Today in fourth grade we did a number talk using a picture from http://ntimages.weebly.com. We’ve used pictures to do number talks before, but not often. I was excited to see what students would have to say about the Lego image below. I was hoping to get more insight on how they are using the properties of operations and if parentheses would come up again.

I decided to start by asking, “What do you notice?” I was expecting comments about colors, or number of bricks. I figured I would get some talk about arrays or groups. But this is not what I got. At first, they all approached this talk with a geometry perspective. They saw parallel and perpendicular lines, 90 degree and 180 degree angles. We are in the midst of a geometry unit now, it was awesome that they noticed some of the things we have been talking about, but it was totally unexpected! I wanted to talk about multiplication so I kept pushing them to notice more.

Next, I asked students “What do you wonder?” They’re were lots of great questions, including “I wonder why you are showing us this…” haha! We eventually landed on the question about how many little pegs were in the entire image. This is when I learned that those little pegs are actually called studs! Apparently, fourth grade boys are Lego experts. So I asked, “If we are trying to figure out how many *studs* are in this picture, show me a representation of what you see.” I passed out white boards. Below are some of their representations.

We talked about how we visualized the studs as a whole group. I tried to record some of their ideas on the Promethean board. I still struggle with recording their thinking…something to keep working on.

As you can see from their personal boards, there were several ways of “seeing” the studs. I decided to focus in on the kids who first figured out the number of bricks (3×4) and then multiplied that by 4 and the kids who saw an array of studs (6×8). The first students said “3×4 is 12. There are 12 Legos. There are 4 studs on each Lego, so I just need to do 12×4. It’s 48.” Another student had thought the same way, but wrote 3x4x4. So I asked, are those equivalent? There was a resounding yes. I wasn’t convinced, “But where is 12 in 3x4x4?” “The 3×4, duh Ms. Miner!” Ok, moving on.

We talked about the 6×8. Some students saw it as (6×4) + (6×4); 24 + 24 = 48. Same number of studs, sweet. Side note: some of the kids, without ANY direct instruction, are starting to “recompose” 6×4 + 6×4 into 6×8. Super cool, you can see it on one of the white boards. Kids were rocking the talk, so I decided to keep pushing. “Is 6×8 = 3x4x4? Can you prove it without calculating?”

This is where my mind was blown! They thought to themselves, then turned and talked and then shared out. And guess what they did!? They factored one of the 4s into 2×2 and saw that they could then do 3×2= 6 and 4×2=8 and so 6×8! I didn’t know if we would get there, and it was one of the first shares, and they all were agreeing with their hands and repeating to one another. It was super cool!

I did have to ask them about parentheses too. It had come up when one students told me they thought of (3×4)x4 instead as 3x(4×4). So I asked students what the difference was. They chatted about how the parentheses represent the groups. And so, by changing where the parentheses are, you change the way the group looks. In the slide below you can see the yellow shows 3x(4×4). The blue shows (3×4)x4. They found the groups, not me.

I know 3x4x4 doesn’t “need” parentheses to be solved. However, students were using them and so it offered us a chance to talk about them (again). I think this is where I want them. They are starting to see that parentheses don’t just mean “do this first,” parentheses also mean something about groups and the arrangement of the group.

I can’t believe how much these kids already know about multiplication. We have not even started the fourth grade multiplication unit! This is ALL carried over from their 3rd grade year. Thank you Mrs. Tweedie and Mrs. Dunphy! I am stoked to start talking multiplication 🙂