Category Archives: Core

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.

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!

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.

 

313. Friendly Functions

Just a quick resource share today!

I’ve been doing functions with my GCSE class as part of the new curriculum and I’ve gone down the algebra route. I could have started with graph drawing like the parallel class did, but I know my class – drawing and accuracy are not their forte. We made brilliant progress with substituting into functions and even composite functions went smoothly. I wasn’t happy with the textbook resources on manipulating functions so I put together a step by step resource, including a basic skills recap:

Manipulating functions (docx)

Manipulating functions (pdf)

I also thought my class needed a little hand holding for inverse functions. There are many ways to do this, but the method I used was designed to allow the class to access the topic with teacher input verbally and on the board.

Inverse Functions worksheet (docx)

Inverse Functions worksheet (pdf)

Hope these help!

Oh and you can even use them as A-Level recap tools.

Updated (19:53): To fix typo on Inverse functions worksheets

312. Class Commentary

I don’t know about you, but going over higher level questions (eg A-Level) after a test can be a frustrating time. The students never seem to fully engage because they think they know it all – even though they do get things wrong! What if I could offer you a way to review the test and incorporate an understanding of exam board mark schemes?

Image credit: www.sri.com

Preparation

  • When you mark the test clearly indicate on the paper which questions students got fully correct.
  • Alternatively get your students to do this.

Set Up

  • List the question numbers on the board
  • Starting with the highest number (usually the hardest questions) students volunteer to answer the questions on the board by putting their name next to a question number. In this way the brightest students who got the tricky questions right can’t volunteer to do the easier questions, allowing other students a chance of success.
  • Long multi-part questions could have more than one student.
  • You can also allocate a calculation checker and algebra checker if you have spare students

Task

  • Bring up each student to go through a question on the board.
  • Whilst they do this you can do a commentary of where marks are allocated by the markscheme, alternative methods and misconceptions.

I did this activity with a Year 12 group whilst reviewing an A-Level paper and it was a such a better use of time. The students were more engaged and I could interact with the class on a much more productive level.

302. Log Proof Puzzle

If you can guess where today’s blog image came from you obviously consume too much damn fine cherry pie and fresh coffee!

log lady

Image credit: Pinterest

You may have guessed that the topic of this post is logs. If you are introducing the rules for adding and subtracting logs or revising them, I have just the resource for you. It’s a basic proof of both rules with a twist. The instructions are in the wrong order and you must rearrange them into the right order.

Easy!

Are you sure?

For those of you who have a student or two who rush everything and don’t read the instructions there is a sting in the tail. One of the lines of proof is a tiny bit wrong. The methodical student will find it, the one who races through may end up changing more than one line – hence breaking the rules.

Have fun!

Proving log rules for addition and subtraction

Answer: It’s the ‘Log Lady’ from the cult classic ‘Twin Peaks’!

297. Crabby Functions

I take no credit for this ‘aide-memoire’ – it comes from a most delightful and hardworking student. To quote a colleague “She is the poster-child for the benefits hard work”.

Let’s call this student Natasha (not even close to her real name). Natasha had been struggling to work out the difference between graph/function transformations, in particular f(x+a) and f(x)+a. Which way did the graph move? How could you tell? Then she had a brain wave:
image

She drew little Y shapes on the brackets:
image

One of the brackets now looks like a little crab:
image

And we all know crabs move sideways – so it most be a horizontal translation!
image

Simple!

Logical!

Genius!

Thank you Natasha!