Maths teachers: I need help with nth number sequences!

[WilfShelf]

I am trying to help DS (11yo) with his homework, not doing it for him but explaining strategies for solving the questions.

He knows what he's doing and I have been getting him to show me how to do it, but we're hitting a bit of a wall with quadratic sequences. He has to work out the formulae for particular number sequences where n = nth number in the sequence...

When he's resolved the first quadratic bit of the sequence, he's left with another sequence, which is (so far) linear. I've been trying to show him that he can then just apply the same rules to this bit, that he did when he was solving the earlier (easier) linear sequences... And then express them as two separate parts of the same equation (thus far they've just been additive so it seems to work fine: my maths stops about here so no idea whether the relationships become more complex!)

Anyhoo. He prefers to do the second bit by trial and error, but I'm trying to explain WHY it works to treat it as a separate linear sequence and apply the rules he already knows. Is this right? And if so, how can explain to him, in simple terms, why this works and is better than just randomly trying different expressions?

Am I making ANY sense?

Yes. Should have realised late on a Friday night NOT the best time to lure maths teachers out from their bottle of wine exhausted on the sofa. Will bump at more civilised hour

Would it help to construct your own sequence instead of trying to decode someone else's?

If you have the formula (a.n2 + b.n + c) and work out the number sequence for n=1,2,3,... then maybe it will be easier for him to see^ the additive nature of it if you have calculated it yourself.

link

All the Y7's we know (all at different schools) are doing this!

I think you should let him use his own methods (if they are working) until he is ready to make the jump. Or arrive there himself.

no no no

The whole point of Maths is logic and order. Guesswork is not Maths!
Besides which, it's much slower. And can only be done in simple situations.

But that is what they do in Primary, until they children are able to see the logic themselves.

he should, without a doubt, write out the additional sequence and use his understanding of linear sequences to finalise the answer

Oooh, I've caused a Maths Spat!

I think he has worked out that he can do the second part using the linear rules, but last night he was tired and we were doing it in dribs and drabs so he'd lost his thread.

I'm still not convinced we understand WHY the second part is linear though. We discussed how the linear ones form a line on a graph, with a constant forming the crossing point, and the direction of the line affected by whether it as a positive or negative n [?]

But it's perhaps jumping to far ahead to imagine what the quadratic line would look like.

God, I'm REALLY enjoying it [saddo emoticon]. I can see I'm gonna have to do all his maths with him because I love it so much... Can you take another GCSE at 43?

at you enjoying it. That's nice.

A quadratic is a curve - if it's a positive x^2 then it's a smiley mouth, if negative then a sad mouth.

It's quite hard stuff for Y7 by the way. Good on him.

He's pretty able at maths [got 100% in his Y6 SAT!] so he's in a fastrack set. And I do actually have a Maths A level, but it has been a LONG time. I loved pure maths but forgot EVERYTHING, and now I deal with stats on a daily basis but it's quite nice to be reminded of how much I enjoyed solving the problems.

And getting him to show me how to do it is working really well, although I have to stop myself competing to do each problem

Right, now we're stuck!

We're on mymaths for his homework, and the sequence is

-1, 1, 7, 17, 31, 49

He thinks the answer is:

2n^2 -(4n-1)

but my maths won't let us enter the final bracket! He's reluctant to submit it if it won't accept the answer as he thinks it must be wrong. But unless we're both being VAIR thick, it does seem to work as the solution. Can anyone help?

Hahaha, yes we're VAIR thick!

We worked it out: I pointed out that minusing a minus makes a plus so actually he didn't need the brackets and instead needed to add one...

then do 2n^2-4n+1

oooooops
did not see final post :)

I tend to teach nth term sequences like this:

Let's say the sequence is -4, -1, 2, 5...

Most students spot straight away that it goes up in threes. I ask them for another sequence that goes up in threes; they (usually) come up with the three times table, which we argue about for a while and decide is 3n.

We write them out next to each other:
3 -4

6 -1

9 2

and ask "how do we get from the times table to the number?" You take away 7.

So the expression is 3n - 7.

===

As for the quadratic, it's not easy to explain why the "remainder" should be a linear term. I'd probably finesse it a little bit with rows of differences (something like:
-1
2
1 4
6
7 4
10
17 4
14
31 4
18
49)

and then say "you need an extra order of n for each column until it's all the same. So this one needs some n^2s, some ns and a number."

Hope that helps - well done for playing an active role!

Hello,

Just to add to this, i also have a maths issue that i cant solve.
My nine year daughter has a question on her homework i cant solve to do with sequences - and i also have an A level in maths -
Can any one help me as i cant solve it and so cant explain to her whats happening in the sequence -

310278 -- 310287 -- 310782
she needs to fill in the 2 spaces.

we have tried everything that i would usually think of, and at this age most of her homework i see the answer a mile off and so can explain, - but maybe im having a slow day because i have spent nearly an hour trying to figure it and cant

Please can someone see if they can solve it?

thanks