Tuesday, 31 May 2016

Why are the orders of operation the way they are?

We want children to understand the 'whys' of mathematics.

We seem to place too much emphasis on the 'hows'.

The 'whys' give purpose, far greater interest and become more memorable to tap into later in their mathematical lives. The 'whys' make the learning meaningful and fulfilling. 

To help us create memory hooks and to discover the whys behind orders of operations (BODMAS), we started with this provocation: 

Individually or with our table partners we chose a question/s to try to solve.

I was hoping that we would come up with different possible answers and we did:

 Our answers:


 910 160

Looking at our ideas, what does it make us think?

- Rules need to exist in maths or else we can get different answers and that would confuse people.

- If there are rules, then there must be reasons for them.

What does it make us wonder?

- Is it alright to calculate them differently and have different answers?

- Is one answer better than another?

- Why do we have brackets?

This seemed like a pretty successful lead in provocation as it conjured up some debate and interesting wonderings.

To help us enquire into the 'whys' of mathematical orders of operation, we imagined that the operations had superpowers.

We personified a number by thinking of the number 3 standing there in the sunshine and along comes an operation like multiplication.

What could multiplication actually do to number 3?

What is its superpower?

How can it change number 3?

We came up with this idea as an example:

In pairs, we then created a superhero name for each operation and thought of what its superpower was.

Some highly imaginative and creative ideas emerged!

We then thought of how we could rank their superpowers.

Which operation can have the greatest change on a number?

Pairs ranked the operations from the strongest to weakest powers over a number:

We really appreciated how this group 'proved' their theories by showing what each could do to the number 11:

This generated some more wonderings that we are curious to discover:

After sharing our ideas, as a whole class we shared connections we could make between the operations:

To help us understand which operations we do first and more importantly WHY, we first identified the operations that 'make numbers grow larger'.

We thought these do:

We then used these to create our number sentence using all of those operations in it and used our sentence to find out both how and why it should be solved that way.

Some samples:


One student wondered if it really did make a difference what order we do things so we tested that out together:


She discovered how it does change the answer significantly!

Pairs then created and shared new questions for their partner to solve and a lot of good discussion and knowledge sharing took place of which we should do first etc.

So, why do we do certain operations before others?

We looked at a number sentence together and discussed our theories of which we would do first etc and why:

We then thought about which would be the next steps and why:

Recapping, we had gained a deeper sense of why we do brackets first, why we would square the number next and now understanding how multiplying is a more 'powerful' operation that addition, we should do that first. 

From this, a student remarked how we had just created a memory hook (we create these a lot ).

She explained how first we need to know the values of all the numbers and then whatever operation is more 'powerful' we do those before we do less powerful operations.

That summed our learning up pretty well.

We needed to stop there, but tomorrow we will continue with investigating orders of operation for those that make numbers 'smaller'.

What I think worked well, was that we were investigating the whys behind BODMAS without even talking about the BODMAS rules yet.

When we help guide children into discovering the whys behind maths thinking for themselves, it does become more meaningful and memorable.

1 comment:

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