IM 6.2.6 Lesson: Introducing Double Number Line Diagrams

Find the value of each product mentally.
[math]\left(4.5\right)\cdot4[/math]
[math]\left(4.5\right)\cdot8[/math]
[math]\frac{1}{10}\cdot65[/math]
[math]\frac{2}{10}\cdot65[/math]
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:
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[/img][br][br][img]data:image/png;base64,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[/img][br]How can we tell that [math]4:1[/math] and [math]12:3[/math] 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 [i]perfect square[/i] 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.[br][br]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?
Here is a diagram showing Elena’s recipe for light blue paint.
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[/img]
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?

IM 6.2.6 Practice: Introducing Double Number Line Diagrams

A particular shade of orange paint has 2 cups of yellow paint for every 3 cups of red paint. On the double number line, circle the numbers of cups of yellow and red paint needed for 3 batches of orange paint.
This double number line diagram shows the amount of flour and eggs needed for 1 batch of cookies. Complete the diagram to show the amount of flour and eggs needed for 2, 3, and 4 batches of cookies.
What is the ratio of cups of flour to eggs?
How much flour and how many eggs are used in 4 batches of cookies? Remember, the original double number line diagram shows the amount of flour and eggs needed for 1 batch of cookies.
How much flour is used with 6 eggs?
How many eggs are used with 15 cups of flour?
On the double number line, label the tick marks to represent amounts of red and blue paint used to make batches of this shade of purple paint.
How many batches are made with 12 cups of red paint?
How many batches are made with 6 cups of blue paint?
Diego estimates that there will need to be 3 pizzas for every 7 kids at his party. Select [b]all [/b]the statements that express this ratio.
Draw a parallelogram that is not a rectangle that has an area of 24 square units. Explain or show how you know the area is 24 square units.
Explain or show how you know the area is 24 square units.
Draw a triangle that has an area of 24 square units.
Explain or show how you know the area is 24 square units.

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