More Fun With BODMAS

We began our maths thinking similar to yesterday.

These 2 questions were posed and in pairs or individually, we tried to solve them:

Before we started sharing our ideas, I asked, "If you saw these number sentences this time last week, what would you have thought?"

Some funny responses were shared and we all agreed that though they are complicated, our mathematical understanding has grown so much that we know how to solve them and why we solve them that way. 

Just like yesterday, we used the same routine:

What do we do first?

Discuss with your partner.

Who can help us?

Why?

Using this routine kept us all thinking out loud and deepening our reasoning abilities especially when we kept discussing why, why, why.

What do these make us wonder?

- Is algebra really that simple?

- What other ways can we use order of operations?

- If someone in another country had this same question, would they also get the same answer?

What do these make us think?

- It's like traffic rules. We have to follow traffic rules when we drive or else people would have so many accidents.  This is the same. We have special rules in maths so that people don't have accidents when solving them.

- It makes me think that maths is just a fun puzzle to try to solve. Even if we get the answer wrong, we still learn a lot.

- Today I've really understood what square root is; it's the opposite of squaring a number.

- It makes me want to create some complicated number sentences like these to see if my Dad can solve them too.

We then continued with our choice of challenge.

Either the 4 4s or creating number sentences that equal our ages.

See yesterday's link

The same high level of engagement and quality mathematical discussions took place.

Students expanded their thinking and understanding of how we can manipulate numbers and there was a lot of creative thinking involved too.

Some samples so far:

I like this idea of 'extras' they found that they could use later:

Reflecting is the most important part of the learning process.

For today's reflection, we used the 'I used to think.......and now I know.....'.

I added a new idea as a reflection choice: ' I used to feel.....and now I feel.....'

After writing our reflections, we shared with the whole class how are thinking or feeling has changed:

Having Fun With Orders of Operations

Today we began with these two questions. We had a choice of trying to solve these individually or with a partner.

The reason we began like this was to:

° tune in

° reflect on what we have already discovered about the order of operations

° extend our thinking for parts of the order we haven't addressed yet. Its important that we give children time to create their own theories and test them out.

° see how we can make mathematical fun

After a bit, we looked at the first question.

What should we do first?

Discuss with your partner.

Who can help us?

- We do what's in the brackets first.

Why?

- Because it is an unknown value.

- We need to know its value so we can use it.

- Brackets have more power than other operations. They can change a number sentence a lot.

What should we do next?

Discuss with your partner.

Who can help us?

- We need to do the square root.

Why?

- It's also an unknown value.

- We can't use it until we know what it is.

What should we do next?

Discuss with your partner.

Who can help us?

- We need to calculate what the squared number is.

Why?

- It's also an unknown value.

So, why do we find out the square root before the squared number?

- They are equally powerful, but when there's an even power we read it like a normal sentence from left to right.

- We could find the squared number first because we still need to know its value. But it's like reading. We don't read from right to left.

What should we do next?

Discuss with your partner.

Who can help us?

We continued with this till we explained not only the how but the whys.

The 'discuss with your partner' routine at each stage is an effective strategy to encourage children to think more about the whys. Having a teacher or just one student sharing their thoughts isn't the most effective way for every child to learn. The more we allow children to talk or visualise their thinking, the better. 

It also energises what could be a fairly drab learning experience.

We did the same routine for the second question.

Children (like adult learners) are far more engaged when we give them choices.

So, to help them solidify and extend their understandings of the order of operations, we had two choices.

We began by looking at the 4 4s challenge. This is a well-known maths problem-solving activity that helps children think about number relationships and what operations do to a number:

To help us see the type of thinking involved, in pairs we tried to find a number sentence that equalled nought and then we shared.

We liked the one circle above until someone pointed out how it has the number 2 in which we cannot use.

We really liked the pair who took the numbers below zero:

4 - 4 - 4 = 0 + 4 = 0

We have recently enquired into positive /negative numbers so its great to see children still thinking and using them.

Looking at the last suggestion:

Someone challenged how it doesn't equal nought.

Oh! We forgot to say it has brackets!

Mistakes are great to make because they help us to learn.

What did we just learn from their mistake?

- Brackets REALLY are powerful!

- They can change the whole meaning of a number sentence.

Together we calculated how and then discussed why brackets have such a great power:

The second choice was:

Normally we have a choice of doing things in maths individually or in pairs / groups.  Today though, I wanted us to think aloud and peer teach each other, so we all did this with pairs.

Soon after some pairs wanted to buddy up with other pairs so they did that.

There was high engagement and a lot of wonderful mathematical discussions taking place.  

Time was nearly up and I asked whether we wanted to continue with this tomorrow or more on.

Everyone eagerly wanted to continue tomorrow.

That's a good barometer that it is engaging and challenging their thinking.

(I'll post pics of their ideas later)


Following Day's Link with Examples of Today

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.

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:

1

13

 50

 50
 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.

Why do we multiply before adding investigation

Continuing our enquiry

Continuing the enquiry