Category Archives: Shape, Space & Measures

358. A spatter of trig

The fabulous Mrs D (@mrsdenyer ) shared this forensics video, by crime scene analyst Matthew Steiner, on Twitter. At eight minutes in the presenter looks at blood spatter analysis. The use of basic trigonometry in a practical situation is a gift of a video for a starter in lesson.

 

My class were absolutely silent throughout and wanted to watch the whole video, however they may have just been trying to avoid work. I shared the video link with them via our digital classroom platform. We are now using blood spatter for 3D trigonometry examples rather then mobile phone masts. Gory, but effective!

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.

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.

 

346. Area & Volume conversion

This is a quick post on how I teach metric unit conversion for area and volume. All you need is a big whiteboard and coloured board pens.

Start by stressing that all diagrams are not to scale/accurate.

Two colours

  1. Draw a square on the board
  2. Pen colour 1: Label it as 1cm
  3. What is the area? Show the calculation
  4. What is 1cm in mm?
  5. Pen colour 2: Label it as 10mm
  6. What is the area in mm? Show the calculation
  7. What is the scale factor between the sides? the area? why?


Three colours

  1. Draw a square on the board
  2. Pen colour 3: Label it as 1m
  3. What is the area? Show the calculation
  4. What is 1m in cm?
  5. Pen colour 2: Label it as 100cm
  6. What is the area in cm? Show the calculation
  7. What is the scale factor between the sides? the area? why?
  8. Repeat in pen colour 1 for mm

Four colours

Well not actually four colours – pens 2,3 & 4 only. Repeat the process for kilometres to metres and centimetres.

Volume – same process, just three dimensions

Why all the colours?

By coding each unit of measurement with a colour students can see the progression of the calculations and the links between area/volume and scale factor. After all, an okay mathematician can reproduce memorised facts, but a great mathematician doesn’t need to memorise – they understand where the calculations came from.

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.

341. Dragon Bridge

Here is a little starter picture for you:

This is the ‘Pont y Ddraig’ at the marina in Rhyl, in North Wales. What mathematical questions could be inspired by this?

‘Pont y Ddraig’ means Dragon Bridge. Find out more about the bridge here