## Playing with Cusineire

Last year after a maths conference session from Pete Mattock (@MrMattock) and a number of other things happening I decided that I wanted to work on developing my own use of manipulatives in my teaching. I bought a box of rods for home and have spent some time playing with them along side my daughter. Last year I wrote this post on our early work with them.Since them we have messed around with them a lot, and have had some nice chats around them. Today I watched a nice twitter live session with Atul Rana (@AtulRana) and Simon Gregg (@Simon_Gregg) about their use (you can watch it back here) and I’ve also been reading a lot about their rods during lockdown. Here I want to share some of the things we’ve done with them over the last few days.Yesterday we were playing with the rods and she asked what she should do. I told her to make a pattern. One of the first things she came up was a staircase:I asked her if it was a staircase why it went up and down and up and not just up and she said, “that would make more sense”, and changed it.We discussed the order and I asked if we could do anything to it to make other shapes and she came up with this:Initially she just put another inverted staircase on top of the one she had done first, then she put the additional rods to make a frame. Before she had put the frame we discussed what each row was showing, and she said it was number bonds to ten. After we had discussed that I asked her what shape she had made, she said it was a ‘rectangle or a square’. I asked her which it was and she said she thought it was a square. I asked if there was a way we could know for sure, and she said there was, we needed to check if the side lengths were the same so she did that and decided ot was intact a rectangle.Today we played with them again, she made a face:Then she made this:Initially she said it was a “roof” but later decided it was actually a basket. We talked about the shape and I asked her if she knew what it was called. She said she didn’t, that she knew it had 4 sides but wasn’t a rectangle or a square, so I told her it was a trapezium and we discussed the properties of them.She then decided to do number bonds to 10:Then we looked at ways we could make 4:This was an activity that I’d seen Atul and Simon discuss and led to some quite interesting discussion. She decided to group the “ways” into distinct ones and repeat ones. I told her that the number of distinct ways to do this for a number was called the partition number. We also discussed the repeats and how she knew they were the same. She was good at describing the commutativity of addition but hadn’t heard the word commutative before. She then wanted to try with some other numbers:For 6 she decided not to do repeats, and initially only found these:Then added another:I find partition numbers really interesting, but after a brief discussion we moved on as I was about to go off on a tangent on Ramanujan’s Congruences and Modulo arithmetic which would have been over hear head just now.We picked a number to play with and she tried to find as many ways as possible to make it using inky red white and green rods:This led to some interesting discussions as to why we could do it all white or all green but not all red.I asked her what numbers we could make with all red and she said all the even numbers, so we looked at this and discussed why that is.We then looked at what other colours we could make those numbers with:She said all numbers could be made with just white so didn’t bother putting those on we talked about why the colours paired up, ie there are 5 reds for a ten as a red is 2 that means we need 2 yellows as yellows are 5. We discussed that this means multiplication was also commutative. I asked what numbers would be made by each rod and she said those in the timestable. We discussed this and the terms multiple and factor.She asked if there was always just 4 factors so I asked her to investigate 12:Obviously there is no rod for 12 so we made one out of an orange and a red and imagined it was a single rod. She was surprised how many factors it had.At this point we had been working a while and she was tired so we left it there. I can see that the discussion could easily have moved onto primes and their lack of factors, and squares and how they have an odd number of factors while all other number have an even number. And this would link back to our earlier discussions around commutativity.*This was an enjoyable way to spend time this evening and I hope we can do some more exploring with the rods in the days and weeks to come. If you have any great activities to do with the rods, or have written/read anything on this, then please let me know.**This has been the 4th post in what is a bit of a series around manipulatives, the others are:*Manipulatives, the start of a journeyFun With CusineireMeaning making with manipulatives

## 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.

## Equal Products

I come across a lot of puzzles and other maths things online and often save them for later, this evening I came across this little puzzle:

*The numbers 2,3,12,14,15,20 and 21 may be divided into two sets so that the product of the numbers in each set is equal. What is that product?*

I had saved this over a year ago, and cannot remember where I got it from, but I can see why. It’s a lovely little question that I intend to use as a starter next week and see how my classes get on.

**How I approached it**

Before you read on have a go at it yourself. Go on, you know you want to……..

Right, good, now you can see if I went about it the same way!

My first thought was that all the fun could be taken out of this by using a calculator, typing all the numbers in and pressing square root. So when I set it I will be adding the line “and which numbers are in each set”, and this is what I set out to find.

Firstly I set out the numbers in terms of their prime factors:

Then I tallied up the prime factors:

From this I knew that the product must be 2x2x2x3x3x5x7 which is 2520.

This, of course, answers the original question but I wanted each set. I looked at the numbers and the first think I noticed was tgat 14 and 21 had to be in separate sets, as they had 7 as a factor. I also needed to split 15 and 20, my intuition suggested that 20 and 21 should be in separate boxes, but it was easy to spot that the 2s and 3s, wouldn’t work out so I placed them together and fit the rest on around them.

A nice little puzzle, I wonder how my classes will find it.

## Mersenne and his primes

On Thursday my further maths AS class and I arrived at the classroom to discover an interesting slide still displayed on the board from a previous lesson.

My colleague had been teaching a lesson on prime numbers to his year 9 class and the slide in question was about finding new primes, how much money you can earn if you do, why this is and the “Great Internet Mersenne Primes Search” (and its unfortunate acronym).

A discussion ensued about cryptography and the uses of primes. It then moved onto the mathematical monk himself and his work in number theory. In particular that he noticed that numbers of the form (2^p)-1, where p is a prime, are usually prime. These Mersenne primes have fascinated me for years. How comes so many of them are primes? Why aren’t the all?!

The class were equally fascinated and we had a great discussion. We also managed to link it to a discussion we had had the previous lesson about p vs np, as trying to factorise (2^11)-1 is fairly difficult, but it is really easy to check if 23 is a factor. The class wondered if they could set a computer to test massive numbers for prime factors. I explained that yes, you could, but it would take so long to check the massive numbers it would be worthless. So if they can find a way to do it quickly they could become very rich.

We lost around twenty minutes of matrices time, but we have plenty of time to make it up. I think all pupils left with a deeper and broader mathematical knowledge and a healthy thirst to know more- which is at least as important.