@teacherbowtie.bsky.social
A braggart, a rogue, a villain that fights by the book of arithmetic!
https://teacherbowtie.wordpress.com
https://teacherbowtie.wordpress.com
I translated it all so that they were distances from the origin, thought about the equation they satisfied, and did an expansion to find the product of its roots. I think you can avoid the point case as there is a factor of z everywhere.
February 4, 2026 at 5:18 PM
I translated it all so that they were distances from the origin, thought about the equation they satisfied, and did an expansion to find the product of its roots. I think you can avoid the point case as there is a factor of z everywhere.
That sounds like we did it in very different ways!
February 4, 2026 at 5:01 PM
That sounds like we did it in very different ways!
I can't remember where I adapted it from - it certainly isn't wholly original.
February 4, 2026 at 4:29 PM
I can't remember where I adapted it from - it certainly isn't wholly original.
I have been saving this one for a while.
February 4, 2026 at 4:14 PM
I have been saving this one for a while.
I should have added #ALevelMaths (Further).
January 15, 2026 at 9:20 PM
I should have added #ALevelMaths (Further).
We teach OCR, and never have to do the correction*, I think for more or less this exact reason.
*Except in Further Maths.
*Except in Further Maths.
January 11, 2026 at 11:30 AM
We teach OCR, and never have to do the correction*, I think for more or less this exact reason.
*Except in Further Maths.
*Except in Further Maths.
It is worth adding that this does not happen for the mean.
Averaging lots of sample means tends to the true mean (which we know from Normal Distribution Hypothesis tests).
Averaging lots of sample means tends to the true mean (which we know from Normal Distribution Hypothesis tests).
January 11, 2026 at 11:29 AM
It is worth adding that this does not happen for the mean.
Averaging lots of sample means tends to the true mean (which we know from Normal Distribution Hypothesis tests).
Averaging lots of sample means tends to the true mean (which we know from Normal Distribution Hypothesis tests).
That is mean!
'These data' in particular, even though we are told it is a sample.
'These data' in particular, even though we are told it is a sample.
January 11, 2026 at 11:27 AM
That is mean!
'These data' in particular, even though we are told it is a sample.
'These data' in particular, even though we are told it is a sample.
I agree!
The real answer is along the lines of:
If we repeatedly do this, and average our answers, the average tends to a predictable amount below the real answer, relative to sample size.
In order to 'unbias' this, we need to multiply by a factor of n/(n-1), and the n terms cancel out.
The real answer is along the lines of:
If we repeatedly do this, and average our answers, the average tends to a predictable amount below the real answer, relative to sample size.
In order to 'unbias' this, we need to multiply by a factor of n/(n-1), and the n terms cancel out.
January 11, 2026 at 11:26 AM
I agree!
The real answer is along the lines of:
If we repeatedly do this, and average our answers, the average tends to a predictable amount below the real answer, relative to sample size.
In order to 'unbias' this, we need to multiply by a factor of n/(n-1), and the n terms cancel out.
The real answer is along the lines of:
If we repeatedly do this, and average our answers, the average tends to a predictable amount below the real answer, relative to sample size.
In order to 'unbias' this, we need to multiply by a factor of n/(n-1), and the n terms cancel out.
Hence, by analogy, sample statistics will almost always underestimate population ones.
Dividing by n-1 makes th result slightly bigger than dividing by n does, which is good enough.
Dividing by n-1 makes th result slightly bigger than dividing by n does, which is good enough.
January 11, 2026 at 11:11 AM
Hence, by analogy, sample statistics will almost always underestimate population ones.
Dividing by n-1 makes th result slightly bigger than dividing by n does, which is good enough.
Dividing by n-1 makes th result slightly bigger than dividing by n does, which is good enough.
Someone once explained it to me by thinking about the range.
If you calculate the range of a sample, you are very unlikely to randomly pick the largest and smallest values, so the range of a sample is almost always going to be too small.
If you calculate the range of a sample, you are very unlikely to randomly pick the largest and smallest values, so the range of a sample is almost always going to be too small.
January 11, 2026 at 11:09 AM
Someone once explained it to me by thinking about the range.
If you calculate the range of a sample, you are very unlikely to randomly pick the largest and smallest values, so the range of a sample is almost always going to be too small.
If you calculate the range of a sample, you are very unlikely to randomly pick the largest and smallest values, so the range of a sample is almost always going to be too small.
n-1 when you are using sample data to estimate the population parameter.
n when you have all the data.
(But whether you have to do this varies by exam board and Single/Further Maths.)
n when you have all the data.
(But whether you have to do this varies by exam board and Single/Further Maths.)
January 11, 2026 at 11:08 AM
n-1 when you are using sample data to estimate the population parameter.
n when you have all the data.
(But whether you have to do this varies by exam board and Single/Further Maths.)
n when you have all the data.
(But whether you have to do this varies by exam board and Single/Further Maths.)
I did something similar?
December 13, 2025 at 12:07 AM
I did something similar?
Could it be along the lines of the numerator having to be always larger than the denominator when added, so there must be an intersection, hence they can't be mutually exclusive?
December 9, 2025 at 10:27 PM
Could it be along the lines of the numerator having to be always larger than the denominator when added, so there must be an intersection, hence they can't be mutually exclusive?
After looking at trig derivatives and the chain rule, I differentiate sin^2 x for my class, ask them to do cos^2 x, and then we look at cos^2 x + sin^2 x, which should obviously be more complicated...
October 23, 2025 at 7:06 AM
After looking at trig derivatives and the chain rule, I differentiate sin^2 x for my class, ask them to do cos^2 x, and then we look at cos^2 x + sin^2 x, which should obviously be more complicated...
en.m.wikipedia.org/wiki/Anamorp...
Is it this?
I remember reading something recently about how in some sports they are now added digitally to look 'wrong' from a particular camera angle, and therefore more realistic.
Is it this?
I remember reading something recently about how in some sports they are now added digitally to look 'wrong' from a particular camera angle, and therefore more realistic.
Anamorphosis - Wikipedia
en.m.wikipedia.org
October 8, 2025 at 5:10 PM
en.m.wikipedia.org/wiki/Anamorp...
Is it this?
I remember reading something recently about how in some sports they are now added digitally to look 'wrong' from a particular camera angle, and therefore more realistic.
Is it this?
I remember reading something recently about how in some sports they are now added digitally to look 'wrong' from a particular camera angle, and therefore more realistic.
The red feels like the odd one out to me.
July 16, 2025 at 5:55 PM
The red feels like the odd one out to me.
Which gives me back your original graph.
Maybe it was right all along?
Maybe it was right all along?
July 16, 2025 at 5:38 PM
Which gives me back your original graph.
Maybe it was right all along?
Maybe it was right all along?
Here is my suggestion then:
Convert it to something parametric (not sure what).
Stretch the two x and y equations separately.
Make it cartesian again.
Convert it to something parametric (not sure what).
Stretch the two x and y equations separately.
Make it cartesian again.
July 16, 2025 at 5:32 PM
Here is my suggestion then:
Convert it to something parametric (not sure what).
Stretch the two x and y equations separately.
Make it cartesian again.
Convert it to something parametric (not sure what).
Stretch the two x and y equations separately.
Make it cartesian again.
I shall continue to think.
July 16, 2025 at 5:29 PM
I shall continue to think.
I think one needs to be 0.5, and the other 2, not both the same.
July 16, 2025 at 5:20 PM
I think one needs to be 0.5, and the other 2, not both the same.
I can reason that out, but it could definitely confuse a pupil.
July 14, 2025 at 9:14 PM
I can reason that out, but it could definitely confuse a pupil.