# Math test grades

Alignments to Content Standards: S-ID.C.9

Ari has collected data on how much her classmates study each week and how well they did on their recent math test. Here is part of the data, showing the percentage of each group (divided according to hours spent studying) who got different grades on the test:

Weekly Hours $w$ Studying C or below on test B on test A on test
$w \lt$ 3 50% 20% 30%
$3 \leq w \lt 6$ 60% 25% 15%
$w \geq 6$ 60% 20% 20%
1. What can you conclude about the percentage of Ari's classmates who got B's? What about the percent who got A's?
2. Looking at the table, Ari decides that in order to do well in her math class, she should not study more than 3 hours a week: she has a 30 percent chance of getting an A as long as she does not study more than 3 hours per week. Is Ari's reasoning valid?

## IM Commentary

The goal of this task is twofold. For part (a) since we are not given how large each of the groups in the table are, the best we can do is to apply reasoning about ratios (in the form of percents) to give a range of possible answers. For part (b), the goal is to recognize a misuse of statistical reasoning. Of the three groups, the one with the highest percentage of A's is the group whose studies were limited to less than 3 hours a week. But the fact that studying a small amount is better correlated with getting an A than studying more does not mean that the students in the first group are getting an A because they did not study much. In fact, this statement makes little sense and so another cause should be investigated, ideally a cause for both phenomena: getting an A and studying very little.

The second part of this task provides an example of confusing correlation and causation. In this case, good grades and little studying are correlated (the teacher may wish to change the given data to make the correlation stronger) but it is certainly not the case that little studying causes students to get good grades. Many examples of this confusion are presented here https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation and some of these are more difficult to detect.

This task supports MP2, Reason Abstractly and Quantitatively, and MP3, Construct Viable Arguments and Critique the Reasoning of Others. For the latter, students are analyzing Ari's reasoning and if they do this in groups they will also have an opportunity to analyze one another's reasoning. For MP2, the students are given some data in a context and are prompted to infer additional information from the data.

The mathematics behind part (a) of the task is quite interesting. If $a,b,c,d$ positive whole numbers with $\frac{a}{b} \lt \frac{c}{d}$ we wish to compare $\frac{a+b}{c+d}$ to $\frac{a}{b}$ and $\frac{c}{d}$. We can show that $$\frac{a}{b} \lt \frac{a+c}{b+d} \lt \frac{c}{d}.$$ Indeed, for the first inequality

\begin{align} \frac{a}{b} &= \frac{a(b+d)}{b(b+d))} \\ &\lt \frac{ab +bc}{b(b+d)} \\ &= \frac{a+c}{b+d} \end{align}

with the second inequality coming from the fact that, since $\frac{a}{b} \lt \frac{c}{d}$, we find $ad \lt bc$. The second inequality can be shown in similar fashion. Beyond saying that $\frac{a+c}{b+d}$ lies between $\frac{a}{b}$ and $\frac{c}{d}$ we cannot say more: if $a$ and $b$ are very large compared to $b$ and $d$ then $\frac{a+c}{b+d}$ will be close to $\frac{a}{b}$. Similarly if $c$ and $d$ are large compared to $a$ and $b$ then $\frac{a+c}{b+d}$ is close to $\frac{c}{d}$.

This task was designed for an NSF supported summer program for teachers and undergraduate students held at the University of New Mexico from July 29 through August 2, 2013 (http://www.math.unm.edu/mctp/).

## Solution

1. In order to accurately calculate the percent of Ari's classmates who got A's or B's we would need to know how many classmates she has and how many of these received an A (or B) on the exam. This information is not available. We can still, however, make some important deductions. Note first that each of Ari's classmates is in one of the three groups listed in the table because the categories are exhaustive. That is, every student must have studied either less than 3 hours per week, between 3 and 6 hours per week, or more than 6 hours per week. Not only are the groups exhaustive but they are also mutually exclusive. It is impossible, for example, to study less than 3 hours per week while also studying at least 6 hours per week.

Suppose we let $x$ denote the number of students who studied less than 3 hours per week, $y$ the number who studied between 3 and 6 hours per week (including exactly 3 hours per week), and $z$ the number who studied at least 6 hours per week. Then, as noted in the previous paragraph, the total number of Ari's classmates is $$x + y + z.$$ Looking at the given information, the number of Ari's classmates who got a B is $$\frac{1}{5}x + \frac{1}{4}y + \frac{1}{5}z.$$ Since $\frac{1}{5} \lt \frac{1}{4}$ we have $$\frac{1}{5}(x+y+z) \lt \frac{1}{5} x + \frac{1}{4}y + \frac{1}{5}z \lt \frac{1}{4}(x+y+z).$$ Note that in order to have inequalities above, it is important to know that $x, y, z$ are non-zero which is true because the percents in the table are all non-zero. Putting all of the information together, the fraction of Ari's classmates who got B's on the exam is $$\frac{\frac{1}{5} x + \frac{1}{4}y + \frac{1}{5}z}{x+y+z}$$ and, according to the inequalities, this is greater than $$\frac{\frac{1}{5} x + \frac{1}{5}y + \frac{1}{5}z}{x+y+z} = \frac{1}{5}$$ and less than $$\frac{\frac{1}{4} x + \frac{1}{4}y + \frac{1}{4}z}{x+y+z} = \frac{1}{4}.$$ Without further information, we can not say more. So the percentage of Ari's classmates who got B's is more than 20 percent but less than 25 percent.

For the percentage of Ari's classmates who got A's, similar reasoning applies telling us that it is more than 15 but less than 30.

2. For Ari's classmates, the largest percentage of A's came from those who studied less than 3 hours a week. But this does not mean that Ari's best chance of getting an A is to study less than 3 hours a week. In the first place, there is no reason to believe that these students received A's on the test because they studied fewer than three hours a week. On the contrary, a more plausible explanation would be that they were very proficient with the material and therefore did not need to study much in order to do well on the exam. In other words, there is a ''hidden'' cause of both the study habits and the grades.

We can rephrase this argument in more concrete terms. In order to get an A on the exam, Ari's best chances come from learning the material well. So her best strategy is to study as much as she needs to in order to feel confident with the material. This might be less than three hours a week or it might be much more. Even if it turns out that Ari only needs to study two hours a week to get an A, the source of the A is her mastery of the material, not the fact that she studied for less than three hours a week.