What is the next step in Maths?

[Lima1]

Hi my DS is 8 and is in 2nd class (in Ireland, equivalent of Yr 3??). He loves Maths and picks it up quite easily. He is always asking me to show him new concepts and to set pages of arithmetic for him. In school they are only just learning their times tables and doing simple multiplication and division.

He can do long multiplication and division, understands fraction including comparative fractions ie 1/3 = 2/6, 3/9 etc he can add, subtract and multiply fractions. He understands percentages and decimal places. He understands a bit about angles ie 360o in full circle, types of angles eg right, obtuse, etc.

I bought him a maths set yesterday with a compass and he is using it to make pie charts, he uses the protractor to work out angles within the circle ...
He is asking about triangles and I was going to start showing him basis trigonometry (ill have to YouTube it as I cant really remember).

We work a lot on really understanding numbers and different ways to work things out. We play games like I call out a number and the kids have to come up with different sums that my number is the answer to eg 20 and they will say 13+7 or 4x5
He knows about negative numbers, I have explained things like
15 + x = 25 and ask him to work out the value of x. He knows the -/+ changes when it moves to the other side of the = sign.

We work on different ways to add numbers by using the nearest round number eg 11 + 8 he knows to add the 1 to the 8 and then add 10 if that makes sense.

He is always pushing for more. I know he needs to really understand the basics as they are the foundation for everything and I think he is pretty good at that now.

My question is what would be good to teach him next. I have used the Khan academy on YouTube and its brilliant but it seems to go from the concepts I have mentioned on to really difficult areas which I don't think DS would understand like linear equations, derivatives and probability.
Is there a middle ground that I'm missing?
Thanks

Irvine I see what you mean, I shouldn't be talking about changing sides, I should be using the term inverse, he knows what that is I just wasn't using the correct term. Thanks for that clarification ;)

DS gets no maths homework so I really don't know what he is doing. His teacher has said that parents don't teach using the correct methods (guilty as charged obviously!!).
He has just completed a couple of pages in a workbook of these inverse sums some of which include brackets and some are written as word problems and he got them all right so he obviously understands what he is doing (despite me confusing the situation)

Fact families are groups of related number facts, rather than all the different ways that a given number can be made.

So if you were to give children a fact such as 3+5=8 and ask them to write the ‘fact family’ or What other facts they can work out you’d expect them to be able to give 5+3=8, 8-5=3 and 8-3=5. And similar for multiplication and division.

Is having a look at whatever textbook/workbook the school are using further up the school an option. If you can find an explanation of the method the school are using that might help you to explain it to him.

Lima1, thank you for the reply to my question. I get concerned that children, particularly girls, get the message that maths is not for them, but you are obviously equally encouraging to them all.

You mentioned car journeys. One game you could try is "what shape am I". One person thinks of a shape then the others guess. Only questions with yes or no answers are allowed and so things like "are you 2d? Do you have more than 4 sides? Do you have any square faces? etc.

Another set of books, also by Letts is mythical maths. Dd used these to revise topics by herself and used to like collecting the stickers for the scene at the end of the book.

To me, maths is all about seeing patters, so the more you can draw things out, preferably with bright coloured pens, the better.

I know I've said it before, but the more you write, the more I think those DragonBox apps I mentioned could be lots of fun for your DS (and perhaps your other children too) (and in fact, yourself!)

The fact that you even consider a problem such as 35+?=62 in an 'equation' way, indicates to me that you/your DS would enjoy the algebraic problem solving and equation manipulating that you learn/practise in DragonBox Algebra. It starts very simply. The goal is always to isolate the 'Box'. To eliminate a 'tile' from the box' side, you have to do the 'inverse' thing e.g. add the inverse tile - but you have to do that on both sides. Etc. It quite quickly progresses to fairly complex equations.

And let me tell you about DragonBox Elements (which is about Geometry). There are no numbers involved, but you learn the 'rules' of geometry and apply them to 'puzzles'. For example a problem solving reasoning path could sound something like this: (The 'problem' could be to find a shape on the screen for which you can 'prove' that it is e.g. a rhombus)
'We know that this is an isosceles triangle (because two of the three sides are the same length). That means that this angle here is the same as that angle there. Accordingly, the angle on the other side of the crossed lines (extending from the triangle) is also the same. As we figured out earlier that those to lines here must be parallel, we can conclude that that other angle, created by our continuation from the triangle cutting the parallel line, is also the same. Ergo, this second triangle here has three equal angles and must be an equilateral. One of that equilateral's sides is also the radiant of a circle, so now we know that every other radiant is obviously the same length as this triangle's side... etc etc.

Honestly, I don't work for them, but these games are fun and it sounds to me as if they would be perfect for you :)