Students are asked to take the words of the story and make a math picture. No question is asked yet. The story is: “Maria saved $24. She saved 3 times as much as Wayne.”
BECCA SHERMAN: I want you to answer these two questions. Finally, we’re asking questions to answer! From her, from Charlie’s way of doing it. So you may have had a different idea, ‘cause we had a couple different ideas, but from Charlie’s way, how much do you think Wayne saved, and how much did Maria and Wayne save together? So answer those two in complete sentences underneath your picture. Or, you can put it in the middle, ‘cause you have space there. Is there , is there a number that would be the same for all of those green boxes, that would work for this story?
BECCA SHERMAN: You don’t think so? Okay. Do you have answers to those two questions from the model from the picture? Do you think you know, do you think Charlie would know how much Wayne saved? I see something in your picture down here, what do you have down here?
BECCA SHERMAN: And how did you get that? Or, what do you think? Was that from your thinking, or was that from this drawing.
STUDENT: From my own
BECCA SHERMAN: Oh, from your own thinking. So his way doesn’t make sense to you? Do you think Charlie would have put an 8 in there? Or do you think he would have put a 24 down there? Still thinking? Okay.
BECCA SHERMAN: Okay. Finish your thought, and let’s share out our ideas on Charlie’s way of doing this. Some things I noticed, for some of us we had a really strong idea from our own picture of what we thought, and then we tried to put it into this one, and it was like, “Wait a minute.” Made us think a little bit, huh? Had to think about what would go where. So. Maria’s money. How much, if we just go back to our picture, how much did Maria have all together? How much did she save? Yeah.
BECCA SHERMAN: And where is that in Charlie’s picture?
STUDENT: On the top
BECCA SHERMAN: Yeah, the 24 is right up here. So Charlie chose to put the $24 up here. Why do you guys think that Charlie made 3 boxes underneath that 24? What do you think? Yeah.
STUDENT: um, he put 24 in each box.
BECCA SHERMAN: So… he has the 24, that’s all of her money. If he put 24 here, and 24 here, and 24 here, then how much would Maria save in this picture? 24 times 3, or 24 plus 24 plus 24… Ron?
STUDENT: Uh, um, 8.
BECCA SHERMAN: 8. 24 plus 24 plus 24 all together, she would have saved 8 dollars?
BECCA SHERMAN: So,
STUDENT: It’s not 24 plus 24 plus 24.
BECCA SHERMAN: Oh, let’s go, so you’re disagreeing, but I’m asking the question, if we put 24 in each one of these boxes, they’re equal boxes, so we have to put the same thing in each one, then we add those all up, I saw people do that, I saw people multiply by 3. We worked on that. What did you guys, when you multiplied, what did you get?
STUDENT: 24 times 3?
BECCA SHERMAN: ‘Cause you got two different answers, and then you fixed it. What did you guys get? Diamond?
BECCA SHERMAN: You got a different answer? What’d you get.
STUDENT: It was, they got 72 and I got 62. So then I multiplied it out, and I made a mistake, so then, um, I, then I solved it? So then I got the right answer.
BECCA SHERMAN: Which was, so the correct answer for 24 times 7 is what? Oop! What, right. What did I just say? 24 times 3, I meant to say. 24 times 3 is equal to 72. So does that make sense in our story? Does Maria save 72 dollars? Does she?
BECCA SHERMAN: Yes? No?
STUDENT: Maybe so.
BECCA SHERMAN: Oh! Okay. So that’s something that we need to understand about our story, and that we need to show in the picture. Maria saved 24. Charlie decided to show that right here with 24 dollars. So, Charlie didn’t choose to put 24 in each of these boxes. Some other people had a different idea. What did you guys think Charlie would have put in each of these equal boxes? Yeah.
BECCA SHERMAN: And why do you think Charlie put 8 in each of those boxes?
STUDENT: because 8 times 3 equals 24.
BECCA SHERMAN: So if this one’s 8 dollars, and this one’s 8 dollars, and this one’s 8 dollars, does that show that in all, Maria had 24 dollars? Down here, for Wayne’s money.
BECCA SHERMAN: What do you think?
STUDENT: 24 dollars.
BECCA SHERMAN: So Wayne had 24 dollars?
STUDENT: 8 dollars.
BECCA SHERMAN: Wayne had 8 dollars. How do we decide? What do you think?
BECCA SHERMAN: Okay.
Because of multiplication errors with 24 x 3, we spent time focused on this problem which lead to the wrong solutions in the moment. However over time I do believe that this encourages kids to listen to logic and sense-making of the math rather than to an “authority” to convince them of a solution.
It was definitely disconcerting that so many students gravitated toward 3 x 24 or three groups of twenty-four. I am certainly left wondering if my choice in student work led them that way. However, if we bring several more story problems and offer time to clarify correct solutions to each problem, I believe a majority of students would have this bar model representation for multiplication and division story problems down.