# 359. Proportional steps

Just a quick resource upload today.

Image credit: https://www.heyn.co.uk/

I’ve written a step by step resource on how to construct algebraic direct proportion relationships, including the answers.

Small steps in Direct Proportion (docx)

Small steps in Direct Proportion (PDF)

I used this with a Year 11 class who aren’t very confident with algebra. They were surprised by how straight forward the work was and were happy to now attempt problem solving with algebra.

# 357. It’s not square!

I do love a little challenge for A-level Further Maths students. They are often confident and very capable mathematicians, but occasionally overlook the small details. This challenge looks into which strategies students use when working with 3D vectors, lines and angles.

The most annoying thing? There is no single correct answer.

What is the investigation?

Students start with two points, create a line, construct two perpendicular lines and then join up the lines – did they create a square? How do you know? Justify it?

Download the instructions here: It’s not square (docx), It’s not square (PDF)

Skills required

• Distance between two points
• Equation of a line in three dimensions
• Scalar (dot) product

Solution/Discussion point

• Students need to use the same direction vector for both perpendicular lines too create a square
• The two new corners need to be n the same direction away from the original line (not one above and one below)
• It’s interesting to discuss what non-squares they made. Technology could be used to plot them in 3D.

# 354. Iced gems

Just a quick idea today. You know the feeling when the multi-pack of sugar paper has dwindled down to just the brown. Great if you want to do trees, bleurgh if you want to do anything else.

I did a tarsia recap with Year 7. There were three different tasks going on and so I photocopied them onto three different colours of paper. The only colour of sugar paper was brown. We went with it. As the class finished their work, we discovered that their work looked like iced biscuits or iced gems. Hence our wall of Algebra Iced Gems:

Some of the cutting and sticking is a bit wobbly, but the class really enjoyed this task and we consolidated a considerable number of skills.

# 352. Functions refresher

We recently finished teaching the AS Maths syllabus to Year 12. My colleague and I decided how to split up the start of the second year of the course. I’m starting with the modulus function.

I took one look at the skills needed at thought “Uh-oh”. The students are going to be out of practice with this. They are a lovely group, with a wide range of ability, but we’ve been very focussed on Applied Maths recently.

Option A: Go for it and patch up the vocabulary as we go (getting very frustrated – they knew this last October)

Option B: Break them in gently, recap the skills and vocabulary and extend them further

Option C: Reteach the work from last October.

Yes, you guessed it. I went with Option C. I found a brilliant task on piecewise function graphs on the Underground Maths website.

Image credit: https://undergroundmathematics.org/

There are four graphs given. The basic task is to interpret the functions relating to each graph, through description or function.

I photocopied the graphs onto card and sliced them up. Each group had a set of cards. One person described a graph and the others had to accurately draw it. Some students went straight onto squared paper, others drafted it out on mini whiteboards. They repeated this until all the graphs were drawn and everyone had had a go at describing (the describer stuck in their card, so that they had a complete set). Whilst they were doing this, I moved around and encouraged the use of mathematical vocabulary.

Note: it was interesting to see how many students had forgotten the significance of open and shaded circles to denote boundaries of inequalities.

The second task was to match up the function cards with the graphs. Once again, accuracy was key as not all graphs had functions and not all functions had graphs. There were also some that nearly, but not quite matched. This activity really brought out the key skills relating to domain, range and function notation that I was looking for. The extension task was to complete the missing pairs.

But, did it work? I can confirm that the following lesson the class made very good progress investing the modulus function and it’s graph, even going as far to solve equations. They knew what the notation meant, how to plot it and how to interpret the graphs.

I really like the Underground Maths website as it has great resources, good support material and always makes students think. Most of the time it gets teachers thinking too!

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.

# 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
• 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.