Lesson video

Hello, my name is Mr. Goldie, and welcome to today's maths lesson.

I hope you have lots of fun.

And here is our lesson outcome.

So, by the end of today's lesson, you should understand that numbers can be added in any order.

For today's lesson, it would be very helpful to have a pencil, some paper, and number cards as well.

If you have number card, numbered from 1 to 9, that would be really, really helpful.

If you don't have any, you can always make them.

And here are our keywords.

So, our keywords are addend, commutative, equation, and sum.

So, an addend is a number added to another.

Commutative means the same answer is reached even if the order of addends is changed.

So, even if you add numbers in a different order, you still get the same sum, still get the same answer.

An equation is a number sentence that uses equals, the equals sign.

And the sum is the total after numbers are added together.

And here's our lesson outline.

So, the first part of today's lesson is addition is commutative.

And the second part is changing the order when adding numbers.

In this lesson today, you will meet Izzy and Alex, and they'll be helping you today and ask you some questions.

So, Alex and Izzy share some cherries.

First, Alex takes 2 cherries, so he puts them on his plate, then he takes 3 more, then he takes 1 more.

First Izzy takes 1 cherry, pops it on her plate, then she takes 2 more, then she takes 3 more.

Who has taken more cherries? So, Alex took 6 cherries, he took 2, then he took 3, then he took 1.

2 add 3 add 1 equals 6.

Izzy also took 6 cherries, 1 add 2 add 3 also equals 6.

2 add 3 add 1 is equal to 1 add 2 add 3.

2 add 3 add 1 equals 1 add 2 add 3.

It doesn't matter what order the addends are in, they both make the same sum.

Izzy and Alex share another bowl of cherries.

Alex takes some cherries.

He takes 4 cherries, then he takes 2 cherries, then he takes 3.

Let's split that up to make it a little bit easier to add together.

So, let's add together first of all, 4 and 2.

4 and 2 equals 6, and all we have to do is to add on the 3.

6, which is made out of 4 and 2, add 3, 6 add 3 equals 9.

Izzy takes some cherries.

Izzy takes 3, then 4, then 2.

Let's again split it up to make it a little bit easier to add together.

3 add 4 equals 7, then 7 add 2, because she took 2 more cherries, equals 9.

They both took 9 cherries.

4 add 2 add 3 is equal to 3 add 4 add 2.

This can be shown on 10s frames.

Alex takes some cherries, so 4 and 2 and 3.

4 add 2 add 3 equals 9.

Izzy takes some cherries, so she took 3 cherries, she took 2 cherries, she took 4 cherries.

3 add 2 add 4 equals 9.

So, we can see the addition is commutative.

It doesn't matter what order the addends come in, it still makes the same sum.

Commutative means you can change the order of the addends and the sum stays the same.

3 add 2 add 4 equals 9.

Here are our addends, 3 and 2 and 4, they're addends and they add together to make our total, our sum, 9.

4 add 2 add 3 equals 9.

Here are addends and our sum.

We could also write here this way round.

So, we could start with the sum 9 equals 2 add 4 add 3.

Here are addends, and here is our sum.

So, Izzy takes some cherries.

First she takes 3 cherries, then she takes 5 cherries, then she takes 2 cherries.

So, she took 3 and then she took 5, so 3 add 5 equals 8.

And then 8 add 2 equals 10.

So, altogether she's taken 10 cherries, that's our sum of the number of cherries she's taken altogether.

Does it matter what order she took them in? So, Izzy takes some cherries.

First she takes 5, then she takes 2, then she takes 3.

5 add 2 equals 7.

7 add 3 equals 10.

What about if she took the cherries in a different order? What about if she first of all took 3 and then she took 2 and then she took 5? 3 add 2 equals 5.

5 add 5 equals 10.

There's a double there, isn't there? Should know that double 5 makes 10.

So, commutative means you can change the order and the sum stays the same.

5 add 2 add 3 is equal to 3 add 2 add 5.

Alex chooses 3 number cards to add together.

So, he chooses the number cards 1 and 3 and 5.

Do I have to start with 1 then add 3 then add 5? And Izzy is saying, "It doesn't matter what order you add them in, you still get the same sum." Is Izzy right? Yes, she is right, commutative means you can change the order and the sum stays the same.

How could Alex find the sum? How could he add together 1, 3, and 5.

So, he could do first of all 1 and 3.

1 add 3 equals 4, and then add the 5, 4 add 5 equals 9.

He could do 5 add 3 equals 8, then 8 to 1 equals 9.

He could do 1 add 5 add 3, so he could do, first of all, 1 add 5 makes 6, 1 add 5 equals 6, 6 add 3 equals 9.

So, it doesn't matter what order he adds them in, he always gets the same answer, always gets the same sum.

"They all give the same answer!" says Alex.

"It's like I said," says Izzy, "It doesn't matter which order you add them in." Alex chooses 3 number cards to add together.

Which calculations give me the sum of the 3 numbers? There are 6 different calculations there, which of them would give the sum of the 3 numbers? So, pause the video and see if you could work out which of them would give the sum of 2 and 6 and 5.

Welcome back, let's see how you got on.

So, our first calculation is 2 add 5 add 6.

That works, it's the same three numbers just in a different order.

Our next one does not work, 2 add 5 add 2 does not give the same sum as 2 add 6 add 5.

One of the numbers has changed, therefore we can't make the same answer.

The next one is 6 add 5 add 5, does it work? No it does not, it does not give the same answer.

One of the numbers again has changed.

Instead of having 6 and 5 and 2, it's 6 add 5 add 5.

The next one is 2 add 5 add 5, would that give the same answer as 2 add 6 add 5? No, it would not, one of the numbers has changed, so it will not give the same sum.

The next one, 6 add 2 add 5, would that work? Yes it would, it's the same three numbers just in a different order.

And the last one, again, same three numbers just in a different order.

Well done if you've got those right.

And here is your task.

So, use the number cards from 1 to 9 and you're going to choose 3 of the cards, so you might choose 1 and 2 and 5 and you're gonna write 3 different ways you can find the sum of those cards.

So, if you chose 1, 2, and 5, you could do 1 add 2 add 5, you could do 5 add 2 add 1, you could do 5 add 1 add 2.

So, play around with the order and see what you get as the sum.

And if you've tried three cards already and you found three different ways of making it, then choose three different cards.

So, pause the video and have a go at your task.

And here is one possible solution.

So, this is using the numbers 5 and 7 and 2.

And these are three possible solutions, three possible ways of finding the sum.

So, we could have done 5 add 2 add 7 equals 14, or 2 add 7 add 5 equals 14, or 5 add 7 add 2 equals 14.

So, while done on your task and hopefully you've got the same sum each time.

Let's move on to the second part of the lesson, so this is changing the order when adding numbers.

Izzy chooses 3 number cards, so she chooses numbers 2 and 3 and 6.

How could Izzy find the sum of the 3 numbers? So, she could do 2 add 3 add 6, she could do 2 add 6 add 3, she could do 3 add 2 add 6, she could do 3 add 6 add 2, she could do 6 add 2 add 3, or she could do 6 add 3 add 2.

Remember, it doesn't matter what order she adds them in, although some ways might be easier than others.

"I'm going to represent my calculation on a number line," says Izzy.

And Izzy chooses 2 of them, so she chooses 2 add 3 add 6 and she also chooses 6 add 3 add 2.

So, Izzy uses number lines to add the numbers together.

So, here's her first calculation, so 2 add 3 add 6.

So, she starts on 2, she adds 3 to get to 5, add 6 to get to 11, so 2 add 3 add 6 equals 11.

Then Izzy tries 6 add 3 add 2.

So, this time she starts on 6, so adds 3, then she adds 2 more so she gets to 11 again, so, 6 add 3 add 2 equals 11.

Which is easier? What do you think, is one way easier than the other? 2 add 3 add 6, is that easier or is 6 add 3 add 2 easier? Well, Izzy thinks it's 6 add 3 add 2.

She says, "I think 6 add 3 add 2 is easier because I started on the biggest addend and added the small addends." Izzy chooses 3 number cards to add together, so she chooses 3 different numbers, a 2, a 4, and a 5.

What's the most sensible way to add the 3 numbers together? Could it be 2 add 4 add 5? So, we could first of all add 2 add 4 together, so 2 add 4 equals 6 and then add the 5, 6 add 5 equals 11.

That's one way of doing it, gives us the right answer.

Is there a better way of doing it? Perhaps you could do it 5 add 4 add 2.

So, first of all we can add the 5 and the 4.

And of course that is a near double, isn't it? 4 add 4 makes 8, 4 add 5 must make 9.

So, 5 add 4 makes 9, 9 add 2 equals 11.

Or you could do it this way, 2 add 5 add 4.

So, add together 2 add 5 first of all, that equals 7, and then do 7 add 4, 7 add 4 equals 11.

All of those ways work, all of those ways give the answer, is there one way which is better than the others? Which strategy would you use? So, Izzy uses a number line to help work out the answers.

So, 2 add 4 add 5, so she starts with 2, she adds 4 to get to 6, adds 5 to get to 11.

Quite a big jump there isn't it? So, 2 add 4, add 5 equals 11.

She also does 5 add 4 add 2.

This time she starts on 5, she adds 4 to get to 9, adds on 2 more to get to 11.

5 add 4 add 2 equals 11.

Which order is easier to add? Which one's easier? Izzy thinks she has the best strategy.

She says, "It's sensible to start with 5 because it's the biggest number." 4 add 5 add 2.

And Izzy is saying, "I know that 4 add 4 equals 8, 5 add 4 is equal to 4 add 4 is 1," so it's using lots and lots of her double knowledge here to help work out the answer.

And all you have to do then is to do 9 add 2 equals 11.

So, Izzy says, "It's sensible to start with the biggest number." Is it always sensible to start with the biggest number? Izzy says, "I think it depends on the numbers you are adding together." So, Alex has chosen three number cards, he's chosen 8 and 5 and 5.

"Should I start with 8?" says Alex.

Well, 8 is the biggest number, is it most sensible to start with 8? What would happen if we did start with 8? So, he's going to add together 8 and 5 and 5.

So, you have together 8 and 5 first of all, 8 and 5 makes 13.

8 add 5 equals 13.

So, 8 add 5 add 5 is equal to 13 add 5.

And then he's gotta do the calculation 13 add 5 equals 18.

Not the easiest way of working out the answer.

Is there a different way of doing it, a better way of doing it? What about if Alex shifts the cards around, he reorders them? What about if he does 5 add 5 add 8? Well, 5 add 5 is much easier to work out.

5 add 5, of course, makes 10.

Double 5 makes 10, or it's also number pairs that total 10, isn't it? And then, 10 add 8 isn't too tricky either.

10 add 8 equals 18.

So, that's actually an easier way of working out the answer.

So, which order is easier to add, and why is it easy to add? Alex says, "I think it's easier to change the order to 5 add 5 add 8.

I know 5 add 5 equals 10." So, sometimes you can look at the numbers and think, "Ah, I know an easy way of working it out.

I know an easy way of reordering the numbers 'cause I've spotted something about the numbers I'm using." And here's one to try on your own.

Which calculation is easier? So, Alex has got to add together the numbers 6 and 1 and 6.

Should he add them together in this order, 6 add 1 add 6? Should he do 6 add 6 add 1? What do you think? Have a go of working out the answer.

How would you work out the answer? Pause the video and have a think.

Can you actually find out what the answer is? Can you work out what the answer is? Welcome back, let's see how you got on.

So, we could have done 6 add 1 add 6 first of all, so do 6 add 1 first of all, that equals 7.

So, 6 add 1 add 6 is equal to 7 add 6.

Get a quite tricky calculation this one, isn't it? 7 add 6 equals 13, bit tricky.

What about the other one? 6 add 6 add 1.

6 add 6 add 1, what do you spot about that calculation? Well, 6 add 6, of course, is a double.

You might know what double 6 is already, so double 6 is 12.

So, 6 add 6 add 1 is equivalent to, is equal to 12 add 1.

All you've got to do then is do 12 add 1, and that, of course, makes 13.

Alex is saying, "I saw double 6." Sometimes you spot things in the calculations about how you work out the answer.

So, the symbol equals shows that equations are balanced.

Can you spot the mistake? Here's your number cards, 6 and 2 and 6.

6 add 6 equals 12 add 2.

Can you spot what is wrong? Well, 6 and 6 does not equal 12 add 2, they are not equal, they are not balanced.

What should the equation look like? 6 add 6 add 2 equals 12 add 2.

That is now balanced, they are now equal.

Rearrange the numbers into the order that is easiest to add.

So, here's a set of three number cards.

How would you add them together, what would you do? Well, you may have spotted there's actually a double there, so it'd be sensible to start with 3 add 3.

3 add 3, of course, makes 6.

So, that is equivalent to, that is equal to 6 add 4.

6 add 4 makes 10.

6 add 4 equals 10.

What about this set of number cards here, 5 add 4 add 4? But again you may spot there's a double there so it'd be sensible to start with 4 add 4.

4 add 4 add 5 is equal to 8 add 5.

8 add 5 equals 13.

And the last three number cards, 1 and 2 and 6, how would you add those together? Sometimes it is easier to start with the biggest number, so we start with 6.

6 add 2 add 1, that is equal to 8 add 1 because 6 at 2 equals 8, and 8 add 1 equals 9.

Here's one to try on your own.

So, three sets of number cards there, can you rearrange the numbers into the order that it's easiest to add? And if you can, can you actually find the sum as well? So, pause the video, how would you rearrange the numbers into an order that's easy to add, and can you find the sum? welcome back, let's see how you got on.

So, the first certain number cards is 8 add 1 add 8.

So, hopefully you thought, "Ah, I spot a double there," 8 add 8 add 1.

8 add 8 makes 16.

8 add 8 equals 16.

So, 8 add 8 add 1 equals 16 add 1.

16 add 1 equals 17.

The next one, start with the biggest number, 7 add 4 add 2 equals, so we add the 7 and the 4 together and that equals 11.

That is equal to 11 add 2.

11 add 2 equals 13.

And the last one is 5 add 3 add 6.

Well, you may have spotted a near double there.

5 and 6 are a near double, they're only 1 away from each other.

And you might know 5 add 5 already, of course.

5 add 5 equals 10, so 5 add 6 equals 11.

So, 6 add 5 add 3 equals 11 add 3.

And 11 add 3 equals 14.

Well done if you spotted an easy way of adding them together, and especially well done if you managed to find some of the sums as well.

Here is your first task.

Can you work out the missing numbers? So, are three numbers being added together, how would you add them together easily and efficiently? So, which numbers would you have together first.

If you want to go back and rewatch any of the video, you may do.

And here is your second task.

Fill in the missing squares using the numbers below.

So, you've got the numbers 2, 2, and 2, 3, 3, and 3, 4, 4 and 4.

You've got to make each row and column add up to the same sum.

So, for example, you could put a 4 here, a 4 here, and a 2 here to make a column, so we've got 4 add 4 add 2.

4 add 4 add 2 equals 10.

Does every row and column add up to make 10 or does it add up to make a different number? So, have a go at that problem.

And if you solved that problem, slightly trickier, can you also make the diagonals add up to the same sum? So, from corner to corner, can they also add up to make the same number as well, the same sum when you have the 3 numbers together.

Now, good luck with both those tasks.

Pause the video and have a go at them.

And here are our answers for the first task, we got there the missing numbers.

So, hopefully you managed to get those ones right.

And if you got onto the second task, here are 2 possible answers.

They're not the only answers, there are two possible ways you've could have solved the puzzle.

So, in our first one, each row and column add up to the same sum.

So, if you add up each column and each row, they all add up to make 9.

And in the second one, each row column and diagonal add up to make 9.

You've also got the diagonals adding up to make 9 too.

Very well done if you've managed to solve those two 'cause they were quite tricky.

Very well done on your work today.

Here is our summary.

Addition is commutative.

This means the same answer is reached even if the order of addends is changed.

Addition can be done in any order.

Finding the best way of ordering the numbers is part of solving the problem.

So, you gotta look at those numbers carefully to try and work out the answer.