Algebra is the arithmetic of unknowns.

I like this a lot … feels so uncomplicated … expanding the brackets (in this situation) just feels overkill. #UKMathsChat

Piecing together thought processes for 9th century algebra … in the absence of a) symbolic algebra and b) negative numbers: now: simplify 14 − (10 − x) then: simplify [14] less [10 less x] minuend: [14] subtrahend: [10 less x] • goal: ... 1/3 #UKMathsChat #iTeachMath #historyofmaths

Pre-algebra simultaneous equations. Clever! 盈不足術 /yíng bù zú shù/ rule of too much and not enough (!) حساب الخطأين /ḥisāb al-khaṭa'ayn/ reckoning from two falsehoods regula duorum falsorum rule of double false position Subtract if both guesses are deficits/excesses. #historyofmaths #UKMathsChat

So I'm reading about the origins of algebra ... and I find this from Abū Kāmil in about 900CE ... and of course Kāmil's algebra (al-jabr!) could solve this. Pretty amazing really. #UKMathsChat #iTeachMath

Euclid Elements Book II Proposition 5 Intriguingly this was used by the Egyptian mathematician Abū Kāmil in the 10th century to solve quadratic equations of the form x² + 21 = 10x #UKMathsChat #iTeachMath

I'm looking for a publisher. 240 pages of old-world trigonometry, geometry focused & algebra free – where it came from, who built it, how they built it, how they used it, and how to do trigonometry in this way today. Any suggestions gratefully received. #UKMathsChat #iTeachMath #MathSky

Semiotic observation (!) ... when you think of the Inscribed Angle Theorem, which arc do you mentally highlight? I've always done (a) but actually Euclid presents it as (b) … which makes sense because the angles are indeed in that segment. Time on my hands? Maybe ;-) #UKMathsChat #iTeachMath

Anyone know what this is called? Is it a thing?! Found it in Bīrūnī from the 11th century ... as one does. #UKMathsChat #iTeachMath

I’m thinking of buying a visualiser … either an IPEVO DO-CAM or an Innex DC500. If you use either of these I’d love to hear your views of them … particularly any shortcomings. #UKMathsChat #iTeachMath

There were no takers for this … but here's the solution for completeness. It's all similar triangles ... with some fancy ratio table moves; the reductions come from common diagonals in the upper pair … and common columns in the bottom pair. Classical moves that we're maybe less familiar with today.

Lovely proportional reasoning from a pupil I tutor. She couldn't remember if she'd been taught similar triangles … but said: since 3 is less than half of 7, then AB will be less than half of 10.5 … and (red lines hers) if we split the 10.5 into 7 parts then AB will be 3 of these parts. #MathsToday

Still messing around with ratio tables. The marking scheme for this question (SQA N5M 2024 P2) focuses on the Sine Rule ... but a similar triangles approach does the job nicely ... and the ratio table bypasses any worries with algebraic manipulation. #UKMathsChat #iTeachMath

Proof of cot θ × tan θ = 1 … using the Out-In Complementary Principle. #UKMathsChat #iTeachMath

This is the geometry in one of the diurnal problems from Brahmagupta's 7th century Khandakhādyaka. I've undertaken some repositioning to aid the aesthetic, and a little obfuscation to add to the challenge. #UKMathsChat #iTeachMath

From @mosababutoha.bsky.social's book, Things You Might Find Hidden in My Ear: bookshop.org/a/862/9780872868601 #poem #books #writing

Still in ratio table mode. I remembered this @mathforge.org ;-)

Some 'aspects/rules' of ratio tables ... collected together as an aid to proportional reasoning. 'El. V–16' = Euclid's Elements Book 5 Proposition 16 #UKMathsChat #iTeachMath

Related ratio tables?? Find what can be found. #UKMathsChat

Nested ratio tables anyone?? Hmmm? #UKMathsChat

A diagram for a diagram-less tradition; gradually making some sense of Brahmagupta's Khandakhadyaka. With a couple of shadow measurements and the angle of the rising sun he could calculate the time of day. Astonishing really.

Where would you give the marks here? It depends how you do your trigonometry of course. But if you teach it via a similar triangles approach I can only find 3 marks. Is there even a 4th mark (with this approach)? (2003 Scottish Credit Paper 2 – 4 marks). #UKMathsChat #iTeachMaths

Apologies for jumping out of the thread … interested in conceptual explanations accessible to pupils. My take: 8/21 ÷ 4/3 8 twentyoneths ÷ 4 thirds So 4 thirds into 8 twentyoneths 4 goes into 8 twice. A third goes into a twentyoneth 'a seventh' times. So 2 × (1/7) So 2/7 @germinalmaths.bsky.social

#poetry Anna Akhmatova, trans. Kunitz and Haywood. Akhmatova lived in Tashkent in the early 40’s. Termez lies to the south, on the border of Uzbekistan and Afghanistan. I go through phases with poetry … it’s been a while … for the last week or so I’ve felt it calling … specifically Akhmatova.

Euclid's Dedomena/Data. 94 'If A is given then B is also given' propositions. The Data proves only whether relationships exist: 'If in a circle given in magnitude a straight line be drawn cutting off a segment admitting a given angle, the line drawn is given in magnitude.' – that's trigonometry!!

Wondering about the parallels (if any) in other disciplines.

Trying to write a question that requires the use of the Angle Bisector Theorem (Eu. El. VI – 3) … but failing … somehow a non VI – 3 solution always presents itself. Still, ended up with a nice enough question. #UKMathsChat #iTeachMaths

Does anyone teach this? Wondering if it's generally known or generally not known. #UKMathsChat #iTeachMaths

An hour in the wind this afternoon collecting an oak forest :-)

#MathsToday Looking through the jotter of a girl I’m tutoring … today’s page … the first two trig identities … below one the word ‘Tesco’ … below the other ‘Scone’. Just when you think you’ve seen everything. I’ll leave you to work out which is which ;-)

When you're teaching integration by substitution, what do you tell pupils is going on (conceptually)? Interested in the phrases people use ... and I appreciate you may just take a procedural route. #UKMathsChat #iTeachMaths

Been thinking more about the merits of the Exterior Angle Theorem and the fluency of pupils' geometric thinking. Here's a proof of one of @karencampe.bsky.social 's angle–arc theorems. Step 4 is beautifully immediate _if_ one uses Exterior-Angle-Theorem thinking. #UKMathsChat #iTeachMaths

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

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

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. 1/6

Idle thought: Explain in a sentence why for any numbers n and s such that n>s, s²/n is closer to s than n is . #UKMathsChat

Idle thought: Explain geometrically why the square-root operator is distributive over multiplication. #UKMathsChat

Something I didn't know. #UKMathsChat

#poetry #poem Christian Wiman

An aide-memoire for right-angled trig?! … maybe not … but it's interesting how orientation affects our ability to remember relationships. In this orientation I find the symmetry of the relationships much clearer. If only all trig questions had horizontal hypotenuses ;-) #UKMathsChat #iTeachMaths

#poetry #poem Louise Glück.

An Ailles Rectangle problem. Find the value of a°. More an exercise in (trigonometry) language fluency than anything else. i.e. do you know how to interpret the terminology to be able to do it in your head. @davidkbutler.bsky.social #UKMathsChat #iTeachMaths

If … you broke trigonometry into smaller pieces … spread it over 3 years … and taught the basics of just sine in early Secondary … then you could play around with problems like this … without everyone collapsing in a heap. Gradual trigonometry. #UKMathsChat #iTeachMaths

Is there a good working definition of 'understanding' (c.f. knowledge) that people regularly turn to? (Apologies, not a #CogSciSky person, unfamiliar with the various schools of thought. Looking for something succinct).

If there was a cot button on calculators … I’d be one of the people that used it.

A sort of proto-trig task. No angle values required. Certainly no sin/cos/tan required. In one sense trigonometry was always two things: 1) building a table of values and 2) learning how to manipulate the results. This task concerns the latter. #UKMathsChat #iTeachMaths

#poetry #poem Mosab Abu Toha, from 'Forest of Noise'. There's a moment in each poem where he gently stops you in your tracks and shows you something you have never had to experience.

Strand curriculum. t.ly/v-JCL. Two different topic tasks per lesson. 15min HW+LSQ, 15min Task 1, 15min Task 2, 5min spare. Keeping to time and the right resources essential. Pros: Pupil focus, spacing. Cons: Time making resources. Intense 50min but I loved it. #UKMathsChat #iTeachMaths

#poetry #poem Anna Akhmatova, trans. Kunitz and Haywood. This grabbed me the first time I read it. O.M. is the exiled Osip Mandelstam. The still shaking earth. The chill of the last four lines.