# Masking patterns

- Cambridge Mathematics
- Mathematical Salad
- Masking patterns

## Masking patterns

**Given that there is a pattern in the sequence:**

**2 ^{2} x 3^{4}, 2^{3} x 3^{4}, 2^{2} x 3^{5}, 2^{4} x 3^{4}, 2^{2} x 3^{4} x 5, 2^{3} x 3^{5}, …**

**Write the next six terms in the sequence, all in prime-factored form. Having accomplished that, write the 200 ^{th} 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…

Followed by…

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 = 3 ^{3} x 5^{2} 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 3 ^{2} x 5 x 7^{3} a multiple of M? Explain.**

**d) Is 3 ^{4} x 5^{2} x 7^{3} x 13^{18} 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) 34 _{five (34 in base five)
}**

**3) 121 _{three}**

**4) 3 ^{100}**

**5) 3 ^{99}**

**6) 2 ^{100} + 3**

**7) 6 ^{71}**

**8) 7 ^{50} x 3^{40}**

**9) 1234567 x 2 ^{40}**

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.