That is the seventh article written within the ‘Dose of Don’ sequence – this one is written by Sudeep Gokarakonda (@boss_maths). Anne Watson posted it on her weblog here and I’ve replicated it phrase for phrase. For the background on this sequence, please see the publish Lines and Angles on Square Grids. My thanks go to Sudeep for giving me permission to share his writing right here.
Dose of Don 7: Symmetry
It is a contribution to sequence of writings, begun by Anne Watson, which delve into the gathering of duties on Don Steward’s blog and pull out threads about key concepts in arithmetic that run by a number of of his duties. Direct hyperlinks to all duties talked about are included under.
Don was very beneficiant together with his duties and it’s hoped that you’ll return this generosity in the best way he requested earlier than he died, specifically to donate to justgiving.com/fundraising/jessesteward.
Symmetry is one thing that permeates arithmetic, and it’s one thing mathematicians can recognise in conditions that aren’t offered in a typical geometric context. Right here, for instance, is such a state of affairs:
You might have the next cash totalling 70 pence. In these questions, “quantity” refers to an entire variety of pence.a) Utilizing solely these cash, what’s the smallest quantity you can’t make?
b) Utilizing solely these cash, what’s the largest quantity underneath 70 pence that you just can’t make?
c) What do you discover about your two solutions?
I bear in mind a lesson involving Pascal’s triangle throughout which I causally talked about its symmetry. One scholar appeared satisfied that I used to be improper to counsel any symmetry right here. On exploring, I realised they had been contemplating the numerals moderately than the numbers, so for them:
This barely constrained conception of symmetry was not completely stunning, realizing that the scholar’s publicity to the thought had been restricted to the geometric contexts within the GCSE specification.
Listed below are some slides from Don’s duties. Not one of many chosen duties is primarily about symmetry. Nonetheless, every provides us alternatives to note and discover symmetry.
However recognizing the symmetry results in an opportunity to generalise. In query 16, if I changed y, w, y, w, y with any symmetric sequence (e.g.) a, b, c, b, a, would the imply nonetheless be 6? If the frequencies had been as an alternative a, b, c, d, a however the imply was nonetheless 6, what may we conclude?
For instance, in query 1, provided that the imply is 5,
- the one rating of three provides a “deficit” of two;
- the scores of 5 haven’t any affect on the imply;
- the 2 scores of 6 give a “surplus” of two; and
- the 4 scores of seven give a “surplus” of 8.
This strategy is, to me, is mentally much less taxing than establishing and fixing the next in my head:
The “balancing out” strategy after all works in all questions, nevertheless it’s maybe greatest illustrated utilizing these questions the place the imply is an integer (i.e. questions 1, 2 and 16), as a result of the arithmetic is stored comparatively easy. I like how this part of the duty will be bookended on this approach.
The road y = x is a line of symmetry on all 4 slides – even the place we solely have the primary quadrant. This line of symmetry might not initially be apparent to all college students. On the very first slide, nevertheless, is the chance to identify that e.g. (1, 12) and (12, 1); (2, 6) and (6, 2) and so forth. are all factors on the hyperbola. Is it essentially the case that if (a, b) is on the curve, then so is (b, a)? What about (–a, –b)?
These questions contain symmetry in a geometrical context, however a chance to think about symmetry in a extra delicate context pops up on slide 4. I see that 3y + x = 12 is simply y + 3x = 12 with the x and y swapped round. With out even seeing the graphs of those, I sense symmetry in these coefficients. Right here is a chance to ask what occurs, normally, to a graph if I swap x and y round in its equation. College students may make predictions, after which verify by making an attempt a number of capabilities utilizing graphing software program. Can they provide you with an intuitive rationalization for what they observe?
This second could also be a pure one to take a detour to go to (or revisit) self-inverse capabilities—and even pattern one other of Don’s duties, comparable to self-inverse and periodic functions.
The thought of symmetry crops up when contemplating preparations, possibilities, and plenty of subjects in statistics. Don’s duties typically embody lovely but easy visualisations illustrating symmetry:
As said earlier, not one of the above duties are primarily about symmetry. Don created different duties that I’ve not detailed right here, the place it’s fairly attainable to work by them – and achieve richly – with out recognizing the symmetries in them. However recognizing them provides lessons the chance to maybe take a “scenic route” by the duties – one which helps construct up college students’ sense of symmetry in arithmetic.