Working with a Growth Mindset

I was just going through some of the other students’ responses to the open ended task – came across this one:

64 X 25 =

How many ways can you work this out? Of course the underlying assumption is that the students understand place value and are comfortable converting units to tens, tens to hundreds, hundreds to thousands.

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My Montessori mind kicked into gear here – I don’t have any Montessori number bead sets (too bad, really because if a student doesn’t understand place value, the bead bars and squares represent the actual quantity, no abstraction, so when you lay out the problem it’s a direct representation of the problem). What I do have is Cuisinaire rods. So I go down to my basement to locate them. Come back upstairs and begin by laying out the problem. I set up a multiplication board with unit/tens along the right side and units/tens along the bottom. I lay out my first number 64, and show it as a 6 and a 4 placing the 4 beside the units and the 6 beside the tens. Next I lay the 25 along the bottom, the 5 beneath the units, and the 2 beneath the tens. (In a real Montessori board, the units block would be completely green, the two tens block would be completely blue, the hundreds block would be completely red, no confusing what the blocks represent.) (Oh, and the board actually has a layout up to ten thousand on each side so that you have a block for the millions – it’s a big board – the units of thousands are green, the tens of thousands are blue, the hundreds of thousands are red, and the units of millions is green – again strengthening the place value distinctions – units, tens, hundreds, units of thousands, tens of thousands….)

Now I lay out the array, I put 5 of the 4 bars in the units block – value = 20 units; I put 5 of the 6 bars in the upper 10’s block – value = 30 tens; I now put two of the 4 bars in the lower tens block – value = 8 tens, and two of the 6 bars in the hundreds block – value = 12 hundreds. For the purposes of this explanation I provide the value of the contents of the blocks: I have 20 units + 30 tens + 8 tens + 12 hundreds.

Next, I begin amalgamating the blocks: I’m only allowed from 1-9 units in the units block, but I have 20 so I convert them to 10s – leaving no units but adding a 2 bar (the small red one) to the tens pile (remember I have two 4 bars = 8 + five 6 bars = 30 ) already in the top half of my second sheet (I’ve drawn this second sheet so I could lay out the way I amalgamate the blocks – but were I using the Montessori multiplication chart, I’d do it directly on the board.) So I now have in the 10’s block – 30 + 8 + 2 = 40. Again, I’m only allowed from 1-9 tens in the 10s column so I convert the tens to hundreds = 4 which I put in the bottom hundreds column; leaving 0 in the bottom tens column. I take the 12 hundreds at the top and convert that to a 2 bar which I put in the bottom hundreds column and place a 1 cube in the thousands column. (As I make the swaps, I’d actually remove the pieces from the board leaving only pieces in the lower set of columns – I’ve left them on the sheet so you can see them – that may be confusing.)

So I’ve actually done the multiplication.

As you can see, after counting out the pieces, amalgamating them, I’m left with a single thousand and a 4 bar and a 2 bar in the hundreds, and nothing in the tens or units. My solution is 1600 – one thousand, six hundreds, no tens, no units. That’s one way of figuring this out. What I may do later is show how this translates into the multiplication of a binomial by a binomial – the Montessori materials for doing that are really elegant!