Math Games Galore
I recently built an activity in my math for elementary school teachers around Oodle, a wordle based game but with numbers.
First let me try to explain the concept of Oodle. Let’s start with a picture:
You can see the concept and numeric options available in this picture. You can play the numbers 1–12 in any of the four boxes.
Let me give you another picture to show a different version.
So you’ll notice that the position of the operations has changed and in this version you have fewer options (1–9).
We’ll go back to the harder version, but it is worth saying that one of the wrinkles of Oodle for casual/first-time players is that you have to pay attention to the standard order of operations in mathematics. Order of operations makes a popular meme in social media. If you search “math puzzles facebook” you will see oodles (I couldn’t help myself) of examples. Note that the last question involves multiplication in a secondary role so that answers will vary based on the understanding and acceptance of the norms of ordering in math operations (in addition to many subtle changes in the final line’s separate images).
Many students below their 40’s will recall the mnemonic device “PEMDAS” (or “BIDMAS” if you are from a British school). The basic hierarchy of operations can be summarized as:
The relevance of ordering in this game is in the second example I provided the multiplication is at the end but it really should be considered first., again assuming (as Oodle does) that the standard order of operations is in play here.
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Now, beyond that warning, the other point to acknowledge is that there is not one unique “solution” to the starting statement. So for instance to solve a x b — c + d = 91, there are many solutions. By my count, only six options make sense to explore as the products:
12x8, 12x7, 11x9, 11x8, 10x9, and 10x8
Keep in mind that any of those products can be switched in order because, for example, 11x9 is the same as 9x11, result wise. Beyond that we can rearrange these six options into a new algebraic statement:
Now in looking at all the cases possible for these six options, some have as few as one possibility and some have as many as eight. According to my back of the envelope work, there are 27 possible (c,d) pairs which means there are 54 possible solutions total because (a,b) and (b,a) would both give the same result.
Guessing 6 times to get the right solution out of 48 possibilities would not be an interesting game, but that is where Wordle comes in. They give you feedback on your guesses, telling you one of three things for each digit you chose.
You can see in my first guess two of the types of responses you could get. A grey box means the number is not in the solution. A yellow box means that the number is in the solution statement but in a different location. A green box (not shown) indicates the right number is in the right place!
The assignment I had my students work on was to first familiarize themselves with the game and then to work with a partner to document their logic in arriving at the correct answer. It turned out that the documentation part of this exercise was a much higher burden than my students could take (granted it was the end of the term).
It made me wonder about student’s habits around explaining things and whether we do a good job of supporting them in developing that kind of habit. The first problem that comes to mind is that if the explanation is “short” or “obvious” it feels like it’s not worth doing. But when you jump up to the level of a complicated four-variable equation, the level of analysis and reflection may already feel too daunting.
I wonder if the memes on facebook actually provide a better starting point. They have a single answer which is right or wrong but there is no “back of the book” to consult, so then you are down to explaining your reasoning.
For the record, I got 144! What about you?
