## Primed

Recently I’ve seen a couple of things on twitter that I’ve thought quite interesting because they have got me thinking about the way I tackled them.the first was this:

It was shared by John Rowe (@MrJohnRowe) and posed a nice question looking at two cuboids that had the same volume asking for the side lengths. It said to use digits from 0-9 without repetition. My immediate thought was to consider prime factors:

2, 3, 2^2, 5, 2×3, 7, 2^3, 3^2

I only had one each of 5 and 7 so discounted them. And I had an odd number of 2s. This meant I’d need to discount either the 2 or the 8. Discounting the 2 left me with factors of 2^6 and 3^4 thus I needed 2^3 x 3^2. I had a 1 I could use too so I had 1x8x9 = 3x4x6. This seemed a nice solution. I also considered the case for discounting the 8. I’d be left with 2^4 and 3^4 so would need products of 2^2×3^2. 1x4x9 = 2×3×6. Also a nice solution.

Afterwards I wondered if John had meant I needed to use all the digits, I started thinking about how this could be done and realised there would be a vast amount of possibilities. I intend to give this version of the the problem more thought later.

The next thing I saw was this:

From La Salle Education (@LaSalleEd). Again it got me thinking. What would I need to get the product 84? The prime factors I’d need would be 2^2x3x7. This gave me only a few options to consider. 1 cant be a number as the sum will always be greater than 14 if 1 is a number. 7 has to be one of the numbers, as if its multipled by any of the other prime factors we have already got to or surpassed 14. So it could be 2,6,7 (sum 15) or 4,3,7 (sum 14) so this is our winner.

These problems both got me thinking about how useful prime factors can be, and they both have given me additional thoughts as to what else I could and should be including in my teaching of prime factors to give a deeper and richer experience.

Prime factors can crop up so many places, and I feel sometimes people let them get forgotten or taught in isolation with no links elsewhere. I always use them i lessons o surds and when factorising quadratics but know lots don’t do this.

*I will write more on prime factors later, but for now I’m tired and need sleep. If you have any thoughts on prime factors or any additional uses not mentioned here I would love to here them. Please let me k ow i the comments or on social media.*

N.b. The La Salle tweet includes this link which takes you to a page where they are offering some great free resources.

## What’s so good about Maths?

The other day Colin Beveridge (@icecolbeveridge) wrote this nice piece about why he loves maths. Today one of my year twelve students asked why I had chosen to study it at degree level. This facts combined have made me decide to share my thoughts here.

I’ve written before about how Maths is a beautiful subject, so I racked my brains and tried to pinpoint when I really decided I loved it. I remember at nursery school learning to add and subtract. I loved doing it and I have vague recollections of doing sums at home as a little one, sums aren’t really maths, in the proper sense, but I guess this was an early indication of what I would like.

I always loved problem solving too, really enjoying logic puzzles and computer games with logic, problem solving and maths puzzles built in (such as “Fun School“, and “Granny’s Garden“. The latter has received a 21st century make over!

My memories of primary school maths aren’t overly strong, on the whole. I was very good at it. I remember not seeing the point in learning times tables because I could work out the answer quick enough anyway, this annoyed my teachers and my mum. (In a slightly off topic, but related note, my yr 3 you teacher, Mrs Bremner, once told me off for not learning my spellings, even though I had got ten out of ten on the spelling test!)

Two of the things I do recall primary maths are from year 5. I have vivid memories of Mrs Hanel teaching me how to find the area of a triangle and how to construct Perpendicular Bisectors. These were things I thought were cool, but I think the real wow moment came in year 6.

During year 6 I finished the maths syllabus that the school was running, and my teacher, Mr Jones, found me a text book in the store cupboard that was left over from when the school had been a middle school. I started working on Algebra. At first I thought it was pretty fun, solving equations to find an unknown, but the moment I think I fell in love with the subject was when I realised the link between equations and graphs.

When I moved to secondary school I spent the first year fairly bored in maths. The classes were mixed ability and the lessons were strange. There was a number of boxes which covered each topic. Every few weeks the classes would swap and get another box. The idea was you would work through the booklets, marking your own and ten do the review tests for the book which the teacher would mark. I just did the reviews.

I still loved the subject, but the lessons did bore me at times, this is evident in my national record of achievement which says “maths is my favourite subject, but sometimes it’s too easy and this can be boring.” This has certainly shaped the way I deal with Gifted and Talented pupils, making sure they are always adequately stretched.

My love affair with maths continued, and still does to this day. Some topics I love (Trigonometry, Coordinate Geometry, Algebra, Calculus, Number Theory, Graph Theory and many more) some less so (Bounds, Numerical Methods, Trial and Improvement, Transforming Shapes by Hand), but as a whole I love it. I love the beauty of it, I love the satisfaction of solving a problem, I love the process of wrestling with a formula, I love that there is not just one “right” way to go about a problem, I love the stories around the maths and the mathematicians, and I love the connectiveness that’s evident with maths.

## A nice little number puzzle

Today I shared this puzzle with my further maths class. I was amazed how quickly they got it, and they asked if they could do another. One of them said “Google ‘Hard maths puzzles'”, so we did and found this nice website which I think warrants more exploration.

The puzzle we selected to discuss was:

*A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:*

* There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?*

The discussion around it was very interesting and I thought worth sharing here.

I tried to take a back seat during the discussion, which went a little like this:

Student A: “There’s at least 1 – The first person will open the first locker and no-one else will touch it.”

Student B: “Likewise, locker 2 will be shut. And 3”

Student A: “4 will be open!” then while inking “5 shut, 6 shut, 7 shut, 8 sht, 9, open” then “Will all the square numbers be open?”

Student C: “Yes! they will!”

Me: “Why?”

Student A: “They have an odd number of Factors!”

Me: “Indeed, but the question asks for how many…”

Student C: “the square root of 1000 is 10 root 10”

Student A: “That’s around 10pi”

Me: “So?”

Student A: “That’s 31 point something so there are 31 open because the next square will be 32!” *(That’s an exclamation, not a factorial)*

Me: “Excellent”

## Aaargh Ruddy BIDMAS

The order of operations is commonly taught using one of the following mnemonics: BIDMAS, BODMAS, BOMDAS, BIMDAS, PEMDAS, PEDMAS or, if you’re Colin Beveridge (@icecolbeveridge) Boodles! (I’m not going to discuss that here, so if you’re interested, do follow the link)

For the first six the letters stand for the following: B: Brackets, P: Parentheses (essentially the same thing) I: Indices, E: Exponents, O: Orders (Again, the same in essence), D: Division, M: Multiplication (Inverse operations) A: Addition, S:Subtraction (Again inverses).

There is a massive problem with these mnemonics, and their use in teaching. I do, however, think it’s down to the teaching rather than the mnemonics themselves, and possibly down to a lack of understanding from some who teach it.

And example of the problem I’m hinting at occurred on Friday with my top set year 8 class. As part of a ten quick questions core skills starter I included the question:

3 – 1 + 2 =. A problem free, easy, question for a class working at level 6-8 one would think, but the uproar when going through the answer was unbelievable. I asked one lad what he had written, he told me zero, I asked if anyone could tell me what he’d done wrong and was met first with blank stares, then with “Sir, it is zero. Have you forgotten about BIDMAS”. The scale of the misconception was enormous. After the first person said it the entire class were up in agreement. I settled them down, jettisoned that days lesson plan and retaught them the order of operations, but properly.The misconception is heavily tied to the mnemonic. A comes before S in BIDMAS, so Addition comes before Subtraction. But this isn’t the order of operations. When I teach it I make a point of telling them all six mnemonics mentioned above and specifically drawing their attention to the way the D and the M are interchangeable. Discussing that because Division is the inverse of Multiplication they are, in essence, the same operation, certainly “of the same order”, and as such take equal precedence in the order of operations so you read from left to right. I then ensure they know this is true for Addition and Subtraction. If I ever write one of these mnemonics I always write it:

BI

DM

AS

Or even

BIR

DM

AS

With the R standing for roots, as this shows the inverse relation for that level too.

The year 8 class now understand the order of operations, despite protests that “(insert name of former colleague here) told us addition ALWAYS comes before Subtraction!” as they all thought this, it is possible. I’ve seen it taught wrong in more that 1 school. I had a real row with another trainee on my PCGE course in a microteaching session when she taught it wrong and tried to say I was wrong. Every year the pupils from one of the feeder primaries get it wrong, so I think their yr6 teacher must teach it wrong. I’ve also been brought in to settle arguments on social media networks that have erupted over those stupid viral questions based around this. All this shows that there is a perpetuation of this misconception in this country, and we need to stamp it out.

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