying to solve has 100.
S1: I know, I know, it’s 100 times 100!
S2: Oh yeah, you’re right! 100 rows of 100.

At this point, the problem was solved as far as I was concerned and I was about to turn my attention to another group, when I heard:

S3: Wait, what is 100 times 100?
S1: I think it’s a thousand.
S2: No, I know it’s not a thousand because I remember my dad telling me that 100 times 100 is not a thousand. I think it’s a thousand ten.
S3: That doesn’t make sense.
S1: Well if it’s not a thousand, then it’s a million.

This part of the conversation continued for a bit longer, but they finally figured it out, with a tiny bit of my guidance. When I recently reminded these students that not so long ago they had an argument over the value of 100 times 100, they had a good laugh.

About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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