We carefully chose the problem 26 x 4 for this number talk (mental math) in order to apply the bar model to student thinking.
I think a key element to this lesson is a deep understanding of division. For this particular class at this time, I might try inserting a lesson before the division bar model problem, that asks students a multiplication bar model problem. Once we tried out both, students might clarify for themselves similarities and differences between the two types of stories and the two opposite operations.
BECCA SHERMAN: So. Who wants to tell us what answer you want to explain, and tell us how you did it! Your strategy. Okay. Why don’t you start?
STUDENT: Um, one?
BECCA SHERMAN: And how did you get it?
STUDENT: Um, I put 26 on the top, and 4 on the bottom, then I timesed it. And then I timesed 6 times 4 equals 24,
BECCA SHERMAN: Okay, and what’d you do with that 24.
STUDENT: I put the 4 on the bottom of the 4, and the 2 on top of the 2, and 2 times 4 equals 8, plus 2 equals 10. And I just put the 10 next to the 4.
BECCA SHERMAN: Okay. Did anyone else do it that way? Okay. Remember this idea, you guys? A picture is worth a thousand words? Do we have a picture for that?
BECCA SHERMAN: Do we have a picture for that?
BECCA SHERMAN: Does someone… if you do, maybe I’ll try your way? How would you draw a picture of that?
STUDENT: You do 26 circles and put 4 in each one.
BECCA SHERMAN: Okay…
STUDENT: Or 4 circles in 26 rows.
BECCA SHERMAN: Either one?
STUDENT: Either one.
BECCA SHERMAN: Okay. Instead of drawing lots of … since 4 circles is easier to draw, I might do that. Instead of drawing, um.
STUDENT: 26 in….
BECCA SHERMAN: 26 dots, can I make a shortcut? And put the number? Okay. And then … this is pretty close to what we were doing down there, so if I wanted to make it like boxes that are all the same, then I could put 26 in each one?
STUDENT: How ‘bout the dots?
BECCA SHERMAN: Okay. Another strategy. Who had a different strategy, not the stack and multiply strategy. Um… what’d you do?
STUDENT: I had… I had one for the drawing.
BECCA SHERMAN: What was your drawing?
STUDENT: 26 plus 26 equals, uh, what you call it? 26…
BECCA SHERMAN: So one person—are you defending the same answer?
BECCA SHERMAN: Okay. So you used 26 and 26. And what’d you get?
STUDENT: I got… 52.
STUDENT: No! that’s not the answer!
BECCA SHERMAN: And… on the drawing, that might look like that?
STUDENT: Bottom of the 52 I put another 52,
BECCA SHERMAN: Where’d that 52 come from?
STUDENT: ‘Cause there’s 2 ways. One you could do..
STUDENT: Oh! I know!
STUDENT: 52 plus 52! Equals 104.
BECCA SHERMAN: Do you want to… do you want to finish your thought? Or do you want someone else to finish your thought? Okay, go ahead.
STUDENT: Yeah, I put a 4 on the bottom,
BECCA SHERMAN: You put a 4 on bottom.
STUDENT: I added 2 plus 2
BECCA SHERMAN: Oh, oh, you were adding, sorry. I thought you just… okay.
STUDENT: Added 2 plus 2 equals 4.
BECCA SHERMAN: Uh huh…
STUDENT: And 5 plus 5 equals 10, so… yeah.
BECCA SHERMAN: Okay, so on the picture could I also say that that equals 52?
BECCA SHERMAN: And… I’m gonna call that.. can we call that, “double?” You doubled 26? And doubled it again? The double double?
STUDENT: A triple!
BECCA SHERMAN: Okay. Is that a triple strategy? I don’t know if that’s the same. It’s a double double… okay. Did anyone… break the number 26 apart?
BECCA SHERMAN: And do that?
STUDENT: No, I did another way.
BECCA SHERMAN: Or do you have a different way. Derek, what was your way?
STUDENT: I have a different way.
BECCA SHERMAN: What’s your way.
STUDENT: I looked, when I saw the number up there, I just looked at it, and, but I didn’t do it the right way. I did…2, I did, I knew the 20 was in the tens place, so I did 20, 40, 60, 80, 4 times. And then, um, after that I did the 6 four times.
BECCA SHERMAN: So how did you do the 6 four times?
STUDENT: I jumped it by 6, is all. So, 4. I did 6, 12, 18, 24. And then I added those to 80, and 24 together….80 and 24 together and made 104.
BECCA SHERMAN: So that’s 24, and that’s 80, and all together.. 104. Whoa. Let’s try that in a picture. And see… 20, plus 20, plus 20, plus 20, hey, that’s like our groups over here. Equal groups? So I’m gonna try drawing that. Here’s 20, then 20 more, then 20 more, then 20 more, and how many did we get all together?
BECCA SHERMAN: 80?
BECCA SHERMAN: 20, 40, 60, 80. Okay. So that is… 80. And then he did the same thing with the 6? Where did this 20 and the 6 come from?
STUDENT: Oh! I got it from the 26.
BECCA SHERMAN: Oh! So wait a minute. I’m gonna go right up here and say, you started with this idea, 26. You did break it apart! Here’s 20, and here’s 6. So 26 is made up of a 20, two tens, and 6 ones. So here’s his… and we had to do what with it? Multiply by…
BECCA SHERMAN: 4? Okay. So let’s see. Here’s our 4 20’s, here’s, maybe, our 4 6’s? and you said that’s 24. Cool. And then 24, my mess, getting kind of messy here! 80 and 24 you said, all together, is…
BECCA SHERMAN: 104? Okay. Pretty cool, you guys.
The connection between “times four” and “double, double” gets us on track equal groups and using the bar model. I wish I had also brought back in their idea of “repeated addition” with this. The “break apart” of 104 into four groups of twenty plus four groups of six furthers the bar model example. For time sake, I skipped the second number talk we planned.