Test question

[xilman]

Originally Posted by lukerichards

I'll post the course spec for GCSE maths and GCSE Additional Maths later. Don't forget they need to cover algebra, number, geometry, statistics and calculus... so depth of study is limited.

Thanks. I look forward to reading the specs.

I also had to cover those subjects for O Level as it then was.

[lukerichards]

Here's the GCSE spec I teach:

http://filestore.aqa.org.uk/resource...00-SP-2015.PDF

and the Additional Maths GCSE:

https://www.ocr.org.uk/Images/457916...from-2018-.pdf

[Dr Sardonicus]

Originally Posted by lukerichards

Also not something they'd have come across in their education yet.

That ain't good. I'm not surprised, but still, the spectacle of students embarking on a field of study without adequate background does not fill me with joy.

I am encouraged, however, that they used trial division. For so small a number as 511, it will work pretty quickly.

My grampa did not have much formal education, and felt the lack later in life. He bought books to broaden his knowledge. One of them was a book entitled Mathematics: Its Magic and Mastery by Aaron Bakst. Chapter 15 has a great title, "Algebra, Boss of Arithmetic."

It seems that this book (which I recommend) is freely available online, though the plain text version I looked at has some errors, probably due to faulty "optical character recognition" (OCR). Amusingly, my own OCR was faulty in reading the title on the book's spine. For years, seeing that book on the shelf, I misread "Mastery" as "Mystery." The misreading was likely prompted by association with the word "Magic."

[lukerichards]

Originally Posted by Dr Sardonicus

That ain't good.

Difference of squares is covered on GCSE (age 14-16). Difference of cubes as a concept or something to be aware of is not covered in age <= 18 education at all in the UK.

[lukerichards]

Here is a list of GCSE topics:

https://corbettmaths.com/contents/

[lukerichards]

Originally Posted by Nick

If your students are familiar with the formula for summing finitely many terms in a geometric progression, you could show them how that gives
\[ x^{n-1}+x^{n-2}y+x^{n-3}y^2+\ldots +y^{n-1}=\frac{x^n-y^n}{x-y} \]
and deduce for all integers x,y,n with n positive that \(x-y\) divides \(x^n-y^n\).

I taught this as an "interesting aside" to one of my post-16 classes a few years ago who were studying A Level (A Level stands for Advanced Level to give some perspective to non-British readers). It was not, however, a curriculum requirement.

It is very removed from the requirements of the Age 14-16 GCSE curriculum however, that I would not get the opportunity to cover this level of algebra.

[lukerichards]

Originally Posted by lukerichards

I'm not sure whether to be pleased or disappointed that, thus far (I haven't finished marking them) they have all used trial division rather than a LL test. Trial division is, of course, the much more efficient way of proving the compositeness of this number!

One student has actually written as her answer:

n has to be a prime number

I'm happy with that!

[Dr Sardonicus]

Originally Posted by Nick

If your students are familiar with the formula for summing finitely many terms in a geometric progression, you could show them how that gives
\[ x^{n-1}+x^{n-2}y+x^{n-3}y^2+\ldots +y^{n-1}=\frac{x^n-y^n}{x-y} \]
and deduce for all integers x,y,n with n positive that \(x-y\) divides \(x^n-y^n\).

I may be misremembering, but I seem to recall the algebraic formulation

x - y divides xⁿ - yⁿ for every positive integer n

being an exercise in mathematical induction in an algebra book I used in junior high or high school. Verify for n = 1 and n = 2, then use

xⁿ⁺¹ - yⁿ⁺¹ = (x+y)*(xⁿ - yⁿ) - x*y*(x^n-1 - y^n-1).

The exercise may have been a specific case, like 19 divides 20ⁿ - 1 or something...

[Dr Sardonicus]

Originally Posted by lukerichards

One student has actually written as her answer:

I'm happy with that!

That gal should go to the head of the class!

[Batalov]

Originally Posted by lukerichards

One student has actually written as her answer:

I'm happy with that!

That is not a proof, though. If that was all that was written, it should get, say, 0.1-0.2 points out of 1. Wasn't the problem statement "Prove that ...."?

Compare: if a problem was "A train leaves Chicago for Detroit going 60 mph... and so on...." and the student simply answers "3" (and it happens to be correct). How many points is it worth? >0, that's fine, but there is no solution, so it could be simply taken from looking over the neighbour's shoulder.

[Batalov]

Originally Posted by Dr Sardonicus

I may be misremembering, but I seem to recall the algebraic formulation

x - y divides xⁿ - yⁿ for every positive integer n
...

But it is important to use n=3 here and x=8, y=1.

( Because x - y divides x^p - y^p for every positive integer, prime p, too, but it does nothing for the case x=2, y=1, because x - y = 1 )