Lesson video

Hi, my name's Mr. Peters, and today we're gonna be thinking about decimal numbers and how we can make and break decimal numbers up using both addition and subtraction.

Let's get going.

So by the end of this lesson today, hopefully you'll feel confident enough to be able to say that you can explain and use decimal numbers with tenths representing these both with addition and subtraction.

So before we start our learning today, it'll be really useful to think about some of the key words that we're gonna be using throughout the lesson.

The first one is decimal number.

The second one is partition or partitioning.

The third one is equation.

And the fourth one is expression.

Let's have a little think about what these mean.

So a decimal number is something we should already be familiar with.

It has a decimal point in the number, and it has some numbers that follow that decimal point, which represent the fractional parts of a number.

When we think about partitioning, that's when we take an object or a value and we split it into smaller parts.

An equation is when we use numbers, calculations, or expressions to show that they're equal to one another.

For example, 3 plus 4 is equal to 7.

The expression 3 plus 4 is equal to the number 7.

And that leads us on nicely to our last word, expression.

An expression is a term we use which contains one or more values and are joined using an operator, for example, 3 plus 4.

So this lesson today will be split into 3 parts.

The first part we'll be thinking about partitioning decimal numbers.

The second part will be representing additive relationships.

And the third part of our lesson will be using number lines to represent this.

Let's get going with the first part.

During this lesson today, you'll also meet Laura and Sam, and they'll be helping us along our way with our journey.

So let's start thinking about our learning then.

Sam has created a number using base 10 blocks, and he's placed it in the whole of our part-part-whole model.

He says that his number represents 3.

6.

There are 3 ones and 6/10 in our number.

This can be written as 3.

6.

We can then start to think about partitioning this number into wholes and fractional parts.

Let's have a look.

We could take the three wholes and place them in one of our parts, and we could take the 6/10 and place them in the other one of our parts.

We can represent this as 3 and 0.

6, and we know that the 3 and the 0.

6, if we were to recombine these together, would make our whole of 3.

6.

Now, watch carefully this time.

What is it you notice about the base 10 blocks when they move out of the whole? What did you manage to spot? That's right, the wholes have been placed into the other part, and the fractional parts, the tenths, have been placed into the part on the left this time.

Hmm, does it matter which way round they are? No, we know because of the rule of commutativity that we have with addition that we can place the parts in either order and they will still make the same sum.

So, 0.

6 and 3 would again recombine this time to make 3.

6.

You might see now that our number that we had, which is placed in the whole, is now represented with place value counters rather than base 10 blocks.

Does it make a difference? No, it's just another representation of the number.

And again, as you can see, we've got 3.

6, and we can partition these place value counters into 3 wholes and 6/10, and together they will make 3.

6 when you recombine them.

One more to look at here.

What do you notice has changed this time? Ah, well, we still have the counters, don't we? But actually the tenths this time, we've changed how we've represented these.

Previously, we've represented them as a decimal number, 0.

1, whereas this time we've represented it using a fraction of 1/10.

And again, we can partition this into 3 wholes and 6/10.

And when we recombine these, they would make 3 wholes and 6/10, and we can record it this way as a mixed number.

Right, time for you to have a quick go.

On the left, I've got my turn, and on the right you've got your turn.

I'm gonna show you an example, and then you are gonna have a go for yourself.

So let's have a look at my example.

My example shows 5.

2 as the whole.

Hmm, have a think for yourself quickly, what could I partition that into if I was trying to partition that into whole numbers and fractional parts? That's right, I could partition it into 5 and 0.

2, or as we said earlier with the commutative law, we could represent this as 0.

2 and 5 and swap the parts around.

Right, now it's your go.

Maybe using a whiteboard or a piece of paper, draw the part-part-whole models and partition the whole into whole numbers and fractional parts.

Can you do it both ways? Good luck.

Great, well done.

How did you get on? Let's have a look.

Did you manage to partition the 7.

3 into 7 and 0.

3, or again, using our commutative law, 0.

3 and 7? Well done if you managed to get that.

Now, onto your first task for today.

I've given you some part-part-whole models here, and I was hoping you could have a go at finding either the missing parts or the missing whole.

Good luck and I'll see you shortly.

Welcome back.

How did you get on? Let's have a look at the first one.

Here, we've got 3.

1 as the whole, and we already know one of our parts, we know it's got three wholes.

It's got three ones as a part.

Hmm, so that's making me think that we don't know the fractional part of our decimal number.

Hmm.

So I'm looking at our number now and it seems to me that it says 3.

1, and I say 3.

1, but I think 3 wholes and 1/10.

So that must mean the other part must be 1/10, and we can write that as 0.

1.

Let's have a look at the next one.

Hmm, we've got 4.

7 this time.

I say 4.

7, but I think 4 wholes and 7/10.

So that must mean we can partition that into 4 and 0.

7.

You may have done the other way around and that's okay too.

Let's have a look at the next one.

Whoa, we don't know the whole this time, but we know both of the parts.

One of the parts is 8 ones and the other part is 6/10 or 0.

6.

And when we combine this together, we write this as 8.

6.

The 8 represents the 8 ones and the 6 represents the 6/10.

Right, onto the next row.

Did you spot what was different this time? Yeah, we've got a bigger whole this time, haven't we? Our whole is 15.

6.

Hmm, how would we separate that? How would we partition that into whole numbers and fractional parts? Yeah, that's right.

We would have to put the 15 as one of our parts, and again, the 6/10 would go as the other part.

And if you did that the other way around again, that's okay.

Onto the last two now.

Ah, our whole's got even bigger again this time.

Did you spot that? This time, it's 103.

2.

Hmm, how are we gonna partition this? What's the whole number and what's the fractional part? That's right, 103 is the whole number, and the 0.

2 would therefore be the other fractional part, and recombining that together, we would make 103.

2.

Finally, what do you notice has changed in this last one? Yes, one of them has been recorded as a fraction this time rather than a decimal.

And again, this time we've got the fractional part first.

Does it matter? No, we know that the fractional part is 4/10 here, and the whole number is 5, and we record that as 5 and 4/10.

Or if you wanted to record that as a decimal, we could write that as 5.

4.

Well done if you managed to get all those.

Now, let's move on to the second part of our lesson.

This time we're gonna be thinking about representing additive relationships.

Sam is saying here that we can represent this part-part-whole model as an addition equation.

Laura has got a sentence stem that she thinks will help us along the way with our learning, and it's one we used a little bit earlier on.

I say 3.

6 but I think 3 ones and 6/10.

Can you see how we recorded that with our equation? We wrote the 3.

6 first and then we placed our equal sign, and then the partitioned parts we placed on the right hand side of the equal sign, and we know that we can join them together with an addition sign, and that shows the equality.

That shows that 3.

6 is equal to 3 and 6/10, 3 plus 6/10.

Using our understanding of commutativity again, we can also write it another way.

Laura's suggesting we use our stem sentence again.

I say 3.

6 but this time we're gonna think of it as 6/10 and 3 ones.

So again, you can see it as our equation.

3.

6 is equal to, and this time we've written the 6/10 first plus the 3 ones.

And we know that no matter which way they're placed around, they will both equal the same sum of 3.

6.

Right, we've represented them as additional equations, and now we can also represent them as subtraction equations.

Let's have a look.

Here's our part-whole model again.

And Laura this time is saying, well, on one side of our equal sign, we have to have the whole minus a part for our equation to make sense.

So on the left hand side of our equation, Laura has written 3.

6 minus 3.

That's one of the parts that she's minusing from the whole.

Therefore the other part would be placed on the other side, and you can see here that we've now put the 0.

6 on the other side of the equal sign.

So to make our equation make sense, we know that 3.

6 minus 3 is the equivalent of 0.

6.

Sam's taken this a little bit further this time.

He's realised that you don't have to subtract the same part each time.

You could have subtracted the other part.

So instead of subtracting 3, which was the part that Laura subtracted, Sam is going to subtract the 0.

6 instead this time.

So this time our equation would be 3.

6 minus 0.

6 is equal to 3.

On the left hand side, we have the whole minus a part, and it's a different part to what we had the first time.

And then the other part is now on the right hand side this time.

So, so far we've formed four equations to represent this part-part-whole model.

We've got 3.

6 equal to 3 plus 0.

6.

We've got 3.

6 as equal to 0.

6 plus 3.

Take a moment to look at those again, and what is it that you notice that's the same and different about them? That's right, the addends have been swapped around, haven't they? The 0.

6 and the 3.

And then have a look at our two subtraction equations at the bottom.

We've got 3.

6 minus 3 is equal to 0.

6, and we've got 3.

6 minus 0.

6 is equal to 3.

And again, take a moment.

What is it you notice that's the same and different about those equations again? Ah, the parts have swapped positions again in our equation.

Laura is saying that she thinks that maybe these are all the equations that we can write.

Hmm, whereas Sam is saying something slightly different.

He disagrees and he thinks that you can swap each expression or each number on either side of the equals sign to the other side.

So let's have a look at what Sam is thinking about in a bit more detail.

Watch each side of the equals sign carefully.

Maybe just pick one equation, for example, the top one.

Watch the top equation where it's got 3.

6 on the left, and on the right hand side, it's got 3 plus 0.

6.

Watch what happens to that equation.

Did you see? The 3.

6 has now gone to the right hand side of the equal sign, and the 3 plus 0.

6 has gone to the left hand side of the equal sign, and it still represents the same thing.

Let's have a look at the second equation.

What will happen to this one? We've got the 3.

6 is equal to 0.

6 plus 3, and watch how they swap places again.

0.

6 plus 3 is equal to 3.

6, and again, it represents the same thing.

Let's have a look at our subtraction equation this time.

We've got 3.

6 minus 3 is equal to 0.

6, and we swap those over, so on the left hand side, now we've got the 0.

6, and on the right hand side, we've got the 3.

6 minus the 3.

And the last one, 3.

6 minus 0.

6 is equal to 3.

What do you think is gonna happen? Tell yourself quickly.

That's right, the 3 is gonna go to the left hand side, and the 3.

6 minus the 0.

6 has gone to the right hand side.

And as we've been saying, all of these represent exactly the same as the first four equations that we had.

So as Sam has pointed out, he feels that we can write eight equations for this.

We can write four additional equations and four subtraction equations.

And in doing so, it might be helpful to think of them as equation pairs, where each of the expressions or the numbers on either side of the equation, when you rotate them to the other side of the equal sign, you then create a pair.

Let's have a look at the first one, for example.

3.

6 is equal to 3 plus 0.

6, but we could also write that as 3 plus 0.

6 is equal to 3.

6, 'cause we've rotated the expressions or the number to either side of the equals sign.

And we can do the same for each other one of those pairs.

So, so far we've managed to be able to write eight equations, four addition equations and four subtraction equations.

But do you know what? I think we can take this even further.

So far, we've only really been using decimal numbers, and I think actually we could replace our decimal numbers with fractions instead.

And we know that 0.

6 is the same as 6/10 written as a fraction.

And you can see that how that's changed in the parts.

And we could change each one of our equations to represent this too.

So let's have a look.

Look carefully at the 0.

6s in each one of those equations.

Now they've disappeared and they've been replaced by 6/10.

And again, these represent exactly the same thing as the equations that we have done before.

So in future, any equation which has a decimal in it, you can replace that decimal with a fraction, if you wanted to record the equation in a slightly different way.

Right, let's check our understanding now.

We've got a part-part-whole model and we've got an equation, and it says this part-part-whole model can be represented using this equation.

The equation is 8.

4 is equal to 8 minus 0.

4.

Have a think for yourself for a moment.

That's right, it's false.

But why is it false though? Take another moment to think for yourself.

Sam's suggesting to have a think about the position of the wholes and the parts.

Does that help you with your thinking? Yes, we know that when we write a subtraction equation on one side of the equal sign, we have to have the whole minus a part, and on the other side of the equal sign, we must have the remaining part.

Hmm, let's look at our equation here.

Well, on the left hand side of this equation, we only have the whole, and on the right hand side, we've got the parts.

One part is subtracting from another part.

Here, what we should have is 8.

4 minus a part, for example, 8 is equal to 0.

4.

Or you could have had 8.

4 minus 0.

4 is equal to 8.

Have another little think about this one.

What's the missing part this time? 5.

2 is equal to 5 plus something.

Take a moment for yourself to have a think.

That's right, it was B.

It was 0.

2.

0.

2 was the missing part.

Why was it not A or C? Well, A represents 2 ones, doesn't it? And 5 ones plus 2 ones would equal to 7 ones, so the place value would not be correct.

And then finally, the bottom one C is 2.

0.

Ha, we know that also represents two wholes, doesn't it? 2.

0 is just another way of writing 2, so again, that would equal 7, so it has to be B.

Right, time for your second task now.

What I'd like you to do is fill in the blanks for each of these equations to make them correct.

Have a look carefully.

There might be a missing part or a missing whole each time.

The second part of your task is to look at this part-part-whole model, and what I would like you to do is write as many different equations as you possibly can to represent this part-part-whole model.

I'll be very interested to see how many you can come up with.

Good luck.

I'll see you shortly.

Okay, let's see how you got on.

The first one says 1.

8 is equal to 1 plus something.

Yep, that's right, it's 0.

8.

The second one says 1.

8 is equal to something plus 0.

8.

Yep, and this time that would equal 1.

Let's have a look at the second one on the top right now.

2.

4 is equal to 2 plus something, and we know that something would be the fractional part, the 0.

4.

And underneath that, 2.

4 is equal to something plus 2, and again, this time that part missing would also be the 0.

4.

Well, did you notice a change that time on those two equations? That's right, the addends had swapped places, hadn't they? But we know it still represents the same thing.

Let's have a look at the bottom left examples now.

3.

6 minus something is equal to 3.

Hmm.

We've also got underneath that 3.

6 minus 6/10 is equal to 3.

Do they help each other at all? Well, for the bottom left one, if we know that the remaining part is 3, therefore the part that must have been subtracted must have been the fractional part, and we know that would be 0.

6.

And underneath that, well, we've taken away the 6/10, 0.

6.

We've taken away the fractional part, which means that remaining part, which would go to the right hand side of the equals sign this time must be 3.

And the last example, 0.

8 is equal to 4.

8 minus something.

Hmm, what's changed about these equations to the previous equations? Ah, we've swapped around the positions, haven't we, of the expressions or the numbers.

We've placed 0.

8 on the left hand side, and this time we've got the whole minus the part on the right hand side.

So 0.

8 is a part, and that is equal to the whole, which is 4.

8 minus a part, and that missing part would be 4.

Underneath that we can see that we were missing the part on the left hand side this time, and on the right hand side, we'd kept the whole we minus a part, and the part that we'd subtracted was 0.

8.

So the part that was missing that needed to go on the left hand side of the equal sign would've been the 4 ones.

Now, for our second task, I asked you to write some equations to represent this part-part-whole model.

You may have written two addition equations and two subtraction equations.

Let's have a look at the ones that we could have written.

You could have written 4.

7 is equal to 4 plus 0.

7.

You could have written 4.

7 is equal to 0.

7 plus 4 using our commutative law.

You could have used 4.

7 minus 4 is equal to 0.

7.

Or again, you may have swapped the positions of the parts this time and written 4.

7 minus 0.

7 is equal to 4.

You may have thought about it in a slightly different way and written it using one of the equations from the equation pairs that you could have used.

If you managed to do that, well done.

You may have written all eight equations.

Fantastic.

And finally, Laura's here reminding us that we could have written each fractional part which was represented with decimal numbers as a fraction instead.

So here, we could have used 4.

7 is equal to 4 plus 7/10.

Well reminded, Laura.

So in total, I think we possibly could have written 16 equations, which can all be represented through our part-part-whole model on the left hand side.

Well done if you managed to get that many.

Okay, onto our last cycle now, and we're gonna be thinking about how we can represent some of this work that we've been doing so far using a number line.

So hopefully now you can see our number line, and we're gonna start using the numbers that we have been using previously of 3.

6 is equal to 3 plus 0.

6.

Firstly, we need to find 3.

6 on our number line.

Now, we know that 3.

6 sits between 3 and 4 on our number line.

And we're gonna bring back out our sentence stem that we used earlier.

I say 3.

6, but I think 3 ones and 6/10.

Look carefully about where we represent these numbers on the number line for this addition equation.

And we're gonna use our sentence stem this time to have a think about how we can record this.

I say 3.

6, which is represented on the left hand side of our equal sign, but I think 3 ones plus 6/10, and you can see that's ended us up at the 3.

6.

So we can represent this equation on our number line.

We can also use number rods to help us represent this on our number line.

I say 3.

6, which again is on the left hand side of our equal sign, but I think 3 ones and 6/10.

And we've written that as 3.

6 is equal to 3 plus 0.

6.

But you also know that we could write it as 3 plus 0.

6 is equal to 3.

6.

So if we start thinking like this, we can start to think about how we could represent missing addend problems on our number line as well.

Let's have a look at this example here.

We've got 2 plus something is equal to 2.

8.

Have a think for yourself of what you think that might be first of all.

Okay, let's use our sentence stem to help us.

I say 2.

8, but I think 2 ones and 8/10, don't I? Ah, so there we go.

Let's have a look at our equation again.

We've got the two ones as one of our parts, but the other part we're missing is the 8/10.

Did you get that? Well done if you did.

We've added on 8/10, and therefore that was the missing addend in our equation.

Hmm, now I've put it into a bit more of a real life context for you here.

Let's have a look at this problem.

It says, "How much shorter is Sam then Laura?" Sam is 1 metre tall and Laura is 1.

2 metres tall.

Look how I've represented it on the right hand side in our part-part-whole model.

Is it a missing whole or is it a missing part that we have? Have a think for yourself.

What do you think that missing part is? Let's have a look and let's use our sentence stem to help us.

I say 1.

2, but I think one 1 and 2/10.

Let's have a look how we can represent that with our number rods.

Well, both Laura and Sam have a number rod of one metre to represent their height.

Sam is exactly 1 metre tall, whereas Laura is 1 metre and a little bit more taller, isn't she? And that little bit extra is the 0.

2 that we realised from our stem sentence.

So hopefully you can see that Sam's height of 1 metre plus the additional 2/10 of a metre would be equivalent to Laura's height.

So how much shorter is Sam than Laura? Well, Sam is 0.

2 metres shorter, or 2/10 of a metre shorter than Laura.

And there we go, as he's pointed out, "I am 0.

2 metres shorter than Laura." Okay, time for you to check your understanding now.

Look at these number lines here.

Which one of these number lines represents 1 plus 0.

6 is equal to 1.

6? Have a think for yourself.

Well done.

It was actually both A and C.

Well, let's have a look at those, shall we? For A, it starts on one and it adds on 0.

6 and gets us to 1.

6, so that's definitely correct.

And if we look at B, it actually starts on 0.

6 and adds on one, which then gets us to 1.

6, which is a slightly different way of looking about it, but you can see that the addends have just been swapped over.

But why doesn't B work then? Well, let's have a look at it.

It starts at 1 and it adds on 0.

6, it says, and at the end of the number line it says 1.

6.

So that seems right to me.

Oh, actually no, each one of our intervals has been representing 1/10, hasn't it? And you can see here that we've got 10 intervals.

So actually the end of the number line would represent 2 and that doesn't represent 1.

6.

Okay, onto task C now.

The first task for you is to complete the part-part-whole models and write an equation, and also represent this on the number line for each example.

So you're gonna start on the top left part-part-whole model, and then you're gonna work across the page to write the equation, and draw a number line to help represent that equation.

Sam is then asking at the bottom, could you create your own example if you managed to finish all three of those? And your second task today is, as you can see here, we have a number line which has got also represented with the jumps and the number of rods as well.

What I'm gonna ask you to do is can you write as many equations as you possibly can to represent what you can see on this number line? If you manage to do that, then think about writing it as a worded problem as well.

Good luck and I'll see you again shortly, see how you've got on.

Okay, let's see how we got on.

So the first number is 7.

4, and we know that can be partitioned into 7 and 4/10 or 0.

4.

In our equation, we could have written this as 7 plus 0.

4 is equal to 7.

4, or you may have swapped around the addend, so it was 0.

4 plus 7.

And on our number line it starts at 7 and therefore we would need to add 0.

4 to reach 7.

4.

Let's look at the middle example.

Well, this time we don't have a whole, but we know both of the parts.

So the parts are 5 and 0.

7, which we know makes 5.

7.

In our equation, we would write that as 5 plus 0.

7 is equal to 5.

7.

And on our number line, we then need to think about, well, where would our equation start? Well, I think one of our parts was 5, so we'll start with the 5, and we know that the end of the number line would be 6.

And because we have 10 equal parts, so that means we can now add on 7 of those 10 parts, which would be 7/10, which would end us up at 5.

7.

And finally, the last example.

Well, we have the whole and we have one of the parts, we're just having a missing part this time.

The whole is 6.

2, one of the parts is 6 ones, therefore we're missing the fractional part, which is 0.

2.

So in our equation, we can write that as 6 plus 0.

2 is equal to 6.

2.

And on our number line, well, we would start at the 6 and end on the 7.

We've got 10 intervals.

By doing that, that gives us 10 equal parts, each one of those parts representing 1/10, and we need two of those parts, so we just jump 2/10 along our number line to 6.

2.

Well done for managing that.

And on the second part, Laura has written some equations that she managed off the back of this number line.

You may have written some additional equations or some subtraction equations.

Let's have a look what Laura wrote.

Laura has written 4.

3 is equal to 4 plus 0.

3.

She's also written 4.

3 is equal to 0.

3 plus 4.

She's then gone on to write that 4.

3 minus 0.

3 is equal to 4.

And she's also written 4.

3 minus 4 is equal to 0.

3.

Well done Laura for writing four equations.

If you think back to previously in our lesson, you could have had those equation pairs too, which therefore means you could have written at least eight equations, or if you've taken it one step further and written the decimal numbers as fractions instead, you could have written up to 16 equations possibly.

Well done if you managed that many again.

Finally, Laura also wrote it as a worded problem.

She said that, "I went for a 4 kilometre run, I then walked the remaining 0.

3 kilometres to get home.

How far did I travel altogether?" And that's really well represented with our number line because it starts at zero and she jumps 4 kilometres while she was running, and then she walked the remaining 3/10 of a kilometre or 0.

3 kilometres, which ends us up at 4.

3 kilometres.

Well done, Laura.

A great context.

I wonder what you came up with.

So, that's the end of our learning for today.

Well done for keeping up, and hopefully you enjoyed yourself and found something new to think about with regards to decimal numbers and representing them additively.

Let's just remind ourselves of what we've learnt about today.

We've learnt that firstly, decimal numbers can be partitioned into ones and tenths.

We've learned that addition and subtraction equations can be used to represent partitioning a decimal number.

We've also learned that expressions can be swapped to either side of the equals sign to make it represent exactly the same thing.

So these equations can represent the same thing but just be written differently.

And finally, we've learned that addition and subtraction equations using partition decimal numbers can also be represented on a number line.

Thanks for joining me today.

Take care and I'll see you again soon.