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Hello, my name's Mr. Peters.
Welcome to today's lesson.
In this lesson, we're gonna be thinking about applying our understanding of column addition and column subtraction and using these strategies when working with decimal numbers, including tenths and hundredths.
If you're ready to get started, let's get going.
By the end of this lesson today, you'll be able to use your knowledge of column addition and column subtraction to calculate with decimal numbers, which include tenths and hundredths.
In this lesson, we've got three key words we're gonna be referring to throughout.
I'll say them first, and then you can say them afterwards.
The first one is hundredth.
Your term.
The second one is tenth.
Your term.
And the third one is regrouping.
Your term.
Let's think about what these mean in a bit more detail.
A hundredth is the name we give to one part of 100 equal parts.
A tenth is the name we give to one part of 10 equal parts.
And finally, regrouping is a process of unitizing and exchanging between place values.
For example, we can regroup 10 ones for one 10, or we could regroup one 10 for 10 ones.
Our lesson today is broken down into three cycles.
The first cycle is to add decimal numbers using column addition.
The second cycle we'll be looking at subtracting decimal numbers using column subtraction.
And the final section, we'll be thinking about solving additive problems with decimal numbers.
Let's get started with the first cycle.
Throughout this lesson, you'll also meet Aisha and Alex.
They'll be here to help us along the way with their thinking and ask a few questions to prompt us along the way with our learning.
So we're gonna start this lesson today thinking about a context that I'm sure we're all quite familiar with, and that's the concept of music.
Over the years, music has been shared in lots of different ways, and these ways have changed over time.
For example, over the years, music has been recorded on either vinyl, a cassette, CDs, or, more common nowadays, is the idea of page streaming, where by just the click of a button on an electronic device, you can be listening to music within a couple of seconds.
So since 1973, the total revenue or the total amount of money made to date on these different formats has been recorded in the table below.
Let's have a look.
The formats are recorded on the left-hand side of the table, which says CD, vinyl, cassette, and page streaming.
And on the right-hand side, we can see the revenue, the amount of money made, and that has been recorded in dollar billions.
So each one of these numbers represents billions of dollars.
So the amount of revenue made by CDs is 3.
67 billions of dollars.
The amount of money made from vinyl was 1.
28 billions of dollars.
The amount of money made from cassettes is 1.
22 billions of dollars.
And the amount of money made through paid streaming is 0.
57 billions of dollars.
Jacob's asking what kinda questions could we ask based on this information? Aisha's got a few.
So here's Aisha's first question then.
She's wondering how much revenue has been made altogether from vinyl and cassettes.
Jacob thinks we can represent this as a bar model.
The blue bar represents the amount of money made from vinyl.
The yellow bar represents the amount of money made from cassettes.
And then, the white bar is the total amount, isn't it? And as a result of that, that means we're going to need to use addition, aren't we? In order to calculate the total amount.
Aisha's had a look at these numbers, and she's realised that she thinks she might need to do one regrouping here.
She thinks that she might be able to do this mentally.
However, we're also going to check it by using a column addition.
Let's start working through it mentally.
28 hundredths of a billion plus 22 hundredths of a billion is equal to 50 hundredths of a billion.
And then, 1 billion plus another 1 billion is equal to 2 billion.
So altogether, the total amount of revenue made from selling music as both vinyl and cassettes is $2.
5 billion.
Let's now check this using our column addition.
You can see, before each of our numbers in our column addition, we've placed a dollar sign, and you can see after them as we've placed the unit, which is billions.
Let's check this now, starting with the smallest place value as always, which is the furthest to the right-hand side, we're gonna start with the hundredths.
8 hundredths plus 2 hundredths is equal to 10 hundredths.
That means we need to regroup 10 hundredths for 1 tenth.
2 tenths plus 2 tenths is equal to 4 tenths, plus the additional 1 tenth, that would be equal to 5 tenths.
And then finally, 1 one plus 1 one is equal to 2 ones.
So altogether, $2.
5 billion, which is what Aisha worked out mentally.
Well done.
Another question Aisha's got is what is the total amount of money made from both CDs and vinyl? Again, we can represent this as a bar model.
So the blue bar represents the amount of money made from CDs.
The yellow bar represents the amount of revenue made from vinyl.
And the white bar, again, will represent the total amount.
Because we know both of the parts and we're looking for the whole means we're going to need to use addition again.
And once more, Aisha's noticed that she's going to need to do a regrouping, so this time she's going to use column addition.
Jacob's asking if we can use the correct mathematical language that we were using in the previous column addition to help us calculate this.
Let's do this together.
7 hundredths plus 8 hundredths is equal to 15 hundredths.
That means we're going to need to regroup, aren't we? We need to regroup 10 hundredths for 1 tenth, which we place under the tenths column, and we place the remaining 5 additional hundredths in the hundredths column.
6 tenths plus 2 tenths is equal to 8 tenths, plus the 1 additional tenth, that would be equal to 9 tenths.
And then, 3 ones plus 1 one is equal to 4 ones, isn't it? So the total amount of money made from both CD and vinyl is $4.
95 billion.
One more question here, "How much money has been made from both CDs and page streaming together?" Again, we can represent this as a bar model.
The blue bar represents the amount of revenue from CDs.
The yellow bar represents the amount of revenue from page streaming.
And again, the white bar represents the whole, which we're unsure of.
So we're going to need to add these parts together again.
So we'll be using addition, and looking at the numbers, it looks like we're going to need more than one regrouping this time.
So we'll definitely be thinking about using a column addition here.
Let's work this out again using similar language to what we did before.
7 hundredths plus 7 hundredths is equal to 14 hundredths.
That means, we're going to regroup 10 hundredths for 1 tenth, and place that underneath the tenths column.
And the remaining additional 4 hundredths will be placed in the hundredths column.
6 tenths plus 5 tenths is equal to 11 tenths, plus the additional 1 tenth is equal to 12 tenths.
So again, we're going to need to regroup, aren't we? We're gonna regroup 10 tenths for 1 whole and place that underneath the ones column and we're going to leave the remaining 2 additional tenths in the tenths column.
Then, we're gonna add the ones together.
3 ones plus 0 ones is equal to 3 ones.
But then we've also got 1 additional one.
So that would be 4 ones.
So altogether we've got $4.
24 billion is the total amount of revenue made from CDs and page streaming to date.
Okay, time for us to check our understanding.
Here, I've got a column addition on the left-hand side, which I'm gonna work through, and then you are gonna have a go at the one on the right-hand side.
So let's start with mine.
4 hundredths plus 8 hundredths is equal to 12 hundredths, which means we need to regroup, as we have done now.
9 tenths plus 0 tenths is equal to 9 tenths.
But I've got an additional 1 tenth here, so that means we're gonna have 10 tenths, which means we're going to need to regroup those 10 tenths for one whole.
We place that 1 whole underneath the ones column.
We didn't have any additional tenths remaining so we place a 0 in the tenths column.
And then finally, we add the ones together.
1 one plus 4 ones is equal to 5 ones, plus an additional one would be 6 ones.
So we've got 6.
02.
Now it's your turn.
See if you can have a go at the one on the right-hand side.
Okay, well done.
Let's work through that together then.
7 hundredths plus 6 hundredths is equal to 13 hundredths.
So we needed to regroup, didn't we? 1 tenth plus 8 tenths is 9 tenths, plus the additional 1 tenth would be 10 tenths, requiring another regrouping.
And then finally, 3 ones plus 2 ones equal to 5 ones, plus an additional one would be 6 ones.
6.
03.
Well done if you've got that.
Okay, your turn now.
Can you have a go at writing out and solving these column additions? And if you manage those, can you have a go at creating your own column addition where there's two regroupings required? Good luck, and I'll see you again shortly.
Okay, here we go.
I've laid them all out for us here, and let's go through the answers then.
The first one is 6.
76.
The second one is 7.
86, where we required a regrouping in the hundredths for 1 tenth.
The third one is 17.
07.
This required us to regroup in the hundredths and in the tenths.
The fourth one, this required a regrouping in the hundredths, the tenths, and the ones.
And finally, the fifth one had three add-ins, didn't it? The answer to this was 54.
10.
And this, again, required a regrouping in the hundredths, the tenths, and the ones.
What did you notice about the regrouping in the hundredths? That's right, we didn't only have 10 hundredths to regroup, we had 20 hundredths to regroup.
So we had to regroup those for two tenths, didn't we? Which is why we have a 2 underneath the tenths column for the regrouping.
Well done if you managed to tackle all of those.
Jacob came up with his own example of one which required two regroupings.
He used the numbers 14.
34 plus 9.
47.
And again, as you work through that, you can see his answer was 23.
81, and it required a regrouping in the hundredths and it also required a regrouping in the ones.
Good example, Jacob.
Well done, you.
Okay, onto the second phase of our lesson now, thinking about subtracting decimal numbers using column subtraction.
Okay, Aisha's got another question this time.
She's asking, "What is the difference between the amount of money made from CDs and cassettes?" Representing this as a bar model then, the white bar represents the amount of money made from CDs.
The blue bar represents the amount of money made from cassettes.
And then, here we can see the difference is represented by this arrow.
And we're trying to find out this difference, aren't we? We want to know what is the difference between the amount of money made from CDs and cassettes.
We know that if we want to find the difference then we need to subtract the blue part from the white part, don't we? To find out the size of the remaining part.
So we're going to use subtraction.
And that's gonna be $3.
67 billion minus $1.
22 billion is equal to something billion dollars.
Let's have a look at how that would be recorded as a column subtraction.
7 hundredths of a billion minus 2 hundredths of a billion would be 5 hundredths of a billion.
6 tenths of a billion minus 2 tenths of a billion would be equal to 4 tenths of a billion.
And 3 one billions minus 1 one billion would be equal to 2 one billions.
So the total difference in revenue made between CDs and cassettes was $2.
45 billion.
The next question is what is the difference in money made between CDs and vinyl? Again, on our bar model, the whole will be represented here by the white bar and this is representing the CDs.
One of the parts that we have is the vinyl, isn't it? We know that $1.
28 billion represents the vinyl in this blue bar here.
And therefore, the difference between the two would be represented by this arrow here.
Let's have a look at how we'd record this as a column subtraction again.
Starting with the hundredths.
7 hundredths or 7 hundredths of a billion.
I'm gonna say 7 hundredths for short.
7 hundredths minus 8 hundredths.
Hmm, do we have enough hundredths here in this column to subtract from? We don't, do we? We're going to need to regroup 1 tenth for 10 hundredths, aren't we? So we have 6 tenths, that leaves us with 5 tenths.
And now we take that 1 tenth and place that in the hundredths column.
We've now got 17 hundredths minus 8 hundredths.
17 hundredths minus 8 hundredths is equal to 9 hundredths.
Now we've got 5 tenths minus 2 tenths, which is equal to 3 tenths.
And then we've got 3 ones minus 1 one is equal to 2 ones.
So altogether, the difference between the amount of money made from CDs and vinyl was $2.
39 billion.
And then one more for us to look at, another difference question.
What was the difference in money made between cassettes and page streaming? So this time, a whole would be the cassettes because it's the larger number.
We then need to subtract the page streaming amount to find out the difference, the remaining part, don't we? Let's have a look at how we record that as a column subtraction once more.
Starting with the smallest place value, the hundredths, let's use that mathematical language again to calculate this.
2 hundredths minus 7 hundredths.
Hmm, we don't have enough hundredths again, so we're going to need to regroup.
We regroup from the 2 tenths, leaving 1 tenth in the tenths column and taking that 1 tenth and regrouping that for 10 hundredths.
We've now got 12 hundredths minus 7 hundredths is equal to 5 hundredths.
1 tenth minus 5 tenths.
Again, we don't have enough here, do we, to subtract from? So we're going to need to regroup 1 one for 10 tenths.
That leaves us with 0 ones in the ones column.
And now, we place that 1, which now represents 10 tenths, into the tenths column.
So we can see we've now got 11 tenths.
11 tenths minus 5 tenths is equal to 6 tenths.
And then finally, 0 ones minus 0 ones, well, that would be equal to 0 ones, wouldn't it? So the difference in revenue made between cassettes and page streaming to date is $0.
65 billion.
Okay, time to check our understanding again.
I'll have a go first on the left hand side, and then you can have a go with the right-hand side calculation.
6.
38 minus 1.
19.
8 hundredths minus 9 hundredths.
Well, we don't have enough hundredths at the moment to subtract from, do we? So we are going to need to do some regrouping.
We regroup 1 tenth for 10 hundredths and place that into the hundredths column, leaving us with 2 tenths.
18 hundredths minus 9 hundredths is equal to 9 hundredths.
2 tenths minus 1 tenth is equal to 1 tenth.
And finally, 6 ones minus 1 one is equal to 5 ones.
The answer is 5.
19.
Now it's your go.
You have a go and we'll go through it afterwards.
Okay, well done.
Let's go through this together then.
0 hundredths minus 7 hundredths.
Obviously, we don't have enough hundredths here, so we're going to need to regroup.
So now we've got 10 hundredths minus 7 hundredths.
5 tenths minus 4 tenths is equal to 1 tenth.
And this time, 5 ones minus 1 one is equal to 4 ones.
Great effort if you got that.
Okay, onto our task again here now then.
Can you write out and solve these as column subtractions? And again, if you've managed to do that, could you have a go at writing a column subtraction which requires one or more regroupings? Good luck, and I'll see you back here shortly.
Okay, the first one was 3.
22.
The second one required a regrouping from the tenths into the hundredths.
So it left us with 3.
18.
The third one also required a regrouping from the tenths into the hundredths, which left us with 13.
18.
The fourth one required a regrouping from the tenths to the hundredths again, so that left us with 12.
88.
And finally, the fifth one.
Well, you may have noticed that the top number, 15.
8, didn't have any digit in the hundredths column, did it? So I'm gonna place a zero in here to help us with this calculation.
It now reads 15.
80 minus 12.
95.
That required a regrouping not only in the tenths to the hundredths but also in the ones to the tenths.
And that left us with 2.
85.
Well done if you managed to get all of those.
Here's our issues example, where she wrote one for herself with a regrouping.
Here, you can see that she had to regroup 1 tenth for 10 hundredths, and left her with an answer of 14.
19.
Well done, Aisha.
And well done to you if you managed to come up with one yourself as well.
Right, onto the final part of our lesson today.
Solve additive problems with decimal numbers.
Let's have a look at our first problem here.
The garden fence at Jacob's house was 18.
5 metres long.
On Tuesday night, a heavy wind blew down some of it.
Now there is only 7.
32 metres left standing.
How much was blown down? Jacob says I can represent this as a bar model.
The whole fence originally was 18.
5 metres long, and then the remaining fence that was left was 7.
32 metres.
So we are trying to find this difference here, aren't we? It's the difference between the original fence and the remaining fence, what was left of it.
The white bar represents the whole fence that we had originally and the blue bar represents the fence that was left remaining.
Therefore, the arrow represents the difference.
And we know, to find the difference, we're going to need to use subtraction.
So let's have a look at this with column subtraction then.
We've got 18.
5 minus 7.
32.
Again, you may have noticed that 18.
5, it doesn't have any digit in the hundredths column, so we're gonna need to place a 0 here to help us with this calculation.
Now we can calculate this.
0 hundredths minus 2 hundredths.
Well, we don't have enough hundredths, do we? So we're going to need to regroup 1 tenth for 10 hundredths.
That leaves us with 4 tenths, and now we've got 10 hundredths.
10 hundredths minus 2 hundredths is equal to 8 hundredths.
4 tenths minus 3 tenths is equal to 1 tenth.
8 ones minus 7 ones equal to 1 one.
And then, 1 ten minus 0 tens is equal to 1 ten.
So the amount of fence that was blown down altogether was 11.
18 metres of fencing.
Looks like they've got a bit of work to do to fix that, doesn't it? Here's another problem.
Aisha is making cakes for the school fair.
She uses 1.
75 kilogrammes of flour.
She needs the same amount of sugar and butter as well.
How many kilogrammes of ingredients will she have altogether? We can represent this as a bar model again, as Jacob is saying.
Each blue bar represents each part, doesn't it? The first bar could be the 1.
75 kilogrammes of flour.
The second bar could be the 1.
75 kilogrammes of sugar.
And the third bar could be the 1.
75 kilogrammes of butter.
And we're going to need to add all of these together, aren't we, to find the total amount required? Jacob says he thinks he could do this mentally but we're gonna calculate this using column addition as well.
So let's work through this then.
5 hundredths plus 5 hundredths plus 5 hundredths is equal to 15 hundredths.
That means we're going to need to regroup, aren't we? Then we've got 7 tenths plus 7 tenths plus 7 tenths.
Again, that's gonna leave us with 21 tenths, plus an additional 1 tenth, that'll be 22 tenths.
That means we're going to need to regroup 20 tenths for 2 ones and leave the remaining 2 tenths in the tenths column.
So there we go, there's the remaining 2 tenths, and there's the regrouped 20 tenths which are now 2 ones.
1 plus 1 plus 1.
That would be 3 ones, plus the additional 2 ones, that would be 5 ones altogether.
So total amount of ingredients used would be 5.
25 kilogrammes of ingredients altogether.
Once you become really familiar with column addition and column subtraction, you can use missing number boxes to really stretch your thinking a little bit further.
Have a look here.
Let's look at the first one on the left-hand side.
4.
5-something minus 1 point-something 6 is equal to 3.
21.
We're gonna start off with working, as we normally would, with the smallest place value, and working from left to right.
So we start off with something hundredths minus 6 hundredths is equal to 1 hundredth.
Hmm, something minus 6 is equal to 1.
Aah, that's easy.
That must be 7.
7 hundredths minus 6 hundredths is equal to 1 hundredth.
Now we go to the tenths.
5 tenths minus something is equal to 5 tenths.
Hmm? Aah, that must be 3.
5 tenths minus 3 tenths is equal to 2 tenths.
So there we go, we've solved that first one there on the left-hand side, now let's look at the addition one.
It reads 5.
26 plus 4 point something 4 is equal to 9.
6 something.
So 6 hundredths plus 4 hundredths is equal to 10 hundredths.
Hmm, that means we're going to need to regroup, aren't we? 10 hundredths or 1 tenth.
So we place that 1 tenth underneath the tenths column.
And that means the missing digit would be a 0, wouldn't it? Because there's 0 additional hundredths remaining.
Then we've got 2 tenths plus something is equal to 6 tenths.
Hmm, 2 plus 4 would be equal to 6, wouldn't it? But, don't forget, we've got the additional 1 tenth underneath as well.
So it's not 2 plus something is equal to 6, it's 3 tenths plus something is equal to 6 tenths.
So 3 tenths plus, that's right, another 3 tenths would be equal 6 tenths.
And then, obviously, 5 ones plus 4 ones in equal to 9 ones.
So there we go, we found out the missing digits here, haven't we, by working from right to left and thinking about it as a normal column addition or subtraction.
Okay, time for you to check your understanding.
True or false? The missing number is 2, representing 2 tenths.
Take a moment to have a look.
And which justification helps us reason that? That's right, it's B, isn't it? If you start from the beginning, 4 hundredths plus 9 hundredths is equal to 13 hundredths.
And by the looks of it, they've forgotten to regroup those 10 hundredths for 1 tenth underneath the tenths column, doesn't it? So, yes, 4 plus 2 is equal to 6, but we actually don't start with 4, do we? 'Cause we've got 4 tenths plus additional 1 tenth underneath, so it would be 5 tenths.
So the missing number would be 1.
Well done if you got that.
Here's another one.
Select the missing numbers for this calculation.
It's A, B, or C.
That's right, it's C, isn't it? 5 hundredths minus something hundredths is equal to 4 hundredths.
And that would be 1, wouldn't it? 5 hundredths minus 1 hundredths would be equal to 4 hundredths.
And then, we've got something tenths minus 6 tenths is equal to 3 tenths.
Hmm, I know that 9 minus 6 is equal to 3, so 9 tenths minus 6 tenths would be equal to 3 tenths.
Well done if you managed to get that one too.
Okay, onto our last task for today then.
Could you find the missing numbers for both of these calculations here? And then, once you've tackled those, have a go here.
We've got a table here showing the heights of the sunflowers that some of the children have grown, and there's three questions I'd like you to have a go at answering.
And then, once you've done that, here's one more for us to have a look at.
It's a column subtraction which has been laid out with lots of missing numbers.
And what I'd like you to do is, using the digits 1 to 9 only once, is have a go at finding a way of making this calculation correct.
Good luck, and I'll see you back here shortly.
Okay, let's have a look at these first two then.
For the first one, 6 hundredths minus 8 hundredths means we can't manage that straight away, can we? So we need to regroup.
So we've done the regrouping.
Now we've got 16 hundredths minus 8 hundredths, which would leave us with 8 hundredths.
And then, I've got 4 tenths minus something would be equal to 0 tenths.
So 4 tenths minus 4 tenths would be equal to 0 tenths.
That's the first calculation completed.
And then, the second one.
Something plus 6 is equal to 3.
Hmm, that 3 can't just be 3 ones, can it? It must be 13 ones.
So that means it must be a number which is gonna help us to regroup.
So what adds with 6 to make 13? That's right, it's 7.
7 added to 6 makes 13.
So we've regrouped now underneath the tenths column.
Then I've got 2 tenths plus something tenths plus the additional 1 tenth.
So we've got 3 tenths and we need to add something to that to make 0 tenths.
Well, that's not gonna work, is it? We're going to need to regroup.
It must be 10 tenths.
So I know that 3 plus 7 is equal to 10.
So 3 tenths plus 7 tenths would be equal to 10 tenths, which then leaves us with that regrouping.
And we can check the ones column, can't we? 4 ones plus 1 one would be 5 ones, plus the additional regrouped one would be 6 ones, wouldn't it? So the missing numbers there were actually both 7.
Well done if you got those.
Let's go through the answers to question two then.
How much taller is Sophia's sunflower than Izzy's? It's 1.
28 metres taller.
Which two pupil's sunflowers combined height is equal to Lucas's sunflower? Well, that's both Aisha and Izzy's.
And Andeep's sunflower is 1.
56 metres taller than Alex's sunflower.
So how tall is Andeep's sunflower, which isn't in the table? That's right, it's 2.
01 metres tall.
Okay, and the last challenge here today then.
This was a tricky one, wasn't it? Had to do quite a lot of thinking about what we could do to make this work here.
Here's one example of how this could be done.
And what I used to help me was the fact that there was a 0 in the tenths column.
We had to have a number which minus another number which left us with zero, didn't we? Therefore, using the same number twice would've worked.
4 minus 4, for example, would work.
But we know that the rules of this problem here didn't allow us to use the same number twice, so that means it must have been a regrouping.
So we can see here we actually, in the tenths column, regrouped from 5 tenths down to 4 tenths, didn't we? Because we needed a regrouping in the hundredths column.
That, therefore, left us with 4 tenths in the tenths column minus 4 tenths, which left us with 0 tenths, which meant we could solve the problem correctly.
Well done if you managed to come up with that or find some other solutions that also worked.
I wonder how many different solutions you did manage to find as well as that.
Any similarities that you found amongst your answers? Was there anything that was consistent throughout the whole time? Okay, that's the end of our lesson again for today.
Thanks for joining me.
Hopefully, you've enjoyed yourself.
Let's just think about some of the key points that we've learned throughout this lesson today.
We know that we can use column addition and column subtraction to solve problems that include both tenths and hundredths.
When using column methods, you should always start again as always with the smallest place value.
And then finally, where one regroup or more is required, it's often best to think about using a column strategy.
Well done for taking part in that lesson today.
Hopefully, you found it useful and you're feeling a bit more confident with your learning around decimal numbers and adding and subtracting these.
Take care, and I'll see you again soon.
