Wednesday 15 October 2014

Another New Way to Teach Dividing Fractions

Remember my post from June 2013 about a new way to teach dividing fractions? Well the other day I came up with another new method!

You might be wondering why I need a new method anyway, when the normal method (flip it and times it) works so well. Here's why: because that method is not intuitive. Well, it is if you understand reciprocals properly, and that the multiplicative inverse of a is 1/a. We don't normally go into the axioms of fields though, when we teach year 8 fractions.

Here's the way my method works. Say you want to do 3/4 divided by 2/3.

The way I usually think of integer division in my head is to make it into a multiplication. So 20 divided by 4 becomes 4 times something is 20. And then I think of what the something is. I think this is the way many students think about division.

So applying that to my question:

But 2 times something makes 3 and 4 times something makes 4 is quite difficult. So what we'll do is find an equivalent fraction for 3/4 so that 2 goes into the numerator and 3 goes into the denominator.

Then we just have to work out 2 times something is 18 and 3 times something is 24. Easy!

I thought of this method because there was a question in the year 9 MEP textbook that my students came across that was something like 3/4 x = 5/7 and you had to solve for x. But not having done algebra recently, my year 9s didn't think to divide both sides by 3/4. This led to them trying to find the answer by the method above. Interestingly, this year 9 class is the same class (then in year 7) that provided inspiration for the previous blog post on this topic!

What do you think, a waste of time, or a nice way in?

Emma x x x

Saturday 11 October 2014

Challenging Gifted and Talented Students Accidentally

This post is about challenging our most able young mathematicians.

I always thought I was good at giving students challenging learning activities. But on Friday I learnt something surprising. Students can be challenged a lot further than I ever thought.

I decided to use the UKMT's Team Maths Challenge resources from last year's competition to run an internal team maths competition with a class of high-ability year 8 students. The Team Maths Challenge is designed to be done by a team of two year 8s and two year 9s, and it takes place in March each year. Running it with just year 8s meant the students would have to work harder as they wouldn't have the older students to help them. Also it is only October, so they have got 5 months less experience than they should have when doing this competition, so that makes it even more challenging still.

I decided to only do the group challenge round and the crossnumber round, as I only had two hours. They started with the group challenge. They responded very enthusiastically, and there was much dialogue between the groups of four. The jottings they were doing were vast and full of impressive maths. When I went over to see what individuals were doing, I had a few comments saying it was hard, but most were too absorbed to even talk to me. One student, about two-thirds of the way through, announced that he had finally got the answer. The other teacher who was with me asked him, "What, you've spent all that time just answering one question?" (as there are ten questions altogether). The student gave the best reply I could ask for. He said, "Yes, but it was worth it".

They got all the way through the forty minutes without giving up, and when I announced there was only one minute to go, the room reverberated with pencil scratchings and the discussions became noticeably higher-pitched. I collected up the answer sheets. It was then that I noticed.

You see, there is a twist to this story. I mentioned above that the challenge would be difficult for the year 8s, for various reasons. But what I hadn't taken into account was that I had accidentally photocopied the wrong materials for them. Yes, I had in fact given them the Senior Team Maths Challenge. Yes, the one designed for year 12 and 13 A-Level students. Yes, the one that even students with targets of A and A* in A-Level maths find difficult. I gave them that.

So these intelligent little 12 year-olds sat for forty minutes tackling problems designed for those who have learnt a lot more mathematical techniques than they. Take a look at the problems I gave them here. They were able to access the questions, give them a good go, and even answer some of them correctly! Yes, many of the groups got one or two answers correct! And what is wonderful is that they weren't put off by the fact that there were several questions that they had no idea how to attempt. They just took the questions they felt they could make a start on, and ran with them. Like the boy who spent 25 minutes on one question - which was worth it. He never even checked with me afterwards to see if he had got it right. The satisfaction came from reaching a conclusion.

We may feel that we challenge our students. But this suggests that students can be challenged more than we give them credit for. What's great about the questions I gave them, was that they don't necessarily rely on knowledge of mathematical techniques (although I think some involved Pythagoras or Trig) so they can be accessed by all ages.

May I suggest you try this with your year 8s and see if you get a similar reaction? It would be interesting to see if they respond similarly or if they give up. The group that I did this with have been taught by a very skilled teacher whose strength is in getting students to be resilient and independent, so this may be why this worked.

Oh, and if you've never entered your students into the UKMT Team Maths Challenge, make sure you do this year!

Emma x x x