Geometry | | DJS


I have put in here:

Pythagoras’ theorem, but I would connect that to the several pages on Squares within Numeracy

Bearings, two pages, included here because when teaching a Y11 class it seemed to me that this very simple topic was thoroughly misunderstood, such that the errors must have come from earlier in school.

Sketching, two pages, coupled also with a similar page in the Sixth form (Topics collection).

Geometry as a drawing skill was something I taught persistently. I say persistently because most of the kids thought I was barking mad to attempt to teach those skills that can be achieved with straight edge and compasses. Since I cannot easily draw these, I refer you to sites linked at the foot of this page, but I specifically taught:

•  Fundamentally, how to use a ruler as a straight edge, how to read a protractor, how to draw a specific length or angle with reliable precision.
•  Bisection of a line
•  Bisection of an angle
•  Construction of angles of 60º and 90º. With angle bisection, a range of other angles can be constructed [and, since subsequent constructions rely upon earlier precision, the errors tend to multiply, showing the students that the precision needs to be high].
•  Construction of a right angle both at the end of a line and from a point to a line.

Combining these skills, several cases can be repeated until the level of precision reaches the desired standard:

•  Construct a right angle triangle of specified size. Measure the hypotenuse and one angle. Check this against calculated values (which may not have been taught, so teacher has a temporary advantage)
•  Construction of the several centres of a triangle from (i) bisecting angles (ii) bisecting sides (iii) joining medians (iv) intersecting altitudes. 3 of these 4 centres lie on a line (named after Euclid?) and a cumulative skill homework (which I set every year) would be to identify which is the exception.

Locus is another topic that follows from this, explored by me towards the end of the second term. I particulalry enjoyed the practical aspects, such as the path of the bottom edge of an up-and-over garage door, followed by asking the kids to criticise such designs and improve them.[For example, so that the door doesn’t hit the person pulling it up, or so that the door doesn’t demand an unecessarily high garage ceiling, or an especially tall operator. Design skills within the maths lesson, overlap of subjects, etc etc - this seemed to be to be valuable, especially in showing maths to be not entirely abstract. Indeed, after a week of work on something particulalry abstract, pushing the abstract to reveal some usefulness always struck me as strongly positive learning. We use that <stuff you thought entirely abstract, boring and unnecessary> for this <thing you reluctantly admit you found interesting>.

Sites that explain and demonstrate construction techniques include BBC BitesizeKhan Academy, Math Open Reference, Maths Doctor, Universal Class. I simply am not prepared to repeat the work involved (any more than I see the point in so doing).

I am well aware that marking at GCSE allows for what I think of as poor precision; as I remember this, measurements to within a millimetre and angles to within 2 degrees. I tried to persuade my classes that they could do very much better, to around 0.2 or 0.25 of a millimetre and to within half a degree. Boys especially tended to persist in using soft blunt pencils; girls tended to have poor poiints on their compasses (yes that may be sexist but it is what I observed across many years); some of the students rose to the occasion if I made the observation towards the end of the first week on topic; some seemed to want to embrace the stereotype.

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