A fun little problem

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Lilly is working on a classic math problem:

A hot tub can be filled with one of two water faucets. The first alone would fill the tub in 2 hours and the second one alone would fill the tub in 4 hours. How long does it take to fill the tub with both of them turned on at the same time?

Lilly thinks this problem is a little too easy and so starts to play around with different ways of solving it. Being a big fan of geometry, she draws a right triangle, with one leg being 2 and the other leg being 4 to represent the time for each faucet.

triangle 1

She then inscribes a square in the triangle so that two sides align with the legs and one corner lies on the hypotenuse:


Lilly then notices that the length of the side of the square is actually the solution to the problem!

  1. Is she right?
  2. Does this work for other numbers? (different times for the faucets to fill the tub)
  3. Does this work for all numbers?

Example vs Proof

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I recently read an old blog post by Kate Nowak.

She discussed a problem I have quite often with my students – getting them to understand the value of math proofs. In particular, I have trouble getting them to understand the difference between an example and a proof.

In the past I have used a silly “false pattern” to make my point. I start the class by writing the following conjecture on the board:

If I double any number, the result will always be less than 100

I then give a lot of examples. This generally gets a number of laughs, and they say it is silly. I then ask them why they do the same thing all the time in class if it is so silly. We discuss how it is actually possible to find infinite examples to support this conjecture, and yet it still isn’t true.

I like the example in Kate’s blog with the circles. However, I feel it is too complicated to really get the point across, just as I feel my example is too simple to really get the point across.

So, what I want to ask anyone who reads this is are there any good examples of the “right” level for 8th or 9th graders?

I want a false pattern where the result is not what it seems, where the first few examples you might try seem to support the idea, but at some point the pattern fails. More importantly, the “correct” pattern should be something accessible to good students with a strong grasp of basic algebra (but preferably not requiring quadratics).

I think these would be good examples to help students understand the value of a proof.

By the way, here is a related fun exercise. What is the next number in this sequence: 1, 2, 3, 4, 5, 6 ?

You can make it whatever you want, but here is a simple option to make it 8 (can you figure out how to make your own function like this? There is an easy trick):

Geogebra image