I have an awesome class of 13 year olds who are starting out on quadratic manipulation. They are great, but there are a significant number who rush their work and skip steps of working out because they ‘know what they are doing’. Really? Let’s see …
I gave the class twelve quadratic expressions and asked them to factorise them, then to spot any common themes. What I didn’t tell them was that all of the factors used were combinations of x, 2x, +/-1 and +/-5. If they were sloppy with their attention to detail, their solution would look like the solution to a different expression. Essentially a difficult easy task.
It soon sorted out those who had at true understanding of factorising a quadratic from those who’d lucked their way through easier questions.
I’ve shared the presentation and pdf version below. I’ve added in two slides where you can cut out the expressions to use as more of a card sort. You’ll notice that there are no 4x^2 expressions – I was focussing on solutions with only one x co-efficient greater than one. Although I used this as a starter, you may wish to use it as a longer activity, depending on your class.
If you teach in the UK and haven’t used the excellent Access Maths site, why not?
Seriously, you are missing out!
I’ve used and recommended to colleagues lots of the Access Maths resources. This is the latest worksheet I’ve downloaded (click on the image to link to the 9-1 GCSE resource page):
Image credit: www.accessmaths.co.uk
I used these pentagonal problems (I believe they are know in pedagogical circles as ‘Fox Diagrams’ – but you try Googling that term and not getting a page of pictures of foxes) with my GCSE class as a two part homework. The first homework was to do the outside skills – if they felt confident they could skip questions, if they needed help they should come and see me. I stressed that they would need to use these techniques to part two and it was their responsibility to make sure they were ready. Part two of the homework was to complete the middle ‘exam’ question in their books in their books, showing the full method.
I actually enjoyed marking this homework as it gave me an insight into how they visualised problems – there were at least four different ways to complete this task. Unusually I made any low achieving student come back and redo their homework in an informal detention. By spending a few minutes reflecting on the skills they’d already practised (or should have practised), every student jumped from 0 or 10% to 100% correct. I did little more than point out where their technique had started to fail them. These students left the extra maths session with big smiles and a sense of achievement.
Inspired by the talented @AccessMaths (you really should follow them on Twitter) I’ve done my own triangular resource on expanding, factorising and solving quadratic equations.
This worksheet metaphorically holds students’ hands as they work through factorising quadratics where the co-efficient of x squared is greater than zero. My students liked this sheet as it gave them a starting point, it stopped them putting their hand up for every question and it would be useful for future revision.
It’s beginning to look a lot like Easter … scrawny plastic chicks and over-priced chocolate eggs everywhere! This little ‘egg’ of an idea was totally inspired by some lovely Tweeters who mentioned ways to use empty plastic eggs.
I bought these two-part plastic eggs from a local craft shop. They are available from lots of places on the high street and online. My pack has 30 eggs in six different colours. You may be able to see that I’ve numbered the top and bottom of each shell – just to avoid arguments.
Activity Now, I used these eggs for revision with my GCSE class. Each colour represents a different topic. There are 30 questions and the answers are the numbers 1 to 30. I hid the eggs in our main hall due to the unpredictable nature of the British weather. You could hide them inside or outside the classroom and give a prize to the person/group who correctly completes the most questions. Points could be deducted for trying to sabotage other groups. If you don’t feel that adventurous or it’s impossible to go outside, you could copy the questions and do this as a desktop activity.
Sometimes we get tunnel-vision on the focus for passing exams. We keep the ‘fun’ stuff for younger pupils. This revision activity is a treat for my hard-working students in KS4. They aren’t the easiest of topics, but they are perfect for students working at GCSE grade C and above.
To an experienced mathematician, factorising a quadratic (with real roots) is a little number puzzle, into which the algebraic terms fall gently into place.
To a secondary (high) school student of middling ability they can be algebraic torments conjured from the darkest recesses of a fevered genius’ imagination. Impossible!
This year I have introduced factorising with no mention of algebra, equations, solving or factorising. My class are at least C grade students who have convinced themselves they are no good at algebra. We started by considering this puzzle:
What values of a, b, c &d make this multiplication grid true?
bc = 1
ad = 30
ac + bd = 13
The solution is fairly straightforward:
Once they got the hang of this kind of puzzle, I compared it to the grid method for expanding double brackets. I asked them to think about what the brackets could be if I gave them the values for bc, bd, ac & ad.
Finally the stabilisers were taken off. I asked them what the brackets were if I gave an algebraic form of the number puzzle. First we considered:
bc = X squared co-efficient
ac + bd = X co-efficient
ad = constant
Once it clicked that this was just a fancy number puzzle they were flying. I was really impressed by their positive attitude and willingness to try.
You can download a set of questions (and answers) here.