28 December 2014

5 Maths Gems #20

Hello. I'm pleased to present my 20th gems post today - a milestone! Twitter has been quiet this week because Christmas took our minds off work for a couple of days, but I've still got some great ideas for you. The Christmas holidays are short and busy so it won't be long before you're back to planning lessons - I hope this post gives you some inspiration for the Spring term.

1. Factorising Quadratics
You know Don Steward's blog is the best thing since sliced bread, right? His catalogue of fantastic rich tasks is updated all the time. I really liked his recent posts on product puzzles. He started with a set of questions like this:
In Question 1 above, you can see that the top left cell has to contain a 3 because it's a factor of both 3 and 6. The rest of the cells can be completed quickly once one common factor has been established. This is a simple example but Don develops the questions to become increasingly difficult, some having multiple solutions. 

The next set of activities extend the same idea to algebra, starting with this:
There's lots of these to complete - excellent practice of factorising linear expressions.

The next stage of this exercise is factorising quadratics.
Conveniently I'm currently planning a Year 10 lesson on factorising quadratics. I want my students to do a lot of practice so will definitely be using this activity. The questions build up to a suitably challenging level of difficulty, ending with this one:
I want to encourage my students to factorise 'harder' quadratics (ie a > 1) by inspection. This is my preferred method (ie 'guess and test') but my students always demand that I teach them a more structured approach (eg 'the Grouping Method') which frustrates me. Their insistence on following an algorithm suggests a lack of confidence. I think the question above turns factorising quadratics into a kind of logic problem. Tackling this question without an algorithm might help my students develop the confidence to factorise harder quadratics by inspection.

One last idea for a lesson on factorising quadratics - I like the problem below from openmiddle.com. There are a number of possible solutions so you could challenge students to find a different solution to the person next to them. 

2. Angle Sense with the Interactive Whiteboard
I've been planning a Year 7 lesson on angles in which I'd like my students to estimate angle sizes. If you were asked to to draw an angle of 210o freehand, how would you do it? I'd think of it as a straight line plus a third of a right-angle. If you have proportional reasoning skills then it's pretty easy to make an educated guess. An angle estimation activity would work perfectly well without technology (read out a series of angle sizes and ask your students to draw their freehand estimates on paper. They then check their estimates using a protractor - another useful skill). But if you want a similar activity for the interactive whiteboard then you might like this fun Estimating Angles Game from Nrich.
Another interactive whiteboard tool is 'How Far Does it Turn?' from MathsPad. This time your students have to estimate the size of the angle drawn - they could do this on mini-whiteboards so everyone is included in the activity.

While looking at these games, I stumbled across a big range of angle tools for the interactive whiteboard here. Some of these angle games are quite funny - Banana Hunt in particular made me chuckle.
If you like these interactive whiteboard games then you'll find loads at Sheppard Software. It's amusing that there's an Absolute Value Number Balls game - this concept isn't covered until Year 13 in the UK but I bet my students would love to play this game - five minutes light relief in a C3 lesson!
FlashMaths.co.uk is another great website for interactive whiteboard activities. Flash Maths was created by Jonathan Hall (@studymaths) who brings us a plethora of fantastic tools on StudyMaths.co.uk. If you haven't seen it before, check out MathsBot.com which is his simple (but brilliant) worksheet generator.

3. Big Questions
Billy Adamson (@Billyads_47) shared a fantastic set of mathematical thinking prompts 'The Big Questions'. Here's a few examples: 

Lovely open questions from Billy to generate discussion and develop understanding. There's some more good examples of open questions here:

4. Trigonometric Problem Solving
Our Year 13s' problem solving skills are tested when they're asked to simplify expressions involving trigonometric identities in C3 (like the example below).
I find that my students get frustrated when they can't spot a 'way in' straight away. They give up quickly. There's actually a pretty standard set of starting points, as described on www.intmath.com (@intmath). 

I struggle to help my students feel confident in tackling these problems, so I really like this activity from @mjfenton. Here's an extract:
The idea is that we start with a lot of structure and gradually give fewer hints until students are able to solve the problems themselves. The steps might seem logical to us, but we're experienced problem solvers. 

It's a good idea for maths teachers to try to solve unfamiliar problems every now and then (like the example below from ‏@dannytybrown) to remind ourselves that mathematical problem solving often requires patience, creativity and multiple attempts. We all experience frustration in problem solving, just like our students do, but we know that the satisfaction of eventually finding the solution is well worth it.

5. Dividing with Decimals
I've mentioned before that I love MathsPad's resources - plenty of them are free and the rest come at a cost of only £3 per month. Whether your school subscribes or not, it's worth registering for email updates in order to keep track of all the new resources. This month, the interactive resources on Decimal Calculations caught my eye. It always surprises me how many students will happily say that 40 divided by ½ equals 20. Activities like the one shown below will help tackle this misconception and encourage students to think before they answer.

That's it for this week. I'll leave you with a video from 1977 - 'Congruent Triangles' by Bruce and Katharine Cornwell (another gem found on @MathMunch). Happy New Year!


  1. Thanks for the reminder to check Don Steward's blog. I agree that factorising quadratics is more of a logic problem and as such usually build up the skills needed so that the students can then tackle the various steps as appropriate, rather than multiple steps at once. They're much less likely to be thrown by something different then!

    It'd be useful if I can work the big questions into my routines, along with some slides from Diagnostic Questions too. I have a class in mind already to try them with.

    The algebra problems in the Y8 Yerwat guide are very useful, with lots of challenge. Some samples here: http://www.mathsisjugglers.com/downloads/free-samples/key-stage-3-samples

    Thanks again for another 'gems' post!

    1. Thanks for the comment Tim. I've not seen mathsisjugglers.com before, lots to explore on there, thank you!