New lesson: Finding roots of quadratics by factorising

This is my biggest set of lesson stuff yet.

The PowerPoint introduces the topic. It sets up some example-problem pairs.

It then leads onto some vary and twist questions, which are available here as Word file.

The PowerPoint then goes onto talking about solving when rearranged and includes a Victorian textbook exercise and some questions where you have to find x, given the area of some shapes (this took ages to do. I posted it to Twitter twice. Both times the questions were written wrong. Doh!)

I’ve also included an open middle problem solving thing and a timed learning check, that graduates in difficulty.

Talking about learning checks…I’ve also written some Timed Questions that increase in complexity every 3 questions. I now have over 6000 questions in my Timed Questions database!

As always, comment and critique.

There’s probably 2 lessons of stuff here, easily.

EDIT: One of the area problems was still wrong! Thanks to Professor Smudge for catching that 🙂

New Resource : Simplifying ratio timed questions

Like the old ten quick questions, but with three levels of difficulty.

Level one (first three questions) : Prime divisor

Level two (second three questions) : Compound divisor

Level three (third three questions) : 1 : n

Level four (last question) : Units

Tempted to extend it to do 12 questions. That last question feels odd on it’s own.

Also the PDF generation is weird. Think you have to click go for it not to be a blank page. And allow cookies.

I messed up the question generation code so had to go back in and manually edit hundreds of answers 🙁

Get it here.

Sometimes if you mess around with changing the id and sid parts of the url you might see stuff I’ve been working on but not blogged about yet. (Try 8,1 8,2 for instance )

New resource : Equivalent and Simplifying ratio lesson

Wrote this powerpoint today.

It’s like my ProjectALesson stuff but I’ve taken A LOT of explanation out. I’ve also got some stuff you need to print out. I’m not sure making them ‘pick up and go’ was that helpful.

It takes a lot of inspiration from variationtheory.com. It’s my new favourite website. It’s ACE.

I also took a lot of inspiration from Jo Morgan’s MathsConf session . She was talking about covering stuff in depth, and not missing an opportunity to go into negatives and fractions.

I made what is probably my favourite exercise of all time based on this. My colleague Mike helped me write the questions (Hi Mike!) and some of them are superb, although they might be a bit much/challenging. As an aside, sitting with my colleague and writing some questions was a great use of our time. Mike came up with some questions that I wouldn’t have. I love the misconception of 2^3 : 2^2 being the same as 2 : 3. Mike came up with that. It also made us discuss stuff really deeply. If you’ve got some gained time, write some questions with a colleague.

See what you think. As always, the discussion is the fun bit.

EDIT : I’ve uploaded this ‘Pixel Picture’ as word file on TES here. It makes the formatting a little nicer.

Updated resource: Collecting Like Terms Powerpoint

Doing teaching resources is hard! By the time you’ve written some, you realise what you’ve written is a little rubbish!

For instance, my ProjectALesson powerpoints need massive improvement. They often contribute to a modern epidemic in teachers : click-itis. I’ve seen this quite a few times. I’ve been guilty of it myself. I have no idea how to get around this, really.

One of the things I did whist updating my collecting like terms PowerPoint was to remove a lot of the writing. I think this was pointless anyway. It can’t have done anything other that overwhelm students with too much stuff on the screen.

I have added a little bit of silent teacher modelling in there, talking about which terms are like.

I’m not sure what to do, really. Update the old resources or make new, better, resources. Hmmmm.

Vary and twist: Collecting like terms

Download the worksheet here

Not sure how I feel about some of the decisions here. I’ve introduced a bit of index laws towards the end of the sheet. Is this madness? I thought I would add it to reinforce the difference between simplifying powers and simplifying regular expressions. Maybe it’s too much.

As usual here’s my little justification for the first 10 questions.

  1. A simple one to start
  2. If you change the letter, it’s the same process
  3. You can have multiples of terms
  4. And it doesn’t matter where in the expression they occur
  5. You can have 3 terms
  6. And it doesn’t matter where in the expression they occur
  7. Introducing a negative for the first time. At the end to make it easier
  8. But the negative can occur anywhere! Here it actually makes you use negatives unless you collect the terms first
  9. Introducing terms like bc. It’s not the same as b + c
  10. We can do some division

Later questions cover stuff like ab being the same as ba.

I quite like the last question

How often do we present students with expressions that can’t be simplified?

If you use it or adapt it, please comment here.

MathsConf15: 6 Takeaways

OK, so yesterday it was MathsConf 15. I’ve got loads of takeaways.

  1. The first session I went to was Jo Morgan’s session on indices. It was phenomenal. Jo puts a lot of work and research into her sessions and it really shows. If you ever get to see her do an ‘In Depth’ presentation, do.
    The crux of the presentation was this: Jo used to cover the 3 main indices rules in one lesson. In fact, so did I! I even have a resource on it on TES! What jo, showed, though, is that by doing this we’re denying students the ability to go properly in depth with the topic and gain a proper understanding. For a start, I’ve been getting the language wrong.

    The ENTIRE thing is the power. The little number is the index.
    We also refer to power in so many different ways that it can be confusing for children
    There’s also loads of subtitles in even the multiplication index law. Maybe by going too fast we’re missing talking about a lot of subtleties that could trip pupils up. (Hat tip to Ben Gordon for these pictures, btw)

    There’s also a lot of practice we can do. Just because we don’t want to talk about what negative and fraction indices are yet, doesn’t mean we can’t have fractional and negative indices in our answers. It makes them less scary and special when encountered later. We can also start using this as an excuse to practice negative number work, or fractions work or algebra work. Hey, why not ask questions like this:

    As you can see, there was loads of content in this talk. But there were a few things I will take into my teaching straight away:
    Atomise. Don’t try an do too much at once.
    Instead of moving on, go deeper
    Don’t miss chances to practise negative numbers at every opportunity.
  2. Next I went to Peter Mattock’s measuring session. Measuring seems so simple, but there’s so much in there. My 1/U borderline year 11 this year really struggled with a question where a pylon was 11.5 x the height of a man. They couldn’t see how you could have half a man. The multiplicative reasoning of measures hadn’t been embedded. Once more, we need to think more deeply about what we do.
  3. I then went to a session PRESENTED BY ME! I’ve never done a session before at something like this, and I was really nervous. I even wore a full suit as a kind of ‘suit of armour’ to protect the nerves. I got even more nervous when I got there and MR BARTON WAS IN MY SESSION. Mr Barton is a maths legend. It’s because of his ‘resource of the week’ that I first started sharing my resources. I wanted to be resource of the week. And I was eventually! It was such a buzz. I can’t be the only one that started sharing my resources because of him. Add to that the impact his book has made and his impact in maths teaching in the UK is incalculable.
    My session went OK. It was on GapMinder. But the best thing about it was the amount of people who came up to me afterwards to share ideas or resources and the amount of people who pointed me to great stuff on Twitter. Some people got in contact to disagree, which I loved. I don’t want to be right, I want to have a conversation.
    If you’re in two minds about doing a session at MathsConf, do. You get so much out of it.
    If that wasn’t enough, I was mentioned on the Mr Barton Maths Podcast! This is genuinely the proudest moment of my professional career. I am still beaming.
  4. Fourth was Mr Barton, Jess (I’m sorry I didn’t catch her last name, which makes me feel bad) and Ben Gordon’s session on variation. It was flipping brilliant. They talked about making student’s think more deeply about the structure of what they’re being asked. Anyone who has read this blog recently knows it’s something I’m into at the moment. And they launched a new website variationtheory.com with loads of variation exercises and stuff. It is, obviously, brilliant.
  5. This last takeaway is something I really debated putting down. But as much as I enjoyed the day, I also found it really hard. I am generally a very socially anxious person, especially in crowds. MathsConf was very crowded. I found the ‘networking’ sections really, really difficult. Mingling with strangers. It’s something that I’ve always found hard, although teaching has certainly helped me improve in this area. You always worry that you’re either boring people, or attaching yourself onto people like a limpet. I also repeat myself a lot when I do this. I can also sometimes be a ‘bit much’. If I did this to you, I apologise.
    However, if you’re someone who is socially anxious, I would still recommend going (maybe bring a friend? Having a friend there has helped me). The people at MathsConf have always been very friendly, and put up with me brilliantly. I would like to thank Jo Morgan and Peter Mattock in particular for both always finding time to talk to me and being really friendly. In fact, everyone at mathsconf is always been friendly. I’ve not had one negative encounter. It’s worth the anxiety.
  6. Follow Paul Rodrigo on Twitter. He’s a good egg.

Vary and twist: Simplifying ratio

Download the word file here

This one was hard. I spent ages rearranging questions and looking at what should be added. Specifically, I had a massive dilemma when it came to introducing fractions. I was trying to point out the ways in which simplifying fractions and simplifying ratio were similar, but I’m not sure that I haven’t just led students down the wrong path thinking they’re equivalent. For instance 5 : 6 is 5/11 and 6/11, not 5/6. Hmmmm.

The variations I used for section A.

  1. An example where you can use a prime divisor
  2. The opposite way around. What happens to our answer. Order is important!
  3. Half one side. 8 : 5 becomes 4 : 5
  4. One that’s already as simple as possible. Time for some questioning? How do you know you can’t simplify it?
  5. It’s not just reducing the numbers down. Here you have to multiply up. Deals with what simple is. I have changed this from the picture to make only one number vary from the previous question.
  6. Needs a non prime divisor. This isn’t really a variation, though. It has nothing really to do with the previous questions!
  7. Again, double one side
  8. Double both. Our answer does not double!
  9. Adding a third part of the ratio. Changes the answer significantly.
  10. Doubling two parts here. Our parts don’t double in our answer!

If you amend this and it works better, please let me know!

Vary and twist : Two step equations

Download the word file here

My thinking

  1. A question to start
  2. Reversing the terms. Does balancing still work?
  3. A take-away. How does this effect our balance.
  4. Does reversing the terms still lead us to the same answer
  5. Increasing the constant by one. What happens? Also: a decimal answer.
  6. We can have a negative answer
  7. Divide x, instead of multiplying it.
  8. Increasing co-efficient of x by one. What happens to our answer?
  9. Doubling co-efficient of x. Not sure about these last two. I think they may be a step back from question 7. This is the problem with presenting these in a linear format. These questions are variations on question 1, not question 7. I might experiment with some kind of spider diagram.
  10. Doubling the divisor from 7. Again, maybe the linear way these are written is a bit rubbish.

Don’t know how I like the order of these questions, but there’s lots to think about and something to tweak.

I have found the transition to asking ‘why have they asked you that question? What are they trying to tell you?’ has been difficult for some students, but I think it’s worth devoting time to it. If students are inspecting questions for things like this, maybe they’re more likely to read the question thoroughly and pick out it’s mathematics. Big hope, I know.

Maths Spelling Tests (Again)

I talked a little about the idea for Maths spelling-type tests here.

I’ve used them this year, but I’m not really sure how effective they’ve been. I’m not sure pupils have fully bought into the idea of using learnt facts.

The other thing I noticed is that I hadn’t really presented them in a way that was useful, so I’ve redesigned them and made enough for an entire half term. You can download the pdf here or the PowerPoint file here.

Now for each week/test you get a version with everything filled in to give to the students to revise (two to a page to save on printing), and on the next page there are two of the tests, with blank sections for pupils to fill in from memory.

As always, feedback appreciated.

Bad Maths

A student sent me this.

I’ve started a ‘bad maths’ category. Does showing students this kind of stuff and asking for them to be aware/send examples to you make them more mathematically aware?