Mediant Madness

Why do this problem?

This problem brings together opportunities to think about the familiar topics of fractions, averages, and gradient. We hope unexpected result in the context of averaging test scores will provoke students to be curious about the mediant of two fractions.

Possible approach

"Kyle got 75% on section A of his maths exam, but he only got 35% on section B.
He says this means he should score 55% overall, but his teacher said he only scored 50% overall! How can this be?"

Give students some time to think about this question, and then discuss their thoughts in pairs or small groups. Then bring the class together - "Who is right, Kyle or his teacher?" 
"We don't have enough information because we don't know if the two sections had equal weighting"

Next, share the extra information from the problem:
In section A, there were 12 questions and Kyle got 9 correct.
In section B, there were 20 questions and Kyle got 7 correct.
Overall, there were 32 questions and Kyle got 16 correct.

The difference in overall scores is because Kyle worked out the average of his two scores, but his teacher worked out the mediant. Define the mediant:
"Given two fractions $\frac{a}{c}$ and $\frac{b}{d}$, the mediant is defined as $\frac{a+b}{c+d}$."

Then share the interactive diagram below. If students have access to computers or tablets, give them some time to explore the diagram. Alternatively, they could use squared paper to create their own examples.




Here are three questions that students could explore:

Can you find some values of $a, b, c$ and $d$ so that Kyle and his teacher both get the same value for the overall score (that is, the average and the mediant are the same)?

The pass mark for an exam is 50%. If I scored 25% on the first $n$ questions, under what circumstances can I still pass the exam?

Is it true that the mediant always lies between the two fractions? How do you know?

 
Key questions

What aspect of the diagram represents the fractions $\frac{a}{c}$ and $\frac{b}{d}$?
How does the gradient of the line joining $(0,0)$ to $((c+d),(a+b))$ compare to the other gradients?

Possible extension

The problem Farey Neighbours invites students to explore the relationship between mediants and consecutive terms of the Farey sequence.

Possible support

Before exploring the mediant of two fractions, students might find it beneficial to work on ordering fractions in the problem Farey Sequences.