Wednesday, 28 September 2016

The WHYS of dividing by 10, 100, 1 000 etc

The WHYS of maths is more important than the HOWS.

Yet, it seems we tend to focus so much on the HOWS and this leaves mathematical thinking as 'doing school', I think, rather than the real learning and discoveries of maths. 

We have been exploring how our number system is based upon 10 and what that means regarding place values.

A benefit we are discovering from having a base 10 number system is that we can multiply / divide numbers by 10, 100, 1 000 etc easily.

Yesterday we explored how and more importantly WHY we can multiply numbers easily by 10, 100, 1 000 etc easily.

To find out about dividing, I wrote the following on the board and the children used the I see / think / wonder routine to record their thoughts:

We then shared what we noticed, theories we had, and our wonderings:

Most of us had discovered (a few already knew) the HOWS of dividing by 10, 100 etc, but the WHYS are more important.

Looking at our gathered wonderings, students chose a question they found interesting and spent time exploring it and then sharing their theories or discoveries with others. Even those wonderings that at first sounded 'simple' became actually quite complex when trying to understand the WHYS behind them.  

Whilst sharing, some students had used strategies which we had previously done when finding the WHYS behind multiplying by 10, 100 etc.

Some students explained to us why this works using a place value grid. They explained how we can see each digit changing its place value:

This helped us to also see why numbers grew smaller when we divide them.

Other examples were explored:

We took our thinking further by exploring how each digit changes with decimal numbers:

Another pair shared how they used our multiplication strategy with the division which helps us see what we actually do to the number.

We then used the same number and had a choice of dividing it by 10, 100 or 1 000. A few students decided they wanted to challenge themselves further by dividing it by 10 000:

By using the split strategy, we felt we could really see what we were doing when we divided by 10, 100 etc.   (Some of us used a calculator when dividing 20 by 10 etc)

When we looked at our answer, we identified the place value of each digit to see how they had changed.

Some great reasoning thinking emerged when we looked at dividing 20 by 100. A student shared how they knew the answer must be a decimal because the place values were 'used up' by the 100. we thought this was a great theory and so tested it out together when we looked at dividing the number by 1 000:

All of these great discoveries and learning was student-led and all had sprung from simply using the 'see-think-wonder' routine.

When we created and tested our partners with questions to divide by 10, 100 etc, another interesting wondering was shared:

When we write large decimal numbers, do we also use spaces or commas?

Some of us thought yes, others no and most unsure.

It was a great question so we tried to reason our theories.

We had learnt recently why we need commas or spaces when writing whole numbers, so it made sense that we should have commas with long decimals. 

The reasoning behind this really made sense.

Someone else though had recently seen the number pi and explained that she didn't see commas or spaces in it.

Why not?

We didn't come to a conclusion on this, but it is on our wonder wall for us to find out........

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