Author: tombennison

Negative numbers are one of the topics that I find hard to explain. I think one of the reasons is that for us teachers they just make sense – I certainly can’t remember finding them confusing at school myself.

When I’m talking about them at school I often use the usual analogies – money (debts and credits), temperature, positional analogies on a number line etc. Whilst these seem to help with addition and subtraction of negative numbers, multiplication is harder to explain. Using the debts and credits analogy seems to help explain the multiplication of a negative number by a positive number: for example if I associate a debt with a negative number and I have a debt of 20, then it makes sense that if I have 3 of these debts then my total debt is 60, that is, \(-20 \times 3 = -60\). However this analogy doesn’t seem to help when explaining why \(-20 \times -3 = 60\). I quite often end up talking about position on a number line, and a negative number being a direction to the left (a positive number being a direction to the right) and a negative sign indicating a change of direction – but I really don’t find this satisfactory.

Of course, in reality this property of negative numbers can be explained by saying that negative numbers need to satisfy all the algebraic rules of the positive numbers – associativity, commutativity and the distributive property (plus the existence of a multiplicative inverse, namely 1). And so, considering the following sum where we evaluate what is inside the brackets first:

\( -20 \times (-3 + 3) = -20 \times 0 = 0 \)

But of course we can use the distributive property to also say that

\(\begin{align} 0 &= -20 \times (-3+3) \\ &= (-20 \times -3) + (-20 \times 3) \end{align} \)

Having already convinced ourselves that \(-20 \times 3 =  -60 \), then by virtue of addition \(-20 \times -3 \) must be \(60\). This of course can be generalised.

My only issue is that this doesn’t really seem an ideal way to explain it to someone first coming across negative numbers. I’m really interested to hear how you explain this….

I had intended for a different post to go live today, but it looks like the post that was scheduled didn’t save properly so I have quickly thought of something else to write about. Looks like I will get it posted just in time so it is probably going to be shorter than some…. 

I enjoyed reading yesterday’s #summerblogchallenge post from Christine (@MissNorledge) concerning her top 5 sites for starters. Unfortunately though I was reading it on my iPad and consequently some of the links reminded me of one of my great pet hates. 

The two things that drive me round the bend with my laptop is the seemingly constant need to update the Adobe Flash plugin and the Java plugin: neither of these update in a “clean” way, they take far longer than you would expect and break my workflow. I’ve hated Flash for a long time; when I first had a Mac, Flash  would regularly crash safari because of some unexpected error. Unfortunately I can’t write as eloquently about my dislike of Flash as Steve Jobs did in 2010 in his famous “Thoughts on Flash” but suffice to say that in general I am extremely happy that neither my iPad nor my iPhone support Flash and I think that the Internet would be a better place without it. 

The only niggle is that there are some great maths resources out there that have been written in Flash and Christine’s post highlighted some of them and reminded me of this. Ideally I look for resources that are robust and will work on whatever device I happen to be using. This is especially true of I am intending on letting students access some of them – I know that se of them don’t have a computer at home but do have an iPad so anything incorporating Flash or Java applets is a no-no. 

I appreciate that recoding resources to avoid Flash is a massive undertaking but with the completely open standards of JavaScript and HTML5 I genuinely think it is worth the effort. More and more people are using iPads as their primary computing device, some schools are adopting a one-to-one iPad policy and so using the more modern and open standards is surely a no brainer. 

I’ve made my pentominoes resource using CSS, HTML5, JavaScript and the fabric.js library for this reason. I have a few other small resources in various stages of development and am hoping that I will be able to get them completed over the summer. Any of the Geogebra worksheets that I embed in HTML pages (such as this on conic sections and this investigating graph transformations ) use HTML5 so that they work on all devices. I also recently became aware of the website amathsteacherwrites from Jeff (@jeff2869). On this site (as well as some interesting blog posts) there are some great draggable maths activities, such as this matching graphs activity that work on any device. The wonderful Times Table Rockstars developed by Bruno Reddy (@MrReddyMaths) also works on iOS devices (though I understand that there will be dedicated apps coming out which will no doubt improve the UI experience even further for these devices). 

I hope that, over time, more resources which avoid the use of Flash come out and I will try to do my bit and add to them myself. 

Just a short post today where I thought I would share a resource that I found almost a year ago and have used with multiple classes since. This resource works as a starter, main activity or plenary depending on the class and where in a sequence you want to do it.

This resource is from William Emeny (@Maths_Master) and was posted on his excellent Great Maths Teaching Ideas website. It is a card sort of famous number sequences as shown below

  
The high resolution pdf version can be found on his site here. I think it is great how for each sequence you are matching the name, a pictorial representation, a way to produce the sequence, a fact about the sequence and the first few terms of the sequence. There are six sequences contained in this card sort

1. Even numbers
2. Odd numbers
3. Square numbers
4. Fibonacci numbers
5. Cube numbers
6. Triangle numbers

Thanks again for sharing it!

Apparently it was my first WordPress Anniversary on Saturday so I thought I would quickly re-share my first 3 blog posts:

1. Hello World – A First Blog Post
2. Why Teach
3. Thoughts on the Draft Maths A Level Content

I wasn’t as regular with posting when I first started ……

Last week Jo Morgan (@mathsjem) posted a picture of a Wordle formed from all her tweets since she joined Twitter. This led me to think about my Twitter history.

Back in 2009 I had experimented with Twitter, but didn’t stick at it and kind of forgot about it until I set up my “professional” Twitter account (@DrBennison) on the 1st of May 2014. I then sent my first tweet on the 7th of may, tweeting a photo of some resources I was cutting up for a Year 8 problem solving lesson.

Inspired by Jo I downloaded my twitter archive and used wordle to create the following Wordle from all my tweets. 

  
  I really value Twitter for collaboration with colleagues across the country (and further afield) and would encourage anyone to sign up and start talking to people. I haven’t met anyone on Twitter who isn’t friendly, encouraging and always happy to provide advice and share resources. As you can see from the above picture, I must have engaged with quite a few people repeatedly for their twitter handles to come out large.

For the next week or so I have pre-written my daily blog posts and the main tweets announcing them as I am going away up north for a break. I may not have such a good reason to not be able to write my posts on the day as @MissNorledge but I have stolen her Twuffer idea and am giving that a go.

It seems timely to write about something that I think is very important as a teacher to give yourself a break sometimes. It is very easy to constantly be thinking about work, marking, preparing resources and planning lessons. And, now there is the added addiction to Twitter that takes more time up…

I love having access to the internet, and to be honest I do often find it a bit of a panic when I haven’t got at least 3G on my phone. However sometimes it is really nice to have a break and be free from it all for a while.

This is of course possible during the holidays, but to a lesser extent I think it is important to try and manage it during term time too. I try to do no work on a Saturday for example, preferring to work more during the week and do a bit on a Sunday. I think having a day completely away from teaching is really beneficial and helps keep me sane – it would be one of my top tips for anyone new to teaching.

I also think it is important to be aware of “the law of diminishing returns”: you can spend hours trying to make a resource completely perfect, but as you spend more time on it you have less and less impact on the actual resource. I once remember spending 6 hours trying to get a piece of Excel to do what I wanted it to do for a 10 minute segment of a lesson – definitely not time well spent!

Please wish me luck with having a bit of a break, I do struggle to turn off to be honest. Good luck to you too if you are doing something similar 🙂

Yesterday’s post was a bit mammoth, so today’s post is much shorter! I’m sure you are all releived by this. 

Back in April I posted on my Mathematical Journey and in that post I mentioned how I felt lucky to have a granny who was a maths teacher. 

Ever since I can remember she would talk about maths with me, set me problems to do and send me coded messages in the post. As she taught at the Royal College for the blind this included teaching me Braille which sadly I can no longer remember. Whilst sorting through some stuff last night I found some old letters etc so thought I would share a photo here 

 
My handwriting hasn’t really improved much since then to be honest.