Lesson video

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Hi, I'm Mr. Peters.

Thanks for choosing to learn with me today.

In this lesson, we're gonna be thinking about how we can calculate with decimal numbers, both within one whole and across one whole.

And we're gonna be thinking about using known number facts as well as some mental strategies to help us with this.

If you're ready, let's get started.

In this lesson today, some key words that we're gonna come across are number facts and bridging.

Use your number facts all the time and there are simple calculations using two numbers.

For example, two plus four is equal to six.

The term bridging is a mental strategy that we should be familiar with now, and we use it when we are adding or subtracting through a number boundary.

For example, going over 10 or going back through 10.

Look out for these key words throughout our lesson to help you with our thinking.

So (indistinct) lesson today, you will have been using your known facts and some mental strategies to help us calculate with decimal numbers when working within and across one whole.

So, this lesson's broke into three parts.

The first part, we'll think about adding and subtracting with decimal numbers within one.

The second part we'll be thinking about adding and subtracting with numbers to one.

And the final part, we'll be thinking about adding and subtracting numbers when bridging over one.

Should we get started with the first part? Throughout today's lesson, Lucas and Izzy will be on board to help us with our thinking as we go.

So let's get started thinking about some of our basic number facts to start off with.

Here, I've represented a basic number fact within our part whole model.

You can see here that one of my parts is two ones, and in the other part, there are five ones.

Lucas is saying that two ones plus five ones is equal to seven ones.

And he's written that as an equation on the left hand side.

Two plus five is equal to seven.

Izzy's now asking, well, how's this gonna help us when we start thinking about decimal numbers? We should have a look Izzy.

Let's have a look at our part, part, whole model this time.

What's changed about it this time? What did you notice? Izzy noticed that each one of the counters, which originally had a value of one, has changed through a value of one-tenth now.

So now, I wonder how we could use what we were doing before with our known facts to help us think about how we can combine these two parts together to make a whole.

Lucas is saying that I know that two plus five is equal to seven.

So Lucas is saying that two tenths plus five tenths is equal to seven tenths.

And you're right Izzy.

We know that two tenths can be represented as 0.

We know that five tenths can be represented as 0.

5, and then we know that seven tenths can be represented as 0.

So we can record it in an equation just like you have there.

2 plus 0.

5 is equal to 0.

So if we can do this for addition, can we do this for subtraction as well? Well, let's have a look.

Have a look at my tenths frame.

How many ones have I got here? That's right, I've got seven ones.

And Lucas thinks we can apply the same thinking here.

Lucas knows that seven minus two is equal to five.

So seven tenths minus two tenths is equal to five tenths.

And again, you're right Izzy, we can write that in an equation.

7 minus 0.

2 is equal to 0.

Okay, quick check for understanding then.

Have a look at the equation I've got here.

4 plus 0.

5 is equal to something.

Look at the expressions A, B, and C.

Which one of these expressions could help me to solve this equation? That's right, it's A.

A can help me to solve this because if I know that four plus five is equal to nine, then four tenths plus five tenths is equal to nine tenths and we can record nine tenths as 0.

Izzy is then asking, well why don't B and C help us with that? Have a little think.

Well, the problem here is that one of the numbers is a one and the other number is representing tenths.

And if we combine that, we would have ones and tenths together and both of those would leave us with 4.

And we know that that's not gonna be our answer here.

And we've worked on four tenths and five tenths and we know that's gonna equate to nine tenths.

Okay, task A for you.

Here, I've got some equations for you to solve, but you're going to need to think about a known number fact to help you with this.

So look at the equation, think about a known number fact that you know that will help you and then fill in the missing numbers.

For task two, I've taken away the scaffold of the sentence to help you.

So here you're going to need to look at the equation straight away and consider what the missing number is by using a known fact that you already know.

If you manage to do that, maybe start thinking about choosing your own known number fact and think about how many different decimal equations you can write, which that known fact would help you to solve.

Good luck and I'll see you again in a bit.

Okay, welcome back.

Let's have a little think.

The first one says three tenths plus two tenths is equal to something tenths.

Hmm! What the known fact I would know to help me here would be three plus two.

I know that three plus two is equal to five.

So if three tenths plus two tenths is equal to five tenths, which therefore would mean that 0.

3 plus 0.

2 would equal 0.

The second one says nine tenths minus three tenths is equal to something tenths.

Hmm, nine minus three feels like the fact I need to know.

Nine ones minus three ones is equal to six ones.

So nine tenths minus three tenths would be equal to six tenths.

And again, we can record six tenths as 0.

Finally, the bottom one, seven tenths minus six tenths.

I think I just need to know seven ones minus six ones for this, don't I? Seven ones minus six ones is equal to one one, therefore 0.

7 minus 0.

6 is equal to 0.

Onto task two then.

Something is equal to 0.

3 plus 0.

Well I can see I'm missing a whole here and I know the parts, so I'm gonna go with a number facts three plus two because I've got 0.

3 and 0.

So if I know that three plus two is equal to five, then something equals to 0.

3 plus 0.

2, well that must be 0.

In the second row, I've got 0.

6 minus 0.

Well, if I use my knowledge of six minus three, I can say I know six ones minus three ones is equal to three ones.

So six tenths minus three tenths is equal to three tenths.

The third example is 0.

8 and I need to minus a part, which I'm not sure of here.

Ah, okay! So I need 0.

8 minus something is equal to 0.

I'm just trying to think about a part whole model.

If 0.

8 was my whole and 0.

6 was a part, then I'm missing another part.

So what goes with six to make eight? Ah, my number fact I know would be two plus six is equal to eight or eight minus two is equal to six.

So if I use that, I know that eight ones minus two ones is equal to six ones.

So eight tenths minus two tenths would be equal to six tenths.

And finally, the bottom example, 0.

8 is equal to something plus 0.

Well, I've used this known fact already I think, haven't I? We were just thinking about this.

We knew that six and two were equal to eight.

So if I know that eight is equal to six plus two, then I know that 0.

8 or eight tenths is equal to six tenths plus two tenths.

So the missing number would be 0.

Well done if you managed to get all of those.

Lucas has written some examples for himself.

He's used the number fact three plus five is equal to eight and he is written two additional equations and two subtraction equations.

He said 0.

3 plus 0.

5 is equal to 0.

He said 0.

5 plus 0.

3 is equal to 0.

He then went on to save some subtraction equations, 0.

8 minus 0.

3 is equal to 0.

5 and 0.

8 minus 0.

5 is equal to 0.

Great work Lucas.

Okay, onto the second part of our lesson now.

Let's start thinking about some of these known number facts and how we can use them to calculate compliments to one.

Before we start that, I think it is a good idea to practise counting up to one whole in tenths.

So let's get started.

Here, I've got my tenths frame.

Each time a place value counter comes, count with me.

Ready? One tenths, two tenths, three tenths, four tenths, five tenths, six tenths, seven tenths, eight tenths, nine-tenths, and ten tenths or one whole.

So look at my tens frame now.

How many tenths are in my tenth frame? That's right, there are six.

And Izzy's now asking, well if I have six tenths, how many more do I need to make 10 tenths or one whole? Lucas says, well I know that six plus four is equal to 10.

So six tenths plus four tenths is going to be equal to 10 tenths or one whole.

There we go.

There's the additional four tenths that we've placed in and Izzy is now written that as an equation, 0.

6 plus 0.

4 is equal to one.

So we use the known fact of six plus four is equal to 10 to help us solve that six tenths plus four tenths is equal to 10 tenths or one whole.

So I'm gonna represent this now, using our part-part whole models and I wonder if you could say it along with me each time the sentences.

I know that six plus four is equal to 10.

So six tenths plus four tenths is equal to 10 tenths.

Therefore 0.

6 plus 0.

4 is equal to one.

So far, we've been adding up to one whereas we can also apply this understanding from subtracting from one.

Have a look at my tenths frame now.

It's full.

I've got 10 tenths.

Lucas is saying, if I've had 10 tenths and I subtract four tenths from this, how many tenths will I be left with? Have a think.

Do you know a known number fact that could help you with this? Izzy says that she knows that 10 minus four is equal to six, therefore 10 tenths minus four tenths is equal to six tenths, which would leave us as Izzy quite rightly says with six tenths.

Lucas has been able to write this as an equation.

One or 10 tenths minus 0.

4 or four tenths is equal to 0.

6 or six tenths.

Let's represent it again using our part part whole models.

And again, please say it along with me as you go.

I know that 10 minus four is equal to six.

So 10 tenths minus four tenths is equal to six tenths, therefore one minus 0.

4 is equal to 0.

Okay, let's check our understanding again.

Fill in the missing number three plus seven is equal to 10.

So 0.

3 plus something is equal to one.

Have a little think.

That's right.

It's C, 0.

3 plus 0.

7 is equal to one.

Okay, onto our second task for today then.

Here are some equations that I need you to have a go at filling in and thinking about a number fact that will help you with this.

But remember, your number fact must either be an addition fact that makes 10 or it must be a subtraction fact where the whole is 10 to start with.

For the second task, I also got some missing numbers I want you to fill out, but you've got two columns here.

I'd like to start down the left hand column first before you go down the right hand column after that.

When you do go through those columns, I want to ask yourself what is it that you notice each time and when you come back we're gonna explore what it is that you've noticed.

Good luck and I'll see you in a bit.

Okay, let's see how you got on.

The first one, 0.

3 plus something is equal to one.

I know that three plus seven is equal to 10, therefore 0.

3 plus 0.

7 is equal to one or 10 tenths.

So look at the next example.

One minus 0.

4 is equal to something.

One, I know that's 10 tenths.

So 10 tenths minus four tenths is equal to, that must be six, six tenths because I know that 10 minus four is equal to six.

So 10 tenths minus four tenths equal to six tenths or 0.

Hmm, something plus eight tenths is equal to one or 10 tenths.

Something plus eight is equal to 10.

Two, two plus eight is equal to 10.

Therefore two tenths plus eight tenths is equal to 10 tenths or one whole.

One whole minus something is equal to six tenths.

Hmm! 10 tenths or one hole minus something is equal to six tenths.

Hmm? What goes with six to make 10? Four so that must mean it's four tenths.

10 tenths minus four tenths is equal to six tenths.

That's right.

So one whole minus 0.

4 is equal to 0.

One is equal to something plus 0.

7 or 0.

7 is seven tenths and one is 10 tenths.

So what goes with seven tenths to make 10 tenths? Three tenths.

One is equal to 0.

3 plus 0.

And finally 0.

1 is equal to one minus something.

Well one is the whole, that's 10 tenths.

I need to minus something from that to make it equal to one tenth.

So if one is 10 tenths and I'm minusing something to make one, that must be nine, nine tenths.

One minus nine tenths is equal to one tenth or one minus 0.

9 is equal to 0.

Well done if you managed to get all of those.

Okay, let's get going here then.

Have a look at the first one on the left hand side column then.

Something is equal to 0.

1 plus 0.

Well that's one.

Have a look at the second row, what's changed this time? Ah, I see, so we've swapped the expression and the number to either side of equal sign, haven't we? So now we've got 0.

1 plus 0.

So that's the same.

So it must just be one.

Okay, look at the next one then.

What's changed this time? Well it's still equals to one and our 0.

1 has changed to 0.

2 this time.

So if that goes up by one tenth, that must mean that the other addend must come down by one tenth.

So if it goes up to 0.

2, that must mean 0.

9 must come down to 0.

So 0.

2 plus 0.

8 is equal to one.

What about the next row? Okay, well again, they equal to one and I've got 0.

3 this time, but hang on, one of my last addends was 0.

2 and now it's 0.

So one of the addends, even though it's not underneath each other, it is diagonal from it now.

One of the addends has increased by one tenth, isn't it? And if that happens, then that means that the other addend must decrease by one tenth again.

So we've now got 0.

3 instead of 0.

And the other addend from the previous example was 0.

8 so that must come down one tenth to 0.

And we know that because seven tenths plus three tenths is equal to 10 tenths as well.

Now have a think, what's happened here this time? Well actually, our sum isn't one anymore.

It's gone up to two.

So how are we gonna work this out? The 0.

3 has remained the same, so we need the seven tenths to go with those three tenths to make one whole.

But I've got an extra whole this time, haven't I? So maybe I need to add one whole and seven tenths this time with the three tenths.

That's right.

If I have one whole and seven tenths and add on the three tenths, that will give me two wholes altogether.

All about the next example then.

Oh again, good spot.

3 has stayed the same, but now we've got three wholes, haven't we? That's right.

The number of ones is gonna increase by one.

So we've gone from 1.

7 this time to 2.

7 plus 0.

3 is equal to three.

Look carefully again what you notice this time.

Well the whole stayed the same this time, but one of our addends has increased by one tenth.

So what's gonna happen to the other addend? That's right, it's gonna decrease by one tenth.

The previous addend was 2.

So this time it's gonna change to 2.

It needs to decrease by one tenth, doesn't it? And finally, what'd you notice about this one now then? Well the 0.

4 has stayed the same but our sum has increased and it's increased by 10.

We've added an extra 10 to the sum.

So that means we need to add an extra 10 to our addend, don't we? So the previous addend was 2.

This time we're going to need to add 10 to that, aren't I? So it'll be 12.

6 plus 0.

4 is equal to 13.

Well done if you managed to get those.

Let's have a look at the right hand column now.

So the first one, one minus 0.

4 is equal to, that must be 0.

6 'cause I know that 10 minus four is equal to six.

So 10 tenths minus four tenths is equal to six tenths.

Look at the column underneath.

What do you notice? Well, the whole has stayed the same, but one of the parts this time which was 0.

4 has changed to 0.

5 has gone up by one tenth.

That must mean the other part has to come down by one tenth for it to be equal again.

So 0.

6 would have to change to 0.

5 to make it equal to one.

So as we can see, one minus 0.

5 is equal to 0.

And look at the next example which you noticed this time.

Well actually this time the whole has stayed the same and the part which is to the right of the equal sign has stayed the same as well.

So, I think it's just exactly the same.

It must just be 0.

5 again.

Ah, and it is.

Have a look carefully this time.

What's changed? Well the whole stayed the same, but the part to the right hand side has decreased by one tenth.

So if that decreases by one tenth, that must mean that the other part has to increase by one tenth.

The previous part was 0.

So this time that part is going to increase to 0.

This time, the previous example we had one minus 0.

6 is equal to 0.

Now we've got one minus 0.

7 and we know that part has increased by one tenth.

So the other part must decrease by one tenth, which would make it 0.

Look carefully this time again.

What do you notice? This time, I've tried to catch (indistinct), haven't I? The whole has changed, isn't it? It's gone from one to two.

And remember what happened last time.

We needed one of our parts to have an extra whole, didn't we? So let's have a look what we've got.

We've got two minus 0.

Well the previous part that went with 0.

7 was 0.

3, but we know we need to give it an extra whole as well.

So that means it's gonna be 1.

2 minus 0.

7 is equal to 1.

Underneath that, oh look, it's changed from two to 12.

Ah, the whole's changed from two to 12, that means it's added on 10 to the whole, that means one of our parts is gonna have to have 10 added onto it.

At the moment we've got 12 minus 0.

So the previous part that we had was 1.

3, therefore we're gonna have to add 10 onto that so it become 11.

And then finally the last one at the bottom, let's have a look what do you notice.

What's the same and what's different? Well some of the numbers have moved around but the numbers were the same.

This time we've got 0.

7 as a part on the left hand side of the equal sign and then we've got the whole of 12.

So we're missing a part, aren't we? And if we've got all those parts, we knew them from the last equation 'cause the numbers are the same.

So we know that that part could also just be 11.

Well done if you managed to spot all those connections.

What have you realised from this? We've realised that we've addition, if you increase one of the addends by a certain amount, you need to decrease the other addend by the same amount in order to keep the sum the same.

And with subtraction, if you were to keep the whole number the same when increasing one of the parts by a certain amount, that means you would need to decrease the other part by the same amount.

Or if you were to decrease a part by the same amount, you would need to increase the other part by the same amount to keep the whole number the same.

Into the final part of our lesson, this time we're gonna be thinking about known facts and strategies when bridging one.

Have a look at this example here.

As you can see in our tenths frame, we have six tenths and we're going to try and add seven tenths to this.

Lucas is asking how many tenths will we have altogether? Izzy saying, well I know that six plus seven is equal to 13.

So six tenths plus seven tenths is equal to 13 tenths.

13 tenths is equal to one whole and three additional tenths.

And we can write this as 1.

Lucas has written this as an equation.

He said 0.

6 or six tenths plus 0.

7 or seven tenths is equal to 1.

3, which is the same as 13 tenths or one, one and three tenths.

In the last example, we used a known fact, whereas this time we're gonna think about using it as a bridging strategy when subtracting through 10.

Let's have a look at this example.

13 minus seven is equal to something.

How would we have done this previously using a bridging strategy? Well, I know that seven can be partitioned into three and four.

On my number line, you can now see 13 and I'm going to minus the three, first of all, from that 13 to get to 10.

And then I'm gonna minus the remaining four from the seven to get to our answer, which would be six.

So 13 minus seven is equal to six.

How does that change then if we change our numbers slightly? Do you notice on my number line I've changed it from zero to 10 to 20, from zero to one to two, and our equation this time has changed.

Instead of 13 minus seven, we're gonna do 1.

3 minus 0.

Both of our numbers have become 10 times smaller or one tenth of the size.

Here's our equation, 1.

3 minus 0.

7 is equal to something.

Well first of all, I'm gonna partition the 0.

7 into 0.

3 and 0.

I then place on my number line, the 1.

3 and I minus first of all, the 0.

3 to get to one whole and then I minus the remaining 0.

4 from the 0.

7 this time to get to our answer, which would be 0.

So 1.

3 minus 0.

7 is equal to 0.

Look carefully at my two number lines now.

What do you notice about them? What's the same about them and what's different about them? What the number lines have the same number of parts, but the numbers on the number lines have changed, haven't they? And the top number line, we have zero, 10, and 20 at each major interval, and on the bottom we've got zero, one and two at each major interval.

Those numbers have changed, haven't they? Yes, they've become 10 times smaller, which is what Lucas is pointing out to us.

And that's the same with the other numbers in our equation, 13 has become 10 times smaller to become 1.

The 0.

3 and the 0.

4, which makes 0.

7 that we've minus altogether is one tenth the size of the three and the four and the seven that we took from the top number line.

And finally our answer of six on the top number line has become one tenth the size or 10 times smaller.

It's become 0.

6 on the bottom number line.

Well done if you managed to spot that.

Okay, a quick check for understanding then.

On the left hand side, I've got some equations for you.

On the right hand side, I've got a known fact that would help you to calculate that.

Can you match up the equation to the correct known fact to help you? Take a moment to have a think.

Okay, so the first one, we've got 1.

4 minus 0.

8 is equal to something.

That's right, that'll match up with the bottom one.

14 minus eight is equal to six.

We'll come back to filling in the gap in a second.

The second one would match to the top one.

7 plus 0.

7 is equal to something and that fact we can use is seven plus seven is equal to 14 will help us won't it.

And the bottom one, 1.

4 is equal to 0.

9 plus something.

That's right, 14 is equal to nine plus five would be the equation that would help us match that.

So Izzy's asking, well what would those missing numbers be then? With the first one, matched to 14 minus eight is equal to six, then 14 tenths minus eight tenths would be six tenths.

So 1.

4 minus 0.

8 is equal to 0.

If the second one matched to seven plus seven is equal to 14, then seven tenths plus seven tenths would be equal to 14 tenths.

And we can write 14 tenths as 1.

And then the last example, 1.

4 is equal to 0.

9 plus something where if that matches to 14 is equal to nine plus five, then 14 tenths is equal to nine tenths plus five tenths.

And we can record five tenths as 0.

Well done if you managed to get those.

Okay, time for our final tasks today.

Here, I'd like you to fill in the missing numbers.

I've given you the sentence to help you to start off with and you can think about those known facts and how you can apply that to the sentence and then also to the equation as well.

For the second task, I've also removed that sentence again.

So you're gonna have to think of a known fact which will help you either to calculate the equation or find the missing number.

And then finally a nice little exploratory task for you as well.

Using number zero to nine, only once in each box, how many different ways could you complete our equation to make it correct? Something point something minus zero point something is equal to 1.

Now you may notice that there is a zero, a one, and a four already been used.

So you can use those digits once more as well.

Good luck and I hope the lesson so far will help you to work on these problems. Okay, welcome back.

Let's get going.

The first one then, I know that eight plus three is equal to 11.

So eight tenths plus three tenths is equal to 11 tenths.

So 0.

8 plus 0.

3 is equal to 1.

I know that five plus nine is equal to 14.

So five tenths plus nine tenths is equal to 14 tenths or 0.

5 plus 0.

9 is equal to 1.

4 16 tenths minus eight tenths.

Hmm, I know that 16 minus eight is equal to eight.

So 16 tenths minus eight tenths is equal to eight tenths.

And we can write that as 1.

6 minus 0.

8 is equal to 0.

And at the bottom, hmm, I know that 13 minus nine is equal to four, but I know that 13 tenths minus nine tenths is equal to four tenths or 1.

3 minus 0.

9 is equal to 0.

Okay, let's so think about the known facts we needed here.

Something is equal to nine tenths plus four tenths.

Well I know that nine plus four is equal to 13.

So nine tenths plus four tenths is equal to 13 tenths or 1.

13 tenths minus four tenths.

Hmm, well we could use the numbers from the previous example to help us, couldn't we? 13 tenths minus four tenths, well I know that nine and four make 13.

So 13 minus four equals nine.

13 tenths minus four tenths is equal to nine tenths.

And we can write that as 0.

And the two examples underneath that.

5 is equal to something plus 0.

Hmm, something goes with eight to make 15.

Oh, that's seven so seven plus eight is equal to 15.

So seven tenths plus eight tenths must equal 15 tenths.

So that must be 0.

7 plus 0.

8 is equal to 1.

5 and underneath that one then, 1.

5 minus something is equal to 0.

Again, we can use the same number fact that we used for the previous one above, I know that eight and seven is equal to 15.

So, therefore 15 minus eight would equal to seven.

So 15 tenths minus eight tenths is equal to seven tenths.

So 1.

5 minus 0.

8 is equal to 0.

And as we pointed out, Lucas has realised that we only needed one number fact to solve both sets of those equations.

And there is the part, part whole model, which should help us to tackle both of those equations as well.

And then finally onto our last task.

I wonder how many different ways you managed it.

Lucas has found one way of doing it.

He realised that he could put 1.

9 minus 0.

5 is equal to 1.

4 and that makes the equation correct.

How many different ways did you manage to find it? Was yours the same as Lucas's or different? If you get a chance share with someone nearby to you and see if they did it a different way.

Okay, and that's the end of our lesson today.

Hopefully you feel more confident using even known facts or some of the mental strategies that we can use for adding and subtracting either, within, to or bridging a whole.

So just to summarise our learning today, we can use known facts to help us adding and subtracting with tenths within one whole.

We can also use known facts to help us with adding and subtracting either two or from one whole.

And finally, we can use known facts and bridging strategies to help us adding and subtracting across one whole.

Thanks for learning with me today.

I've really enjoyed myself and hopefully you have too.

Take care and I'll see you again soon.

I've finished the video