# 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.

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

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.

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

Number four turns over all his multiples…

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….

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.

# 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!

# 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……

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

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

3, 2 times.  Or back to the bar model

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

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

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.

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.

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.

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

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

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.

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.

# Teaching Biff and Chip to read.

When I was at home on maternity leave with my 19 yer old son- I mean- with my son who is now 19 years old- my husband came back from a school he had been visiting as part of his PGCE where everybody in year 2  could read- where everybody passed their reading sats and many got level 3. They did this by teaching them phonics.  It is difficult to convey quite how shocking this was. At that point (1996), the idea that you could teach something out of context- without some meaningful frame of reference that made sense to the child- was seen as almost tantamount to child abuse. Children were said to learn to read by being immersed in a rich world of high quality children’s literature, by bathing in wonderful stories and glorious poetry. Not by dissecting language into individual atoms and labelling them with differently voiced grunts. It was like being told they taught witchcraft. Surely, I said, the children had sprouted horns as a result, or hair on the palms of their hands or something?  But husband- usually the sort of person to champion wonderful stories and glorious poetry- was adamant that  this atomistic sound grunting business had something to it. So I went to visit.

Within 5 minutes it was obvious these kids were streets ahead of mine. About a year ahead, I reckon. So although if I were God I would have created things so that children learnt to read best through exposure to rich language as that appeals to my liberal tendencies- I conceded that unfortunately the seemingly less inspiring, dry and dull phonics method was annoyingly superior. Witchcraft was now in. We adopted it, found that children actually really enjoyed it because they loved the success that went with learning to read. Our results soared.  Every child except some of those with statements for thorough going global delay learnt to read. The headteacher of that school was Ruth Miskin and we’ve been using her products through their many iterations ever since.

At this point, the phonics champions are whooping, poised to retweet. But fear not, oh phonics denier- there is more to this tale. Actually SPOILER ALERT, both purist phonics advocates and deniers are going to be disappointed as I will come to a ‘sit on the fence- we need both phonics and other strategies’ moderate conclusion.

Anyway, time passed and we realised that while we were really good at getting almost everyone a 2c or higher at the end of year 2 ( if you remember the old days when we used to have these levels), the ones who got 2c – and some who got 2b, limped through ks2 and might just scrape a 4c on a good day, with a fair wind behind them. They might be able to read, but they couldn’t always pass that damn test.  Pot holing at Dingley Dell in 2011 was a particular low point- with an able reader scoring N, yes N, in the reading test. Reading for meaning was a real issue for some children. In fact for some, their tolerance of non meaning was truly shocking. They just didn’t expect reading to make sense- it was  just an exercise in barking at print. The most proficient print barkers would get quite cross when you tried to slow them down and enquire what a sentence actually meant. They know they were good at decoding and saw reading as nothing more than that. Yet here we were, calling their hard earned prowess into question. For some children, this may have been exacerbated by the way they were taught to read Arabic at Quran school, where as far as I am aware ( do correct me if I am wrong) learning to read the Quran means learning to decode it fluently- sometimes beautifully- but not actually understanding what the text says.

We’d been using ‘accelerated reader’ for years to keep tabs on whether our children were actually reading the books they took home from our library and this clearly showed that a stubbon minority of children – let’s call them Biff and Chip- just don’t read at home ever. And if we force feed them reading at school- they still don’t really understand what they’ve read.

Deep breaths. Yes the red blobs are annoying- but actually- I sort of get it.  Say Biff and Chip had scraped a 4c? (I’m ignoring Wilma and assuming that was down to exam nerves- she’d got solid 4’s in all practice tests.) I would have patted myself on the back, smug in the knowledge that all was well with reading. But seriously – 4c…! You can ( or rather could)  be pretty poor at reading and scrape a 4c. Hardly setting you up for success at secondary though- is it?  Not good enough. When I looked back at our results over past few years realised most years there was at least one Biff……what was going on? Why weren’t we picking up that these children were so weak and doing something about it?  I could hear David Didau tutting.

‘But Biff and Chip don’t read at home’, decried my SLT. ‘Mrs Biff took Biff on holiday in term time…..in year 6!  And neither Biff nor Chip came to February or Easter booster classes! ‘ they continued. True, but we need to be good enough to enable our children to succeed even without parental support. With support- well just look at those other disadvantaged but well supported children with all those level 5’s. But we can’t rely on it for all children. What were we doing wrong? Or what weren’t we doing? Tracking showed that the children were 3c by the end of yr4, 3a by the end of yr5- so 4b here we come. 4a should be possible.

Sure when they actually entered yr6 the teacher ranted about them not actually being able to read properly…..but then she always does this about every pupil at the beginning of the year. ‘ ‘Woe is me…there is a mountain to climb….they are all barely level 2….what am I supposed to do.’ That’s just what year 6 teachers say ever year- it’s tradition. I don’t take any notice of that!  Especially because look, the tracker is all green- they are on track to make good progress. And usually- almost all of them do. Except Biff. And Chip.

Then I read Daisy Christodoulou’s piece about reading ages versus national curriculum levels and I felt sick. Daisy shares the evidence that shows that national curriculum level  2 reading assessment can span a seven year range of ability when the same children were assessed using reading ages. Now we’d always poo  pooed reading ages as when we used the bit of accelerated reader that assesses reading age- the results were so out of kilter with our national curriculum judgements that we just simply didn’t believe them. Sats held such sway that those results just had to be infallible- didn’t they?

Anyway the team needed convincing- so I read the  boring small print bit in accelerated reader about how they ensure their test is reliable and it may not be perfect but it’s got heaps of trials behind it and is way, way more reliable than the old nc test.  So we are using that now. Alongside some other stuff.  And each time it throws up a couple of kids in each class that are doing less well than we had thought. Not bad enough for us to already have them flagged for phonics intervention but a good year behind where they should be.

So that’s where we are at now. Screening all ks2 using reading ages and trying to catch all the Biffs and Chips for whom 3 years of high quality phonics in ks1 still hasn’t been  quite enough to enable really fluent reading or, unsurprisingly, a passion for reading.  A literature rich curriculum, a well stocked library, author visits and teachers who are passionate about passing on a love of reading still hasn’t inspired them to really want to and so to voluntarily put the work in.  Not even the lure of being able to read ‘ The Recruit’ ( parental permission required because a character says ‘shit’ or something) has been sufficiently tempting- because it’s just a bit too hard for them to read by themselves.   So now we are overhauling our lower ks2  for children who are not yet fluent.  We’ve bought in ‘ Project X Code‘ for them- rwinc is great but they need a change after 3 years on the programme. And they like the stories. Then in upper ks2 trying to  find enough staff to give up their lunch break  to help children become fluent and eager readers even when we’ve deprived them of football, using real books  via the Chatterbooks project.  As they remind us, being someone who reads for pleasure is a better indicator of long term success than  education or social class.  But will try and supplement with yet more phonics (rwinc’s fresh start) at some other point in the day. Just don’t have enough staff to do this at the moment. Am in school tomorrow trying to squeeze budget for solutions.  And we are really well funded compared to other local authorities. No idea how the rest of the world copes!

So that’s my solution. Don’t choose between phonics and other approaches. Do it all!

# 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.