This number string is inspired by an investigation (Comparing and Scaling, Investigation 1.2) from the *Connected Math Project* curriculum for 7th graders. It can be used as a way to extend and solidify ideas that will develop as kids investigate, or it could be used to launch the task. In the investigation, students are asked to compare the following orange juice recipes, according to their “oranginess.”

In our number string, we rely heavily on images of the juice recipe (or mixtures of water and concentrate) as a way to invite students to a) draw upon their mathematical intuition and b) reason about quantities in a proportional way without *necessarily* naming it as such. In other words, we are using the construct of “oranginess” as a way to reason about proportionality.

Start the conversation with kids about how juice can be made from a mixture (or combination) or frozen concentrate and water. Let students tell the class what they already know (social knowlegde), including stories about when they have made juice or lemonade themselves.

Remind them, if necessary, that same container (or cup) is used to measure these quantities. That usually makes sense to kids who have either made juice (or lemonade) from concentrate or seen someone else do the same.

Next, you might say….

*When making juice at home, some people in my house like the flavor really strong, really “orangey.” And other people like the flavor not so strong, less “orangey.”*

*Using the recipe below, tell your partner two ways to make the flavor “more orangey” and two ways to make the flavor “less orangey.” How would you do it?
*

Here, we hope kids talk about increasing (or decreasing) the cans of water and increasing (or decreasing) the cans of concentrate.

With that idea explored, you can begin to **compare recipes**:

*So, which of these recipes would taste “orangier”? And how do you know?*

Kids should quickly reason either about the part-to-part relationship (concentrate to water) or about the part-to-whole relationship (water or concentrate to juice), but it is meant to be friendly so that we can build more complicated recipes from it.

Where it makes sense you can extend their ideas by saying things like:

*So, one out of four parts (1/4) and two out of four parts (2/4) are not equivalent and won’t taste the same? Is that your idea? What do other people think? Are you convinced of this?*

Then you might go here:

*So, what about this one?*

Give kids a minute to think, though this recipe should also draw upon some mathematical intuition:

*Renny: The recipes are basically the same, except the second one has more water.*

*Jeremiah: The second recipe will taste “more watery” or “more diluted” or “less orangey.”*

*Laila: The one can of concentrate is mixed with 3 cans or water and 4 cans of water, so 3 cans of water makes it taste stronger or “orangier.”*

*Okay, what about this one? *

You might hear:

*Joanna: The second recipe is 50/50 (50% water and 50% juice) but the first one is less than 50/50 (less than 50% concentrate in the entire mixture), so the first one is stronger. ( using landmarks or benchmarks to compare)*

*Amaya: The first recipe of 2 out of 6 parts concentrate, and the second one is 3 out of 6 parts concentrate. I know that 3/6 is greater than 2/6. ( thinking about part-to-whole relationships)*

*Jonathan: So the first one is 2 to 4 and there’s a lot more parts of water for each concentrate. And the second one is 3 to 3, so there are like equal parts concentrate and water, like they are matched up, so that will taste “orangier.” (thinking about part-to-part relationships)*

Listen for how students are thinking about the ratio — as part-to-part (concentrate to water) or as part-to-whole (water or concentrate to juice).

If it comes up, note the difference and chart kids’ ideas for the whole class to see:

*Jonathan, do you hear how you are thinking about the recipe differently than Amaya? Everyone take a minute to describe how Amaya and Jonathan are thinking about the problem differently. And decide if you are convinced by both of their claims. *

For the next two recipes you could still ask:

*So, which of these recipes would taste “orangier”? And how do you know?*

Or, you might change the question slightly to ask:

*Would these two juices, when all mixed up, taste the same or different? How do you know?*

And later….

By doing so, you are subtly linking the idea of proportionality to the idea of taste. In other words, when cooks and chefs make bigger or smaller batches of the same recipe they are scaling up or down the relationships between the ingredients proportionally *so that the entire recipe tastes the same*.

Throughout the conversation, promote the idea that students are making claims to convince each other (not you) of their ideas. Focus less on the construct of “right/wrong” and more of the idea of reasoning:

*Are we convinced? Could this be true? Does it make sense to you? What would it take to convince us?*

You might also design a situation that gets at the heart of a major misconception, as we did here:

*My friend Will thinks that if you have one recipe and you add equal parts of juice and concentrate to it, the flavor will essentially stay the same. What do you think of his idea?*

We are eager to hear how a number string like this unfolds in your class. And how you design new strings from it as a result.

Many thanks to our colleagues at Wagner Middle School (MS 167) and MS 22 for helping us develop these ideas.