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
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.
4 thoughts on “Hooray for an array!”
Some great opportunities in this post to organise discussion, analysis and reasoning by learners. Getting ‘underneath’ any concept and seeing the connections between concepts is crucial for relation understanding (Skemp MT77 1976) to occur. Thus working with arrays not only enables commutativity but also understanding inverse processes – about factors (or ‘guzintas’ i.e. what goes into 20). Rules for divisibility will not be far away and so evermore relational understanding can occur. Really like all your ideas about using spotty wrapping paper – a lovely exploration to be had here for learners. I think your post chimes well with Boaler’s work in number sense. Regards Mike
only just seen your comment for some reason – thanks!
Ben Orlin’s post on arrays is good too: http://mathwithbaddrawings.com/2014/12/03/the-sixth-sense-for-multiplication/
Yes, this is an excellent post, thanks for sharing