# Constructing Equivalent Ratios

I spent a good deal of time racking my brain for a more open-ended activity to do for equivalent ratios—which make their appearance in the Ratio Tables math lesson app. Below is what I came up with. It is not so much an activity or a puzzle as it is an environment in which puzzles can be created according to some limitations. The physics (thanks to PhysicsJS) is just a little extra to make it play-worthy.

 = 0.4 = 1 = 2 = 3 = 4 = 10

One puzzle I thought of that has some interesting potential is to ask students to create different structures representing different ratios. For example, can you create the part-to-part ratio 1 : 2 with 2 blocks, 3 blocks, 4 blocks, 13 blocks—using at least 2 different types of blocks? What if there was an added restriction that the structures be symmetrical? Are there any numbers of blocks which are impossible for a given ratio?

So, for example, here is 1 : 2 with 2 blocks. It seems desirable to keep the blocks representing the numerator and denominator respectively separate.

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With 3 blocks, keeping the blocks separated into numerator and denominator and keeping symmetry—shape symmetry at least—maybe this:

Smallest and Largest Towers

There is some weakness in my implementation which shows some instabilities when stacking blocks (after 3 or 4, it gets weirdly wobbly). But using the "shelves" blocks can be stacked into tall towers. What is the tallest tower you can build to show 4 : 15? What about the smallest? Symmetrical?

Build and See (and/or Ignore the Masses)

Certainly one way to go about an activity is to just build something and then report the ratio that is represented. With the restriction that I use only two kinds of blocks, the numerator and denominator separation can be evident without having to make them visually evident. So, I can make a staircase to show 1 : 15 = 4 : 60, for example.

Also, ignoring the masses at first might be a better way to ramp up. The structure on the right can represent 4 : 6 or 6 : 4 instead.

Still noodling on all of it! If you have suggestions for puzzles or equivalent ratio constructions you'd like to share, email us at qanda[at]guzintamath.com.