Feeling positive about negative numbers

Now sats have finished yet term stretches out before us – what shall we teach year 6 in maths?  The test and the demands of the curriculum behind it dominated the landscape for so long, now that’s all done and dusted, it can all feel a bit anchor-less – what now? What shall I teach them now I am free to teach them anything I want, without anyone checking up on me? It’s all a bit disconcerting.

We usually opt – as I am sure many do – for doing lots of problem solving. I bet the website analytics at nrich show a peak in June and July. And there is nothing wrong with that.  It was difficult to yield precious curriculum time to their more open ended problems when there was so much  stuff  to be got through. But now – now we have 7 whole weeks to fill and spending time on actually applying all that maths we’ve been learning seems like a great idea. Consolidation and application; isn’t that what mastery is all about?

And, to a degree, it is. But 7 weeks of nothing but…I’d go crazy – never mind the kids.  I know maths isn’t all about getting the right answer but its nice sometimes to be able to get the right answer fairly quickly – and be pretty sure that you’ve got it!  And while some activities are great – this is my all time favourite -others are just boring number crunching.  Also – you actually have to be pretty good at maths reasoning yourself to teach these well. To use these effectively, you have to know yourself what would be an effective solution so you can prompt the children appropriately in the right direction.  nrich and other maths problem solving sites are written by mathematicians who find such matters trivial. They don’t supply the correct answers – what would be the fun in that? In the example I dismissed above as boring number crunching – Im sure if I actually sat down and spend a few minutes I would see that  – of course – if I did such and such that would be better than something else. It’s not something I immediately know and frankly would prefer some pointers to save me time.  The same goes for the wonderful Don Steward. His website is aimed at secondary teachers who presumably solve the problems he poses in almost instantly. He doesn’t write for primary practitioners who mostly stopped learning maths aged 16. If only he would put the answers!

Also – all that maths we’ve taught year 6 – it’s amazing how quickly it begins to dribble out of their ears once you stop the highly focused daily practice that marked lessons before sats.  There’s a sort of ‘match-fitness’ to a lot of maths that rapidly declines once you stop.  Consider times-tables facts. Remember how rusty yours were when you first started teaching and how much better they are now? Once term ends, they will probably have 6 maths-free weeks. If we tack on another 7 weeks when they don’t really learn anything new or practice much old, then no wonder secondary school maths teachers sometimes think sats ‘levels’ (50p in the swear box) are inflated. They may have been a 4B/at the expected level/secondary ready in early May; by September they have probably slumped way below that. Spaced repetition doesn’t like spaces quite that big.

So, what to do?  Giving them a flying start to secondary school obviously. So if they don’t yet know their tables or understand place value or can’t do the four operations in their sleep – then you should do those. But I’m taking it as read that nearly all the children can. There is no point in accelerating them through ks3 content because  *mastery* and also because if you have ever looked at a ks3 maths textbook, part from the algebra it’s just just the yr6 curriculum again. Seriously – we bought some k3 textbooks for the level 6 children last year – when level 6 still existed and we had to jump to halfway through year 8 to find anything sufficiently challenging.  I am sure secondary school teachers are well aware of this and use such text books selectively.   They probably even have new ones for the new curriculum.  Although I am slightly haunted by the experience of my first son who spend the first half term (yes the whole half term) revising what happens when you multiply or divide by 10 and multiples thereof. But that was some time ago, I really must move on.

So the obvious contenders are those things we whizzed through prior to sats in the mad dash to cover the new curriculum and suspect the children do not really understand deep down. They just know a few tricks.  As I worried about here.  So possibly we could revisit fractions, especially multiplying and dividing which are so easy to teach as procedures and so difficult to understand conceptually. Hey, we could even do some nrich investigations on them. This is the only one I could find that focus on multiplication. Most focus on equivalence but I am assuming the class is pretty solid on that.  This one from Don Stewart is good – but see what I mean about answers being useful – took me a minute or so before I realised that drawing a bar model made these ridiculously easy.

The other candidate is negative numbers which we whizzed through at the start of the year and then realised some children were confused about when we did algebra. There wasn’t time at that point to go back and address those and actually, the algebra questions in the sats paper were far easier than those we had been practising. However, what  gift to the child and their future teachers to have a rock solid understanding of negative numbers on starting secondary school. Including, of course, a firm grasp of why when you subtract a negative number, you end up adding. Algebra gets really tricky if you are not secure in your understanding of positive and negative numbers – so let’s give our leavers something that will really set them up well for the coming year.

Now when I started writing this post, I fully intended to explain at length about using integer counters to teach negative numbers.   If you don’t know what integer counters are, watch this. (There’s the added bonus the teacher sound a bit like Officer Dibble from Top Cat – the original series obviously).  But then I did a bit of pre-post googling, to se what else was out there and stumbled across this from Tess India which is simply brilliant and uses integer counters as well as various other good ideas.  I particularly liked the bench number line  where you use PE benches to make a number line with children describing how they are going to move from say -4 to +3, and the ‘feeling positive’ idea where you ask students to think about things that make them feel positive and things that make them feel negative.  If you add yet another negative thing you become even more negative whereas if someone takes away a negative thought you become more positive. Simple – but brilliant! So there is no point in me explaining much more about negative numbers – read the lesson plans in the link.

Indeed the whole Tess India resource is a treasure trove of wonderful ideas – well the bits I’ve managed to look at so far are anyway. And it is humbling to read the plans and realise they are aimed at a context where a class size of 60 is not uncommon and interactive whiteboards don’t exist.  Alongside the maths there are handy tips about how to make flashcards from old cardboard…rather puts things into perspective. There are English and science resources alongside the maths and it covers primary and secondary. It’s really well worth a look.

But back to integer counters. These don’t seem to be well known about in the UK. Indeed, we use the wonderful Primary Advantage maths scheme in key stage 2 and while they bang on about CPA everywhere else – for negative numbers they state that no concrete materials are possible and go straight to number lines.  Now number lines are all fine and dandy but some students get so confused using them and it all seems a bit arbitrary why you are moving forwards or backwards.  I  love  integer counters because I love being able to see why the maths works.  When I found out you could even model  why -4 x-2=+8, I beamed for days. I kept on showing people my newest party trick. (I didn’t get invited to many more parties after that.)  And here’s a great link showing how to divide negative numbers.  And no I am not suggesting you teach year 6 how to multiply and divide negative numbers. I’m just banking that if you’ve got this far, this sort of thing brings you great joy.

Using integral counters is the final idea in the Tess India resources.  To be able to understand them, children need to understand the concept of a zero pair, made from one positive counter and one negative counter.   These can then be added to any equation without changing its value. I could try and do some badly drawn graphics to explain it properly – but others have done so with so much greater flair I suggest you look at these instead. Unfortunately Officer Dibble’s video on this seems to be missing. However, this from Learn Zillion  is perfectly serviceable although this is a bit more fun, if a bit more complicated. Can’t quite place the accent.

With a bit of practice, children soon learn to just draw themselves + and – signs if they want to check a calculation, rather than need to counters.

One of the problems children face with this topic is that we never make it clear that all numbers have polarity –  that they are either positive or negative – and that strictly speaking we should write 3 as +3 etc. It’s a shame that the polarity signs are the same as the operator signs – I’m sure it would be a lot easier if they weren’t.  When we write 5-3 do we mean

+5 + -3 or +5 – +3?   I’m sure if we did a lot more work with counters showing that they give the same answer but actually represent something different, that might help. No wonder children get muddled and think -3-5 equals 2  or possibly 8 (because you have got 2 minuses and they half know something about two minuses making a plus. How much better to act it out with counters and see the maths before you very eyes. Works for me.

Then we can return to where this post started and have children investigating negative numbers.  Back to nrich.  And here’s a great reasoning activity from maths pad. If they can articulate why certain statements are or are not true then that’s job done. Secondary schools- here we come!

 

 

 

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Feeling positive about negative numbers

The light switchers solved- with actual photos!

In my previous post (silly carpets and the light switchers) I mentioned an interesting problem and promised to post the solution – I’m sure it was easy enough the work out especially since the title also mentioned prime and square numbers. I’ve never actually done it practically before. But for you, dear reader, I’ll go that extra mile. Really glad I did because I’ve just appreciated that that this is a really great problem for upper ks2 ( possible ks3. – what do I know?) and a fun way to rehearse those tables yet again. I wanted to video it but lack of something to fix my phone/ iPad into eluded me. Too many shots of me in my dressing gown (hey it’s the weekend, you don’t expect me to actually get dressed?) So just what is probably one of the most boring sets of photos ever- but what was strangely satisfying to do in practice. I needed an object I could turn over to signify the light being on or off.  Having no plastic cups I resorted to nespresso coffee capsules which makes this the most middle class poncy maths investigation ever….in class maybe use those mini ketchup cups you get at McDonalds or biscuits like Jaffa cakes which have very different sides and which we now all know are zero rated for vat, thanks to MP Stella Creasy.

So first of all all the lights are off so my capsules are all the right way up.

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Then number one switches all her multiples on – i.e turns all of the capsules over.

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As every number is a multiple of one, all the capsules are turned over- all the lights are now on.

Then along comes number two and turns over all,of her multiples, making this obvious pattern.

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Then number three turns over all of his, making this interesting pattern.

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Number four turns over all his multiples…

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And I’m losing the will to live inserting all these photos, so I will cut to the chase. Here is what it looks like after number 25 has done her switching. Drum roll please….

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So the lights left on are 1,4,9,16 and 25 ( and following- I decided modelling it to 25 was quite enough) because, of course, square number have an odd number of factors whereas all the other numbers have an even number so return to their original ‘off’ state.

The light switchers solved- with actual photos!

Silly carpets and the light switchers- investigating prime and square numbers

This is just a short post about a very simple idea I did with my yr2 class a long time ago when I was still a class teacher. It sort of follows on from my previous ‘ hooray for an array’ post. It was a way of children investigating prime and square numbers. Our class carpet was made from carpet tiles.  We were learning area by counting squares. I had a few spare carpet tiles….not quite sure why. So I showed them that with some numbers like  8 you could make a sensible (rectangular) carpet, and some numbers like 9 you could even make square shaped carpets. But some numbers, like 11, no matter how you tried,  you could only make a silly carpet that was very long and very thin. There was something about the notion of the class trying to fit onto a long thin carpet, one behind the other, that the class found hilarious- and with squared paper quickly found all the squares and primes to 50. Made a great maths display. Hardly an original investigation, but I hope an interesting context that your children will enjoy doing.

For older children (upper ks2, secondary, adults) here’s an interesting problem. With primary children I’d use something like two coloured counters or paper cups for them to model this- basically anything that could be used to indicate if a switch is on or off. The problem starts like this…..

There is a long corridor in school with fifty light switches- numbered 1-50.  There are 50 pupils also numbered 1-50. All the lights are off, so pupil number one helpfully goes along the corridor and flicks every switch on.

Pupil number two,comes along and thinks- what a waste of electricity, I’m going to flick all the switches for lights that are multiples of two. So she turns off lights number 2,4,6,8 etc.

Pupil number three comes along and wonders why some lights are on and some off. She  flicks every third switch ( so if it were on she flicks it off and if it was off she flicks it off).

Pupil four flicks every fourth switch, pupil 5 every fifth and so on, until finally pupil number 50 just flicks switch 50.

At the end of this, how many lights are left switched on? Which ones? Why these?

I will post the solution tonight, but the title should help a lot!

Silly carpets and the light switchers- investigating prime and square numbers

Hooray for an array!

Now I maybe wrong, but I was taught that strictly speaking mathematically 3×2 means 3 taken 2 times or in other words 3+3 rather than 3 lots of 2 (2+2+2).  Which means that 3y means y lots of 3 rather than the easier to understand 3 lots of y…. But that’s what I understood the notation to officially mean. Not that it matters with the commutative law and all- but a teacher in America marked some poor kid down for saying 3×5 was 5+5+5 instead of 3+3+3+3+3 (i.e disagreeing with me). The teacher even marked the array the child had drawn for 4×6 wrong because it was orientated the’wrong’ way. The whole point of arrays is that they can be ‘read’ either way and hence are very useful for teaching the commutative law. I love an array! Do you know they are also brilliant for showing the associative and distributive laws, link to the bar model, the grid method, prime factorisation and multiplying by 10!  You just need some spotty wrapping paper……

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Here is 2+2+2 or 2, three times- what I’m going to call 2×3.  Like this it is also a bar model. But like this

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it is an array that could be seen as 2, 3 times or

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3, 2 times.  Or back to the bar model

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3+3

Now I’m sure we’ve all done similar with our children in yr2 and yr3- I used to do it with pegboards and exploring 24. Dotty paper works brilliantly though- get the children to do the cutting and experimenting.

But I’d never thought about using it to teach multiplying by 10, until the new curriculum. The new curriculum stresses multiplicative reasoning. In yr3 they are supposed to understand that 4x12x5=4x5x12=20×12=240. (Associative law and all that).  Which got me thinking. Obviously we NEVER tell children ‘ just add a zero’ when multiplying by 10 and tell then about moving digits one space to the left. My colleague gets the whole class to stand up, jump to the left while chanting ‘multiply, go large, left’ ( for division it’s ‘division, reduce, go right). The local secondary teacher came to see a lesson and finally understood why a quarter of her class would jump up out of their seats and do his routine everytimemshe asked about multiplying or dividing by10! But even though this helps children remember what to do – does it help them understand?  But what if we taught it ( alongside the dance) with arrays? Make an array of 6, then make 10 of these arrays to make an array of arrays eg 2x3x10=6×10=3×20

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Which they can play about with and establish that this also equals

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5×12 and even perhaps after a lot of cutting 2x3x2x5.  Hence prime factorisation. Remembering of course to make the link with factory… The factors are what the 60 factory uses to make 60.

Then photocopy or scan the 10×6 array and by cutting and pasting make another array ten times bigger by putting 10 copies together and establish you know have 600 because 60 can be written as 6×10, so 6x10x10= 600. Then photocopy again to make 6000 and- well maybe leave it at that. When I showed my yr6 that 60×10 could be factored into 6x10x10 they were enthralled and wanted to know if it worked for multiplying by 100 – which they quickly proved it could. This kind of multiplicative reasoning not having been common until recently, they were entranced by it. Last year I showed the level 6 group how you can easily work out eg18x6 by halving the 18 and doubling the 6 and they thought I was teaching them some kind of dark magic- arrays of course to the rescue. Indeed the bottom two photos above show precisely this.  Much experimentation later they got the general rule that you can multiply one factor by any number as long as you divide the other factor by the same eg 96×15=12×120=3×480. By now, they can ditch the arrays and experiment with factorising numbers and then recombining in ways that make multiplying easier. Multiplying even numbers by multiples of 5 being an easy win eg 24x 15= 4x6x5x3=2x2x2x3x5x3=2x5x2x2x9=10x90x4=360. Well,maybe it’s not easier but  I find it kind of satisfying. And they are going to have to understand this if they are going to be able to reason about questions like these from NCETM’s brilliant assessment document. To be honest it did take my yr6 teacher and I a few moments, quite long moments, to get our head around these at first. The last one still gets me every time.

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Have you ever used the dotty wrapping paper method as a prelude to the grid method? It works a dream. The photo below was taken in bright light so it’s hard to see the pen annotations dividing it into 10×10, 10×9, 10×7 and 9×7- try zooming in perhaps.

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Again, get the kids to do the dividing up and cutting, finding the arrays that will make the calculation easiest (the multiples of ten).

This method can form a bridge to the column method. In the class I’d do this with actual paper we had cut up and made into a grid first but I’ve drawn it ( very badly). So put the actual arrays where I’ve put the drawings.

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Finally the distributive law. Easy Peasy when you just split you array into two pieces

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10×6 distributed to make 6×6 and 4×6. Cue investigation- how many ways can we distribute this array. I had to model this to a primary teacher who didn’t believe you were allowed to split numbers up like this in the middle of a multiplication when I did a staff meeting on multiplicative reasoning using the post on Shanghai maths from resourceaholic. This is what we were trying to work out mentally

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He got the bit where you divide 45 by ten and adjust by multiplying 1.58 by 10, but didn’t agree that 4.5x 15.8 + 5.5x 15.8 was the same as 10×15.8. Until I drew this.image

Then the penny dropped. So hooray for arrays and remember CPA all the way . (That’s concrete pictorial abstract, in case you haven’t encountered this yet). If I haven’t converted you yet to the joys of arrays, read this from NRich.

Hooray for an array!

Fractious fractions

Imagine that you’ve got two pizzas…mmm…but you’ve got to share them between three people (boo), and share them fairly, how much does each person get?  With this apparently simple question started the ‘big maths cpd ruckus of 2015.  Teacher A said of course you got two thirds of a pizza whereas teacher B said you got a third of the pizza and apparently it all got quite heated and they ended up having to get some paper plates and cut them up in order to prove who was correct. Which of course they both were.  It’s a determiner problem ( Oh no, not grammar and fractions…is this the new curriculum gone mad?)  Teacher A said two thirds of a (i.e one) pizza, whereas teacher B said one third of the pizza ( i.e one third of two pizzas). Which just goes to prove that fractions are tricky. They are tricky because they aren’t really numbers at all ( ok ok maths peeps, I’m playing fast and loose with definitions here, but I’m talking from the perspective of primary kids ok).  Up until we introduce fractions, when we talk about numbers, we mean natural numbers……the counting numbers. If I show a child a 2 I can get 2 pencils or 2 chairs to illustrate what I mean. Whereas if I show a child 1/2…well I could show them half a pizza or half a triangle but 1/2 is so much more loaded than that. Back to determiners. If I show them half of a pizza, I am actually saying – look- here is one pizza. Now let me divide it into two pieces. Each of those two pieces can now be called ‘ a half’. The difficulty being that if I were to have two pizzas to start with, then a half would be one pizza. So  sometimes a half is half a pizza and sometimes it is one pizza or 8 pizzas or…..well pretty much any number could be half of the pizzas.depends on how hungry you were in the first place. Fractions are not naturally numbers at all- quite often they are an instruction to do something: veritable bossy verbs. 1/2 really means ‘divide 1 by 2.’  It also means, ‘ how much you end up with when you divide 1 by 2.’ Which is an adjective apparently ( although it feels like a noun to me).  The answer and the operation are the same, which seems a little tautologous.  And when you cut three things into four equal pieces then the answer  is 3/4 of one thing- even though you started with three things.

The technical term for this is that 1/2 and other fractions are not natural numbers but rational numbers (ratio-nal numbers; they describe ratios). You have to know what the whole is for them to make sense. The whole is not always one. But the final answer is always expressed with reference to one, regardless of the size of the original whole. But we don’t make that clear. Or appreciate that clearly ourselves. Hey, I’m confusing myself writing this!

My better half (little fractions joke there) was recently at some training where they said that children learn about fractions more effectively if taught it in the context of division. Now a reference would be nice here, but husband can’t remember. Did I say he was my better half? It was probably Nunes, it always seems to be Nunes. Anyway – that got me thinking.

A while back resourceaholic posted something by some other secondary teacher (which of course I can’t find) about teaching  addition of fractions with different denominators by using Playdoh . Each pot of Playdoh was divided into different amounts and rolled into little balls of equal size. Because the different denominators are obviously different sizes, the students are less likely to make the error of adding different denominators together. So that made me think that at primary level maybe we should start teaching fractions by just investigating different denominators – and not mention numerators AT ALL until children clearly get that larger denominators means smaller pieces. In terms of writing, we would just write the / line ( that’s probably got a special name…) and the denominator underneath.   I imagined telling the children that the vinculum ( thanks google) aka the fraction bar or slash was a knife that cut the Playdoh ( or whatever) into a certain number of pieces and that we could record how many pieces by writing that number underneath the knife slash. Writing nothing at all above the line.  Then, after many happy hours slashing Playdoh into different amounts and becoming really fluent at writing these (sans numerator) and at naming these,  I finally pose the question- as if it has just occurred to me- ‘ but what if I wanted two of these things called fifths, how would I write that?’ And hopefully someone would suggest we could write that number above the knife/slash/fractions bar/vinculum. And the entire class would henceforth have a rock solid understanding of what a denominator is and how it is logically prior to the numerator- which is a mere adjective to the denominator’s  verb.

Because playdo is sticky, it also makes it easy to divide say  3 tubs of playdo into 4, by firstly amalgamating the 3 tubs worth into one great big ball and the dividing it into four, and then also by dividing each tub into four and then amalgamating one piece from each- and realising this is the same.  And of course it allows us to see that half of one tub is half but half of 4 tubs is 2 and finally that 1/3 of 2 tubs is 2/3 of 1 tub.

Edit: March 2016. My colleague Keeley Warren was much taken with this idea and spent a couple of days with her year 3 class and some play dough dividing it into different denominators….or ‘doughnominators’ and recording these saying the vinculum was the knife and not putting any numerator above it.  As a result, the entire class quickly grasped that the bigger the denominator the smaller each piece is.  Then when she returned a few weeks later to do fractions again- they had no problems whatsoever and grasped everything really quickly. She was amazed at how well it worked and it is definitely going to be standard procedure here from now on.

But that was before I heard about the ‘learn fractions as division’ thing. Which made me want to adapt this lesson further and start with bigger numbers. In fact, to teach division as sharing whole numbers using the vinculum thingy from the off- again saying it’s the knife that divides thing up – or pushes them into piles…..two for you and two for me. Sticking to even numbers initially but them moving onto dividing one thing by two, four etc.  Meanwhile division as grouping could be represented using the other sign ( the obelus, apparently) which keyboards  don’t tend to let you do- certainly not iPads on WordPress- but you know the one I mean. Not quite sure how and when you’d tie the two together yet. Via arrays probably. Love a good array.

Haven’t actually taught it like this- but I think it might work well. One problem though is that the numerator means slightly different things in each case and teachers would need to be very clear in their explanations. In traditional fractions teaching, the denominator indicates the number of equal parts into which a whole was cut and the numerator indicates the number of parts taken.  In division, the numerator refers to the number of items being shared and the denominator refers to number of recipients: if 2 chocolates are shared among 4 children, the number 2 refers to the number of chocolates being shared and the number 4 refers to the number of recipients; the fraction 2/4 indicates both the division – 2 divided by 4 – and the portion that each one receives.  It’s two quarters of one chocolate. Not of two chocolates. Our fraction is now an adjective, or arguably a noun, with its own special space on a number line. Hey it could probably grace the back of a football shirt.

Ps. Here’s that reference. It was Nunes! 

Fractious fractions