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

[PhilODox]

What about simple algebra?
And maybe some factors, powers etc?

[noblegiraffe]

I think you need something more structured than random topics on YouTube. Trigonometry is a huge step up from angles in a triangle, way too hard. If you want to teach him then buying a proper workbook would probably be a good idea. It would hopefully also avoid dubious methods like change sides change sign for equations.

[norbert23]

Look on "nrich" for ideas. Focus on understanding why / the way maths is connected rather than next steps. Lots of great ideas on there & the ncetm site x

[GeorgeHerbert]

Buy him some Murderous Maths books - my ds loved them at that age and they introduce quite complex ideas in a very fun way.

[user789653241]

Khanacademy has tutorial videos for all the maths topic, so he can work thorough his interested topic himself.

Also these sites are very good and useful.

nrich.maths.org/
wild.maths.org/
mathigon.org/
diagnosticquestions.com/UKMT
artofproblemsolving.com/videos
www.mathantics.com/
www.mangahigh.com/en-gb/

[user789653241]

Oh, sorry I totally missed you used Khan already.

My ds uses Khan quite systemically and also by interest. So he works to complete each grades and each topics, but also he does something totally off like matrix and complex numbers out of interest.

We do use paid site just follow NC and make sure he doesn't have any gaps in knowledge though.

[Lima1]

Thanks for those replies and for all those links, I hadn't heard of most of them.
I am conscious that he understands what he is doing and that there are no gaps in his knowledge and that what he is doing are natural progressions.
We mainly work on looking at topics from different angles rather than rush to the next thing.
I actually just found a maths workbook at home with topics like area and probability so ill try him with those too.

Noble - am I doing something wrong in telling him to change the sign if the number changes sides? I don't want to be teaching him something wrong but that's what I remember being taught?? Or s it that I'm just oversimplifying it? Thanks

[user789653241]

It seems like as a first steps, they don't teach change sides and change signs these days.
If you are doing something like 3x-5 = 7, they add 5 to both sides like 3x-5 + 5 = 7 + 5, to make it 3x = 12. Khan seems to emphasize on if you do something on one side of equation, you need to do to the other side.
Once the child understand the concept, they can eliminate that step.

[user789653241]

Also if you want to do the trig, I would suggest you do start from pythagorean theorom first, and work though it.

www.khanacademy.org/math/geometry/hs-geo-trig

[macbethh]

Could you speak to his teachers? You could ask if they could tell you what the next following topics will be, that way he learns more and is ahead but without it being completely far away from the current stage

[Lima1]

Thanks Irvine - actually started Pythagoreus therom with him today and basic trig sin, cosine and tangent. I looked at the Khan video on basic algebra and it was tracing about doing something to both sides of the equation.
It's so long since I did maths I have forgotten most of it.

Macbeth his teacher gives him more advanced word problem questions in line with the current curriculum and he loved those. She lets him do his own sums so if the other kids are adding 2/3 digit numbers she lets him do ones in the tens/hundreds of thousand (but it's just the same concept).
My DD is two years ahead so I know the curriculum and he would find her maths work easy (I have caught him doing her homework!).
He is at least 3 years ahead of the curriculum at this stage.

[Lima1]

Pardon all the spelling mistakes I'm on my phone.

[noblegiraffe]

am I doing something wrong in telling him to change the sign if the number changes sides?

The number isn't changing sides, it's teaching a trick that doesn't actually lead to understanding of what is going on. You aren't picking up terms and moving them around, you are doing the same thing to both sides of an equation - the inverse of something which is already there in order to get rid of it from one side - and thus keeping the equation balanced.

Change sides change sign won't help if want to solve 2^x = 5, taking logs of both sides will. If you want to solve sin(x) = 0.5 (if you want to do trig) then you need to take the inverse sine of both sides. It's not a trick that scales up.

[thecatfromjapan]

It seems incredible to me now but I was taught the whole 'changing the sign, change sides' thing all the way through secondary. needless to say, my understanding of maths was cloudy.

It wasn't until I was learning Maths alongside my children - with them teaching me, really! - that I discovered what was actually going on. It was a revelation and suddenly a whole area of Maths became logical, rather than a strange form of magic.

I know I've been mocked for saying this before but I do think Maths is taught far better (for most students) these days.

[user789653241]

Yes, I totally agree, cat. I was taught the same. And despite having parents who were strong with numbers, I really didn't get maths in depth.
Watching videos now makes a lot more sense.

[JustRichmal]

When I taught dd maths at this stage, I used the Lett revision guides, so I could be sure of covering the curriculum.
I kept things visual as much as possible. For instance, for a sum like
5x+3=8 I would draw scales with 5 boxes and 3 dots on one side and 8 dots on the other.
Introducing trig, I drew the x axis and y axis then imagine a line one unit long laying along the x axis, fixed at (0,0). As you rotate it up, the "shadow" on the x axis gets shorted and the shadow on the y axis gets bigger.
Could I ask why you only teaching maths to the younger child?

[brilliotic]

Hi,
you could look at the Dragonbox apps, especially the Algebra (perhaps start straight with the 12+ one) and the Elements (Geometry) one.

They are basically just fun logic puzzles/games, but sneakily teach a good basic understanding of geometry and algebra.

[Lima1]

Thank you for all those tips and recommendations.

So how do I teach him how to do these sums:

45 / ? = 9, or 7 x ? = 42

If the sum is 25 + ?=31, I would have taught him to bring the 25 to the other side of the equals sign and change it from a + to a -. How is this done properly? Thank you.

I just watched the khan academy video on linear equations and it there was an example like this:

-5/6x = 7/8 and you multiplied both sides by -6/5 to get the value of x so I understand that

Justrichmal, I do maths with all my kids, DD is 10, DS2 is 8 and DS1 is almost 7. They all love mental maths and will pass car journeys doing this so I get in a lot of extra practice this way!

DS2 will ask me to write him out pages of sums, when I am doing this I ask the other two if they want me to do it for them and usually DS1 will say yes but rarely DD will. DS1 and DD get bored quickly and want to do something else so I don't force them but DS2 will ask me to show him other things and he will happily watch the Khan academy videos on YouTube.

I am going to order those Letts revision workbooks and see how he gets on with them.

[user789653241]

45/ ? = 9 , 7 x ? = 42 should come with inverse operations, or fact family.

25 + ? = 31 should come from knowing inverse as well, 31 - 25 = ?, I would imagine, at least my ds did, since before he started school, if you expect your child to get the maths intuitively.

[HarveySchlumpfenburger]

7 x ? =42 he should just know if he understands his tables properly, the same with 45/ ? = 9 . As irvine says it’s just fact families.

Has he done any work on fact families at all?

Something like ‘If you know 3+4=7, write the 3 (or 7) other related facts you also know.’

[HoneyWheeler]

Check and see if your school has a document called 'progression in calculation'. This will set out the methods of working out sums for each age group. So instead of moving on to different topics, your son could perhaps learn a more complex but efficient method of working things out? Eg when they first learn division they talk about 'sharing' and then they move on through bus stop method and then onto long division. It's a big thing in England where I teach, as its supposed to develop 'mastery' of topics, ie going deeper into a topic rather than covering more topics. Depth vs breadth. Hope that makes sense!

[Lima1]

Yes he does know, sorry the examples I gave were too simple. He understands that 10+ 3= 13 can be written as 13-10=3 and that 3x3 =9 can be written as 9/3=3.

I had told him a while ago that he could do the sum like this
25+?=31 , move the 25 to the other side of the equals sign it then becomes a - (multiplication -division and vv) and can be written
?=31-25.

Simple sums like this he can work out the subtraction in his head and wouldn't need to go to this effort but I was showing him this for larger numbers eg 2538+? = 3127 or 45 x ? = 540 but I thought from the comment above that this method was incorrect and he should be using a different method.

I haven't heard the term fact families, but do you mean this:
10 = 9+1, 8+2, 7+3 etc?
I play games that incorporate this, like I might ask them to give me 3 numbers that add up to 20, or tell me 3 ways of coming to the number 20 eg 4x5, 18+2 and 22-2. They take it in turns and cant use the same sum that the others used. Is this what you mean?

BTW we went onto nrich and he had great fun with the puzzles this afternoon, thanks for the recommendation.

[user789653241]

I don't think you should teach it as moving to the other side.
At my ds's school, they were doing lots of A + ? = B, A- B = ? B + ? = A as a fact family operation in ks1. (yr1-2/ 5-7 years old)
Once they are secure with concept, they shouldn't struggle with big numbers.

[Lima1]

Honeywheeler I had never heard of that but I just googled it and read through it. They would do the addition and subtraction like that in school, breaking numbers into tens and units. He is in year 3 so is doing the multiplication and division as per that document but obviously hasn't gotten to the year 4-6 part yet.

If I asked him to divide 3 into 18 he would know to count in 3s until he got to 18 (or the nearest number and work a remainder) which is what they show on this doc.

The way they teach maths according to that seems awfully long winded. dividing numbers into tens and units and working them out separately and then adding it all together!
I asked DD who is 10 and she said they learned multiplication like this:
T U
2 (3) 4
x 8

1 9 2
multiply 8 x4, put down the 2 carry the 3 (which is a ten), then 8 x 2 = 16 plus 3 = 19

I suppose the method in the doc is a more in depth way of doing it so that kids understand how the tens and units work.

Thanks for all the advice and info.

[user789653241]

For multiplication, they teach them grid method, lattice method, expanded method before teaching column method in ds's school.
It's tedious for those who get it easy, but good for understanding those who don't straight away. By yr6/10 years old, they will be doing column methods.