Apples to Apples

Task

Alice and Claire go apple picking. When they are done, Claire has 3 times as many apples in her basket as Alice has in hers. All of the apples are whole.

1. What are three different possibilities for numbers of apples that could be in the baskets?
2. What is the ratio of Alice's apples to Claire's apples? 

Alice and Claire's mom measures each of their heights in inches, rounded to the nearest whole inch. She remarks, "Wow! Alice's height is exactly three fourths of Claire's height!" 

3. What are three different reasonable possibilities for their heights?
4. What is the ratio of Claire's height to Alice's height? 

IM Commentary

The purpose of this task is to connect students' understanding of multiplicative relationships to their understanding of equivalent ratios.

Creating tables of equivalent ratios isn't required for this task, but it is a convenient way to organize them. Students should eventually come to view ratio tables as collections of equivalent ratios, so this might be a good opportunity to suggest such a tool.

In part (b) and (d), we ask "What is the ratio?" The meaning of this question can be misinterpreted by teachers, because experienced doers-of-math usually default to lowest terms, so it would be easy to mistakenly conclude that 3:1 is the only acceptable answer for (b) and 4:3 is the only acceptable answer for (d). However, any ratio equivalent to those is acceptable.

In part (c), reasonable height values aren't determined by the task statement, but they are suggested by the context, and students should explain the thinking behind the reasonableness of their values. Paying attention to the meaning of numbers in a context and evaluating them for reasonableness is important to the standards for mathematical practice, especially MP2, MP4, and MP8.

In this task, the units for each quantity are the same and students are essentially given the values of the unit rates and asked to determine possible ratios. The task is meant to come before students have studied unit rates in order to provide a precurser to the idea that unit rates characterize sets of equivalent ratios, but without having to deal with the complexities of the units. If students have already studied unit rates, they might notice that in this case we could say, "There are 3 of Claire's apples for every 1 of Alice's apples" (so there are "apples per apple") and There are 3/4 as many inches in Alice's height to every 1 inch in Claire's height (so there are "inches per inch").

Solution

1. Here are some possibilities for the numbers of apples in each basket. (For clarity, these are organized in a table, but they don't necessarily have to be. Any response in a ratio of 3:1 is acceptable.) 

   Apples in Claire's Basket	Apples in Alice's Basket
   6	2
   12	4
   30	10
   18	6
   3	1

2. The ratio of Alice's apples to Claire's apples is 1:3. Any equivalent statement is also acceptable, for instance, "There are 4 apples in Alice's basket for every 12 apples in Claire's."

3. Here are some possibilities for their respective heights. (For clarity, these are organized in a table, but they don't necessarily have to be.) Example reasoning about reasonable heights: the average height (usually called "length") of a newborn is 20 inches, but since Alice and Claire were presumably walking around and picking apples, we'll take 33 inches, the height of a short two-year-old, to be the least possible height. (These heights are easily googleable.) Since they are being measured by their mom, they are likely young children, so we'll say that 5 feet or 60 inches is the greatest possible height. (Students may set different bounds on the heights based on different reasoning.)  

   Claire's height in inches	Alice's height in inches
   44	33
   48	36
   52	39
   56	42
   60	45

    

4. The ratio of Claire's height to Alice's height is 48:36 (or equivalent).