The following is just a suggested scenario for kicking off a class investigation of Ratio Triplets, and mimics how we used the task with our classes. Read a description of the Ratio Triplets task, and also get some printable copies of the three Versions needed for use in your own class.
Materials:
· Copies of all three Versions (the procedure below gives an idea of how many copies to make)
· Calculators (optional)
Activity Procedure:
· First, we had students get into small groups of two, three, or even four. Each group had only one Version given to every member of that group. For example, in this article you’ll hear from:
Line Student Comments
1 Jennifer “This [for Alisha] is dollars per ounce…[&] this [for Mark] is the cents per ounce? That doesn’t make any sense”
2 Heather “How are they…Is it correct? ‘Cause on this one [for Alisha], they’re different”
3 Cameron “I’m confused”
4 Jennifer “‘Cause she [Alisha] divided them differently”
5 Heather “Yeah, but then…”
6 Cameron “How do you divide something differently if you use the same numbers?”
7 Daniel “Well, it’s like money per ounce…This [for Mark] is how much money you pay per ounce, and this one [for Alisha] [is] how much you get per, how many ounces you get per dollar.”
8 Jennifer “So, ounces per…what? Ounces per cents?”
9 Daniel “Money per ounces”
10 Heather “So she’s [Alisha] correct because…”
11 Jennifer “So like, cents per ounces, or dollars per ounces, or what?”
12 Daniel “Dollars”
13 Heather “Well, wouldn’t this one [64-oz pkg for Alisha] be the better buy then?”
14 Jennifer “You make it difficult, because that’s [.10609375] not a dollar, that’s .10”
15 Heather “For down here [Alisha], wouldn’t this [64-oz Pkg] one be the better buy?”
16 Daniel “No”
17 Heather “Why?”
18 Cameron “How in the heck does she [Alisha] get $9.43 for ice cream?”
19 Jennifer “I’m even more confused than I already was”
Line Student Comments
1 Mrs. Peters “How did you determine which was the better buy here?”
2 Jennifer “Because just because…you still get more ounces in this one [64 oz pkg] but it’s more, and the difference is only like one or…point-one or two or whatever”
3 Mrs. Peters “What did you mean, ‘It’s more’ ?”
4 Jennifer “Like, this is 9, that’s 10. That’s more than that. So, even though this [48 oz] is a little bit less – you still get a little bit less ounces – but the price is like two, almost three dollars less. For like 15 less ounces.”
5 Mrs. Peters “Okay. Have you talked to all your group members here?”
6 Jennifer “Yah.”
7 Cameron “Yah.”
8 Mrs. Peters “Daniel too?”
9 Jennifer “He helped us, but, like, I don’t get what he was saying.”
10 Cameron “He helped us in a…”
11 Heather “[Didn’t] make sense…”
12 Jennifer “He said this was money per ounces, but then I asked him if it was like cents per ounces or dollars per ounces, but then I don’t get how it’s dollars ’cause it’s point-[something]”
13 Jennifer “I don’t get how she can divide this and get an answer for what the question is”
14 Cameron “I don’t get it, period.”
Line Student Comments
1 Jennifer “If I wanted to get…Do I multiply it by 48?”
2 Cameron “No, you divide it”
3 Jennifer “Uh-uh, I’m trying to get the answer”
4 Cameron “What the heck did you do, multiply?”
5 Jennifer “What do I do, divide it by what?”
6 Heather “You ran out of numbers”
7 Cameron “You probably, uh…I don’t know how you get the answer.”
8 Heather “You ran out of space”
9 Jennifer “You know how this [48] divided by this [4.69] equals this [10.23454158] ? So, I multiply by something. Yah, see? I get 47.999”
10 Jennifer “So, I almost got 48.”
11 Heather “It would round up”
12 Jennifer “So it is the Unit Price, yah. It rounds to 48.”
13 Cameron “I’m confused”
14 Jennifer “Ok, you know what the Unit Price is?”
15 Jennifer “How 1 ounce is point-ten or whatever?”
16 Cameron “Yah”
17 Jennifer “And for this [ALISHA], 1 ounce is ten point-something?”
18 Cameron “How in the heck do you get $10.29 per ounce…?
Line Student Comments
1 Cameron “‘Cause this is saying, if it’s after…If there’s something in front of the number, that means it’s dollars, and if something’s after, that’s the cents.”
2 Jennifer “I don’t care.This, though [10.23454158], times 4.69 equals almost 48, so…”
3 Heather “It would round up to 48, so…”
4 Jennifer “So I think if I did this [9.42562592] – put this in the calculator, multiplied it by that [6.79], then I’d get almost 64.”
5 Jennifer “See? 63.999, and you round that up…”
6 Cameron “Times what?”
7 Jennifer “Watch.”
8 Cameron “Isn’t it times…?”
9 Jennifer “This number [10.23454158]…”
10 Cameron “Yeah, 47.999997.”
11 Jennifer “…times 4.6[9]…yeah. Equals, what is it? 47-point…?”
12 Cameron “Point-99999…”
Clip 1.
After watching and reflecting on Group A’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- For all except Daniel, it’s not immediately apparent what Alisha’s calculations mean –
- Jennifer initially wonders if Alisha is showing dollars per ounce (Line 1)
- Cameron wonders how can someone divide differently using the same numbers (Line 6)
- Daniel gives a reasonable interpretation of dollars-per-ounce for Mark and ounces-per-dollar for Alisha (Lines 7, 9, and 12)
- Jennifer doesn’t initially conceive of the “.10609375” in Mark’s calculation as a fraction of a dollar (Line 14).
- Cameron has a similar difficulty interpreting the “9.42562592” in Alisha’s calculation as ounces (Line 18)
The difficulty in making sense of the calculation results – such as “.10609375” for Mark or “9.42562592” for Alisha – was shared by many other people who worked on Version 1.
Clip 2.
After watching and reflecting on Group A’s Clip 2, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Jennifer’s response to Mrs. Peters’ first two questions show glimpses of what we call additive thinking.
- In Line 2, Jennifer seems to be looking at the difference of Mark’s results of “0.10609375” and “.0977083”, without actually identifying those numbers as prices per one ounce.
- In Line 4, Jennifer is noting the difference of actual prices and amounts of the original containers. She is estimating these differences rather than giving precise results.
- In Line 12, Jennifer reiterates her difficulty in reconciling the “0.10609375” as part of a dollar.
- Line 13 is particularly revealing. Similar to things Cameron had said earlier (Clip 1, Line 6 & 18), the primary difficulty we found in Versions 1 & 3 were in making sense of Alisha’s strategy. And Jennifer is quite explicit in expressing her confusion.
Mrs. Peters, the host teacher who allowed us to adopt her classroom for the period in order to lead the activity and do the filming, did a great job at prompting Group A: Note how all her comments in this Clip (Lines 1, 3, 5, and 8) are in the form of questions, keeping the interactions student-centered rather than teacher-centered.
Clip 3.
After watching and reflecting on Group A’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- A common thread through this Clip is what it means to “get the answer” (Lines 3 & 7) –
- Jennifer seems to want to check Alisha’s calculations for accuracy. In other words, the issue no longer seems to be explaining what Alisha has done, but whether in fact the provided results for the 48-oz container (“10.23454158”) is correct.
- Although initially unsure how to check the result (Line 5), she eventually realizes that to multiple “10.23454158” by 4.69 should give 48 (Line 9).
- The main difficulty at this point seems to still be in making sense of Alisha’s computation. Notice how Jennifer refers to Alisha’s work in connection with a “Unit Price” (Line 12). However, Alisha is actually providing a Unit Rate: The “10.23454158” is really the amount of ounces for 1 dollar, but it’s not at all evident that the three students in this Clip understand this yet.
- Cameron’s comment in Line 18 is very appropriate: If 48-oz cost $4.69, then how indeed “do you get $10.29 per ounce” ?
We found it interesting that the written prompt “Is Alisha correct?” on Version 1 seemed to serve as a mandate to determine if the calculations were accurate as opposed to a request for making sense of Alisha’s work.
Clip 4.
After watching and reflecting on Group A’s Clip 4, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Cameron expands on his previous question (Clip 3, Line 18) about Alisha’s “10.23454158” as dollars per ounce. When he talks about “something in front of the number” (Line 1), he might have been meaning to the left of the decimal being dollars. It would then make sense that “after”, or to the right of the decimal, would indicate sense. He clearly is trying to make sense of Alisha’s work, and is possibly confounded by Jennifer’s earlier references to Alisha’s computations results as showing a “Unit Price” (Clip 3, Lines 12 & 14).
- However, Jennifer continues with her theme of verifying Alisha’s calculations. Having done so for the 48-oz package (Line 2), she also speculates that she could verify the Alisha’s calculations for the 64-oz package (Lines 4 & 5). Cameron and Heather go along with this plan of verifying the calculations.
In the written work for Jennifer, Heather, and Cameron, it again comes through that Alisha is supposedly correct in showing a “unit price for one ounce” (Heather’s work for Alisha).
Line Student Comments
1 Alicia “And then, you do, ok, 64, and then under that, you’d put 6.79.”
2 Matt “Nice”
3 Dustin “So why’d you put x here [under the 48] instead of the price?”
4 Alicia “To see if they’d equal the same amount as it’s supposed to.”
5 Dustin “Oh”
6 Alicia “Because if it did, if it [x] was the same amount as up here [4.69] then they’d be like, they’d equal.”
7 Dustin “Okay”
8 Alicia “They’d have the same price
9 Dustin “So we’d just have to solve for x, and if it’s equal, then we…If it’s less, then we want the other, er…You want the…”
10 Alicia “Yah”
11 Dustin “I get it”
12 Alicia “Actually, if it’s more, then you want the other one”
13 Dustin “Yah”
Line Student Comments
1 Matt “Alicia, there’s one thing I want to make [sure of]…With that one it’d be, 5.09, that means you’d want that one [48-oz. pkg], want to buy that one?”
2 Alicia “Yah. Because if it was, like, had the same price as this, then…”
3 Matt “Then it wouldn’t really matter which one you bought.”
4 Alicia “No”
5 Matt “If they had the same price”
6 Alicia “If they had the same price then this [5.09] would be that [4.69], get it?”
7 Matt “All right”
8 Dustin “Wait a minute. So if we’re solving for x, then that means these are going to be the same, but they’re not”
9 Alicia “What?”
Line Student Comments
1 Dustin “If we’re solving for x, right, that means that this [5.09] is equal to this [price of 48-oz pkg.] . But it is not.”
2 Alicia “No, it would, like…Okay, it’s saying, if it’s equal – which it isn’t – it’s saying, okay, like, if these were the same…”
3 Dustin “Yeah?”
4 Alicia “Like, equaled the same, then the 48-oz would cost $5.09”
5 Dustin “Okay”
6 Alicia “But it’s less, it’s cheaper, so it doesn’t cost that, it costs [$4.69]”
7 Dustin “So what is all this [work] down here?”
8 Alicia “Solving the proportion”
9 Dustin “Oh.”
10 Alicia “So, just solve it.”
11 Dustin “Okay. Ah, I’m so lost. Do you get it, Matt?”
12 Matt “What?”
13 Dustin “Her…whole thing.”
14 Matt “No. Did you get mine?”
15 Dustin “No.”
Line Student Comments
1 Matt “In order to get an accurate reading…accurate results, one of the numbers has to be the same. So instead of dividing to get the price – the ounces – down to one, I decided to multiply to get them up to the same number…AKA, use the least common multiple.”
2 Alicia “So like, whichever you multiplied the least times is the more expensive one?”
3 Matt “Not necessarily. See, after I did that, I had to multiply the numbers by a certain amount to get to there. So, it’d be 48 x 4… and 64 x 3. But to get the prices, to have – to be accurate with the ounces, you’d have to multiply them by what I multiplied the number of ounce by. Otherwise they’d still be in relation to 64 and 48 …Not 192.”
4 Alicia “Oh”
Clip 1:
After watching and reflecting on Group B’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Alicia explains her proportional equation for Alisha’s strategy in this Clip, which can be seen in her written work. She has set ( 64 / 6.79) = ( 48 / x ) and solved for x. But how to interpret the result?
- Dustin’s question in Line 3 shows a tenuous grasp of proportional reasoning, since the ratios would only be equal if neither package of ice cream was a better buy than the other.
- Alicia comments about how “they’d equal” (Lines 4 & 6), and rephrased in Line 8 (“They’d have the same price”) can be interpreted as follows: If the answer to the proportion (“x”) was in fact equal to the price of the 48-oz package, then the unit rates for the two packages would be identical.
- But, the answer to the proportion is more than the price of the 48-oz package. Alicia notes (Line 12) that “if it’s more…you want the other one” by which she seems to mean you want the 48-oz package.
Following Alicia’s spoken explanation seems difficult for us without referring to her written work, but using her verbal and written work together reveals a good understanding of proportional reasoning. Dustin says “I get it” in Line 11, but by the time of Clip 3 he sounds less confident.
Clip 2:
After watching and reflecting on Group B’s Clip 2, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Matt wants to clarify the interpretation of Alicia’s result for solving her proportion (Line 1). If x = 5.09, what does that mean in terms of finding the better buy?
- In Lines 2 – 7, Matt & Alicia seems to have consensus that if the 48-oz package cost $5.09, then “it wouldn’t really matter which one you bought” (Line 3).
- Dustin questions the meaning of solving the proportion (Line 8). When he says “…that means these are going to be the same…” , it’s not initially clear what he is referring to. But he expands on his question in Clip 3.
Watching these three students articulate themselves is quite rewarding from a teaching perspective. If we value communication and reasoning in our classrooms, then it pays to learn how in fact students negotiate amongst themselves to understand a mathematical task. What language would be helpful to each of these participants at this point? How could they better express themselves mathematically?
It certainly is laudable that Matt & Dustin are persistent in their attempts to understand, and that Alicia continues to try and explain herself and answer her peers’ questions.
Clip 3:
After watching and reflecting on Group B’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- At first, it seems that Dustin is moving towards an understanding of Alicia’s proportion. Lines 1-5 promote the notion that if the packages were the same in terms of a Unit Rate, then (64 oz / $6.79) would equal (48 oz / $5.09), and “then the 48-oz would cost $5.09” (Line 4).
- Tying in to her earlier explanations, in Line 6 Alicia drives home the reason she thinks the 48-oz package is the better buy, since “it’s cheaper” than $5.09.
- Dustin, however, still isn’t sure of what Alicia’s work in “solving the proportion” (Lines 7-10) really means. He then looks to help from Matt (Line 11)
- Meanwhile, there is question about whether Dustin understood Matt’s way of thinking (Lines 14-15).
In Clip 4, we get to see Matt explaining his strategy for how Alisha could have found the better buy, which turns out to be very different from the strategy used by fellow group member Alicia.
Clip 4:
After watching and reflecting on Group B’s Clip 4, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- It’s useful to see Matt’s written work for both Mark and Alisha. For Mark, Matt computed (6.79 / 64 = .106 ) for the 64-oz package, for example. Matt correctly identified this process as finding a price in dollars per ounce, and this process is what Matt refers to in Line 1 for “dividing to get…the ounces… down to one.” But now, for Alisha’s strategy, Matt’s will find the LCM for the ounces.
- Alicia’s question in Line 2 is interesting, since on an intuitive level it has certain appeal to think about “whichever you multiplied the least times is the more expensive one”.
- Matt continues with his explanation in Line 3, and note that the 48-oz package got multiplies by 4 while the 64-oz package got multiplied by 3. And in fact, the 64-oz package was more expensive, as Alicia conjectured. Yet once the ounces are both equal to 192, the key is then to also compare the prices accordingly.
To see the rest of what Matt did requires looking at his written work , and he did discover a lower price for 192 ounces using the 48-oz packages ($18.76) than for 192 ounces using the 64-oz packages. It is not clear how well his fellow group members have understood Matt’s work, and in Group M Clip 4, Matt can be seen asking for feedback from another class member.
Line Student Comments
1 Analiese “I got it.”
2 Lily “Is it… 4 times?”
3 Analiese “Uh-huh.
4 Lily “Okay. So…”
5 Analiese “Well, 12 goes into 48 by 4 times, so…”
6 Lily “Yeah, because 12 times 4…”
7 Analiese “Okay, so let’s try 16 divided by…”
8 Lily “16 divided by 48?”
9 Analiese “3 times”
10 Lily “It’s 3 times”
11 Analiese “[3 times] it goes in”
12 Lily “It does.”
13 Lily “Okay, so let’s write those two things down. So 64 oz is four 16 oz. So, 4 pounds. And then the 48 oz, would be 3 pounds.”
14 Analiese “Yeah”
15 Lily “And then, what should we do for the money?”
Line Student Comments
1 Analiese “What we did is, we got, what we thought we could do is, we would divide 16 – There’s 16 oz in a pound – and we divided 16 into 64 and 48, and we got 4 pounds for 64 oz and 48 oz we got 3 pounds…”
2 Analiese “And for 4 pounds, we divided it into 6.79, and so we’d figure out how many – How much money it was per pound.”
3 Analiese “And we did the same thing for the 48 oz package, and we got $1.56 per pound.”
4 Lily “So then we figured out, if there was 4 Lbs, and it would cost $6.79, then we tried adding $1.56 – which is 1 Pound – to 3 Lbs, to see much money we would get.”
5 Lily “And so it ended up being $6.25, And we found out that … the first set of data, which is originally 4 Lbs, was 54 cents more than if you bought the 3 Lbs and got another pound”
Clip 1:
After watching and reflecting on Group C’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- This Clip focuses on a strategy for Mark (see the students’ written work). Since this Version 3 offers the same ratios as in Version 1 (without providing the results of the division operation), we found it interesting that this Group C did not merely do the division calculation, but instead wanted to convert the ounces to pounds (Line 13).
- It’s not clear if they initially intended to use pounds, or just wanted any large number that divided evenly into both 48 and 64. We noted that they first looked at “12” as a candidate. Analiese saw that “12 goes into 48 by 4 times” (Line 5), and later she suggests they “try 16” (Line 7).
- Lily’s phrasing of “16 divided by 48” (Line 8) is something we think is common in the middle grades: Students considering how many times X goes into Y may tend to speak of “X divided by Y”.
These two students seem to work well with each other, and have succeeded in converting the ounces to pounds according to a rate of 16-oz per pound. But what will they do with the prices (Line 15), and how will they analyze which is the better buy? We’ll see what they do next in Clip 2.
Clip 2:
After watching and reflecting on Group C’s Clip 2, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Although they’ve merged with other students, our focus is still on Group C (Lily and Analiese) as they explain to others what they’ve done. Analiese starts off talking about the ounces-to-pounds conversion in Line 1.
- In Lines 2 and 3, Analiese shows what they did to find “How much money it was per pound” for the two packages. In particular, they stress how the 48-oz package cost “$1.56 per pound” (Line 3).
- As Lily picks up the narrative (Lines 4 and 5), we learn that rather than simply compare the prices for one pound, they added a pound to the 48-oz package and figured the relative price, which “ended up being $6.25”. The “first set of data” mentioned in Line 5 is a reference to the 64-oz package.
Thus, although Group C did find a price-per-pound for Mark’s strategy, they actually chose to compare prices for four pounds. Having displayed such a sound mathematical analysis for Mark’s strategy, we were intrigued at Group C’s confusion over a strategy for Alisha. Although not captured in our Clips, both Lily and Analiese concluded that Alisha was incorrect in thinking (64 / 6.79 ) and (48 / 4.69 ) could help her find the better buy. This is shown in Group C’s written work.
Our directions to these groups were for them to solve the problem individually at first, and to write down their thinking about why it made sense for them to solve it the way they did. Then they were to explain their solution approaches to other in their group.
· Second, after these “unified” groups (in the sense that all groups members were working on the same Version) were finished, we shuffled students into new “mixed” groups, so that there were different Versions within each group. Again we asked students to share their thinking with each other. In this article, you’ll hear the thinking of
Line Student Comments
1 Alex “Ounces per cent, not cents per ounces.”
2 Steve “I thought it was ounces per dollars”
3 Beth “Yeah”
4 Jennifer “Yeah, so did I”
5 Alex “Whatever.”
6 Beth “Cents per ounces? Whatever.”
7 Alex “Whatever.”
8 Beth “How many cents per ounce.”
9 Jennifer “Okay, I did the version three whatever, and it already told you what the answer was, like…”
10 Jennifer “So 6.79 over 64 is point-ten or whatever, so that’s basically ten cents. And then the other one was 4.69 over 48, and that was 0, or .09, and that was like nine cents.”
11 Jennifer “And then it says ‘Explain how these rates can tell Mark which ice cream is the better buy’, and I chose the 4.69 over 48, because…The difference is only about, like, .2 , and you’re only paying for 15 oz less, so that’s why I chose that one.”
12 Jennifer “And then, on the bottom, ‘Alisha claimed that she had a different way to solve it’, and she just found the ounces per dollar…”
Line Student Comments
1 Jennifer “But I wanted to see if she [Alisha] was correct, so what I did was I multiplied 9.425 like as far as I could get, um, by 6.79 and I got 63.9999 and it rounded up to 64, that’s what the total ounces was…”
2 Jennifer “Then [I] did the same thing and got 47.9999, and round it up, you get 48, and so she [Alisha] was correct.”
3 Alex “That just proves that her division was correct, though.”
4 Jennifer “Yeah, I know. I said I…”
5 Alex “But it already says that the division is correct. It just wants to know if she, uh, did it the right way. If this [calculation] tells her something.”
6 Jennifer “Yeah, I know. That’s what I was doing, I was seeing if she did it the right way. See, that’s how I found out she did it ounces per dollar.”
7 Alex “Okay.”
8 Jennifer “‘Cause I wasn’t sure.”
Line Student Comments
1 Beth “So, was it the 48 oz, or the [64 oz] ?”
2 Alex “All three of ’em say it was 48 oz.”
3 Beth “That’s what I thought. But like…I would have gone with the 64 ounces…”
4 Jennifer “So would I”
5 Beth “Because it’s like a tenth of a cent cheaper, and you get a lot more [ice cream]”
6 Jennifer “That’s what I said, but then I changed it, because I found another reason for it to be 48.”
7 Alex “Well, people could’ve done it the long way, and then found out that if you bought the same amount of ice cream, how much it would cost, and the one that was 48 oz would cost less…If you bought the same amount of ice cream”
8 Beth “I don’t get it”
9 Jennifer “That’s what Daniel was trying to explain to me. He said something, how much ice cream you get per ounce or whatever – I didn’t know what he was talking about – He must’ve did it the long way”
10 Alex “Yeah. You just find the lowest common multiple. The least number that they both go into.”
Line Student Comments
1 Alex “Matt, what did you need?”
2 Matt “First of all, I just wanted you to look through that and see if you can understand what I did. Most people don’t.”
3 Alex “Yeah, you did [it] the long way. That’s what we were just talking about.”
4 Jennifer “Yeah, we were just talking about it. Can I see?”
5 Beth “You spacing out there, Kara?”
6 Alex “Well, it just says ‘Alisha claims she could do it a different way: What could she have done?’ – And he said that she could have done it the long way.”
7 Beth “I don’t get it”
8 Alex “Basically you just want to find how – If you buy the same amount of ice cream, how much will each one cost”
9 Jennifer “Where would you start, though, to find the same amount of ice cream?”
10 Alex “How – What’s the littlest number that both 48 and 60, uh…64 have in common”
11 Jennifer “[Sixty] four have in common…That’s like 190-something”
12 Beth “It is?”
13 Jennifer “192 or something, I think.”
14 Beth “Yeah, that takes a long time, though. How would you do that?”
15 Alex “Exactly”
16 Jennifer “So then what do you do after that?”
17 Alex “You can either make trees, or…”
Clip 1:
After watching and reflecting on Group M’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- In the first eight lines, it’s not clear to us whether the students are discussing Mark’s work (which for Versions 1 & 3 could be seen as money per ounce) or for Alisha’s work (which similarly could be ounces per money). Regardless, it seems that there is no need for a distinction of monetary units between cents versus dollars. Is this because they recognize “0.10…” as 10 cents or 0.10 dollars?
- Jennifer’s “…version three…” that she refers to in Line 9 is what have called in this article Version 1. Note how her view is that “it already told you…the answer”, which is not quite true, since the question asks for an explanation. She does note, in Line 10, how .09 is “nine cents”, but she is still not explicit that she is looking at nine cents per ounce. Can we assume she knows this ?
- In Line 11, Jennifer looks at the difference in prices-per-ounce (although it’s not entirely clear she recognizes them as such), and also the difference in total ounces for each container.
- The idea that Alisha has found “ounces per dollar” (Line 12) was prompted earlier by Daniel in Group A, Clip 1.
It is interesting to hear Jennifer explain these ideas without her earlier Group A partners around, and she gets a fresh perspective on these ideas in the next Clip.
Clip 2:
After watching and reflecting on Group M’s Clip 2, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Jennifer tells her new group members what she did to see if Alisha “was correct” (Lines 1 & 2), which involves checking the calculations. As Alex notes in Line 3, this only shows that Alisha’s “division was correct.”
- Alex goes on by assuming that the provided calculations and results for Alisha are in fact correct, and suggests that the point of the task is to determine is those calculations “tells her something” (Line 5). Thus, Alex’s articulation seems to capture the essence of the task, which is to explain how Alisha can use those calculations to find the better buy.
- In response to Alex, Jennifer explains that she wanted to find out if Alisha “did it the right way” (Line 6), which may still translates to “performed the calculations correctly” in Jennifer’s mind. Jennifer does talk about finding out if Alisha “did it ounces per dollar” (Line 6), but it’s not evident just how clear this concept is for Jennifer.
Many teachers, like us, might have finished watching Group A’s Clip 3 & 4 and thought: Why are they just checking the calculations? What do those calculations show? Here in this Clip, it is Alex who put these issues on the table. Thus, the value in this exchange is in students questioning their peers.
Clip 3:
After watching and reflecting on Group M’s Clip 1, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears): The first 6 Lines are very instructive. Beth is still unsure of which container is the better buy. Although Alex states how all three Versions show the 48-oz container to be the better buy, Beth makes an interesting argument for why she would have chosen the 64-oz container (Line 3). Her argument is akin to Jennifer’s earlier reasoning in Group M (Clip 1) and Group A (Clip 2).
- Alex describes another method – “the long way” (Line 7) – whereby one could consider the costs “if you bought the same amount of ice cream”.
- While Beth doesn’t understand Alex’s idea (Line 8), something about the way Alex has phrased his thoughts (in Line 7) connects with what Jennifer heard earlier from Daniel in Group A. She now feels she might understand what Daniel had been trying to explain.
- Alex explicitly ties his earlier idea (in Line 7) to finding the LCM.
In looking back at Daniel’s written work, we don’t find evidence of an LCM approach. So perhaps Daniel explained this approach in addition to his existing (and correct) written strategies. Or, Jennifer might not have been to clear on whatever Daniel might have been explaining (as she points out in Line 9).
Clip 4:
After watching and reflecting on Group M’s Clip 4, we’ve summarized just a few highlights (the Line #s refer to the place in the transcript where the student comment appears):
- Now we have a new interaction for Group M, coming from Matt (formerly of Group B), who wants Alex to look over Matt’s written work. In Lines 1-6, Alex identifies Matt’s work for Alisha as being consistent with an LCM approach.
- Because Beth still didn’t understand (Line 7), Alex again mentions finding costs for the “same amount of ice cream” (Line 8).
- As Jennifer and Alex talk about finding the LCM for the ounces of ice cream (Lines 9 -13), it seems clear that Jennifer has seen this before: She correctly identifies LCM(48,64) as 192.
- As Beth wonders how that LCM(48,64) can be found (Line 14), Alex’s reference to making “trees” (Line 17) may be a reference to the prime factorization trees often taught, and then using the prime factors to help construct the LCM of two numbers.
Again, this Clip shows a positive outcome of small-group discourse, whereby students are able to ask each other their questions and practice explaining their thoughts.
· Third, we held a whole-class discussion about their reactions and responses to Ratio Triplets. We’ve included some Class Discussion Prompts that we found useful in debriefing the class and motivating their collective sharing of ideas. There are also some further questions of interest that we as teachers considered afterwards about the use of the Ratio Triplets task; these questions and other ideas are shared in our Short Debrief for Teachers.
