Using Benchmarks To Compare Fractions - Illustrative Math Tasks

IM Commentary

This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, $\frac{1}{5}$ and $\frac{2}{7}$ in this case, by finding a benchmark fraction which lies in between the two. In Melissa's example, she chooses $\frac{1}{4}$ as being larger than $\frac{1}{5}$ and smaller than $\frac{2}{7}$.

This is an important method for comparing fractions and one which requires a strong number sense and ability to make mental calculations. It is, however, a difficult ability to assess because the method is only appropriate when there is a clear benchmark fraction to be used. In part (c) of the problem, for example, students may see the denominator of $25$ and think that $\frac{1}{5}$ or $\frac25$ would be potential fractions to use for comparison. In this case, it turns out that $\frac{2}{5}$ is an excellent choice which works well. However, if the numbers were different (for example $\frac{8}{25}$ and $\frac{14}{39}$) then there may be no fifths between them and students might spend a lot of time spinning their wheels trying to make $\frac{1}{5}$ or $\frac{2}{5}$ work. In addition to $\frac{2}{5}$, suggested by the denominator 25, both fractions are less than $\frac{1}{2}$, so identifying $\frac{1}{3}$ as a possibility for comparison might also come from the students and could be suggested if they struggle.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task is linked very intentionally to Mathematical Practice Standard 3, critique the reasoning of others. Students are asked to explain and critique the reasoning of their classmate, Melissa.  This type of task provides students with an opportunity to distinguish a reasonable explanation from that which is flawed.  If there is a flaw in the argument they can further explain why it is flawed. Learning how to argue whether a claim is true or false concisely and precisely becomes a routine part of a student’s mathematical work. This task is further extended by directing students to explore Melissa’s strategy with 2 additional examples. Their exploration may spark a conversation about when this strategy is most effective and what other strategies may be more effective and why. (MP.5)