Alex ~ VIC Maths Notes

Because it's for implicit equations as opposed to explicit equations of the form y=f(x)

Great etymology video from @robwords.bsky.social

Inflation hitting so hard, it's affecting the percentages

YouTube video by EpsilonDelta Do You REALLY Understand Derivative? | Symmetric Derivative and Generalized Pseudoderivative

Literally just came across this the other day
youtu.be/oIhdrMh3UJw

The main other ones I'd drawn attention to are probably sin(0)=tan(0)=0 and cos(0)=1 for y-intercepts and because they show up in integrals regularly, and I want to make sure students don't mistake e^0=cos(0)=1 with being 0 which happens frequently.

We usually have already got that sin(45)=cos(45)=1√/2 by osmosis, but making that explicit (useful for intersections too)

We use the fact that tan is increasing from 1/√3 to 1 to √3 for 30 to 45 to 60° to help differentiate those.

That basically leaves sin(60)=cos(30)=√3/2 and the multiples of 90

After they've been getting familiar with those and we're trying to improve efficiency, we start memorising the rational ones in tri's: sin(30)=cos(60)=½, tan(45)=1 (because rational tends to feel easier to remember)

I'm in the same boat. I call them the half-square and half-equilateral triangle to help Ss remember how to construct the tri's if they forget.

I also use the unit circle definitions (x, y coordinates for cosine and sine and gradient of radius or length of tangent to x-axis) for the multiples of 90

We only do x^2, log x, and 1/x, so it's a fairly short ladder. But it was /so/ much easier to remember how to label the circle.

Yes, I can do it by considering similarities to the respective graphs, but this subject doesn't go through that.

circle of transformations with possible transforms listed. signs indicate the sign of the correlation after transformation (without compensating for the reversal of 1/x)

the bulging rule where the ladder is labelled as a scale across each axis

ladders going from x^-3 to x^-1 to x^0 treated as log(x) to x^1/2 x^1 then through x^5

Got to test something new in #mathstoday. Normally, to decide which transformation to linearise a data set, I'd just show a circle of transformations where each quadrant has the options listed. I recently stumbled upon Tukey and Mosteller's bulging rule and ladder of powers and it's /so/ much easier

I think I love it even more than discorectangle

I've discovered some people call it a convex angle, which makes some sense, although it suggests a reflex angle should be called concave.
Incidentally, did you know that angles that add uo to 360° are called explementary?

Explementary is such a handy word

Giving a child a calculator too early doesn’t help them learn maths; it helps them avoid learning maths.
Great piece from @amandavande.bsky.social on why fluency must come before function. springmath.org/fluency-befo...

easy change

differentiating x^2, x^3, and x^4 by factorising (x+h)^n - x^n using the factorisation of a^n - b^n and letting h → 0 without expanding the second factor.

differentiating x^n by factorising (x+h)^n - x^n using the factorisation of a^n - b^n and letting h → 0 without expanding the second factor, where there are n terms of x^(n-1)

the (hopefully) fully corrected version and x^n

First principles, yeah, heaps. By factorising rather than expanding? Surprisingly, no!

This was wrong too, lol #OnARoll

differentiating x^2, x^3, and x^4 by factorising (x+h)^n - x^n using the factorisation of a^n - b^n and letting h → 0 without expanding the second factor. *corrected x^4 working

Doesn't surprise me

differentiating x^2, x^3, and x^4 by factorising (x+h)^n - x^n using the factorisation of a^n - b^n and letting h → 0 without expanding the second factor.

Can't say I'd thought to do this before. #mathstoday

Would appreciate any reposts to try and get to the bottom of this

16 principles of task design with Nathan Day

Loved discussing some excellent maths tasks in @nathanday.bsky.social’s session. Great to have the opportunity to dig into what makes them so good and how they could be adapted. #MathsConf39

Queries:
1. Are the two methods actually distinct and have been inadvertently conflated somewhere along the line?
2. Which (if either) should be favoured as an introductory/straight-forward/trend-agnostic method?

average percentage method applied to data over 2 years to determine seasonal indices and deseaonalised sales; sales in millions of dollars

simple average method applied to data over 2 years to determine seasonal indices and deseaonalised sales; sales in millions of dollars

differences in seasonal indices: -0.0093950283 0.0278223623 -0.0145401985 -0.0038871355 differences in deseasonalised values (in millions of dollars) 0.032382259 -0.04141642 0.025299911 0.017857768 0.019279611 -0.035355481 0.015528388 0.011324438

Additional example over 2 years, quarterly.

They're out JUST enough that you could get marked incorrectly from an assumption of one method or the other because it looks like a rounding error

seasonally adjusted time series showing original data, and deseasonalised values using average percentage and simple averages methods which are generally equivalent but have deviations

same graph without the original data to exaggerate discrepancies

differences between seasonal indices 0.001 0 0.001 0 -0.001 -0.002 -0.002 -0.001 0 0.002 0.002 0.002

Here's the catch: they're not equivalent (nuisance 2). They produce VERY similar results, but not exactly the same (algebraically distinct). And the difference is enough that it could look like a rounding error to 2-4 decimal places occasionally.

text refers to average percent method: data as percentages of the average for the year which are then averaged

text refers to simple averages method: calculates average for each quarter across all years, then written as a percentage of the overall average

text lists average percentage and simple average as the same

Some texts describe the "average percentage" method but call it "simple average", but "average percentage" is pretty consistently as above (feels very self-referential tbh).

When I've looked into other methods for seasonal adjustment to see what else is out there, different texts tend to refer to them in place of each other or as being equivalent.

Simple averages is much more efficient calculation wise, even if you have to first average the seasonal values for each season. For reference, 8 years of monthly data goes from 116 calculations for average percent to 25 calculations for simple averages to get the averaged seasonal indices.

simple averages method applied to the same data over 8 years, monthly; conveniently the average of the seasonal averages is equal to the average of all the data values

Past exam question that refers to long-term monthly average heating costs

past exam question referring to long-term quarterly average visitor numbers

Some questions we've gotten give the average values for each season (assuming across years) and ask to compute the seasonal indices from those. From what I've found, this is the "simple averages" method.