Lesson 6: Introducing Double Number Line Diagrams

Let’s use number lines to represent equivalent ratios.

6.1: Number Talk: Adjusting Another Factor

Find the value of each product mentally.

$(4.5)\boldcdot 4$

$(4.5)\boldcdot 8$

$\frac{1} {10}\boldcdot 65$

$\frac{2} {10}\boldcdot 65$

6.2: Drink Mix on a Double Number Line

The other day, we made drink mixtures by mixing 4 teaspoons of powdered drink mix for every cup of water. Here are two ways to represent multiple batches of this recipe:

A discrete diagram for two quantites labeled "drink mix, in teaspoons" and " water in cups." For the number of teaspoons of drink mix, there are 8 small squares. For the number of cups of water, there are 2 large squares. Two groups of 4 small squares and 1 large square are indicated and one group of 4 small squares and 1 large square is circled.
 
A double number line with 6 evenly spaced tick marks. The top number line is labeled "Salt, in teaspoons" and the numbers 0, 4, 8, 12, and 16 are indicated. The last tick mark is blank. The bottom number line is labeled "Water, in cups" and the numbers 0, 1, 2, 3, 4 are indicated. The last tick mark is blank.
  1. How can we tell that $4:1$ and $12:3$ are equivalent ratios?
  2. How are these representations the same? How are these representations different?
  3. How many teaspoons of drink mix should be used with 3 cups of water?
  4. How many cups of water should be used with 16 teaspoons of drink mix?
  5. What numbers should go in the empty boxes on the double number line diagram? What do these numbers mean?

6.3: Blue Paint on a Double Number Line

Here is a diagram showing Elena’s recipe for light blue paint.

"A discrete diagram of squares that represent the amount of paint. The top row is labeled "white paint, in cups" and contains 2 large squares. The bottom row is labeled "blue paint, in tablespoons" and contains 6 small squares."

  1. Complete the double number line diagram to show the amounts of white paint and blue paint in different-sized batches of light blue paint.

  2. Compare your double number line diagram with your partner. Discuss your thinking. If needed, revise your diagram.
  3. How many cups of white paint should Elena mix with 12 tablespoons of blue paint? How many batches would this make?
  4. How many tablespoons of blue paint should Elena mix with 6 cups of white paint? How many batches would this make?
  5. Use your double number line diagram to find another amount of white paint and blue paint that would make the same shade of light blue paint.
  6. How do you know that these mixtures would make the same shade of light blue paint?

Summary

You can use a double number line diagram to find many equivalent ratios. For example, a recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” The ratio of cranberry juice to soda water is $5:2$. Multiplying both ingredients by the same number creates equivalent ratios.

A double number line with 6 evenly spaced tick marks. The top number line is labeled “cranberry juice, in cups” and the numbers 0, 2, 4, 6, 8, and 10 are indicated. The bottom number line is labeled “soda water, in cups” and the numbers 0, 2, 4, 6, 8, and 10 are indicated. There is a circle drawn around the 5 and the 2 and another circle drawn around the 20 and the 8.
 

This double number line shows that the ratio $20:8$ is equivalent to $5:2$. If you mix 20 cups of cranberry juice with 8 cups of soda water, it makes 4 times as much fizzy juice that tastes the same as the original recipe.

Practice Problems ▶

Glossary

double number line diagram

double number line diagram

A double number line diagram is a pair of parallel number lines with the numbers 0 aligned. Each number line is marked in equal increments and numbered. The tick marks are aligned, but the numbers on the two lines are often different. A pair of aligned numbers on the diagram represents a ratio that is equivalent to every other pair of aligned numbers on the diagram.