This lesson develops the idea of equivalent ratios through physical experiences. A key understanding is that if we scale a recipe up (or down) to make multiple batches (or a fraction of a batch), the result will still be “the same” in some meaningful way. Students see this idea in two contexts, taste and color:
- A mixture containing two batches of a recipe tastes the same as a mixture containing one batch. For example, 2 cups of water mixed thoroughly with 8 teaspoons of powdered drink mix tastes the same as 1 cup of water mixed with 4 teaspoons of powdered drink mix.
- A mixture containing two batches of a recipe for colored water will produce the same shade of the color as a mixture containing one batch. For example, 10 ml of blue mixed with 30 ml of yellow produces the same shade of green as 5 ml of blue mixed with 15 ml of yellow.
The fact that two equivalent ratios yield the same taste or produce the same color is a physical manifestation of the equivalence of the ratios.
Students see that scaling a recipe up (or down) requires multiplying the amount of each ingredient by the same factor. For example, doubling a recipe means doubling the amount of each ingredient (MP7). They also gain more experience using a discrete diagram as a tool to represent a situation.
- Draw and label discrete diagrams with circled groups to represent multiple batches.
Create two separate drink mixtures. Container A has one cup of water and one teaspoon of powdered drink mix. Container B has one cup of water and four teaspoons of powdered drink mix. You might have to stir the mixtures vigorously for a minute or more to ensure all the powder dissolves.
Get 6 small paper cups. Do not mark the cups. Put a small amount of mixture A in three of the cups and a small amount of mixture B in the other three cups. (Keep track of which is which, as you will give each of three volunteers one of each cup.) Discard the rest of the mixtures for now. (You will do a dramatic performance creating each mixture during class.)
During class, you will need three empty mixing containers with at least a 2-cup capacity each. One marked A, one marked B, and one marked C. You will also need a supply of water, a supply of drink mix, a measuring cup, and a teaspoon.
- I can explain the meaning of equivalent ratios using examples.
A ratio is an association between two or more quantities.
For example, the ratio \(3:2\) could describe a recipe that uses 3 cups of flour for every 2 eggs, or a boat that moves 3 meters every 2 seconds. One way to represent the ratio \(3:2\) is with a diagram that has 3 blue squares for every 2 green squares.
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