# Lesson video

In progress...

Hello, my name is Miss Robson.

In this lesson, we will be working on our understanding of commutativity.

That is that we can add parts together in whatever order we like, but the whole will still remain the same.

We will start by combining objects in a set.

Then we will look at writing addition equations, before we apply our knowledge of addition to an independent task.

And then finally, there's a quiz to complete.

For this lesson, you will need a pencil, some paper, a number line to 10 and some objects for counting.

Pause the video here to collect the resources that you need and when you're ready, press play.

We have on the screen some beads of different colours, or counters, they're circles, and they show us the number bonds to six.

So the first number bond is six red circles and zero yellow circles.

All together, there are six.

The second number bond is five red circles and one yellow circle, and all together there are six circles.

See if you can pause the video here, and complete the rest of the equations and fill in all of those missing numbers.

When you're ready, press play and we'll go through the answers together.

How did you get on with your number bonds? Let's read them all together with the answers filled in.

Ready? Six and zero make six.

Make sure you're joining in.

Five and one make six.

Four and two make six.

Three and three make six.

Two and four make six.

One and five make six.

Zero and six make six.

Excellent work.

We're going to quickly recap counting on and addition.

In the context of a story, looking at our big picture again with Snow White and the Seven Dwarfs, I can see that Snow White has set the table for dinner.

And she's used a variety of different plates.

I can see that some of them are round, they're circular, and some of them are rectangular.

There are four circular plates, four circles, and there are two rectangular plates.

I'm going to represent those using cubes.

So here I have four orange cubes to represent my circular plates, and two blue cubes to represent my rectangular plates.

I'm also going to show you those numbers in a part whole model.

So let me pop four and two in the part whole model.

Altogether, there are six plates.

There are two rectangular plates and four circular plates.

If I put four in my head and count two on my fingers, four, five, six.

But I can also swap that around and put two in my head and four my fingers.

Two, three, four, five, six.

Both times I ended up at the same spot.

Both times I ended up with six as the whole.

Here are my cubes.

Two and four make six, or I could turn them around this way and see that four and two make six.

Today we are exploring commutativity.

That means that when we are adding things together, it doesn't matter if I add them this way, or if I add them this way.

I'm always going to have six cubes all together.

I can add whichever part first that I like.

We are also going to explore how that can make us more efficient counters and I'll show you how to do that now.

Here is the equation on a number line.

On the blue line, you can see four plus two is equal to six and the red line shows you two plus four is equal to six.

When we jump using the blue line, we jump all the way to four, and then we count on from four.

So four on my head, two on my fingers.

Four, five, six, or jump all the way to four.

Jump all the way to four, five, six.

With the red line, I'm putting two in my head and four on my fingers.

Two, three, four, five, six.

So starting at jumping to two, and then counting on four.

One, two, three, four, and ended up at six on the number line.

One of these is more efficient than the other.

It's a faster way of counting.

Pause the video if you need to look at the two equations and to look at the two jumps, and decide whether you think blue is more efficient or red is more efficient.

Which one do you think is more efficient? Which one's the faster way of counting? The blue way of counting is faster.

Jumping to the biggest number first is faster, because it means we have to do less little hops afterwards.

So, when we are adding two numbers together, because we know that no matter which way we add them together, we will always have the same total.

We should always start with the biggest number in our head and count on the smaller number.

Rather than wasting time, putting the little number in our head and counting on this big number.

We know that we'll end up with the same number each time.

Each time here, we've added the same numbers together and landed at six.

But one has involved a lot less jumping and a lot less work.

It's time for you to do the talk task.

We're going to do the first one together.

It's about creating math stories to do with a campfire and a tent.

I like to think that there are bears at the campfire.

So, I've got six green cubes and they're going to represent the bears.

I had the campfire going and it was lovely and warm.

But some of the bears decided that they weren't ready for bed.

So, two of the bears stayed and sat by the campfire.

And four of the bears went to bed in the tent.

My total was six.

My whole was six.

And I have two as a part and four as a part.

Which number do you think I should put in my head to count on from? Should I put four in my head or should I put two? Which part should I put in my head? I should put four in my head because it's the biggest of those numbers.

Four and two on my fingers.

Four, five, six.

I am, for the talk task, going to check by putting the smaller number in my head too.

So, two and four on my fingers.

Two, three, four, five, six.

What you need to do is get a part whole model so you could draw that on a piece of paper if you need to, or you could just draw a campfire and a tent, and you need your countable objects.

You can have as many as you would like up to 10.

Try telling you some different math stories about the bears going to a campfire, or going off to stay in the nice warm tent.

And see if there is a difference between adding the biggest part first or adding the smallest part first.

We're also going to be practising counting on in our heads.

So have a go at telling some interesting math stories, and when you're finished, press play.

I just told math stories too whilst I was doing my talk task, and one of my favourites was about all six bears being around the campfire.

And they all looked up at the sky, and they saw lovely stars.

One of the bears decided that he was too tired to keep watching all the stars so he went to bed.

So altogether, I had six bears.

Five of them stayed out by the campfire and one of the bears went to bed.

So five is a part, one is a part.

Here is that written as an equation.

Five as a part, five bears stayed around the campfire looking at the stars, and one is a part.

One of the bears decided that he was too tired to stay up any later.

So he went to bed.

Five is a part, one is a part.

Which of these numbers do you think I should put in my head to count on? Five or one? Which one's the biggest number? I should put five in my head and count on one.

I know that one more than five is six.

So all together, there are six bears.

Remember though, if I had decided to put one in my head and count on five, it would have come to the same total.

So, I'm going to put one in my head, five on my fingers.

My whole is still equal to six.

Five plus one equals six, and one plus five equals six.

Can you have a go at saying those equations with me? Five plus one equals six, and one plus five equals six.

Remember, commutativity means either way that we add them, the total will remain the same.

We need to think about which way is more efficient.

And we'd like to start with the biggest number in our heads and the smaller number on our fingers.

That way we're counting on quickly.

We can also represent it like this, five plus one is equal to one plus five.

So, if I show you using my cubes, here, I have five blue cubes and one orange cube.

Five to represent the bears that stayed inside and one to represent, sorry, five to represent the bears that stayed around the campfire and one to represent the bed that stayed inside.

And then here, I have one and five in the opposite colours, but you can that the bars are the same size.

So, both of them have six, but the top one has one blue cube and five orange cubes.

And the bottom one has one orange cube and five blue cubes.

They still both have six cubes all together, but they just have different parts around the other way.

So because five plus one is the same as saying one plus five, we know that five plus one is equal to one plus five.

An equation is like a scale that we balance.

So on either side of our equal sign, the things have to be the same.

So whether I've got five orange cubes and one blue cube, or five blue cubes and one orange cube, on either side I still have six.

So, my scale is balanced.

Let's look at one more example of commutativity.

I'm going to have four as a part and two as a part.

Four is a part, two is a part.

My whole is six.

So this is the story I was telling you earlier, where I had circular plates and rectangular plates.

Now, I could count on from four, or I could count on from two, because I know those are my two parts.

I've got four as a part and two as a part.

What number do you think I should count on from? I should count on from four, because it is the biggest number.

It's the greatest of my two parts.

So, everybody's putting four in their head and two on their fingers, ready? Four, you've got to count with me, four, five, six.

All together, there are six.

I can see on the screen that is represented in a part whole model.

It is also represented on a bead string.

So first there are four red beads and two white to beads.

But, it would be a lot faster and quicker to count on the two beads to the four instead.

So instead of having four, instead of having two and going three, four, five, six, it would be a lot easier to swap the parts around and instead have four and count on two.

One, two, five, six.

We can look again at the jumps.

In both cases, we are adding the same parts together.

We're adding four and two, or two and four.

Adding the same parts together to make six.

But they are two different representations of how we could count on to get to six.

For your independent task, there's a bit of a puzzle for you to complete.

I can see on the left hand side, there are some different cards.

There is six plus three, and three plus six.

These are a partner.

These are a pair, sorry, because they are the same parts.

So here we have six and three and then the other way around, we have three and six.

All together, the total is going to be the same.

But one of these is more efficient than the other.

So you would find six and three and three and six, and put them together because they are the same equation just with parts on different sides.

You also need to find the number lines that represent those parts.

So here I can see this number line has jumped from zero all the way to six, and then one, two, three little hops.

And then this one here is three plus six.

So instead this time we've jumped to three first, and then jumped on six little hops.

One, two, three, four, five, six.

So, you need to match the equation to the number line.

Match the two equations to each other.

And think about which one you think is more efficient and why.

When you're finished, press play.

So to finish, we're going to look at an example of a child who completed this activity, and then explaining their reasoning.

And you have to decide whether you agree with them, or if you think that there's a different way they could have counted.

They say, I calculated two plus six by finding the number two on a number line and then jumping on six times.

Pause the video to consider, do you think this is an efficient way of counting? Or do you think there is a more efficient way that they could have calculated this? So, what do you think? Thumbs up, it was efficient.

Thumbs down, there's a different way they could have worked this out.

There's a different way they could have worked this out, if they were having six plus two.

We have six plus two.

I can see that one of those numbers is a lot bigger than the other.

Six is the biggest number.

If I were them, I would have jumped to six first, and then jumped on two more times, as opposed to jumping to two and then jumping on six more times.

Remember, it doesn't matter which way we add them on to each other, the total will remain the same.

But jumping to the biggest number first is the more efficient way of counting.

And then jump on two times.

One, two.

I land in the same place.

Both of us have ended up with eight as our total.

Six, seven, eight.

But my counting would have been a lot faster, so a lot more efficient than this child's counting.

Thank you for joining me today.

I've really enjoyed exploring commutativity with you and I hope that you have too.