## 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”!)

## Quadratics

Currently my year 11 class are working on quadratic equations. I have always loved quadratic equations and as such i am quite excited to be teaching them. I have recently gone over them with my year 12 class so I had some good stuff on and I have built on that to create a good set of lessons around solving equations. I’ve shared them on TES here. Most of the stuff is mine, but a couple of the worksheets and the tarzia came from elsewhere, but i have included them for ease.

As part of my teaching with the group I have been trying to expand their matrhematical thinking. I want them to be able mathematicians who can think mathematically and solve a mathemnatical problem put before them, rather than pupils who are good at answering questions phrased a certain way. They all already have C’s and B’s, and to ensure they hit the A’s and A8’s they need they must be able to think mathematcially. For tomorrowa lesson i have come up with this starter (It is in one of the notebooks on TES):

I’m hoping that they will be able to make the jump and combine the ideas to come up with the right answer. Even if they don’t, I’m sure they will produce some good maths to help them on their way!

I also intend to show them this photo (also included in the lesson):

It provides a good talking point as the discriminant is replaced by a delta in the picture and next to it the delta is than defined as b^2 – 4ac. Why the quadratifc formula shows up in Peppa Pig is another question entirely. It seems stranbge how many cartoons involve higher level maths problems, and I may highlight Simon Singh’s new book “The Simpsons and their Mathematical Secrets”, as something they can potentially add to their christmas list! (Its certainly on mine)!

## Similarity, and other stories

Recently I have been looking into a variety of things. One thing is “Inquiry Maths” and another is something I found on the #matheme site through the explore the MTBoS project called “Notice and wonder”.

These got me thinking about how I could introduce some of their elements into some of my lessons. I had just introduced similarity to my year elevens and I was going to move to similar area and volume problems. So I came up with this starter:

I put it on the board and gave them ten minutes (I think) and let them get their teeth into it. A few were a little confused at first, but the discussions on each table enabled all pupils to make their own way to the correct answer. I didn’t know what they would notice or wonder, but I was pleasantly surprised to hear some of their comments:

“I notice that the area has gone up by four, not two. Does that mean you double the scale factor for area?”

-I loved this one, and refused to answer it, instead I asked him to enlarge the shape sf3 and see if the area was enlarged by six. I then got:

“It’s nine, not six. Why’s it nine? Stupid thing. Oh hang in, it’s squared. Oh, of course it’s squared! you times each side by it [the scale factor] and you times them together! Duh!”

Others I particularly liked were:

“I wonder if there’s a way of doing Pythagoras on triangles without right angles” (I told her that we would be meeting the cosine rule soon enough).

“the angles are the same! Wait, that’s how this SOHCAHTOA thing works isn’t it, cos it’s ratios an that.” (I said “very good, but can we use the proper name please!” then another pupil interjected with “Trigonometry”)

The lesson goes on to pose question prompts similar to those I’ve seen on inquiry maths in which we discussed similar volume and then I included a set of questions for then to attempt. I have uploaded the resource to TES:

http://www.tes.co.uk/teaching-resource/Similarity-and-other-stories-6374388/

## Drawing Axes

Today I’ve been thinking about drawing axes (as in the plural of axis, not as in two discworld dwarves preparing for a duel). The reason for this has been this:

This is the question on yesterday’s edexcel exam paper on drawing linear graphs. On the face of it a rather straightforward question, but there is one slight difference to its predecessors. The axes were not already drawn.

I’ve written before (here), about drawing axes. I always used to give out paper with axes drawn on as that was how the exams were, but then I observed an A-Level physics lesson and was baffled by the amount of time it too them to draw a set of axes, so I have started getting classes to draw their own. It is a skill that they should be able to do.

While thinking about this exam question I can’t help wonder how many pupils around the country Lost marks because they didn’t read the question properly and drew the graph in one quadrant.

I am liking the way the exam boards are trying to ensure the papers are less predictable, and I loved the triangle question (8 or 9). I’m excited about the changing curriculum and I can’t wait to see what the exam boards will do with it.