And, of course, a knowledge of the Laws of Mathematics is central to this understanding.

So, we started with the Commutative Law.

With this provocation, I wanted to see what the kids knew about addition, the relationship between addends and how the Commutative Law works.

So I asked them to show that this statement was true using 3 different methods.

This was the first time I had done something like this with the kids this year. It was probably the first time some of them had ever done "proofs" or tried to explain their ideas using multiple strategies.

Here is what they came up with:

*Coloured counters are used here to represent*

*the Tens and Units columns, an unusual way to show place value. It doesn't really answer the question for me and tells me I need to spend some more time exploring how we can represent numbers.*

*This student has chosen to use*

*paddle pop sticks to show the two numbers: 57 and 32. The line across the middle is to show that the numbers above the line are exactly the same as the ones below the line, therefore the sum when added will be the same.*

*A bit like the previous example, this student is showing by modelling that each number has a corresponding number on the other side of the question, therefore two sides will be equal.*

*Abandoning calculations altogether, this student has written a story to explain why the provocation is true.*

*Here is a very nice use of a number line, showing that if you start at 32 and add 57 you get to the same end point (89) as you would if you started at 57 and added on 32.*

*And here is a really well-displayed sample of work showing multiple strategies and explaining a bit of the thinking behind the process.*

So, a great start to the year. Some interesting ideas and evidence that there is some thinking going on.

Can't wait till we have a look at the Associative Law tomorrow!

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