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Segar Rogers @segarrogers.bsky.social · 1d

Zeta CfE Third Level.

Segar Rogers @segarrogers.bsky.social · 1d

‘Do they really?’: Bizarrely yes. Department courses. Textbooks. Not everyone but it’s surprisingly common (in my experience) . Identifying triangles is a lot easier than identifying corres. & alter. angles … which is probably why.

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Segar Rogers @segarrogers.bsky.social · 1d

Absolutely. Options are good!

I’ve never seen the Exterior Angle Theorem taught in Scotland … that’s why I’m interested (options!). Many courses put the triangle-interior-angle-sum before corres. & alter angles … which makes it tricky to do of course.

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Segar Rogers @segarrogers.bsky.social · 1d

Yes I agree. Maybe that’s why the right is good … it reinforces corres. & alter. angles … which are more fundamental yet harder to ‘see’ … and in the long run it’s quicker. (I’m aware the purest in me is at work here … when answering an exam question it really doesn’t matter left or right).

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Segar Rogers @segarrogers.bsky.social · 2d

Left: Requires triangle-interior-angle-sum and straight-angle-sum. 3 steps. Slower?

Right: Exterior Angle Theorem (Euclid 1–32). Requires corresponding and alternate angles. 1 step. Faster?

Right feels cleaner to me ... I wonder what pupils would think?
#UKMathsChat #iTeachMaths

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Segar Rogers @segarrogers.bsky.social · 2d

Hmm. I know it might feel like that but ... do most history teachers write history papers, do most English teachers write novels, do most music teachers perform publicly, are most PE teachers in a team? I would say probably not.

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Segar Rogers @segarrogers.bsky.social · 2d

Yes in principle ... but you have to work with the class in front of you. I can think of classes where your language would be well received ... and I can think of classes which would immediately be lost and would switch off ... and one has to be mindful of that.

Segar Rogers @segarrogers.bsky.social · 2d

Yes, it’s something that is almost never mentioned … even when expanding brackets. We sometimes shy away from naming what it going on. I went through a phase of explicitly naming ‘quotative’ and ‘partitive’ division with S1 classes … I could see me saying ‘by the distributive law’.

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Segar Rogers @segarrogers.bsky.social · 2d

Not quite sure what you mean. Could you give an example?

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Segar Rogers @segarrogers.bsky.social · 3d

Gotta love James Taunton … an amazing method – thanks :-)

Segar Rogers @segarrogers.bsky.social · 3d

I did not know this! Brilliant. Thank you :-)

Segar Rogers @segarrogers.bsky.social · 4d

I agree with you yes; there’s usually a ‘proof by demonstration’ … or a ‘proof by what’s the pattern’. I was interested in thinking about the things we gloss over because the pupils don’t have the mathematics to properly prove it yet. And of course it’s natural to climb on the shoulders of others.

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Segar Rogers @segarrogers.bsky.social · 4d

True. I was just meaning on a very general level .. like using the formula for the volume of a sphere with say an S3 class … which would then need S6 Advanced Higher calculus to prove it.

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Segar Rogers @segarrogers.bsky.social · 4d

lol … I don’t recall a ‘by symmetry’ argument in the Elements ;-)

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Segar Rogers @segarrogers.bsky.social · 4d

As an aside, have you ever taught integration first? ( … as accumulation … area under a curve … it’s lovely!). You can then just about teach differentiation as anti-integration … er, admittedly by avoiding proving the fundamental theorem of calculus ;-)

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Segar Rogers @segarrogers.bsky.social · 4d

Fair point … I think Gottlob Frege decided that we’d made them up … but don’t quote me on that ;-)

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Segar Rogers @segarrogers.bsky.social · 4d

What things in maths do we teach before we have taught the maths that shows it is true? #UKMathsChat #iTeachMaths

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Segar Rogers @segarrogers.bsky.social · 4d

Maybe repeat but in clockwise direction. Hmm? Thinking ;-)

Segar Rogers @segarrogers.bsky.social · 4d

So …

AB rotates through 50°.
∴ arcBB’ is 50°
∴ arcAA’ is 50°
∴ angleAOA’ = 50°

… then repeat (?) … hmm, not sure how to show A’ lands at C without invoking Eu.III–20?

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Segar Rogers @segarrogers.bsky.social · 6d

This one is a little harder to see because the vertex of the 55° angle coincides with the end of the 110° arc. But the 110° arc is still being subtended from a point on the circumference, so Eu III-20 still applies.

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Segar Rogers @segarrogers.bsky.social · 6d

... and the same again, this time starting at the 40°

1 → 2: Eu. III, 20
2 → 3: Supplementary arc.
3 → 4: Eu. III, 20

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Segar Rogers @segarrogers.bsky.social · 6d

Start at the 20° and follow the sequence of steps.

1 → 2: Eu. III-20
2 → 3: Supplementary arc.
3 → 4: Eu. III-20

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Segar Rogers @segarrogers.bsky.social · 6d

Euclid's Elements Book 3 Proposition 20 (called by some the 'Inscribed Angle Theorem') will look like this: the arc is double the angle (on the circumference) that subtends it.

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Segar Rogers @segarrogers.bsky.social · 6d

An arc of 30° will have a supplementary arc of 150°.

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Segar Rogers @segarrogers.bsky.social · 6d

A very short primer on mixing degrees of arc with degrees of angle for circle theorems.
#UKMathsChat #iTeachMath

Historically degrees measured arcs and arcs measured angles. This might feel uncomfortable but remember how we measure an angle today … with a protractor … measuring around an arc.

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