Perfect Purple Paint I

Task

Jessica gets her favorite shade of purple paint by mixing 2 cups of blue paint with 3 cups of red paint. How many cups of blue and red paint does Jessica need to make 20 cups of her favorite purple paint?

IM Commentary

The goal of this task is to provide a good context for engaging students in reasoning about ratios. There are many approaches available, ranging from concrete (tape diagram) to more abstract (setting up an equation). The teacher may wish to have physical manipulatives on hand so that students can simulate mixing the paint: for example, blue and red interlocking cubes (or tiles) could be used to model the cups of blue and red paint. Colored pencils (red and blue) should also be made available, either for making strip diagrams or for students to draw their own model of the mixed paint. 

The teacher may wish to use this task to demonstrate or introduce some of the different representations of ratios (ratio table, double number line, graphing points in the coordinate plane). The numbers are small so that the focus can be on the methods and not performing arithmetic. It is worth noting, in this direction, that the ratio table and the double number line present essentially the same information laid out in two different ways. The main difference is that the double number line provides an additional geometric representation of the sizes of the different quantities. Because of this, the double number line may be helpful for students as they move toward graphing points (as shown in the fourth solution).

This task is appropriate for a variety of formative assessments. Early in a unit on ratios, it can provide ideas the teacher can build upon introducing the different methods for representing ratios. At this point, students would likely draw pictures which could form the basis of a tape diagram representations. Students might also use the idea of multiplication as scaling: there are 5 cups of purple paint in each batch so 4 batches of purple paint will be 20 cups. The idea of multiplication of scaling is the foundation of thinking about ratios and this will provide an opportunity to assess student knowledge in this direction.

The task can also be used in the middle of a unit. At this point, the teacher may want to suggest multiple solutions and could even suggest the methods (tape diagrams, double number lines, ratio tables) if desired. This will provide insight into which methods the students are most or least comfortable with and so could help the teacher decide where to put extra emphasis for the rest of the unit. Similarly, if given at the end of the unit, the task could help the teacher decide how to make changes to the unit for the next time through the material. If, for example, students struggle with or rarely choose to use double number lines, the teacher may wish to integrate this method more throughout the unit.

This task was developed with the assistance of a group of teachers from Illinois and Washington in connection with an SBAC digital library project. In the videotaped lesson related to this task, the statement was:  ''If Perfect Purple paint is made by mixing 2 cups blue to 3 cups red paint, how much of each color would be needed for 20 cups?''

This task was written as part of a collaborative project between Illustrative Mathematics, the Smarter Balanced Digital Library, the Teaching Channel, and Desmos.

Solutions

Solution: 1 Tape Diagram (6.RP.3.a)

Below is a picture showing 4 batches of purple paint making 20 cups total, along with the blue paint and red paint that make up the purple paint:

    

We can see that in the 20 cups of purple paint, there are 8 cups of blue paint (4 groups of 2 cups) and 12 cups of red paint (4 groups of 3 cups).

Solution: 2 Ratio Table (6.RP.3.a)

A ratio table is a more abstract approach than the tape diagram. A ratio table shows the numbers of cups of blue, red, and purple paint rather than physically modeling these quantities:

Blue Paint (cups)	Red Paint (cups)	Purple Paint (cups)
2	3	5
4	6	10
6	9	15
8	12	20

The rows show 1 batch, 2 batches, 3 batches, and 4 batches of purple paint. To get 20 cups of purple paint, the table shows that we need 8 cups of blue paint and 12 cups of red paint.

Solution: 3 Double Number Line (6.RP.3)

We can solve the problem using double (or triple) number lines. This is similar to the ratio table although the information is presented in a different, more geometric format resembling the tape diagram. For example, to find out how much blue paint there will be in 20 cups of perfect purple paint, we can use the double number line below:

   

A similar method will show that there are 12 cups of red paint in 20 cups of perfect purple paint. Or, we can use the fact that there are 8 cups of blue paint and then a second double number line tells us that we need to mix this with 12 cups of red paint to get 20 cups of perfect purple paint:

    

The picture below shows how to combine the blue, red, and purple paint with a triple number line (as above, the units for the paint are cups):

   

If we mix no blue paint and no red paint we get no purple paint as shown on the left. The next vertical line shows that one batch of 5 cups of purple paint is made out of 3 cups of red paint and 2 cups of blue paint. We are interested in 20 cups of purple paint which is 4 groups of 5. This means that we need 4 groups of 2 or 8 cups of blue paint and 4 groups of 3 or 12 cups of red paint. 

Solution: 4 Graphing Points (6.RP.3.a and 6.EE.9)

Instead of the double number line which shows both the blue and red paint on different horizontal axes, we can represent the situation using a horizontal axis and a vertical axis as shown below:

    

The given information (2 cups of blue paint and 3 cups of red paint is shown in black). The other dots show two, three, and four batches of purple paint. Four batches make 8 cups of blue paint and 12 cups of red paint, for a total of 20 cups of purple paint.

Notice that the 4 points all lie on a line and the line goes through the point (0,0). The point (0,0) is the case where there is no blue paint and no red paint, combining to make no purple paint. Note too that blue paint could be placed on the vertical axis and red paint on the horizontal axis.

 

Solution: 5 Division and Multiplication with Units (6.RP.3.d)

If Jessica  mixes 2 cups of blue paint with 3 cups of red paint this will make 5 cups of her favorite purple paint. She wants 20 cups of purple paint and this will require $$20 \text{ cups} \div 5 \,\frac{\text{cups}}{\text{batch}} = 4 \text{ batches}.$$ So Jessica needs to make 4 batches of her purple paint. This will require 4 $\times$ 2 = 8 cups of blue paint and 4 $\times$ 3 = 12 cups of red paint.

Solution: 6 Setting up an equation (6.RP.3.a and 6.EE.7)

Each batch of 2 cups of blue paint and 3 cups of red paint makes one batch of 5 cups of purple paint. We let $x$ stand for the number of batches of purple paint. In $x$ batches of purple paint there are $x \times 5$ or $5x$ cups. Since we want 20 cups of purple paint, we need to solve $5x = 20$. This means $x = 4$ so Jessica needs to make 4 batches of purple paint. She will need 8 cups of blue paint and 12 cups of red paint to make 20 cups of her favorite purple paint.