## Vary and twist : Inverse proportion

Another ‘vary and twist’. This time on inverse proportion. Download it here.

I have added a section at the top making explicit that students should make an expectation of the answer before working it out, as I think this is where the power of these types of activity come from.

My thinking

1. A question to start
2. Does swapping x and y matter? If not, why not?
3. Doubling x, what happens to y? What you expect?
4. Doubling y in our initial relationship. Why does this differ in it’s result from doubling x from before?
5. As above, but for x
6. What happens if we half the value of x we want? Maybe this should be moved up in the list to question 4.
7. A multiplication of 10 from the original. Does it hold up to more than halving?
8. Now we’re finding x not y!
9. We’ve doubled the y. What happens to x? Does the relationship work both ways?
10. A decimal answer

As always, feedback would be lovely. I’m not sure I’ve correctly hit the sweet spot between making links and confusing the pupils. I’m also not sure that they flow the way they should.

I guess since I’ve done dividing in a ratio, proportion and inverse I should finish off the module. Expect simplifying a ratio and recipes (which will be a challenge to lay out) to come next.

## Vary and Twist: Proportion

I’m going to call these Vary and Twist.

Section A is some variation practice. Here’s my thinking behind the variations:

1. Simple example
2. Context change, numbers stay the same
3. OK. Now double what we require. answer doubles
4. Amount needed triples, answer triples. Also 6 x first one
5. We’ve doubled the number of pens in the first one. essentially having the price…
6. Now doubling the price
7. Halving the price. Link between 5? Maybe 6 + 7 should be swapped around
8. Less pens then initially. Answer will get smaller for first time
9. Rounding introduced.
10. Don’t just quadruple your answer from 9! You’ll introduce a rounding error!

Sections B and C add a little extension, but they use the idea of spaced practice to make students think a little more than standard blocked practice.

Think I might be using this format for more exercises. I’ve done the table layout now. Took me ages.

I’ve found it’s a little limited, though. Tried to do a sheet with probability trees and it quickly became unmanageable with images.

Comment etc.

## SSDD Problems

I’ve been using Mr Barton’s SSDD Problems with my year 11 class. I think they’re really good. They do a really good job of making students pick out what they need to do.

Here’s a great one.

This is great because it makes pupils read the question. It was fantastic for revision because we got to discuss 4 topics in one lesson.

There are loads of great SSDD problems. I recommend Socks in a drawer, this nice circles one and this difficult formula one.

Some are less good, though. This one for instance:

I made this one. I’m not sure it’s a proper SSDD. It’s just 4 different questions based on the same image. Not sure it quite fits the SSDD criteria. I’m also not quite sure I can put my finger on exactly what the SSDD criteria is.

As ever, thoughts appreciated.

## An attempt to combine two ideas: dividing in a ratio

A worksheet attempting to combine Craig Barton’s ideas on variation theory (only changing one part at a time) and Dani and Hunal’s ideas around making students make choices. I’ve tried to build up to that.

Maybe by trying to combine both I miss the point of each.

Would love criticisms and thoughts.

## #Mathsconf14 Takeaways

I thought I would use a similar format to my ATM Don Steward takeaways. These are just some quick thoughts. They’re by no means things I’ve thought about for a great deal.

1. My first session was Peter Mattock’s session on averages. It was great, although in sessions like this I worry I talk too much. On our table we kind of basically agreed we should stop using the term ‘average’ as it’s unhelpful. Mean, median and mode are also different skills. ‘Levelling off’ or sharing, centrality and frequency respectively. Maybe they should be taught as different units. I showed Dudamath, which I love as it has a lovely stats module that does ‘levelling off’ well. Here’s a little video:
2. I was in Hinal Dhudia and Dani Quinn’s on ‘Making it stick’. They talked a lot on variation theory and questions that make students think. For instance a set of five questions, instead of looking like
$1. 2a + 3 = 7$
$2. 3a + 10 = 13$
$3. 7a + 5 = 19$
$4. 9a + 1 = 10$
$5. 6a + 3 = 15$You might get something looking more like
$1. 2a + 3 = 7$
$2. 4 + 3a = 16$
$3. 10 = 2a + 6$
$4. 8a + 1 = 5$
$5. 5a + 3 = -2$
I’m not sure how I felt about this. I get that we need student’s thinking more. Students often treat work as their doodle. Something to keep their hands busy. And we can play into that by giving them ‘busywork’. Learning is the residue of thought, and all that jazz. I’m just not sure if this is the way to do it. I worry that too much intervention is required between each question, and I’m not sue where this fits in with variation theory.NOTE: This is not to say I didn’t enjoy Hinal and Dani’s session. Quite the opposite. It gave me lots to think about
3. At lunch I did Martin Noon’s session on braiding. I found it extremely difficult but fascinating.
4. After lunch came Jemma Sherwood’s session on building effective learning. Lots of good practice in this. I use Corbett Maths 5-A-Day with every class, every day. I really think it helps build up long term memory and it’s effective at building routines.
Jemma talked about the idea of a three-part lesson being an issue. We fit things into an hour, even if that’s the appropriate amount of time. Not everything needs a plenary after every lesson. I agree with this generally, but I think the idea of ‘episodes’ of learning is useful. Something to ponder.
5. Last, but not least was Craig Barton’s packed session. The man’s a flipping’ celebrity.
He talked a lot about only varying one aspect of a question to make students predict what was going to happen. Having predictions either confirmed or rejected is powerful. I am thinking of building this into stuff that I do.
He also launched two new websites! https://mathsvenns.com with rich tasks and the amazing SSDD problems, which I think are PHENOMENAL. He’s looking for more. I’ll get writing one this week.

As usual MathsConf was ace. I love the conversations you’re able to have and the enthusiasm of everyone there is inspiring and wonderful.

I’ll be going to MathsConf15. I’ll see you there!

## Maths in the real world – The Sun’s Brexit percentages

This piece of maths in The Sun was all over Twitter this week. It was retweeted and praised by MP Jacob Rees Mogg. The maths in it is wrong. Wrong, but interesting.

A lot of people tried to correct it on Twitter, but Twitter is not the place for explaining mathematical content.

The Sun has tried to calculate the cost of items – the tariff percentages. I’m not going to talk about things like tariff calculations being based on other things apart from the retail price, or the fact that some of these items are from tariff-free countries. I am merely going to talk about the common percentage misconception and why maths matters.

The easiest way to see the error The Sun have made is by looking at the calculation for butter.

This seems to make sense. 50% of £2 is £1, so take £1 off the price, and presto. Except, working backwards we can see that this doesn’t work.

If butter cost £1, and we applied a 50% tariff to it, the final cost would be £1.50. The maths doesn’t stand up when working in reverse.

How should we do it?

I think the easiest way to think about this is to call the original price 100%. After the 50% tariff has been applied our new price (£2) represents 150%. To find 50% of this new amount, we actually have to divide £2 by 3, as $\frac{150}{3} = 50$. We can then take away our 50%. $2 - \frac{2}{3}$

The new price should be £1.33 (and a third).

Does this matter?

Well, kind of. I don’t think anyone would have been less convinced by the new, accurate figures. I do, however, think that we need to be careful about promoting sloppy mathematical thinking in the public arena. At the very least, it is indeed an example of ‘real life’ maths.

Several people on Twitter have suggested it as a nice activity to do in lesson, to correct the figures. However, I would be careful about the surface of the problem distracting from what is intended to be learnt.

I think The Sun could do well to offer a correction, along with a page explaining the mathematics.

## Don Steward ATM session: 5 thoughts

I’m trying a new kind of post. I went to the ATM/MA Don Steward presentation on Saturday. It was excellent, and if you get a chance to see Don speak, go. Don has uploaded his slides here. Rather than the daunting task of writing a full follow-up, I thought I’d just write my thoughts down in a quick bullet-point format.

• It was nice to do some maths.
Part of the session involved doing some of the activities that Don had devised. It was nice to sit with fellow professionals and actually just do some maths. Lots of interesting conversations were had.
• Don is a master of task design
I love the way that Don designs tasks, in both meanings of the word. His worksheets are always beautifully presented, but also very well designed to introduce concepts of generality. I love the way he might introduce something simple like

and twist it like
this was in an introduction to the difference of two squares.It was nice to see these tasks introduced by Don, though. Sometimes it’s difficult to tell what someone means from a task by looking at it.
• I was struck by the importance of subject knowledge.
Don’s activities go off on interesting tangents, often only made possible because of Don’s subject knowledge. Subject knowledge is always something to work on. We’re all always learning.
• Sometimes, a nice visual is useful. Here’s a lovely one showing the $(n-2) \times 108$ rule.

## Everyone should read ‘How I Wish I taught Maths’

This isn’t a review. I’m about halfway through Craig’s book, so I can’t offer a judgement as to if the book is good or not. This is simply a suggestion: everyone should read this book, and everyone should read it ASAP.

The book chronicles Craig’s mathematics teaching journey, from being the king of tarsia, to the complete about turn he has taken.

I’m not sure I agree with everything I’ve read so far, but that’s what makes the book great. It’s sparked discussions. Several of us are reading the book at work (I’m trying to get the entire faculty to do so. I’ve persuaded 5/12 so far) and the discussions around the book have been, as Craig would say, flippin’ brilliant.

It’s the book’s willingness to get into the weeds that I really have appreciated so far. This is a book for maths teachers. It’s laser focused, which is what makes it so great. Ever sat in a CPD and though ‘Yes, but how do I do this in maths?’, well this book is the exact opposite of that. It’s full of wonderful examples I wouldn’t have come across (ie : $\frac{6}{35} \times \frac{35}{3} \times \frac{11}{14}$. These discussions have been gold dust. They’re enthusiasm generators.

One of the things that I think can get lost in teaching maths, is that we are learning along with our pupils. I am trying to teach my pupils maths, but I am also trying to learn how to become a more effective and better maths teacher. Every chapter so far has had me walking into class the next day ready to try something new, and reflect on why it did or didn’t work.

This book is, so far, a wonderful addition to the conversation and I really recommend you read it.