In this activity, students continue to use double number lines to reason about equivalent ratios. Students revisit the green water recipe from an earlier activity. This time, their attention is directed to amounts of blue water and yellow water that are less than the ratio given in the original recipe but would produce the same shade of green. Students begin to find and use ratios containing a 1.

One key idea to convey here is that finding a ratio associated with 1 unit of a quantity can be very helpful because it can help us determine any equivalent ratio. Another key idea is that the intervals on double number lines can be partitioned to help us find such ratios.

As students work, monitor for those who use division to determine the amount of yellow water for 1 ml of blue water, and then use multiplication to determine the ratios for 8 ml and 11 ml of blue water. This is a key insight for a type of reasoning that is broadly useful and will be developed further.

This is the first time that Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured pair meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . . ?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words that they got from their partners to make their second draft stronger and clearer.

Focus discussions on how students determined the amount of yellow water for 1 ml, 8 ml, and 11 ml of blue water. Ask previously selected students to share their reasoning.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question on why it is useful to know the amount of yellow water to be used with 1 ml of blue water. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

Consider displaying these prompts for feedback: 

• “The part I understood best was . . . .”
• “What do you mean when you say. . . ?”
• “Can you say more about . . . ?”
• “Can you describe that another way?”
• “How do you know . . . ?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “It is helpful to know how much yellow water to use with 1 ml of blue water because . . . “
• “If I know how much yellow water to use with 1 ml of blue water, then I can . . . 

If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, highlight the relationship of blue to yellow using phrases such as “for every 1 ml of . . . ” or “per milliliter of . . .”

• “There are 3 milliliters of yellow water for every 1 milliliter of blue water.”
• “There are 3 milliliters of yellow water per milliliter of blue water.”

The word per means “for every.” Ask students to think of any other situation in which they may use the word “per” as it is used here (for example, price per bottle of water, cost per ticket, and so on). Discuss why knowing the value of one item would be helpful.

Prior to this point, students were given blank double number line diagrams and were responsible only for labeling them to match the situation. In this activity, students follow instructions for drawing a double number line diagram from scratch. Then, they use their diagram to answer questions about the ratios in a recipe that they encountered in an earlier lesson. During discussion, students reflect on elements that are important when creating a useful double number line diagram. 

Select students to explain how they used their double number line diagram to answer the last question. Ask students how they can indicate the number of cups of corn flour per pint of glue on the diagram.

If incorrect or imprecise diagrams are observed in students' work, consider inviting students to take a closer look at diagram construction. Ask questions such as:

• “Do the tick marks on each line need to be equally spaced? Why or why not?”
• “Do the tick marks on the top line need to match those on the bottom line? Why or why not?”
• “Does it matter what number we use to start each line? Why or why not?”

In this activity, students revisit equivalent ratios in a previously introduced context. They see how double number line diagrams are helpful for answering more questions about quantities in a recipe. 

Students also see that a double number line diagram can be adapted to include a third number line that represents another quantity. This third quantity forms a ratio with each of the other two quantities in the recipe, so the diagram can be used to find equivalent ratios involving any two quantities or all three of them.

Invite students to share their responses and display their diagrams. Discuss how using a double (or triple) number line diagram helped them find equivalent ratios in this situation. Consider asking students:

• “How is reasoning with three number lines like reasoning with two number lines?” (We’re still looking for values that are in a ratio and that line up vertically. The tick marks still represent different numbers of batches.)
• “How is it different?“ (We can use the same diagram to reason about the ratios of different pairs of quantities, such as the ratio of radish to rice flour, rice flour to green onions, or radish to green onions. We can also use it to reason about the ratio of all three quantities.)
• “Can we use the double (or triple) number line diagram to find the amounts of ingredients for a fraction of a batch of radish cake? How?” (Yes, we can partition the first interval of each number line into equal parts.)

If time permits, solicit new questions that could be asked about the ingredients or the number of batches and could be answered by using their double (or triple) number line diagram.