Open ended investigation: Exploring fractions in different bases

This project is related to the repeating decimals project I’ve outlined before (see this post) but with a twist.

What do fractions look like in bases other than 10 (for example: base 7)?

Instead of decimals, what would septimals look like? When we write 10.356, what we really mean is 10 + \frac{3}{10} + \frac{5}{100} + \frac{6}{1000}. In septimal notation, 7 + \frac{4}{7}  would be written 10.4. Can you extend this example?

Here’s a trickier question: how is 0.5 in decimal notation represented in septimal notation?

Is there a relationship between the number of digits it takes a fraction to repeat when it is written in expanded (in base ten, we call these decimals) form and the base in which it is being represented? Is it possible to write numbers in any rational base, for example 1.5? What about base \pi?

Open ended problem: Scoring investigation

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Using shoes for keeping track of score

The problem: Design a scoring system for a game that can be used with shoes. Note that this problem may actually be different depending on what game you are playing.

This problem has  a lot of possible solutions, and might make assessment challenging, but my recommendation is: play a game with each of the scoring systems so that students can see them in action, and then decide which one they think makes the most sense.