How to Construct Loci

The power of student’s intuition is something I try to draw on in all of my lessons. Learning how to construct Loci is something I think lends itself to this way of teaching. Here’s how I do it.

At the beginning of the lesson I make sure every student has the following pieces of equipment:

  • Sharp pencil
  • Pair of compasses
  • Ruler
  • Eraser
  • Mini-Whiteboards
Setting the tone for how to teach loci

I explain I have plumbing issue at home which has prevented me from having a shower for the past two or three weeks. Quite honestly, I stink. I mean I really stink. So bad in fact that if I was to stand in the middle of this classroom and flap my arms my odour would actually knock-out anyone within a metre of me.

So, if we take the classroom to be a 4 m by 3 m rectangle (which I sketch on the main whiteboard) and I stand in the middle of it, ‘Shade the region in which it is safe to be.’

At this point the class begin to realise that maybe I don’t smell as quite as much as I am pretending to and this may be a genuine mathematical point. Either way they’re hooked. All the students draw a 4 m by 3 m rectangle on their whiteboard and have a go at sketching the safe region as a circle in the centre. I ask to see their work and notice some have kept to some sort of scale but most hadn’t. More than one student drew a square as the locus rather than a circle. I feedback to the class the construction of the correct region and discuss the meaning of the word locus and loci.

How to Construct Loci

Next, I move to the corner of the room and explain to make my odour even worse I begin to flap my arms like a chicken on steroids. This causes the knock out distance of my odour to increase to 2 metres instead of one. This time I ask them to shade the region which is safe. By this point the vast majority of the class are much more accurate with their scales. Some however, continue to draw the locus as a quarter of a 4 m by 4 m square rather than a circle of radius 2 m. To address this common misconception I tie my pen to a piece of string and rotate it about the point of the rectangle drawn on the whiteboard.

How to Construct Loci

More Loci Constructions

To introduce the locus about a line I ask the class if they have seen the new X-Men film, Days of Futures Past. Some have, some haven’t. I refer to the character QuickSilver who can move extremely quickly. Once again I stand in the middle of the room and like QuickSilver I pretend to run a metre forward then a metre back from the centre. I run so fast to and from these points my odour lingers. However, the radius of my odour is now only 50 cm. ‘Sketch the region in the classroom which is safe.’ From looking at their whiteboards it’s interesting to note most are now struggling to keep their sketches to scale. A quick review on the main whiteboard illustrates how a pair of parallel lines is created either side of my path with a semi-circle on the ends.

How to Construct Loci

To progress on to combining loci I place my supposedly very smelly jacket which has a knock out radius of 1 metre in the middle of the room and pretend to run very quickly around the perimeter to create a knock-out radius of 50 cm from the edge of the classroom. The class are asked to sketch the radius that is safe.

How to Construct Loci

Teach this lesson

Extending transformations beyond shapes on a grid

While I was teaching a higher GCSE class about  reflections, rotations and translations  I wanted to explore extending transformations beyond shapes on a grid to include transforming straight line graphs.

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