First up, this post is NOT going to be about this:
This is what came to mind when joining Derek Ball’s session at the ATM conference this week, entitled “Recurring Decimals”.
In fact, we were going the other way: converting fractions to recurring decimals, specifically looking at fractions where the denominator is prime. It was a fascinating session, great for deepening subject knowledge. This blog post is my attempt to reflect on what I learned and my thoughts about how I might use this in the classroom, probably Year 10 or 11, but really any group that is confident with bus stop division could investigate this.
I was already aware of some pretty cool things that happen with sevenths, mainly from Don Steward’s blog:
Why should this happen? Why do we only see these six digits with sevenths? The process of bus stop division helps us see why and this is where I feel I would start with a class. This is good practice of a technique that should be secure but often isn’t in Year 9 / 10. It helps learn the seven times table and I don’t think it is too tedious to ask students to perform these six calculations manually:
Some learners might want to find 2/7 by doubling 1/7. And then maybe find 3/7 by adding 1/7 to 2/7 and so on. Even if they stick with the bus stop division, they will find that they are essentially doing the same six calculations:
10 ÷ 7 = 1 rem 3
20 ÷ 7 = 2 rem 6
30 ÷ 7 = 4 rem 2
40 ÷ 7 = 5 rem 5
50 ÷ 7 = 7 rem 1
60 ÷ 7 = 8 rem 4
The ones digit is always zero and the tens digit must be less than 7 so there are only these 6 options. Can we extend this rule to other fractions with denominator less than 10? Of course, all others except 3, 6 and 9 will terminate – why is this?
There are also some interesting things to notice about the 1/7 “wheel” before moving onto higher primes. I won’t spoil it for you, but suggest you include the fractions around the outside of the wheel to spot some of the patterns.
We then moved onto looking at 13ths. 11th are interesting too for different reasons, so I can see why we went onto 13th because there are some surprising relationships between 7th and 13ths. At this point, we started using calculators and I would do the same with a class. Or even better, open a spreadsheet, which I am always keen to do!
I’m not sure of the value of kids typing 12 calculations into their calculators, so I might give this image above as a print out for them to write on. Hopefully they will soon spot that there are two sets of recurring digits, i.e. 076923 in 1/13 + 5 others and 153846 in 2/13 + 5 others and which, this time can be written as two wheels:
Ideally at this point you’ll have the class hooked and they would be asking all sorts of questions. Well, maybe enough of them to get everyone else thinking. I would try really hard to encourage the students to come up with these questions. This is what I hope they will ask, but I hope they will also ask questions I hadn’t thought of, something which is a really special moment in any lesson.
- Do we see the same patterns in the sevenths wheel as we do in the thirteenths wheel?
- Is there something special about 6 points around the wheel?
- Why are there 2 wheels? Are there any patterns across the two wheels?
- What about other fractions, do they have multiple wheels?
On the image above, I have shown that as you move clockwise around the wheel, you are effectively multiplying by 10. This is obvious going from 1/13 to 10/13 but actually occurs for all other hops if we ignore the whole number part. i.e.
So if go 3 hops we have multiplied by 1000. I think this goes some way to explaining why the fractions that are opposite each other must sum to 1.
The next prime fraction: 17ths recur after its maximum of 16 digits, so effectively we have one, rather large wheel. Unfortunately, Excel gives up after 16 decimal places (please let me know if there is a way around this) but you can still see some patterns in here:
Beyond that, we found that the following fractions recur after the maximum number of digits so have only one wheel: 3,7,17,19,23,43,59,61 – someone had a much better calculator than me that showed 32 digits!
Other fractions worth exploring are:
- 31ths – contain 2 wheels of 15
- 37ths – contain 12 wheels of 3
- 41th – contain 8 wheels of 5
With the fractions that recur after a relatively few numbers of digits, we can find factors.
Because 1/41 is a decimal that recurs after only 5 digits, if follows that 41 must be a factor of 99999.
And in the larger wheels there are all sorts of patterns – pentagons and triangles in the wheel of 31ths apparently.
I’m not sure how far I’d go with a class, there could be several lessons worth in here. It would depend on how they responded, of course. Maybe 13ths would be sufficient unless…
This post captures only some of what we worked on in this session and highlights to me the depth of subject knowledge that can be gained by attending the ATM conference. Other sessions were more pedagogical in nature, but this one was pure fascinating mathematics and I was grateful to be surrounded by so many knowledgable and friendly people!