teaching of Maths

[teachersaspirations]

hi
my dc has come home with exercises in how to work out the HCF (highest common factor) & LCM (lowest common multiple)
when I asked them what they were used for, they didn't know
and the teacher has not explained why they are useful
(this has happened with more than 1 dc in different years)
why is maths being taught like this?

[SirKumPherence]

It's not always easy to give a meaningful, "real world" context for a topic.

Sometimes it might be better to teach an efficient way of finding the LCM and then pose a problem using it - I like the decorating example mentioned. Alternatively, it is possible that a teacher will teach LCM, then will link back to it when talking about addition of fractions.

[richmal]

@FirstVix Another thanks here from me. I home ed and will be going over quadratics with dd again in the next few months. This seems alot more methodical than "Just keep trying different numbers".

[noblegiraffe]

Good question, contortionist, why don't we use Euclid. I don't think I've seen it in any secondary textbook and had forgotten about it.
We normally teach HCF via listing factor pairs, LCM by listing multiples and then teach prime factors, then HCF and LCM via prime factors. Maybe being able to list factor pairs and find prime factors are important so those are taught and the HCF is a by-product? Certainly it doesn't seem to come up in applied questions (witness us struggling to think of any!). If it comes up, it's just 'find the HCF of..'. As mentioned, you wouldn't actually calculate the HCF to cancel a fraction, you'd just get on with it.

I don't know. I'll certainly file it away for future use!

[contortionist]

Or perhaps I'm running a joint activity involving class A (18 pupils) and class B (24 pupils). I need to divide them both into teams of equal size, each with members of only one class. And I want as few teams as possible so we can get the whole thing over with as soon as possible.
It's probably a good thing I don't make these things up for real!

[contortionist]

And here's my attempt at a gcd question. I have a field 180m x 240m around which I want to put up a fence. I'd like the fence posts to be equally spaced, and of course I need a fence post in each corner. What's the largest spacing I can use?

[contortionist]

For LCM, I would have thought it would be quickest to do a.b/gcd(a,b), finding the god using Euclid's algorithm. E.g. for 18 and 24:
gcd(18, 24) = gcd(18, 6) = 6
Then lcm(18, 24) = 18.24/6 = 18.4 = 72
It's a long time since I did this at school so I don't remember what method I was taught or what exercises we did.

[FirstVix]

You're welcome noble. I have to admit that I ran around the department 'showing off' my 'new' method when I 'discovered' it as I was proud of myself!!
[They weren't that impressed!!]

teachers, if anything I'd say that students in general don't use calculators effectively enough. They are a tool to be used and one that they will have access to for most of their lives. They should obviously be taught non-calc methods too though, not least because these techniques will be tested in their exams.

In terms of explaining what an LCM could be used for, I mentioned before the time/usefulness now constraints that we have.

Seriously, try telling a bunch of 13 year olds how it's useful in industry to work out blah blah.....and watch their eyes glaze over or have them pipe up ' but I'm not going to be a [insert job title] so why do I need to know?'. You can't have these conversations when introducing EVERY topic!!

Of course, if the child ever asks then I will try to explain where/when it's used. Also, as far as possible make it real life - but not for every topic.

[teachersaspirations]

girafe

my main point was that it should be explained what an LCM is used for, is that unreasonable?

is there an over-reliance on calculators? (you advocate the fraction button), does it compromise mental agility? you should know you are the teacher

you don't like my method, fine

[noblegiraffe]

Ah, I like it, Firstvix. The problem with the ac method was when ac is big as C1 is non-calc, but that's great. Thanks!

teacher in this thread you appeared to be slating your DD's teacher for not showing them your more 'expedient' method that you describe as simple and easy, and suggest that to not show it is to demonstrate a lack of mental agility, and possibly over-reliance on calculators. It's not, it's because it's not very good. Why swap a 'fool-proof method'for an unreliable one that it's very easy to make mistakes with?

[teachersaspirations]

girafe
its just an alternative method, which you don't seem to like/agree with
maybe you are right and I am wrong

[FirstVix]

[Obviously 'endless' in a non-literal sense!!]

[FirstVix]

noble it's when finding the factor pairs . It's much easier to pair up smaller primes together. It's something I literally came up with a couple of years ago when there was a really nasty question and no-one could see the right factor pairs (I don't mean I invented it, just that I hadn't seen it before).

I found the prime factors of ac and used (played with) these to work out how to make b (ie how to split the x term).
So, didn't have to find all the factors of (eg) 378 that added to -13, but used 2x3x3x3x7 and played with various combinations to get -13 (which I found a lot more user friendly and fool-proof).
The students liked the method too as you didn't have to trawl through endless lists of factor pairs (and possibly miss some).

[noblegiraffe]

I've just done a load of LCM calculations using your factor method, teacher and conclude that it is a pain in the arse. I will not be showing it to my pupils as it will only confuse them.

[teachersaspirations]

clay
the point is to look for the LCM, but not necessarily via primes, but common factors
using the LCM means you end up with smaller numbers which is generally easier
if you don't get the LCM then you just have to do a bit of extra simplification after
but if you look and learn to look then you begin to see them if they are there

[teachersaspirations]

paperclip
the original comment was about LCMs and why there was no explanation of their use
I didn't think that that was too much to ask

but you must be right, curriculum constraints

[ClayDavis]

I know it can be used to find 1/18 + 1/24, teacher. But I thought your point was that it was better to use the LCM when adding fractions not any multiple and that teachers should be pointing this out to children.

[paperclipsarebetterthanstaples]

Curriculum restraints mean that your child's teacher has probably had 1 or 2 short lessons to teach LCM etc to 25+ children - they're not doing a maths degree nor participating in a deep thinking debate about the subject. We don't have time to ponder every avenue and need to keep certain things short and simple.

You seem to have some funny advice for teachers even though you're not one.

[noblegiraffe]

Teacher, in your 18,24 example, there are 4 combinations of factor pairs with an overlapping factor, only one gives the correct LCM when you multiply.
27 and 35 for example, don't give any overlapping factor pairs. Then the LCM is 27x35, but I'd rather do 3x3x3x5x7 in my head, tbh.

If you are going to faff around listing factors (make sure you don't forget any!) checking for overlaps, checking the other numbers are co-prime then multiplying, remembering to ignore one of the overlapping numbers, if a suitable combination of factors exist, then is that really better than doing a couple of prime factor trees (which are fun!) and the fool-proof method?

[noblegiraffe]

How do you use prime factors to do that, FirstVix? I use the method where you multiply a by c and split into a factor pair that adds to b, but the kids don't seem to like it.
Agree that it's hard to answer the 'when will I use this?' question when the answer is 'it's handy for this bit of maths you don't know yet and wouldn't understand if I showed you'. Like 'listing factor pairs is really useful for factorising tricky quadratics but you don't even know what a quadratic is"

My main problem with calculator use isn't kids relying on them too much (like they even bring them to lessons unless you tell them to!) it's them being so bloody used to not using them that they forget to use them in the calculator paper. Gah.

[FirstVix]

I use prime factors when factorising quadratics with coefficient of squared term >1. Because I can never do it in my head. I was very chuffed when I realised that I could do this..

I also encourage calculators to be out so that students can check their work instantly (and stop asking me if it's right) but to try without first. They get used to it, use the calculators as the tools they are and only ask me for help when they can't work out where/why they went wrong. Thus they learn to assess their own work more naturally.

However, I agree that you should try to make the maths you're teaching more relevant to the real world, but as you change topics quite often, and as some of the topics aren't that real-world helpful until a much higher level, it is hard sometimes.

[teachersaspirations]

well if calculators were set aside (this does not mean bring out the log tables etc..)
it may force students to look for simplifications, etc...
to become comfortable with simplifying every time before they grasp the calculator
it may force them to be comfortable with surds

am very happy for you to give me any example in which you suggest that expediency shouldn't be tried; there may not be a more expedient way, but sometimes there will and if you don't look you will never see it

[noblegiraffe]

Yes, let's confiscate calculators and go back to using log and trig tables and pounds, shillings and pence for that matter. Let's make everything more complicated than it needs to be

What don't we agree on btw? That your 'expedient' method is in fact not very good? I could explain some more if you like?

[teachersaspirations]

girafe
it doesn't look as if we will agree
'and I would encourage use of the fraction button on their calculator for unwieldy calculations' unfortunately doesn't shock me
perhaps calculators should be confiscated from teachers

[TeenAndTween]

vivacia .

You know when you have a thinish slice of cake, you can't really cut it in half from the middle of the curved edge to the point because it will fall apart, not cut properly etc?

You can approximate that the curved edge is really straight.

Then you cut from one corner where the curved edge is, to the middle of the opposite side. The two resultant triangles (looking at the top of the cake) are very different shapes, but will be the same size. Due to area of a triangle being half base times height. They stay together better, and DD2 (who doesn't understand the maths) is always happy as she gets to choose the 'larger' bit.

I really need a diagram.

Doesn't work if there is yummy chocolate icing round the outside of the cake though, as one person gets all of it.

[Vivacia]

Or areas of triangles (very useful to understand when halving the last slice of cake fairly)

What way do you mean TeenAndTween?