Any maths teachers around?

[Verycold]

Please can you give me your thoughts on this hw questions:

Can every cube of a number be written as the difference of two square numbers?

How many different ways can you make a sum that is equal to 20?

But it is true for all cubes greater than 1!

My proof is not very elegant;

The difference between the squares of consecutive numbers gives the odd numbers, the cube of any odd number is itself an odd number and hence will appear somewhere in this list

The difference between the squares of numbers that differ by 2 gives the multiples of 4 (starting at 8). The cube of any even number is a multiple of 4 (actually, it will be a multiple of 8)

Bit tough for most 15/16 y/o in my opinion

I would be wanting to see how they tackled it, not just whether they got the "right" answer.

I've just been reading a book about learning theory. One very interesting feature was that we learn MUCH faster by testing knowledge we haven't yet learned. This ties up with normal experience, if you think about how much we learn from doing puzzles and riddles, quizzes, etc. It just sounds anarchic because we have so many received opinions about education.

So, yes, do it as a puzzle: she'll learn stuff she never thought she knew :)

It just seems awfully difficult to do anything other than write down examples

For the sums to 20, start simple.
How many ways can you make a sum to 3, then 4, then 5 etc.

Look for way in which you can make sure you have all of them.

Look for pattens.

Extend the patterns

Describe the patterns.

I have just been having a go at the sums to 20. I started simple, but it soon got very messy.

Yay, I've just done the number of sums problem and derived the formula!
Cheers, Mumsnet

(Hint: both questions are about triangular numbers.)

Not sure how much that hint is helping

Didn't want to do it for her! But you can find the answers if you google triangular numbers.

Here's a little example using a sum to 5 :)

One
1+1+1+1+1
1+2+1+1
1+3+1
1+4

Two
2+1+1+1
2+2+1
2+3

Three
3+1+1
3+2

Four
4+1

Get DD to count how many sums there are for each starting number.
See the pattern yet?

That has to have been one of the least helpful posts on mumsnet for a while. Even if you include AIBU.

Fair enough.
The last was helpful.
Patronising also.
But helpful.

Might be patronising for you, thecat, but I did it and derived the formula correctly.

Why don't you just call me thick and have done with it?! As it goes, I'm quite pleased with myself; I learned arithmetic before new maths came in and never did this kind of thing.

That wasn't what I meant.
I read your initial post as quite smug: the OP posted for help. Your reply essentially said: "I've done it! And you can fuck off." The second post came across as: "OK, I'll give you a hint - but I won't tell you the answer, because that is norty."
I do think neither response is ideal, given that the OP posted on mumsnet asking for help. Not on teats net asking for a good load of condescension.
But,hey, given your response to my post perhaps all we can say is that nuance gets lost on the internet.

goodness knows what"teats net" is.

I guess nuance often gets lost in written posts!

Teats net has its own bag of nuances, though

Garlic

One
1+1+1+1+1
1+2+1+1
1+3+1
1+4

What about 1+2+2 ?

Oh, bugger!

... aaand, THIS is why I'm doing my maths GCSE, despite having an advanced qualification in statistics!

Oh, I got it (correct if I'm wrong)
1+2+2
is the same as
2+2+1
which we've already done.

It's a matter of listing the sums in a logical way - I think.

Keep going (imagining you are y7 student) how far away does 20 seem when you already have more than 20 ways to add up to 9.

I'm not sure the teacher thought either of these problems through.

Interesting as they are to an adult, they are likely to create confusion and frustration in most 11/12 y/o's imho

Imagine 20 sheep in a line - divided into two fields by building an infinite fence like this S S S I S -> 3 sheep in field 1 & 1 sheep in field 2. How many places can this fence go? What if you had two fences - how many positions for the fence are there? And so on. That would give you all possible sums - wouldn't it?