As her students work to describe and classify their problems, identifying which are more like Jesus’s problem (multiplying fractional quantities) and which are more like Camila’s problem (finding a fraction of a whole), Erika circulates around the classroom questioning her students and probing their thinking.
Two of the problems she has given her students do not have visual representations, so she invites them to create drawings of those problems if it would be helpful to their process.
STUDENT: Randy is making a kite. He will need six pieces of string. Each two-thirds of a foot long. How much string will he need all together?
STUDENT: So he will need six pieces...
STUDENT: This is kind of like Jesus's [problem].
STUDENT: Because of how much he will need total.
STUDENT: So Ruchita decided to make cookies for all of her friends, so she tripled the recipe. If the original recipe calls for one and three fourth cups of sugar, how much sugar did Ruchita actually use? Um, probably I think this is like Camila's because, like, it already says how much does she have but she tripled it, so how much more does she need?
ERIKA ISOMURA: Now we know?
ERIKA ISOMURA: We have some ideas of what to do and go on and talk about another problem, or you can try to solve that one to see what you get. It's up to you. Oh, I like how you're labeling that. Can you tell me what you guys are doing there with those labels on the top? I see whole and pieces, pieces, whole. What does that mean?
STUDENT: Like, the one at first is what they have.
ERIKA ISOMURA: Okay, so that's what they already know and what they already have. And then the other part is what they're missing?
ERIKA ISOMURA: So are you noticing that some are similar to Jesus's and some are similar to Camila's?
ERIKA ISOMURA: Okay. Clever. I like how organized that is.
STUDENT: I think this is like Camila's again because, like, it already says, like, eight rocks and then just in one bag. So each rock is, like, so it's probably half...
ERIKA ISOMURA: Okay, so what are you still not positive about?
STUDENT: Because, like, I'm not positive of how much sugar did Ruchita have to use.
ERIKA ISOMURA: Ah. Okay, so that is what we're looking for. Is there a reason why you're not sure how much sugar she actually used?
STUDENT: Because we didn't actually, um...because it said one of one [and] three fourth cups. We didn't actually add it together yet. So.
ERIKA ISOMURA: Okay. So if you were going to add it together, how many times would you be adding together?
STUDENT: Three times.
ERIKA ISOMURA: Why?
STUDENT: Because it says triple the recipe.
ERIKA ISOMURA: I thought I can trick some people with that word. Clearly I did not trick you. Good job of paying attention to the word. It's a really important word to know there. So you guys know that you don't actually have to solve them yet, right?
ERIKA ISOMURA: You can, that's absolutely fine. But our first task is just thinking: Are any of these problems like Jesus' problem, where we already know the parts and we're trying to find the whole, or are the problems like Camila's, where we already have the whole thing and we're trying to think about this [inaudible]. Okay?
STUDENT: Seven halves. She needs seven halves. Seven halves, yeah.
STUDENT: Twelve divided by three, which is four.
STUDENT: Wait, say that again.
STUDENT: So first you do the six. You do the six.
STUDENT: Yeah, I already did six times two third.
STUDENT: Times two. Six times two, which is twelve. And then you do twelve divided by three.
STUDENT: Four. So twelve over four?
STUDENT: Yeah. No because you're doing twelve divided by four...twelve divided by three.
STUDENT: Yeah but then after that, this would equal three wholes, right?
STUDENT: The denominator, yeah.
STUDENT: So this would equal three whole.
STUDENT: One and one third.
ERIKA ISOMURA: When I was doing this, the planning and the stories and all of those things, I really wanted to give them an opportunity to really experience. Some multiplication is this amount that we're taking a part of and it does feel often a bit like division, even though it's actually a multiplication problem. Then other times, it's very intuitive that this is a multiplication, I'm repeating, and repeating, and repeating. If it's repeating then I've got all the parts, and I just repeat them a certain number of times then I get that final quantity, versus I have something and I'm taking the fraction of it, which means I'm taking the smaller bit which is going to feel more like division.
After they had decided what kind of problem, they had the option of starting to just play with it on their own. I told one student, "That really does, to me, show that you’re really thinking about what's happening in the problem as far as the quantities.”
What I asked her was, "Now, let's actually be in real life. If you had that box of 8 crayons, it makes sense that you could take half of each crayon, but would you? Would you walk around snapping crayons in half and have all these broken crayons?" Her response was, “No."
Then we talked about in the real world, if you've got this box and I said, "You're only going to take out half of the crayons, what would you do?" She said, "I'd take out 4." Then we said, "That's so interesting because the answer you got by snapping crayons in half was also 4.” I told her specifically that there are 2 different ways that we could be thinking about those numbers, but when we go back to the actual problem, the story, maybe one of them makes a little more sense.