Ten-frames to explore decimals

Chris Hunter writes on his blog about a student explaining how they would express 0.500 using ten-frames:

One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.

Showing 500/1000 or 0.500 using ten-frames

 

Have you seen examples where students come up with innovative ways of representing numbers?

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?

Investigating repeating decimals

A number of years ago, I had a 9th grade math class where I decided we would investigate when the decimal representation of a fraction repeats, and when it terminates. We also decided to investigate to see if there was a pattern in the number of repeating digits when the decimal representation of a fraction does repeat. It took us about two weeks to conclude that we weren’t going to find a pattern without a lot more work, and so we abandoned the investigation, but I still remember the process quite keenly.

If you decide to try this investigation with your own students, you may find this arbitrary precision division calculator (we quickly ran into the limits of our computer’s calculator when doing this investigation) useful: http://davidwees.com/divider/