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

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

Confession: I Hate Drawing Graphs

Shocker! Emma, the girl who loves maths, hates drawing graphs. This includes: cumulative frequency curves, histograms, graphs of parametric equations, roots of unity, and those log ones from C2 chapter 10.

Students are always surprised to find out there's an aspect of maths I find boring. They exclaim, "but you're a maths teacher!" Well, the nicer ones do. The not-so-nice ones exclaim, "but you're a massive nerd!"

I have a huge problem with this. Think of your music teacher from school. They loved music, didn't they? Do you think they loved all types of music? Classical, rap, indie, ska, dubstep, One Direction? Probably not. You wouldn't be surprised if you found out there was a genre of music they hated.

Think about an art teacher. They love art. But in their house, would they hang impressionists, surrealists, er... etc... on their walls? Would they love every medium, every era, every subject? Of course not. If they did, you would probably think less of their love of art. "Proper" art lovers usually know what they like, and are dismissive of, or even angered by, artwork they don't like.

So why shouldn't mathematicians be the same? And more importantly, why shouldn't our students see this? I let my students know when we're studying a bit of maths I'm not interested in. Maybe this puts them off it. But who cares? Of course, when my favourite bits come up, I make that known too. I also encourage students to have favourites in maths. I like my students to feel they have a specialism. It makes them feel good when that topic comes up. I want them to see mathematics as being similar to music and art: subjective.

Do you admit to students that there are things in maths you don't enjoy?

Emma x x x

When Will I Use Algebra in Real Life?

There is one question my students ask me that I hate far more than all others. Far more than "Why are you just a teacher?" and "Do you have a boyfriend? What's his name? What's his job..." etc.

That question is: "When will I use simultaneous equations/ the laws of indices/ completing the square in real life?"

A good maths teacher, of course, would have a bank of answers to such questions. "Satellite dishes are in the shape of a parabola!" etc. Urgh.

Even though I do happen to know a few applications of algebraic principles to "real life" (picture me making air quotes with my fingers), I never tell these to my students. I refuse. 

First of all - real life? I'm sorry, are my maths lessons not real? When you enter room 204, are you entering some kind of alternate universe? Is a  maths lesson merely a state of mind? Some kind of lucid dream that looks real and feels real, but can't possibly be real because instead of English the teacher is speaking in an alpha-numeric jumble?

Secondly, where did students (and, for that matter, teachers) get the absurd idea that everything one learns has to have some kind of practical "use"? Can't we simply enjoy learning maths for its own sake? Does everything we do have to have a useful purpose? What kind of depressing life would that be?

Maths is beautiful. It is deep and interesting. It is a language. It is an art. It is a self-contained world with its own rules, patterns, and mysteries. So don't try to spoil my beautiful maths with your ugly "applications". 

I will leave you with some profound quotes from mathematician G H Hardy: 

"Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics". 

“Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.”

"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

Of course G H Hardy was wrong with that last one - his work was actually widely applied to genetics and thermodynamics. But the point is, that wasn't why he did it. And he didn't need these applications as motivation for producing this work.

Do you agree with me or are you someone who sees maths as a "tool" with which to get things done?

Emma x x x

7 Habits to Get Your Year 7s into

(Substitute "year sevens" with "seventh graders" if you're American, "S1s" if you're Scottish, or "first years" if you're posh/old/a wizard).

Year 7s are so cute, aren't they? So eager to learn, so willing to please. Some are actually shorter than me, which is nice. By Christmas they'll have had their growth spurts and be taller than me. That's why I like this term. (FYI if you want a mental picture of me, I'm 150cm tall, 50kg, and look a bit like Garth from Wayne's World but slightly more feminine).

The thing with year 7s is that they are lovely little blank slates. They're a fresh batch of play-doh just waiting to be moulded. At my academy, we keep our classes throughout their school career. So it's important that you get your year 7s into good habits early on, to make your life easier later.

Here are the good habits I'd like to get my year 7s into:

1) Leaving the answer as a fraction

To some students, an answer of 3/5 doesn't look finished - because you haven't actually carried out the division. They would much rather put 0.6 because it looks like a proper answer. We need to stamp this out! Fractions are infinitely superior to decimals. The use of fractions should be encouraged from day one. Don't you just hate A level students who convert all their fractions to decimals? Think ahead, teachers!

2) Lining up the equals signs

Some teachers are very anal about this and I admit I'm not really one of them. But it does make algebra look a lot more beautiful when there is neat line of =s down the page.

3) Drawing margins

Why do maths exercise books not have margins pre-printed like all the other exercise books?! It drives me mad having to remind students to draw margins every day. It's amazing how some of them still forget - even my top set year 11s! We need to try some Pavlovian conditioning to get them to automatically reach for a ruler and pencil as soon as they open their books.


4) Drawing diagrams

To solve any geometrical problem, the first step should be to draw a diagram. This is something good mathematicians do automatically. I don't know which is the cause and which is the effect, but it's worth getting our students into this habit.


5) Resilience

Perhaps the most important characteristic of a mathematician is resilience. Try something. If it doesn't work, try something else. Don't tippex out your first attempt. Don't sit there with a blank page because you're scared of writing something that's wrong. If we can instil this attitude into our youngest students, they will grow up to be good mathematicians, whatever their attainment level.


6) Using a calculator properly

Calculators are great. I can honestly say I haven't done bus-stop division with pen and paper for a good 10 years. Because I own a calculator. My computer has a calculator. My phone has a calculator. Even my tape measure has a built-in calculator. Don't diss calculators. However, some students become instantly stupid as soon as they pick one up. They don't question whatever answer it spits out. Students need to be taught to estimate the answer first to check if it's roughly right. Also, calculators are really sophisticated these days, and for example you can type in the entire quadratic formula in one go without pressing equals in between or using complicated nested brackets. Teach students how to do this. Teach them about the magical s<=>d key. Explain how the fraction key works. Get them to make frequent use of the "ans" key. And most importantly, get them to buy their own and bring it in every lesson. But make sure it's a Casio! (Sorry Sharp, but you make my life so difficult. Please stop making calculators.)

7) Taking pride in their exercise book

That's "jotter" if you're Scottish. Or "notebook" if you're American. (Or "parchment" if you're a wizard).
When an exercise book gets filled up, students are supposed to keep it. What do most students do? Throw it away. How awful is this? The problem is that many students see their exercise book as the place where they do work, not the place where they write down things to help them understand. Also some books are just horribly messy! I find that if your work is neat, you take more pride in your book, and hence you put more effort into your work. I know presentation is not about learning and hence presentation-focused comments is considered ineffective marking, but I think good presentation does lead to better learning.

Those are my personal picks. Are there any you would like to add?

Emma x x x

What is x?

How do you introduce "x" to your students?

I'm guessing you start teaching informal algebra in this kind of way:

10 + ? = 20

Where the question mark obviously represents the mysterious-sounding "unknown". Except it's not really unknown, because it's obviously 10.

Eventually you replace the question marks (or empty boxes) with letters. Not just the letter x, obviously, but you have to admit x is a popular one.

In this number sentence, what is x? I don't mean what is it's value, I mean, what is it?

2x + 4 = 12

It's an unknown. It is a number that definitely exists and has one particular value which at this very moment is unknown to us but in a matter of seconds will be completely known. Two quick steps and we and x will be on first-name terms.

In this number sentence, what is x?

2x + 4 = y

Suddenly, x is no longer an "unknown". It is a "variable". Meaning, its identity is still a secret, but it's not one specific number, it could be any (any any?) number in the world.

2x + 4 = y and y = 5.

Now, suddenly, although x is a variable, it has been forced to stop varying, and simply be unknown.

Can you see how the dual nature of x (or any letter really) could be very confusing for students? If students think of a letter as representing one particular number (even if they realise that number can change on a daily basis), this might hinder them when it comes to studying linear graphs, or functions.

Maybe we should try to introduce x as a variable instead of an unknown. Think about how you could do this, perhaps with your brand new, untainted year sevens with their clean-blanket-of-snow brains.

Let me know how you get on,

Emma x x x

What is a Regular Quadrilateral?

Have you missed me? It's been over a whole glorious month since I last posted. I've been lying on my sofa reading nineties teen romance novels and leaving the house only to visit Ikea for free tea and to pretend I live there. 

But anyway, I'm back and have I got a mathematical ponderance for you!

First, grab a piece of paper and a pen (or open a notepad file, for the more evolved among you) and write down (type) the definition of regular (as in, a regular polygon). 

Done? OK. Hands up who wrote this:

"All the sides are the same length".

And hands up who wrote this:

"All the sides are the same length and all the angles are equal".

Did any of you just write this:

"All the angles are equal".?

A more important question, perhaps, for teachers, is this: what do you tell your students?

Now another little exercise for you. Is this statement always, sometimes, or never true:

"If a polygon has all equal sides then all of the angles must be the same size".

Let's think for a moment. We know it is true for triangles, although you might not be able to prove it, or even justify it, beyond the fact that your Year Five teacher told you it was true during a particularly soul-crushing numeracy hour. We have also always assumed it was true for "big polygons" like octagons, decagons, etc. But is it true?

There is one type of polygon for which the statement is definitely not true. Quadrilaterals. Sorry if the title spoiled this major reveal for you. A square is equilateral and equiangular. However, a rhombus is equilateral but the angles are not all the same size. 

OK, here's another one for you:

"If a polygon has all equal angles, then all of its sides must be the same length". Always, sometimes or never true?

Again, think about triangles first, and then think about hexagons etc. Can you prove your conjectures?

Again, the statement is clearly false for quadrilaterals. A rectangle is equiangular but not equilateral. 

So what is a regular quadrilateral? Is it a square, a rhombus, or a rectangle?

My initial thought (because I admit, I didn't actually know the correct answer), was that a rectangle is regular. I thought this because geometry mostly comes from Greece, and in Greece, they're mostly bothered about angles. Hence the word polygon: "poly" meaning many and "agon" meaning angles. So a hexagon is literally a shape with six angles. In the UK, we're more likely to say a hexagon is a shape with six sides. So I thought a regular polygon would mean a shape with regular angles. 

However a quick tussle with my favourite search engine revealed that a regular polygon must be both equilateral and equiangular. 

I'm ashamed to admit I think I might have taught students that regular just means equilateral. Or, even worse, I think I might have even implied that equilateral shapes were always equiangular! It's funny how such a simple little definition can be messed up because you think you understand it perfectly (after all, I was taught it in year five). Maths teachers like me need to make sure we are completely clear about these things. Maths doesn't leave much room for error, and these definitions need to be water-tight. 

On that note, have a good first day at school!

Emma x x x 

PS I am of course talking about convex regular polygons. Non-convex (star) polygons are of course a whole other kettle of fish!