square numbers

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.

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!

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The light switchers solved- with actual photos!

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