Given that there is a pattern in the sequence:
22 x 34, 23 x 34, 22 x 35, 24 x 34, 22 x 34 x 5, 23 x 35, …
Write the next six terms in the sequence, all in prime-factored form. Having accomplished that, write the 200th term in prime-factored form, and describe a method that will provide the prime factorization of the nth term of the sequence.
(Brown, 2002)
I found this problem in a paper that began by suggesting that what I go on to read would be more meaningful if I stop and solve the problem first. This sort of thing seems to happen in ‘mathematical’ reading quite a bit – so much so that it often seems to constitute a very special sort of reading which requires having a pen and paper handy!
I ask you now to do as I did, and spend some time thinking about that problem. When you feel satisfied that you have ‘a sense of’ the presented sequence and have ‘solved’ the particular questions posed, then please read on…
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My first question now is: did you follow the instructions? Or did you take a sneaky peak at the rest of this blog and return to the problem later?
This can be rephrased to ask: how comfortable, or not, were you to tackle a problem that is presented ‘naked’: without any scene setting, context setting, or proposed mathematical lenses through which to gaze?
I am often interested to reflect on my own and others’ responses to this kind of task. There are times when I feel a nagging concern that I might be ‘wasting’ time on something that is not useful (or immediately so) to me, in my context. This feeling can be mediated by the presentation of the problem itself: if I trust the person asking me to do this then I am more willing to comply. It is also affected by the appeal of the problem itself, and I think that is a very personal thing. I found the problem above immediately intriguing and didn’t need much persuading to dig out some paper and explore.
On my first look at the problem I felt that slight panic that comes with seeing something that I can’t immediately see a way through. I’m not used to seeing sequences presented in prime-factored form and initially this felt as though it was masking the underlying pattern – a part of me wanted to return to the safety of ordinary decimal form. But, instead, I copied out the sequence carefully into my notebook, forcing myself to start thinking about what stayed the same and what changed as I wrote down successive terms.
This sort of thing happened…
![](data:image/png;base64,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)
Followed by…
![](data:image/png;base64,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)
And it got me thinking about how easy it is to compartmentalise prime-factorisation as a piece of mathematics associated with particular procedures that are learned and then applied in very specific circumstances. When teaching I don’t think I placed much emphasis on noticing the underlying multiplicative structure of natural numbers and using the prime-factored form as a transparent representation of number with respect to questions of, for example, divisibility. The questions below feel very different in nature to those that I have experienced being used with students.
Consider the number M = 33 x 52 x 7
a) Is M divisible by 7?
b) Is M divisible by 5? by 2? by 9? by 63? by 11? by 15? Explain.
c) Is 32 x 5 x 73 a multiple of M? Explain.
d) Is 34 x 52 x 73 x 1318 a multiple of M? Explain.
(Brown et al. 2002)
This type of question perhaps helps to reveal some of the underlying multiplicative structure of natural numbers and provides opportunities to develop a greater conceptual understanding of standard procedures (such as finding the highest common factor or lowest common multiple of a pair of numbers). I wish I had made use of such questions sooner!
Taking a step back to those ‘basic’ concepts of odd and even numbers I am now more acutely aware of the importance of recognising that this is more than a property of the last digit of a number. Consider, for example, these questions:
For each of the numbers listed below, decide whether it is odd or even:
1) 1234567
2) 34five (34 in base five)
3) 121three
4) 3100
5) 399
6) 2100 + 3
7) 671
8) 750 x 340
9) 1234567 x 240
(Zazkis 1998)
Making explicit connections between divisibility, prime-factorisation and ‘oddness’ and ‘evenness’ has helped me to ground my thinking about otherwise isolated ideas in a more complex web of prior experiences. I suspect that this would improve my ability to communicate and facilitate learning of these things in the classroom too. This is part of a greater pattern in the work that we are doing on the Cambridge Mathematics Framework: exploring and paying attention to rich connections between ideas in mathematics, and considering the implications for teaching and learning.
What do you think? You can comment below or tweet us @CambridgeMaths with your ideas, comments or questions.