## Matrices

I love matrices, I loved them at school from the very basic matrices that I learned at high school to the more hardcore matrices that were involved in my A Levels I always enjoyed them. Their associative, but non-communicative properties for multiplication fascinated me. As for the things that happen at university with tensors…..

So recently, I have been really enjoying teaching the chapter on 2×2 matrices to my FM AS class. From basic arithmetic with matrices to using the inverses to solve simultaneous equations. I think its fascinating to look at the transformations matrices geometrically, and as think it is certainly one of the topics that shows the links between the different strands of mathematics really well. My class have enjoyed the topic, and have been equally as fascinated as I have by the algebra and geometry involved.

I have uploaded the resources I used to TES here If you do use them , please let me know how they went.

The lessons on the notebook/exported PowerPoint presentation follow the chapter from the Edexcel textbook. I think the order is quite good, although I taught the bit on simultaneous equations prior to reversing transformations.

## 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.

## Multiplication Methods

Last night I saw an intriguing tweet from @mr_chadwick . Mr Chadwick is a primary teacher and he was worried that his daughter who is in year 8 was still multiplying using the grid method. This caused a fascinating conversation regarding the different methods of completing multiplication tasks and the pros and cons of each and it got me thinking quite a lot about the subject.

There are many ways to complete multiplication problems, the main ones being Grid, Column (aka Standard or Long), and Chinese Grid (or Spagetti Method or Lattice Method). (Also, I have recently been shown this ancient method by an A Level pupil who grew up in the Democratic Republic of Congo).

I was wondering if any one was particularly faster than the other, so I tried some out:

Each method took me a little over a minute to compute the product of 2 three digit numbers. (Except the grid method which took about 15 seconds longer). From experience I know that people tend to prefer the method they were shown first, and I don’t see a problem using any of them as long as there is an understanding of the concepts and that the person in question is proficient at using the method they have chosen.

I did think my prefered method for teaching someone who didn’t know any methods would be the Lattice Method, as I see a much larger potential for silly errors in the other methods than i do for this, but last nights discussion has got me thinking a little differently. The discussion moved onto applications in algebra. I know a lot of people prefer to use the grid method to expand double brackets, I personally prefer crab claw method, but i teach both and allow my students to decide, and some much preefer the grid. It also works quite well for larger polynomials, as shown here (in a video which rather confusingly calls it the lattice method!). The grid method can also be applied to matrices, as I have written about previously.

I’m still unsure as to which i prefer. The Lattice method gives a far lower chance of making silly errors, and I think it is the best one fro ensuring the decimal point ends up in the correct place when multiplying decimals, but the grid does have the benefits of being applied to much higher levels! I’d welcome your views on the subject.

I feel that i should also include another multiplication method which I discovered a few years ago, I was introduced to it as Japanese multiplication, but I’ve recently heard of it referrred to Gorilla Multiplication. I think it may have its origins in india and if you want to know more then Alex Bellos has written about it in his book Alex’s Adventures in Numberland (Another on my christmas list… and a book who’s american title is the most amazing title for a maths book I’ve ever heard: “Here’s Looking at Euclid”!)

## Grid Method Matrix Multiplication

This week I was “tweeted” at by one of my “followers” (@bt2bn) with the “hashtag” #MTBOS. I thought, “I wonder what that stands for?” and promptly looked it up. It stands for “Math(s) Twitter BlogO Sphere” and I thought “that sounds like something I’d be into.” So I signed up. (Sorry for the unworldly amount of quotation marks in that opening stanza…)

On further inspection, the #MTBoS seems to be a maths based #blogsync. This week’s prompt was to choose a tweet you’ve favourited and write about it. And this tweet arrived in my feed today and fits the bill entirely.

I retweeted this, and it turns out that quite a few people did already know about it and teach it this way. I’ve never encountered this method before, but given that the preferred multiplication method for most of my pupils is the grid method, it would seem to make sense to offer this as an option next time it comes up.

I was intrigued to see this new method, and I was also intrigued to know so many were using it. If you do, I’d be interested to here your views, and if there are any draw backs. I’ve got a while before it comes up this year, so I’m going to play around with it and get a real feel for any positive or negative points, and to see which method I feel is better for understanding.–