Wednesday, 1 June 2016

Why multiply before adding?

Continuing with our enquiry into why we have rules for the order of operations, we began by recalling what we had learnt and wondered about yesterday using our superpowers analogy:


One of our wonderings was: 'Why do we need rules for order of operations in maths?'

It's a great question and so we investigated that today.

When children discover the WHYS of maths (instead of just the HOWS), deeper learning and appreciation is grown.

It's admittedly much easier to simply explain the BODMAS / order of operation rules, but we should want children to understand why this is the way it is.


We began by using our calculators to answer:



Some of our calculators said the answer was 19 and others said 35.

How can this be?!?!?

We were complexed by this.

Some theories were shared.

So what is the right answer? / Is there a right answer?

To find out, we thought of ways we could interpret the number sentence if there were no rules.


By remembering that multiplication is repeated addition, we discovered it makes more sense that we multiply before adding because multiplying is repeated addition. To make sense of the number sentence, we can break it all down to addition:







A student suggested we should try another one and so this number sentence was proposed by a classmate:


We thought it was interesting how we came up with so many different possible answers.

What does this tell us about mathematics?

- There must be special rules or else there is more than one answer and that can't work with maths.

- Imagine if these numbers were representing money. We would end of with different amounts!

- Maths must have some sort of grammar like languages do. We can jumble up words, but we need to follow grammar rules or else we might be misunderstood. I think in maths there must be rules so we can all make sense of what we are doing.


Which should we do first?

- the brackets

Why?

- Because we don't know its value. We need to find that out first before we can do anything in the number sentence.



So, after the brackets, what should we do next?

- Calculate 3 squared.

Why?

Because it is also an unknown value. We need to know its value.

But why not solve that before solving the brackets?

- I think its because when we look at 3 squared, we can gain a sense of its value, but with the brackets, we really need to think more about what value it represents.

That theory made sense to us.

So, what should we do next?

- Multiply the 4 and 7

Why?

- Because multiplication is more powerful than addition. We always do what is more powerful first.

Any other ideas to add to that?

- It's repeated addition so we can break down to an easier way to understand. It's another way of finding out its true value.

Interesting.



We could then see how we had found all the unknown values in our number sentence.

So then?


- Now we can add them altogether.


Let's look at it a different way. Can we see this entire number sentence as completely being addition?

Pairs played around with this and we then shared our discovery:


We felt this helped us to understand why we do some operations before others.

As both a formative to find out who was understanding the whys, and also to help the children solidify in their own minds what we were learning, we created our own number sentence and explained WHY we do the order we do.

Some samples:






We then shared our ideas with different table partners and thus, used our discussions as a reflection to our learning.



I think this approach has been successful so far.  It's day 2 and we haven't even discussed BODMAS yet, but we have a deepened awareness for the need of rules and reasons why we should do some orders of operation before others and that is the key to learning.














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