I have a new favourite problem!! It looks like this:
Without counting one by one, determine how many shaded squares are in this 10×10 grid.
This is not a new problem. I think it’s appeared in many books and possibly even textbooks throughout the years. My first exposure to it was actually in Jo Boaler and Cathy Humphreys’ book “Connecting Mathematical Ideas”. I loved watching the video of middle school students explaining the way they saw the problem and then eventually generalizing the methods for an nxn square, but I always wondered what it would look like in a high school class. Luckily one of my good friends allowed me to come in and try this problem with her class. It was awesome! Like exactly what I was hoping it would be! Maybe even a bit better! Here’s how it went:
1. We put the graphic above up for a moment and asked kids to give us a thumbs up when they knew the number of shaded squares. I asked someone to explain how they were thinking and then wrote that method on the board. “So is this how you thought about it?” I asked. They agreed. “Did anyone think about it differently?” It kept going until we had a list like this:
2. I asked students to imagine they had a 6×6 square now. “How would Jessica figure out how many shaded square there are?” I asked them. I gave them a moment to talk about it with their partners. When I did this again with Angelo’s method, the conversation got interesting. The kids started doing funky things with the numbers “6 x 10 – 6” was one of the ones I heard. I could tell that they weren’t thinking about the visual, and definitely not thinking about the expression as a function. I asked them how we could check and they told me that they could add up the numbers and see if it equaled what Jessica’s equaled. Not a bad method – definitely was a step towards getting us to talk about equivalent expressions which was one of my learning goals. I kept referring them back to the “imaginary 6×6 grid” and we eventually ended up with a list like this:
3. I asked students if we could do this for any sized grid. They all told me we could for any number… even for a 100241 x 100241. “So instead of making a list for every single possible grid, could we just say that we have like a say, nxn, square and n could be any one of those numbers?” I asked them to talk about that in their partners. These students all have had lots of exposure to variables in the past, but I find they often have started to forget what they really mean by the beginning of grade 9. They start to think they are some fixed thing and then start doing all sorts of weird things with them. The students all agreed that that would be alright. They also said that we could pick any letter as the variable. I asked them what Jessica’s method would look like for an nxn grid and gave them some time to talk about it and write it on their whiteboards. As I circulated to see what students were doing I saw a bit of this ” n+n+(n-2)+(n-2)” but a lot of this “n+n+p+p” or something to that effect. I was expecting this, but I still didn’t know what the best way to deal with it was. I didn’t say anything to individual groups, but once I saw every group had something written down I asked them to put up their whiteboards and look around the room at what other people had written. They did, but there weren’t really any lightbulbs flashing, so I started the discussion “I see two different types of answers (wrote them on the board). Are these both the same?” Most students were nodding their head. “Okay, so in the n+n+p+p, what does n represent?” A student raised her hand and said it was the dimension of the grid. “Okay, so what does the p represent?” No hands. And then one, “the p is like the inside part of the other two sides that the n didn’t count”. I asked students to talk about this – is this true? how do we know that this is what p represents? Some good discussion started happening. Some of the ones that had the (n-2)’s already written became more convinced, others who had written the n+n+p+p’s started looking over to the (n-2) people. I brought them back, “what did you talk about?” One student told me that she had converted, “we need to write the n-2 so that we know that it is a row minus the 2 squares we already counted.” Most students seemed convinced about this, others seemed a bit shaky. I erased the n+n+p+p and left the other expression on the board. Then we moved on to Angelo’s method. As you can imagine, similar conversations ensued. After some good discussion and some synapses firing we ended up with a list like this:
4. We had already touched on the idea of equivalence with the 6×6 list (the fact that all the expressions should equal the same number), so I brought it up again with the nxn list. “How do we know that these expressions all are equal?” No volunteers, but I was out of time. I left them with that challenge for homework – to pick two of those expressions and show to me how you know they are equal. My purpose was to hopefully get kids to start thinking about equivalent expressions and also like terms, the big idea being that even though expressions look different they actually mean the same thing, and we can manipulate them to make them look differently.
Because I wasn’t able to be there the next day, I’m not sure what kids did to show that the expressions were equal. The classroom teacher told me that the next day they pulled the algebra tiles out to start modelling these expressions with them and proving equality. I think that is a great way to prove this, and also a great lead in to combining like terms, distributive property, etc.