I am currently helping a friend's child with his schoolwork. He is currently in primary school and being taught the topic of speed. I would like to give him the following problem as practice but I am not sure how to explain the solution to him at his level:

A and B competed in a race. Every time A ran $7$ m, B ran $3$ m. If B took $14$ minutes to complete the race, how long did A take?

My proposed solution: the ratio of A's speed to B's speed is $7:3$. So A will take a shorter time than B. Since distance $=$ speed $\times$ time, we have distance covered by A $=$ A's speed $\times$ A's time $=$ distance covered by B $=$ B's speed $\times$ B's time, i.e. $7$ $\times$ A's time is equal to $3$ $\times$ B's time. Hence the ratio of A's time to B's time is $3:7$. So, $7u=14,\ 1u=2$ and hence $3u=6$.

How do I rephrase my solution to make it more student-friendly? I am trying not to use any formula.

I think many students have difficulty grappling with the fact that speed is inversely proportional to time. If A's speed is $x$ times B's speed, where $x$ is whole number, then most students can easily see that A's time is $1/x$ times of B's time. But if $x$ is not a whole number, most will get stuck.

I appreciate all advice. Thank you.

• Are you opposed to giving him the problem to see how he broaches it - without having a pre-planned digestable solution? Alternatively: Can you give an example of his work to indicate the current stage of his proportional reasoning? I am not sure what would be "student-friendly" without more info on what the student knows/can do. – Benjamin Dickman Nov 14 '15 at 22:44
• You say you gave this problem. Are you sure the problem is at his level? There are a wide range of problems related to speed and I think this may be harder than what he is learning. When my 11 year old students learned speed, a typical problem was given two of the three paremeters (speed, time, distance), find the third parameter. This is much harder than that. – Amy B Nov 15 '15 at 21:28

My teaching philosophy is to start at a level below the learner's assumed mathematical maturity. I will assume that the student has not mastered the concepts of ratio or basic algebra. But I will assume that the student is familiar with the concepts of speed and arithmetic equations.

Review the definition of speed as the quantity equal to the distance traveled divided by the time it took to travel. Then, noting that most problems like these involve equations (statements that two quantities are equal), ask the student what quantities are equal in this situation. The student should be led to discover that the distances traveled by A and B should be equal. Then note that the distance traveled is equal to the speed multiplied by the time.

We are not given the speeds of A and of B. But note that the speed of A is $7$ meters for some unknown but fixed unit of time, and the speed of B is $3$ meters for the same unit of time. Thus, the distance traveled by A is $\frac{7\mathsf{~meters}}{\mathsf{unit~of~time}}\times\mathsf{?~minutes}$; and the distance traveled by B is $\frac{3\mathsf{~meters}}{\mathsf{unit~of~time}}\times 14\mathsf{~minutes}$.

Noting that $3\times 14=3\times 2\times 7=6\times 7=7\times 6$, it can be seen that A would take $6$ minutes. (For brevity, I skipped a lot of steps here. You should supply the missing steps.)

Note that we only dealt with a specific case, that is, we did not use a generalization. Have the student solve, say, ten more specific cases (exactly the same as this problem, but with different numbers). (You should carefully choose the numbers so that the answers are whole numbers.) By the end of these "drills," the student should be able to construct the generalization by himself/herself.

Also note that we did not use division or the concept of inversely proportional. Actually, we did not even use the terms ratio or proportion. We just used multiplication (and factoring) and the concepts of speed, distance, and time.