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.](https://staging-cms-im.s3.amazonaws.com/6KKTeh5tL6u3p7vnU5rjp4N5?response-content-disposition=inline%3B%20filename%3D%226-6.2.C1_Image_2.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.C1_Image_2.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093139Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a0e6cb0f0a72ec5cc2b119b1a72bfb3f7adb71311a93e3380524336569f4381c)
![A double number line. Salt, in teaspoons. Water, in cups.](https://staging-cms-im.s3.amazonaws.com/7gjEaJkTZjRjMToHakZxNkbQ?response-content-disposition=inline%3B%20filename%3D%226-6.2.C1_Image_3.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.C1_Image_3.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093139Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=617b1075995edbd780ddb957cfefd78bbb702f97effce7cc6519791396ea77fd)
- How can we tell that \(4:1\) and \(12:3\) are equivalent ratios?
- How are these representations the same? How are these representations different?
- How many teaspoons of drink mix should be used with 3 cups of water?
- How many cups of water should be used with 16 teaspoons of drink mix?
- What numbers should go in the empty boxes on the double number line diagram? What do these numbers mean?
Recall that a perfect square is a number of objects that can be arranged into a square. For example, 9 is a perfect square because 9 objects can be arranged into 3 rows of 3. 16 is also a perfect square, because 16 objects can be arranged into 4 rows of 4. In contrast, 12 is not a perfect square because you can’t arrange 12 objects into a square.
- How many whole numbers starting with 1 and ending with 100 are perfect squares?
- What about whole numbers starting with 1 and ending with 1,000?
6.3: Blue Paint on a Double Number Line
Here is a diagram showing Elena’s recipe for light blue paint.
![Two large squares and six small squares.](https://staging-cms-im.s3.amazonaws.com/U6L65gXWqX3S7LK5nmdnE1s4?response-content-disposition=inline%3B%20filename%3D%226-6.2.A2_Image_6.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.A2_Image_6.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093139Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e09b4a01ce9e4bca0a5d96217d2a101f34a4c3bee076ae1010102d98e9e92e9c)
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Complete the double number line diagram to show the amounts of white paint and blue paint in different-sized batches of light blue paint.
- Compare your double number line diagram with your partner. Discuss your thinking. If needed, revise your diagram.
- How many cups of white paint should Elena mix with 12 tablespoons of blue paint? How many batches would this make?
- How many tablespoons of blue paint should Elena mix with 6 cups of white paint? How many batches would this make?
- 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.
- 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.
![Double number line. Cranberry juice, cups. Soda water, cups.](https://staging-cms-im.s3.amazonaws.com/wZ4TX6R12Jax9NNWAgJ98mQo?response-content-disposition=inline%3B%20filename%3D%226-6.2.C1_Image_7.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.C1_Image_7.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T093139Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dd077105d91a453b91aa98ed825071876289fbcc62429802686effca8c524bcf)
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.
Glossary Entries
- double number line diagram
A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.