Category Archives: Algebra

350. Quadratic factor puzzle

Back in posts 95. Quadratic puzzles and 322. Quadratic puzzles I’ve looked at how to approach factorising and solving quadratic equations/expressions in a ‘gentle’ way.

Time to take off the kid gloves!

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.

Solving quadratic expressions (PPT)

Solving quadratic expressions (PDF)

349. Circumcircle Investigation

The A-level textbook we use has a nice picture of the circumcircle of a triangle and a definition, plus a brief description of how to work through them. For those who are pondering what a circumcircle is, click on the image or link below

Image credit: WolframMathWorld

I’ll just stick to basic vocabulary in this post, rather than the formal circumcentre and circumradius.

Back to the book – not exactly inspiring or memorable stuff!

I looked at the class and off the cuff changed the lesson plan.

Equipment

  • Plain paper
  • Pencil
  • Ruler
  • Compasses
  • Calculator

Step 1

Draw a decent size triangle on the paper. Label the corners A,B,C.

Step 2

Using geometrical constructions, find the centre of the circle that your triangle fits in. Check by actually drawing the circle

Step 3

Discuss what techniques gave the best results – hopefully you’ll have perpendicular bisectors. There is a nice comparison between bisecting the angles (which some students will do) and bisecting the sides. The angle bisectors always cross inside the triangle, the side bisectors don’t.

Step 4

Randomly generate co-ordinates for A, B, & C. Get the students to pick them and then they can’t moan if the calculations are awful.

Step 5

Discuss how you are going to find the centre and radius of the circumcircle. We decided on:

  • Only use two sides
  • Find the midpoints
  • Find the gradients and hence perpendicular gradients
  • Generate the equations of the lines through the midpoint
  • Find where they intersect
  • Use the point and one corner to find the radius

Step 6

Review their methods, looking for premature rounding in questions. I’m still instilling an appreciation for the accuracy of fractions and surds, over reaching for the calculator.

Step 7

This is how my solution looked – I numbered the picture and the steps so students could follow the logic. I was answering on one page projected on screen.

 

348. A-Level colouring (Updated)

Those of you who follow this blog will know I have a thing for explaining with colours. This isn’t just a gimmick for younger students, it also works for 16-18 year olds.

In the picture below we were looking at proving a statement involving reciprocal trigonometric functions and fractions. A common source of misconception with this kind of question is that students split the question into working with the numerator and denominator separately, then make mistakes when they put them back together. They can’t see the big picture.

Image credit: Mathssandpit

When I discussed this on the board I used separate colours for the expressions in the numerator and denominator. The class could follow the logic so easily. It’s probably my most successful introduction to this topic. I saw that some students used highlighter on their notes after I’d gone through it, so they could track the solution.

The second type of question we looked at was solving a trigonometric equation. The straight forward expansion was all in one colour, but the roots of the quadratic were highlighted in different colours. The reasoning behind this was that students often solve half the quadratic and neglect the other impossible solution. Our exam board likes to see students consider the other solution and formally reject it. It makes the solution complete. By using a colour, the impossible solution stands out and reminds students to provide a whole solution.

Image credit: Mathssandpit

So when you are planning for misconceptions at A-level, remember that coloured pens aren’t just for younger students.

Update: 22nd October

The brilliant Mr B has shared how he uses colour to identify the forces in perpendicular directions in Mechanics.

342. Revision jotters

With the exams looming large, I thought I’d share how my class have been revising. To give you some context roughly a third of the class are doing Foundation GCSE, aiming for at least a Grade 4. The rest are doing Higher and aiming for a Grade 5 or better. We have three, one hour, lessons a week. I’m rotating between doing an exam paper, a whole class revision activity (eg a revision clock) and tiered revision.

I know if I tell the students to revise independently the results are going to be mixed. Some will be brilliant, some will be more laid back. To resolve this I pick a topic (or two) from each tier that I know they need to improve on from or that they have requested. It’s helpful if there is a theme to the work. I’ve recently done things like y=mx+c (F) with plotting inequalities (H).

Now the genius part: PixiMaths revision jotters

How to run the session

Photocopy a big stack of revision jotters. If you are doing black and white copying, use the b&w version. We requested the b&w version and, because PixiMaths is awesome, it is now on the website.

Clearly put on the board which topic each tier is revising

Eg Foundation: exact trig values, Higher: trig graphs

Give students 5-10 minutes to fill their revision jotters with everything they know. Have textbooks or maths dictionaries available to fill in the gaps. You may find that Higher students want to do the Foundation topic too – no problem, just make sure they have two jotters. Due to the complexity of the Higher topic, they will need more time to make initial notes.

My students are allowed headphones in revision sessions. At this point it’s headphones in for Higher and out for Foundation.

Do a skills recap on the board (exact trig values), with maybe an exam question too. Students can ask questions on the topic and add to their jotter. Then have a worksheet for students to do eg Corbett Maths or KeshMaths GCSE exam questions booklets. They can refer to their revision jotter or scan the Corbett Maths QR code for extra help.

Swap over. Headphones in for Foundation and out for Higher.

Repeat the process for Higher, with drawing trigonometric graphs. Issue an appropriate worksheet.

Once you’re done, make a judgement call. Are there students who could push it further? Maybe transform a trig graph or problem solve? Go for it. Foundation are busy, Higher are busy, spend some time stretching your most able. Every mark counts.

A huge thank you to PixiMaths for the revision jotters (and everything else).

Examples of students’ work

Shared with permission of students. You can see that they have personalised them to meet their needs and some are a work in progress. Also, the b&w jotter photocopies so nicely.

338. Grappling with graphs

Have you noticed that textbooks are okay with graphs, until you need some interpreting graphs questions?

Image Credit: trustedreviews

I thought that mobile phone tariffs would be a good starting point for comparing fixed charges and rates. Using the iPhone X as a starting point, I’ve put together a discussion starter and couple of additional questions. All the tariffs are actual offers available at the time of writing.

You could start by looking at the graph and asking students what they notice, you could give them the tariffs and ask them to generate graphs or you could give them the data and ask them to plot the graph and derive the tariffs. It’s up to you!

The graph is deliberately vague so that students can discuss trends without getting obsessed by the detail of the numbers. Everything is downloadable below.

iphone X tariff graph

Iphone X mini investigation

Interpreting graphs

 

335. The power of colour

As Mathematicians we appreciate the importance of getting the basics right and building a firm foundation. With this in mind I’ve been an absolute harridan with my Y8 students regarding presentation and technique for solving equations. If they can nail good algebraic presentation now, their future studies will be be much easier.

When we started there were students doing everything in their head, not always correctly. Some insisted on working backwards, which is great for basic cases but not for unknowns on both sides. Most frustratingly some students were breaking up the logic by putting extra working out between steps and losing track of what they were doing.

For example:

2x – 10 = 5x + 8

5x – 2x = 3x

3x – 10 = 8

So we had a really good discussion about logical presentation. We decided to write down what we were doing in the margin, try and keep the = sign lined up in the working and put any extra working out on the right.

This worked really well for most of the class, but I had a small group of students who just lost track of what they were doing and why. They knew things had to balance, but struggled to cope with equations with an unknown on both sides.

While I was talking things over with them using a mini whiteboard, I noticed they had a profusion of coloured pens and highlighters. Bring on the colour!

By highlighting the key point of each line of algebra and matching it with the balancing step they started to build the structure of good solutions. It was slow work to start with, but a couple of lessons later and these same struggling students are now hitting the extension work every time. And most of them no longer feel the need to highlight key information.

333. Resource of the week

Just a quick resource for you today and apologies if you are already using this!

Plickers

Not some new ‘youth slang’, but an amazing online tool. Students have an individual card with each side labelled A, B, C or D. You ask a multiple choice or True/False question, they hold up their card with their answer at the top, you scan the class set of cards.

Image credit: Plickers.com

It really is that simple and here is what to do to get started:

  1. Create a free account at www.plickers.com
  2. Download the app to a portable device with a camera (phone, tablet etc)
  3. Print out the cards
  4. Allocate the cards to your class on the website
  5. Stick the cards in your students’ books
  6. Set a question
  7. Scan the cards

I have a tablet device that I use for school purposes as I keep my phone for personal use. The only problem I had was my android tablet doesn’t have a light source or as high quality camera as my phone, but we sorted that by having students move to a brighter part of the room for scanning. Instant feedback with no handheld devices!

Finally I have to say a huge thank you to Mr L, our trainee teacher, for introducing this to the Department.