Preparation for Fraction Multiplication Transcript
Card: TCH Teaching Channel TeachingChannel.org
Card: Mount Erie Elementary Anacortes, Washington
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Olivia: You split it in half like this and then you go three, six, nine, twelve.
Student: So we did 12 divided by 12 equals 1.
Student: I don’t get this at all.
Alicia Farmer: We’re actually working with two fractions and we’re going to be talking about this is a new type of thinking.
Card: Illustrative Mathematics: Preparation for Fraction Multiplication
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Alicia Farmer: All right. We’re going to start with one-half of 16.
Alicia Farmer: Started the class with the number talk so all the students come to the floor and we follow some procedures for our number talks where kids are showing me that they can do mental--
Card: Alicia Farmer, 5th Grade Math Teacher Mount Erie Elementary, Anacortes, WA
Alicia Farmer: -- arithmetic but it’s with conceptual understanding. So it’s not just I’m multiplying in my head or I’m adding in my head, but it’s what am I doing to those numbers that allows me to do that.
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Alicia Farmer: All right. Who has a solution for one-half of 16? Freddie?
Freddie: I believe the answer is eight.
Alicia Farmer: You got eight. Give me a shake if you also got eight?
Alicia Farmer: All right. Everybody got eight. All right. Go ahead and tell us how you got eight?
Freddie: Okay. So I got eight plus eight would equal 16 and also eight times two would equal 16, so I knew that.
Alicia Farmer: Okay. So then half of that would be one of these eights?
Freddie: Yeah.
Alicia Farmer: Okay. All right. So let’s try-- you want to try a harder one?
Students: Yes.
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Alicia Farmer: So we were really just taking a half of 16 and then a fourth of 16 and then an eighth of 16. And then I went to three-eighths. I wanted to see what they could do with non-unit fractions because in fourth grade, it’s all unit fractions with whole numbers.
Alicia Farmer: All right. Here we go. Three-eighths of 16.
Students:
Alicia Farmer: You always know it’s a challenge when you hear afterwards.
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Alicia Farmer: Tuscan, what did you get?
Tuscan: I got six.
Alicia Farmer: Okay. Give me a shake if you got six.
Alicia Farmer: All right. Why don’t you start by telling us what you did?
Tuscan: It’s kind of like 16 divided by two is eight. However --
Card: Tuscan, ½ of 16 is 8. 16 / 2 = 8
Tuscan: then I know that with division, you could switch the two and the eight in that and it would still be correct.
Tuscan: So two is one-eighth of 16 and then--
Tuscan: -- two times three equals six because it’s three-eighths.
Alicia Farmer: Equals six?
Tuscan: Yeah.
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Alicia Farmer: So should we be nervous when the numerator is not a one?
Students: No.
Alicia Farmer: Okay.
Alicia Farmer: Nice job guys.
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Alicia Farmer: After the number talk, they went back to their seats and I introduce the situation, which was we were having a fundraiser and we’re selling square pans of cornbread. And normally when you buy cornbread, it’s usually cut up already for you. That’s nice. But we’re mathematicians, so we’re going to actually challenge ourselves.
Alicia Farmer: All week we’ve been talking about fundraising for our mountain school. You guys remember that?
Students:
Alicia Farmer: All right. And we’ve decided that we’re going to have this Chili Cook-Off for the district. So we’re going to be selling square pans of cornbread at this Chili Cook-Off, right?
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Alicia Farmer: When you start, the first fraction tells you how much of the cornbread is in the pan still. And then you have to go to that cornbread is now your new whole. Instead of just the pan, it’s now just the cornbread and you take part of that. But then the question actually makes you go back and find out of the whole pan, how much did that person actually buy.
Alicia Farmer: So let’s talk about our learning target for today.
Card: Learning Target, We can… solve real world problems involving fractions, Success Criteria, Success means I can:
Alicia Farmer: We can solve real world problems involving fractions.
Card: Common Core Standards, Solve real world problems involving multiplication of fraction.
Alicia Farmer: Notice I haven’t told you if it’s addition, subtraction, multiplication or division.
Card: Common Core Standards, Solve real world problems involving multiplication of fraction.
Alicia Farmer: That’s part of the learning today, okay?
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Alicia Farmer: It’s really the first time that students are taking a part of a part of anything. In fourth grade, they look at repeated edition with whole numbers and fractions.
Alicia Farmer: It requires you to change your thought process about what is the whole that I’m actually talking about.
Alicia Farmer: All right. We’re going to start with some private think time. I’m going to give you five minutes on the timer and you will do a go-around at the end. So the expectation is you have something to share at that end of the five minutes. Does that mean you have to have an answer?
Students: No.
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Alicia Farmer: Can somebody tell me what you might say in a go-around if you don’t have a solution yet? Erik?
Erik: You might say where you started or what you understand and don’t understand.
Alicia Farmer: Yeah, that’d be a really good thing. I mean I started this and then I got stuck because this wasn’t making sense and you want to be specific about what those things are. You guys can go ahead and begin.
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Alicia Farmer: I’ve tried to create a culture where all the students understand that their opinions and their thoughts and their reasoning, that’s all valued. Even missteps. Even flaws. Even mistakes that you make. Those are all valued because they’re all learning opportunities from old standards to the Common Core standards, getting the right answer isn’t enough. That’s not preparing our students to be literate math citizens in the world.
Alicia Farmer: They have to understand the concepts that underlie those procedures. So there is this shift to understanding what you’re doing along the way, being able to explain that, communicate that.
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Alicia Farmer: We’re going to start a go-around now. After all four people have shared, I would like you to work as a table to come to a final solution that everyone agrees with.
Student: I wanted to turn that, the one that she had left of the one-fourth into a third.
Alicia Farmer: If you have to come to a consensus, that means you have to be able to explain what you’re thinking is.
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Alicia Farmer: So they’re not only having to show the work, but they’re having to construct those mathematical arguments and say them out loud, as well.
Olivia: How much do you guys think that she bought out of the whole pan?
Student: 13, one and 13.
Student: Why do you get-- why did you--
Student: Think that.
Olivia: Yeah.
Student: I found some up here that said one-half of $12. So I knew that one-third cost Ms. Farmer $13, so that’s how I knew.
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Olivia: You guys think that much could be one-twelfth because if you go like this, you need one more because it says one-fourth--
Student: Yeah. Yeah.
Olivia: -- of the thing of cornbread.
Student: And you only got--
Student: And then--
Student: You had to cross off all of it.
Olivia: Yeah, because--
Student: That stuff.
Student: Okay. Now, okay.
Olivia: Now do get it?
Student: That was the--
Student: Yes.
Olivia: Yeah.
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Alicia Farmer: I’m walking around and just taking notes, both mental but also often physical notes of where are kids. Like where were they able to enter the problem or were they no access. I’m learning which ones can read the context and then apply it and have a strategy to solve it.
Alicia Farmer: Also I’m looking and I’m choosing pieces of work for kids to bring up.
Alicia Farmer: So I would like to start with Angie. Can you bring up your diagram there?
Angie: Over here?
Alicia Farmer: Uh-hm.
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Angie: What I did is, I realized that I can’t subtract three from four, so I did the bar model and what I did, I made it into fourths, then cut it into thirds and colored in one-third and then circled one and counted all the little boxes and I got one-twelfth.
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Alicia Farmer: I had brought up and then I followed her up with Erik because I wanted to show that Angie’s was what we had been doing in class already with equivalent fractions. And then Erik’s was a little bit different.
Erik: I tried and failed a lot but I finally got it. And so what I did is I split my diagram here into fourths and then I shaded in this one-fourth to remind myself that only one-fourth is there. and then I split each fourth into three and I shaded this one in darker so that you know that that’s the one-third that I’m using.
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Alicia Farmer: How is this one different or similar to Angie’s? Olivia?
Olivia: His model is like set up a little differently. So like those are set up into thirds.
Alicia Farmer: So he has two--
Olivia: Yeah.
Alicia Farmer: -- rows of--
Card: Erik, Angie
Alicia Farmer: -- of six pieces and she actually three rows of four. All right. Thank you, Erik. Have a seat.
Alicia Farmer: Then the students went back and they tried a new problem with a non-unit fraction.
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Alicia Farmer: So it had a numerator other than one. And it’s a little bit more complex because you’re talking about more pieces of the cornbread that are actually there. and they tried to apply the strategies that they saw from the students that presented or ones that were working in their group already to a new situation.
Student: I crossed out this part of the cornbread right here and then I had the five left over here and I know there’s 12 total, so I got five-twelfths.
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Alicia Farmer: And that piece was where I could really see are students able to apply what we’ve been learning, what I’ve been questioning about, what they’ve been answering about. And so it was practice but also deepening that understanding.
Student: I circled around the five ones that he bought and this is what he left and that is how much he took with him.
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Alicia Farmer: The procedures, kids call them everything from tricks to shortcuts, those things can be memorized and then quickly forgotten. What sticks with you is that understanding. Now I mean if you can, in your head, picture taking a part of a part then you can figure out the solution no matter what the context is.
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Alicia Farmer: Nice job today, guys. I’m very proud of you. you were able to use what we did in the first part to apply it to the next part. That was really impressive work, okay?
Card: TCH Teaching Channel
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