Objectives

The goal of this string is help students construct the ratio table as a generalizable model for reasoning about equivalent ratios.

Placement

This string is the first of four ratio table problem strings found in my book: Algebra Problem Strings. Use this Problem String to remind students of or to build proportional reasoning and relationships.You can use this Problem String to introduce other Problem Strings that deal with proportional reasoning. If your students have experience using a ratio table to solve problems, you may want to use a different Problem String that begins with a non-unit rate. You can find examples of these in my book. 

Guiding the Problem String

This string should proceed fairly quickly. It is intended to link a context (packs of gum) to a model (the ratio table) while reasoning proportionally. Encourage flexibility, reasoning about new combinations of packs and sticks of gum. Encourage students who are over-reliant on the unit rate to take up other interesting relation-ships that the string makes available to them. If we are not purposeful, ratio table strings can become quite cluttered with students’ strategies. It is important to record strategies, but you want the class to be able to follow the progression of ideas in the string and keep the table as a useful tool. Erase before the table gets too smothered with arrows and calculations. It is vital to use ratio tables flexibly, pointing toward efficiency—modeling how to scale up and scale down ratios, and to reason about rates even when the unit rate is not given, stepping back to ask which strategy we wish our brains would gravitate toward next time or without the other problems first. Too often students see ratio tables that begin with the unit rate and proceed only by “chunking” these quantities to make new equivalent ratios. This form has been codified for some students as the signifier of a ratio table (e.g., “It must begin with the unit rate.”), although this isn’t true.

About the Mathematics

A ratio table is a special paired number table where all of the pairs are equivalent ratios. In this scenario, the unit rate is given, 1 pack to 17 sticks, but a unit rate is not always given.

If you are not familiar with using ratio tables as a tool for computing, study the dialog to find examples of using the distributive and associative properties with the table to find equivalent ratios.

Sample Interactions

Use the following as you plan how to elicit and model student strategies. This is not meant as a script, but as a view into the relationships involved and the intent of the problem string.

Teacher: All right, let’s warm up with a short Problem String today. I’m thinking about packs of gum. You know how they come in different sizes—some have 5 sticks, some have 10 and then some are like those jumbo packs?

[Optional: Quickly display an image of various popular forms of gum to invite students into the context. Find out who chews gum and what kinds/flavors of gum they enjoy.]

Teacher: Great. So let’s imagine a pack with 17 sticks of gum inside. And I’m going to make a table to keep track of our ideas here. So far, so good?

Teacher: So, how would you find the number of sticks in 2 packs? Brief think time.

[Here the teacher writes the "2" as they ask about 2 packs. During the rest of the sample interaction, the teacher continues to write the value as the teacher asks the question, but the sample model will show that and what happens next.]

Teacher: Did anyone just double?

Student: Yeah, that's 34. I doubled 10 to 20 and 7 to14, so it's 34 together.

Teacher: So, if you double the number of sticks, you double the number of packs? Yes? That's an example of the distributive property, where you break up one of the factors into chunks that make sense.

Teacher: What about the number of sticks in 4 packs? What's a nice way to find that?

Brief think time.

Teacher: What is the number of sticks in 4 packs? And how?

Student: 68. I doubled the 34. Double 30 plus double 4.

Teacher: So you doubled the packs to get double the sticks again. Nice

Teacher: Okay. Let’s go big. How about one of those jumbo bags with 20 packs inside? Can you picture it? Enough gum for a year. Maybe, you tell us. How many sticks is that? How can you use what you know? Think time. Turn and tell your partner two ways you might have solved this.

Students turn and talk while the teacher listens in.

Teacher: Tell us about your thinking.

Student: My partner and I did different things. I tried to take the last problem, with 4 packs and multiply that by five, but that was taking longer. My partner just took the 2 packs and just multiplied by ten—340 sticks of gum.

Teacher: Nice. We know something about times 10, yeah? That can be handy.

Teacher: Anyone try anything else? Was using the original pack of gum—the unit rate—friendly here?

Student: Not really. I guess unless you think of 17 times 20 as 17 times 2, times 10. Then it’s a bit easier.

Teacher: So you can think of 20 as 2 times 10? So, 17 times 20 is equivalent to 17 times 2 times 10. There’s a big idea here. Why we can do this?

Student: You kind of doubled the 17 and halved the 20. I remember doubling and halving.

Teacher: Yes, and that's called the associative property. Does anyone notice anything connected between the first strategy up here, where you multiplied the 2 packs times 10 and the last strategy, where we've been talking about the 17 times 20?

Student: That's cool. They use the same 2 times 10 is 20 relationship. If you already have 2 17s, then you can scale that up by 10. If you are thinking about 17 times 20, you can think about 17 times 2 and then times 10. Same thing!

Student: And in this case, it's also doubling and halving.

Teacher: Maybe that could help us again sometime. Next question. How many sticks in 5 packs?

Brief think time.

Teacher: How many sticks in 5 packs?

Student: 85.

Teacher: Did anyone use the 4 packs to find 5 packs?

Student: Yes, I added one pack to 4 packs, so 17 and 68 is 85.

Teacher: Did anyone use the 20 packs to find 5 packs?

Student: You can divide 20 packs by 4 to get 5 packs. 340 divided by 4 is like 320 divided by 4, which is 80, and the leftover 20 divided by 4 is 5, so 85.

Teacher: What do we think? Give me a signal if you follow this thinking. Okay.

Teacher: Did anyone use 10 packs? Yeah, I thought I heard you talk about that. Tell us about that.

Student: I know how to find 10 easy, that's just 170. Then halve that, it's 85.

Teacher: So, you added a row to the ratio table? At least in your mind? That seems handy too!

Teacher: Great, the next problem is 15 packs. Look for a clever way to find the number of sticks.

Think time.

Teacher: Turn and tell your partner and decide who had the more clever way to do it.

The teachers asks for and models adding the 10 and 5 packs and multiplying the 5 packs times 3.

Teacher: Alright, let’s end this gum scenario with a twist. What if we have 153 sticks. How many packs? Not 153 packs, mind you. How could you find the number of packs that hold 153 sticks?

Think time.

Teacher: How many packs have 153 sticks?

Student: 9 packs.

Teacher: Did anyone use the 4 packs to help?

Student: I did. I looked for numbers that could get me to 153 and I saw that the 68 and 85 would add to 153, so 4 packs and 5 packs is 9 packs.

Teacher: Did anyone use the 10 packs?

Student: Yes, the 153 is close to the 170. And so I figured out it's just 17 under, so 1 pack under 10 is 9 packs.

Teacher: That sounds like a really important question that you both just asked. Once you looked at the 68 sticks, you had to find out how much more you needed. Once you looked at the 170 sticks, you had to find out how much too much you had. Finding the difference between what you know and what you're looking for seems like a really important idea. How could we make a note of that?

Student: Asking, "How far away are we?"

Teacher: Which of these strategies do you like? Is there one that might be easier if we didn't already have some of the packs?

Student: The four and five were pretty easy, but only because we had already figured them out. I know what ten 17s is,170; I could've used it even if I didn't have anything else.

Teacher: That also sounds like an important idea. Someone who is making sense of that strategy restate it for us, so we can hear it in a new way. Go ahead.

Student: We can look for things to use in the table and that's fine. When we don't have stuff already, we can think about what we know.

Follow Up

Teacher: Okay, last question...Earlier I referred to this table of packs and sticks as a ratio table because all of the ratios in the table are equivalent. In each of these rows lives the same relationship of one pack to 17 sticks. We could write these relationships as a statement of equivalence.

Teacher: What kinds of things did we just do with a ratio table? Turn and talk with your partner.

The teacher leads a brief conversation about the multiplication problems and division problems that the students just solved using the relationships in the table, bringing out these points: what you do to the packs, you also do to the sticks; the table does not need to go in order; you could insert values that you find helpful.

Teacher: How would you summarize some of the things that came up in this string today?

Elicit the following:

• Every combination on the table makes an equivalent ratio (1/17 = 2/34 = 4/68 = 20/340 = 5/85 = 15/255) and could be written as one statement of equivalence.
• The 17 sticks in each pack show a relationship between every value on the left (packs) and every corresponding value on the right (sticks)—multiplying or dividing by 17.
• Even though this ratio table had the unit rate given, 1 pack to 17 sticks, that is not necessary for the table to be a ratio table.
• We can use ratio tables to multiply and divide.
• Helpful ideas: Use what you know and the question "how far away are we?"

Sample Final Display

Your display could look like this at the end of the problem string:

Facilitation Notes

This version of the Problem String lists short notes for important teacher moves during the string. After you’ve done the string yourself and studied the relationships involved, you might make similar notes for the things you want a reminder of or deem important.