60 − 20 − 8 − 6 = 26
Ah yes ... so it is ... after a little rearranging. I'd never actually seen it in use before. Thank you!
Anyone know what this is called? Is it a thing?! Found it in Bīrūnī from the 11th century ... as one does.
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I meant that the two different approaches can make the problem seem to be about different things. Contrast Paddy MacMahon’s solution to the last Catriona Agg Puzzle and mine. His is a double application of Pythagoras. Mine is more about symmetry in a square. Makes the puzzle hard to categorise!
Not necessarily an easy task! I’m always struck by the split between those who throw algebra at the problem and those that stay within traditional geometry.
Great. Thanks so much for taking the time :-)
I remember thinking that establishing a culture of neriage could take … well, months rather than weeks with many classes … and I just didn’t have the energy for it at the time.
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.
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Be interested to hear how you get on with the neriage part … I was never brave enough to fully embrace the spirit of it … but I love the idea of it.
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.
I seem to be ever late to the Agg puzzle releases. The way I see it the symmetry of the overlapping red squares ensures the 4-small-squares area equals the yellow-square area. I like the way the purple square dictates the size of all the other squares.
An under-used approach I’ve always felt.
The only exception to this that I can think of is along the lines of Kris Boulton's 'What Before Why'. At 9:28 sec he explains how he used to teach indices; at 13:55 he explains what he changed to. I've certainly had classes that preferred the latter.
www.youtube.com/watch?v=iqEc...
… okay … so ‘mathematics before mathematics-as-symbolic-manipulation’ … that sort of argument? … even ‘mathematics before semiotics’?! (… which is probably a slight misuse of the term semiotics but hey). I think we agree but might frame it in different terms.
‘formalised notation’ … algebra?
It was an SQA question … they’re rarely able to draw anything to scale. Sigh.
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
You should subtitle them in a small font with ‘It would be remiss of me not to share your cultural heritage with you.’ ;-)
… reminds me that a circle of radius 1 has circumference 2π but an area of only π. I alway feel uncomfortable about this!
These things always mess with my head: so many of a linear unit are equal in cardinality to just as many square units. I can never decide if it’s profound or not!
Nice. Very nice :-)
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.
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... meant to say: the scale factor of the small right-angle triangle to the larger right-angled triangle is 3 (by inspection of the hypotenuses) ... hence the ×3².
For the top right diagram the Chinese out-in complementary principle says that:
- the area above the diagonal equals the area below the diagonal …
- and since area A=area B … and area E=area F …
- it must be true that area C= area D.
Hence I can move area D into area C.
Proof of cot θ × tan θ = 1 … using the Out-In Complementary Principle. #UKMathsChat #iTeachMath
Why ‘relational thinking’? … as opposed to ‘proportional thinking’?
Oo … you posted icebergs in 2023 (when it was still very quiet here!) … but that is an interesting question!
Sigh. Commentary from the luxury of the side lines.
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.
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