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Multiplicative Comparison: Amy is 8 years old; Sally is 4 years old. Amy is twice as old as Sally. Sally is 1/2 of Amy’s age.
Composed unit: Jessica’s Purple Paint. Jessica mixes 2 cups of red paint and 3 cups of blue paint to make the perfect shade of purple paint. If she needs 20 cups of paint, how many cups of red paint and how many cups of blue paint will she need to mix to keep her perfect shade of purple?
I’ve done both of these with my students, the first one to have the discussion between multiplicative and additive thinking; the second to introduce the idea of using the ratio as a unit which can be iterated to solve a problem.
They’re both similar in the sense that they use relative thinking– you have to look at the relationship between the two quantities being compared; they are not separate amounts independent of each other.
They’re different in that the ratio of ages is like looking at a snapshot of two quantities, kind of like they’re a fixed point. The paint ratio lets you take that relationship and expand it– you can scale it up or down to find different amounts for different situations. Is she painting a wall? is she painting her whole house?