Common Issues with Proportional Relationships Transcript

Speaker 1: [in class] Okay, if I can have your attention, we’re going to get started. Yesterday we looked at some situations similar to the ones that we have been looking at throughout this whole week-and-a-half instruction. So today, we’re going to look further into those things and hopefully, it’s going to help you work through your thinking.

Speaker 1: [to camera] Today’s lesson is about determining whether relationships are in proportion or not; they’re in a proportional relationship or not. The first thing we did was look at a problem together. Are they in a proportional relationship?

Speaker 2: [in class] 10 ounces of cheese costs $2.40. Ross wants to buy blank ounces of cheese. Ross will have to pay blank dollars.

Speaker 1: So 10 ounces of cheese costs $2.40. I want you to come up with some numbers that’ll work there on your whiteboard. Just one set -

Speaker 1: [to camera] I asked them, is this a proportional relationship and why? To kind of get them in the frame of mind, as if I’m looking at two quantities, how do I tell if they’re proportional, what can I use?

Speaker 3: [in class] Multiply 0.24 because every 10 pounds is 2.40. So if you divide 2.40 by 10, you get 0.24 to get 1.

Speaker 4: [to camera] The lesson creates the opportunity for the kids to get in pairs. And through sort of wrangling, productive struggle, grappling with complexity, conflict, whatever you want to call it, the lesson portion is designed to get the kids to resolve the issues that they have.

Speaker 1: [in class] Okay, here’s my question, is the relationship between the ounces bought and the total cost a proportional relationship? Kelly?

Speaker 5: It forms a straight line, if you graph it.

Speaker 1: How do you know?

Speaker 5: Because you continue to go up and over by the same route.

Speaker 1: So you’re going up, over 1, up -

Speaker 5: 0.24.

Speaker 1: I love it. So we got this. So that’s what we’re going to do today. We’re going to look at a bunch of proportional situations, and look and see if we think that they’re proportional, okay?

Speaker 1: [to camera] I handed out the cards sets and said, with your partner, you’re looking at two quantities, and you have to set, you know, they had to set the parameters.

Speaker 1: [in class] Here are the rules for working on the cards with your partner. Choose one of the cards to work on together. And you’re going to choose some easy numbers to fill in the blanks. You’re going to answer the question you’ve written and write all of your reasoning on the card, working together on this. Then you’re going to choose harder numbers to fill in the blanks and answer your new questions together writing all your reasoning on the card. Decide whether the quantities vary in direct proportion or in proportion. Write your answers and your reasoning on the card. When you have finished one card, you’re going to choose another. Okay, you can begin with your partner.

[students partner up and talk]

Speaker 1: [to camera] The way that the collaborative portion is set up, it gives you an opportunity to ask students about their understanding and to really move their learning forward. That’s one of the reasons why it’s really important to have students in homogenous groups.

Speaker 1: [in class] Two slices of toast and I push down the buttons at the same time, what does the toast – does the toast pop up at the same time probably?

Speaker 6: Yeah.

Speaker 1: Okay, what if I put one slice in?

Speaker 6: Probably [?] in four minutes.

Speaker 1: So it’s faster if you only do one?

Speaker 6: Oh, I get what it meant.

Speaker 1: Does that change how you’re thinking about it?

Speaker 6: Yeah.

Speaker 1: Okay, talk about it. You didn’t have an “ah” moment there. You were just like, “Huh?”

Speaker 7: Yeah, I was confused -

Speaker 1: When you guys talk, involve Matthew in the conversation.

Speaker 6: So it would still be at 8. No, no, no, it would still be at 16.

Speaker 8: If you order 4 times, each of them 5. That’s 5, 10, 15, 20. But each of them is $2, so it’d be 2, 4, 6, 8. And that’s why I times it by 2.

Speaker 1: What’s my cost going to be for 10 minutes?

Speaker 7: $7.50.

Speaker 1: But you’re forgetting about, they’re charging $5 per month.

Speaker 7: So do you want us to add that on -

Speaker 1: I do, I do. You have to because you’re going to get charged. I like the work. I thought that was nice. I would like you to do some of the hard numbers. Okay, so maybe switch roles. But now you have to decide, is this a proportional relationship.

Speaker 8: Okay.

Speaker 1: To wrap up for today, maybe you didn’t get through all of them, and that’s okay because we’re all at a different level, that’s fine.

Speaker 1: [to camera] The takeaways from today were, they’re starting to figure out what the quantities are. They’re starting to say what’s the relationship and how am I going to check this. I’m going to look at the cards to see what everyone got. Just kind of see in the end what I want to make sure I clarify or talk about tomorrow.

Speaker 1: [in class] See ya.

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Speaker 1: [in class] Okay, so today we’re going to finish up the activity that we started yesterday. I’m going to give you about three or four minutes to make sure that you’ve looked at that information, so that when we go to work again today in partners, that we’re ready for the work. There you go.

Speaker 9: I got 2.0 and then –

[crosstalk]

Speaker 10: I just want to do a more complicated number. So like -

Speaker 1: To see if it still works.

Speaker 10: Yeah. It’s not equal to these, but it’s not a proportional relationship either.

Speaker 1: So you and your partner have had a chance to work through most of these cards. Now you’re going to have a chance to let someone else try the situation that you created.

Speaker 1: [to camera] And I had them replicate their problem and give it to another group. And they had to choose one proportional relationship and one that was not.

Speaker 11: [in class] I don’t think this is proportional because in the cell phone bill it starts as $5.00 per month, so that’s already $5.00, so it can’t go through 0:0.

Speaker 12: It can’t be yes because this one’s already yes and they have to choose no.

Speaker 1: [to camera] At one point I realized that several of the groups had received two non- proportional situations. And the students who received that work were a little confused.

Speaker 1: [in class] I want to raise a question about some of the cards that I’m seeing being passed. And let’s just kind of throw it out to the group for a moment. So they were handed these two cards: cell phone and internet. The question is which one is the proportional relationship on those two? Damian?

Speaker 13: Neither.

Speaker 1: You think neither?

Speaker 13: Yeah, because they both have an added variable at the beginning.

Speaker 1: So you’re saying this internet service provides whatever amount. So you’re having that amount that you’re adding on in the beginning. That you’re getting for free really. Okay, did you want to react to that Willie?

Speaker 13: I agree. I would say that neither one of them are proportional.

Speaker 1: Why, what makes them not proportional?

Speaker 13: Because they don’t start at 0. And if you graph them, they wouldn’t go through the origin.

Speaker 1: Gabrielle?

Speaker 14: About how Willie said it doesn’t go through 0:0, it actually would, because if you use, say it gave you two free gigabytes and you didn’t use any extra, your total cost would be 0 dollars, because you wouldn’t have to pay for the extra gigabytes.

Speaker 1: Woah. Questions about that? Matthew?

Speaker 15: How can you tell if something’s proportional or if it’s [?]

Speaker 1: That’s a good question. Who’s answering it? How can you tell if these two quantities are in a proportional relationship? Sydney?

Speaker 16: They have a constant ratio.

Speaker 1: Okay, they have a constant ratio between what?

Speaker 16: Between the, for example, the costs and the gigabytes used.

Speaker 1: [to camera] Everything that I wanted to happen, started to happen with those problems because I really wasn’t answering questions, I was asking questions, and the kids were answering them.

Speaker 1: [in class] If I used one gigabyte, it’s how much? 0. If I use 2? It’s 0. If I use 3, it’s whatever the cost is. See how that’s not constant.

Speaker 17: Oh, because there’s 2 free ones.

Speaker 1: Right, exactly.

Speaker 1: [to camera] I knew that the toast problem was a really tough one for them to understand.

Speaker 1: [in class] You had your hand up as it wasn’t proportional, right? Why?

Speaker 18: Well, because it’s a ratio of 1 piece of bread per minute. And if like you don’t cook any bread, well then it’s 0:0. So it’s starts at 0, then it goes up by 1 each time.

Speaker 1: Kelly?

Speaker 19: If you drew it in a graph, then like it would go up, it would stay the same.

Speaker 1: [to camera] We started drawing the graph, and it looked like a step. So it was kind of stepping up.

Speaker 1: [in class] What if I had 3 pieces of toast?

Speaker 20: 4 minutes.

Speaker 1: Why would it take 4 minutes?

Speaker 21: You’d have to put more bread and what amount.

Speaker 1: Yeah, okay, all right. So Willie, what about 4 pieces?

Speaker 13: It would still be 4.

Speaker 1: It would be 4. So now we haven’t really seen a graph like this, have we?

Speaker 22: Wait, Ms. Walker, for it to be proportional, wouldn’t it depend on how many pieces of toast you’re putting in the toaster each time? Like if you put in 2 each time, then it could be proportionate.

Speaker 1: You just blew my mind. Oh my gosh, that was awesome. Holy moly, that was awesome. I never really thought about this. Say what you just said.

Speaker 22: Couldn’t it be proportional if you just put in just 2 pieces of toast every time?

Speaker 1: [to camera] I never thought of asking that question, how could I make it proportional, but he just led us right into it. It showed me how much more they understood about the pattern.

Speaker 1: [in class] Matthew, is that one proportional?

Speaker 23: Yes.

Speaker 1: Why?

Speaker 23: Because it goes through the origin.

Speaker 1: And?

Speaker 23: It has a constant ratio.

Speaker 1: What is that ratio?

Speaker 23: 2:2, 4:4, 6:6.

Speaker 1: 2:2, 4:4, okay. I think you guys have done an awesome job. I mean, it gave me the chills today.

Speaker 1: [to camera] They did a phenomenal job in actually grappling with the math and looking at the relationships and expressing their opinion. And they were applying what they knew. And I think they grew in their understanding of it as a result.

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