**Reduced
Fat Milk and Milkfat**

Here’s an activity that my friend Nic Vitale and I wrote for MTMS‘ Math for Real and then found out that they had published something similar (Bu & Marjanovich, 2017) so they decided not to publish our article. It’s funny that Nic contacted me about this situation and we wrote it up on a whim and apparently this also happened to others. I think ours is a bit different from the one published in MTMS (check that one out here) so I thought I’d share it with you all. đź™‚

#### Zandra and Nic’s Task

Understanding percentages and their applications is an important topic in middle school mathematics. One aspect of this topic that is particularly important is that percentages, like fractions, refer to a particular whole. In instances in which the whole is not clearly stated, it can be difficult to make sense of situations. For example, if I said that I got an 80% on quiz A and 75% on quiz B, one may think it possible to conclude that I answered more questions correctly on quiz A. This is not necessarily the case; quiz A may have had 5 questions and quiz B could have had 100 questions.

The importance of understanding percentages and their related wholes came to light in the real world during a conversation one morning between Nic and his son Levi. Nic and Levi were drinking milk when the following conversation happened:

Levi: â€śDaddy, whole milk has forty percent fat!â€ť

Nic: â€śWhat do you mean?â€ť

Levi: â€śLook!â€ť (pointing to label on milk container, *Figure 1*)

**Questions for Students**

1.Maria told her mother that she got an 80% on her first quiz and 75% on her second quiz. Her mother concluded that this meant Maria got more questions correct on her first quiz than on her second quiz. Is Mariaâ€™s mother correct? Explain your thinking.

2. How do you think Levi arrived at his conclusion? Explain any interpretations or assumptions he might have made in his reasoning.

3. Decide whether Leviâ€™s thinking is correct. Justify your answer.

4. Based on the label (*Figure 1*), what is the percentage of milkfat in whole milk? Justify your answer and include a math drawing.

5. If whole milk did have 40% milkfat as Levi concluded and reduced fat milk still had 2% milkfat as shown, what percentage should replace the 38% on the label? Justify your answer and include a math drawing.

6. Redesign the label so the information is clearer to the buyer. Explain your reasons for your redesign.

**Answers**

1.No, Mariaâ€™s mother is not necessarily correct. The second quiz may have had more questions than the first and so, although the percentage correct on the first quiz may be greater, the number of questions correct may be greater on the second quiz.

2. Answers may vary. A probable answer is that Levi simply added the 38% and the 2%, interpreting â€ś38% less thanâ€ť as an additive comparison between two fixed quantities of the same unit (similar to: Wendy had $2, which was $38 less than what Mary had, therefore Mary must have had $40).

3. No, Levi is not correct. The 2% refers to the percent of reduced fat milk that is milkfat. The 38% describes how the 2% compares to the percentage of milkfat in whole milk. Because they refer to different wholes, you cannot add those percentages together to get a meaningful answer.

4. Based on the given information, whole milk is about 3.2% milkfat. Students may arrive at this in different ways. The 2% milkfat on the label is 62% of the milkfat in whole milk (100%-38%). Thus, students can set up a proportion to find how much milkfat is in 100% of whole milk as follows .

5. The label would then have to read 95% less fat than whole milk. 2% is 1/20^{th} or 5% of 40% so 2% is 95% less than 40%.

6. Answers will vary.