I was doing a math moment with the kiddos at Ethel Streit this week. We have a couple who have had negative experiences with math, some who have only experienced math in traditional public school settings, and some who are quite gifted in mathematical thinking but have little facility with math facts or algorithms. To try to set the stage for the idea that we all explore big math ideas together, I made a “simple” drawing on the whiteboard:
I asked the kids how many squares there were.
The answers were immediate: 9. No, wait! 10! As the kids were thinking, I heard L say”9 times 5…?” I have to admit: I had no idea what she was talking about. She tried to explain to me that there were 9 on top and 9 on the right and 9 on the bottom and 9 on the left. She said that wasn’t quite it, and I chose to move on.
Eventually, the kids figured out there were 14 squares (for those of you following along at home, that’s 9 1×1 squares, 4 2×2 squares, and 1 3×3 square).
L piped up again with “times 5?”
I still had no idea what she was talking about. Luckily, my incredible co-teacher, Sara, looked at L and asked if she meant like a cube. Yes! L’s face lit up. She explained that we had only counted the squares we could see, not all the ones we couldn’t see. She was seeing the visible squares as the front-facing side of a 3-D cube…
I didn’t want to lose the other kids, so I decided to stick a pin in that idea. I told L we would talk more about that later, erased the drawing, put up a set of triangles, and moved on with the lesson. L clarified, “We should only count the triangles we see, right?” Right!
On the way home, L piped up again. “It’s not times 5! It’s times 6!” She had realized that you had to count the squares on the face you COULD see in addition to those on the faces you couldn’t. I realized that clearly, L wasn’t done with this yet.
When we got home, I printed and cut out a cube net.While I was doing that in the office, I gave her a piece of scratch paper and asked her to represent what she meant mathematically. This is what she created:
I then gave L the cube to represent her work in three dimensions. She asked me to draw the 3×3 grid on each face and then proceeded to complete her calculations.
She began by counting each small square, realizing that her total was 54. I had her check using multiplication to see that 9×6 is 54.
She then said, “But there are 14×6 squares,” remembering that we had found 14 on each side. I gave her a high five and we decided to multiply it out to find the total. We broke 14 into 10 and 4, giving us two problems: 10×6 and 4×6. She immediately knew that 10×6=60. 4×6 was harder for her. She did know that 4×2=8, so we used that fact three times to compose 4×6=24. We then joined the 60 with the 24 to learn that there were 84 squares on the six faces. There are our written notes from the process:
Because L is L, though, we weren’t done yet. She wanted to figure out how to build the cube out of smaller cubes. As she was considering the build, she gasped, “The cubes are all hidden in plain sight!” When I was done laughing, I told her she was right.
We pulled out Cuisenaire Rods to build. I wouldn’t usually use cuisenaire rods for this build, but we had recently taken our base ten blocks into the school and I could tell we needed to do this build right now. I was particularly aware of the fact that we don’t have 27 1-unit rods. I decided to let it ride, though, and see what she did with it.
L began to construct a 3×3 base out of the 1-unit cubes. I went outside to collect cherry tomatoes from the garden (yes, the garden is still crazily producing). L ran up to me and said, “I don’t have enough cubes!” I said, “Oh.” As I was thinking of what to say, she spurted out, “But it’s ok! I can get one that’s 3 long and use it in place of three ones!” She turned around and ran into the house. Ok. I thought. She’s got a plan.
I came back in to find her build complete.
As she began to figure out how many 1×1 cubes were in the whole cube, I kept quiet and watched. She sorted the white cubes from the purple rods. There were 12 white cubes and 5 rods. She counted the white cubes one by one, moving them over as she counted them. When she got to 12, she began touching the first purple rod. She touched it on the end, the middle, and the other end as she counted, “13, 14, 15.” She moved that rod over with the white cubes and repeated the same process until she got to 27 as her sum.
The look of pride on her face was palpable. I loved that this was all self-driven and to solve a question that she had been puzzling over. She’s also been talking about the math on the board and the cube daily since then. Hidden in plain sight, indeed!