In this activity, a double number line (a new representation) is presented and interpreted alongside the more familiar discrete diagrams and in the familiar context of recipes.

Students learn that, just like discrete diagrams, double number lines represent equivalent ratios. They see that alignment between the numbers of the two lines matters, and that it is through the alignment that the association of two quantities is shown. Students notice that pairs of numbers that “line up” vertically are equivalent ratios.

Because double number lines are quicker to draw and can be extended easily to show many more equivalent ratios, they are more efficient than discrete diagrams are, especially for dealing with larger quantities.

As students work, monitor for those who contrast the two representations in terms of using graphic symbols versus numbers, and those who think about equivalent ratios in terms of the alignment of numbers in the double number line diagram.

While the double number line diagram is given here, some students may not feel comfortable with seeing the same numbers (the 4’s) in different positions. Remind students that each number line represents a different quantity, and that the two 4’s have different meanings.

Select students to share their observations about how the two representations are alike and how they differ. As students discuss solutions to the questions, circle pairs of associated quantities on the double number line. Help students connect information as it is represented in the different diagrams.

On the last question, ask students how they knew that 20 was the next number on the line representing teaspoons of drink mix? (Skip counting by 4, or multiply the next number of cups of water by 4.)

Ask students to think more generally for a minute about the representations at hand:

• "What is a double number line diagram? What do they do? What do the numbers on the tick marks represent and how should they be scaled?"
• "What might be some benefits of using double number lines instead of diagrams?" (We can use them to show many more batches; they are quicker to draw.)

In this activity, students apply their understanding of different-sized batches of a recipe to label the tick marks on a double number line diagram with equivalent ratios. The context is familiar so students can focus on further making sense of the new representation.

Monitor for the different ways in which students interpret the discrete diagram and represent the ratio of white paint to blue paint on a double number line diagram. Here are some approaches that students may take, from more common to less common:

• Label the tick marks on the line representing white paint with multiples of 2 and those on the other line with multiples of 6.

  

• Label the tick marks on the line representing white paint with multiples of 1 and those on the other line with multiples of 3.

  

• Use the given diagram to represent a single batch (by labeling the third and sixth tick marks on the line representing white paint with 1 and 2, respectively, and labeling the tick marks for blue paint in increments of 1).

  

During partner discussions, students may need guidance in recognizing that each of the listed approaches is valid. Encourage students to explain the rationale behind their labels. 

The last two representations may be less common and may affect students’ responses to subsequent questions about the number of batches for 12 tablespoons of blue paint or for 6 cups of white paint. 

Select students to present their solutions. Sequence the discussion of students’ completed diagrams in the order listed in the Activity Narrative. 

Connect students’ representations to the ratio of 2 pints of white paint to 6 cups of blue paint and to the discrete diagrams. Highlight the different ways in which this ratio (or its equivalence) is visible in the diagrams, extending or annotating the number lines as needed. Ask questions such as: 

• “Where do you see one batch in each diagram?”
• “What information do we gain when we label the number line for tablespoons of blue paint by 3s or by 1s instead of by 6s?”
• “How do we show larger batches using drawings? Using a double number line?”

Emphasize the importance of labeling everything clearly so our representations are easier to interpret.

In this optional activity, students apply their understanding of equivalent ratios to solve a problem they might encounter naturally outside a mathematics classroom. The reasoning is more involved, as the question is open-ended and the response needs to be validated by mathematical reasoning.

Students may suggest adding 2 more cups of milk since Andre accidentally added 2 more tablespoons of cocoa. Ask students what the ratio of milk to cocoa would be in that case (3 cups of milk to 5 tablespoons of cocoa) and whether the mixture would taste the same as the original recipe. Encourage them to create a diagram (or amend the given diagram) to represent their proposed fix and reason about its taste compared to the hot cocoa that follows the recipe.

Invite students to share how they would make the hot cocoa taste like the recipe. Record the strategies for all to see. After each explanation, ask the class if they agree or disagree and how they know the resulting hot cocoa will taste the same. Highlight appropriate use of ratio language (such as “3 tablespoons of cocoa for every cup of milk” or “ cup of milk per tablespoon of cocoa”) and explanations in terms of equivalent ratios (such as “the ratio 1 cup to 3 tablespoons and 2 cups to 6 tablespoons are equivalent”).