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Hello, my name's Mrs. Hopper and I'm going to be working with you today in this lesson from our unit on using knowledge of part-part-whole structures to solve additive problems. I hope you're ready.

Pencil poise to draw some bar models and to have a think about some problems. So let's get started.

So in this lesson, we're going to be using our knowledge of part-part-whole structures to solve additive problems in a range of contexts.

So you've probably heard about parts and parts and wholes before.

You might have used part-part-whole models, bar models maybe.

So we're going to be looking at using those to solve additive problems. So problems involving mainly addition and subtraction.

And in a range of context, so lots of different ways that we can use our problem solving to solve problems involving different sorts of numbers.

So if you're ready, let's make a start.

We've got just one key word today and that's represent.

So my turn, represent your turn.

Great, and I'm sure you've heard the word represent or representations before.

Let's just remind ourselves what it means.

So to represent something means to show something in a different way.

So we're going to be looking at how we can represent or show the problems that we're solving in different ways to help us make sense of them and to help us to solve them.

So there are two parts to our lesson today.

In the first part of the lesson, we're going to be looking at number puzzles and thinking about how we can use representations to help us to solve them.

And then in the second part of the lesson, we're going to be looking at wider contexts.

So beyond just numbers, thinking about times when we might need to use our number knowledge and our understanding of part-part-whole to solve other sorts of problems. So if you're ready, we'll start.

And we've got Izzy and Sofia helping us in our lesson today.

So in this number puzzle here, each row and column sum to five and we are asking how can you find the missing numbers.

So let's just have a little look at this problem.

Can you identify the rows? So those are the ones that go across, and the columns, the ones that go up and down.

Because we are told that each row and column sums to five.

So just have a little look at the puzzle.

Can you see the rows and columns? Hmm, well there's lots of rows and lots of columns.

I wonder where we can start? Let's see.

I wonder if Izzy's gonna give us a hand.

Ah, Izzy says each row and column is made up of three parts.

So you can see each row and column has three circles.

Some of them have numbers, some of them don't.

We've got to fill in those missing ones.

Izzy says, if I know two of the parts, I can work out the third part.

So Izzy saying if we know two of the parts and we know the whole, the sum of each row or column is five, we can then work out the missing part.

So Izzy's saying, let's start here.

So she's identified this sort of top row I suppose to have a look at.

So we know the numbers two and a half and one and a quarter, but we've got a missing one to fill in as well.

And Izzy says we know the whole, which is five because we know that the rows and columns each sum to five and we know two of the parts, two and a half and one and a quarter.

She says, let's use a bar model to represent this.

So here is our bar model and we can see the bar at the top represents the whole, which is five.

And then the bars underneath represent the three parts.

And we've put in the two parts that we know, two and a half and one and a quarter.

Izzy says I can now add the known parts and subtract them from the whole.

So we know when we're thinking about parts and wholes that if we know parts then we can subtract the part or the parts we know from the whole and work out the missing part.

So here we need to find a way of subtracting those two parts that we know from the whole to find out our missing part.

So we could add them together.

So two and a half add one and a quarter is equal to, well, let's have a think the whole numbers two plus one is equal to three and then I can visualise a half and another quarter and that gives me three quarters.

So two and a half plus one and a quarter is equal to three and three quarters.

So those are the parts we know.

Now we can subtract the parts we know, that three and three quarters subtract it from the whole, which is five.

There's different ways you might think about this.

You might think about subtracting the whole number and then subtracting the fraction.

Or you might think about counting on from the known part up to the whole.

But whichever way you do it, you'll find that that missing part is one and a quarter again.

So three and three quarters, if we added on one quarter, we get to four, add on another one, we'd get to five.

Or if we said five, subtract three, that's two.

And if we subtract three quarters from two wholes, we would have one and a quarter left.

So whichever way we think about it, our missing part is one and a quarter.

So we can put it into the bar model and most importantly we can add it to our puzzle.

Now I wonder where you'll go next.

I wonder where Izzy will go next.

I wonder if it's the same place.

Let's have a look.

So Izzy says, what else can we work out? Well, we've got some choices here, haven't we? We've got two columns that we could look at.

The final row still has two missing parts, but the two columns there both have one missing part.

So Izzy's gonna start with the column with a three in it.

So can you have a think? I think what we did last time to work out that missing part of one and a quarter, what do you think we're going to do this time? That's right.

Again, Izzy says, again, we know two parts, but one part is missing and we know that the whole is going to be five.

So we can draw another bar model.

What do you think the bar model's going to look like? You might want to have a go at sketching it yourself.

So a whole is still five.

This time our parts are three and one and a quarter.

Izzy says, so I think I'll subtract each part this time.

Well, she's got a whole number there to subtract.

So I think that makes sense, doesn't it? So five, subtract three is equal to two, and then two subtract one and a quarter leaves us with three quarters.

So this time our missing part is three quarters.

Well done Izzy.

So we can put it into our bar model and of course we can add it into the puzzle itself.

So now there's one remaining missing value and there's possibly more than one way you could have a go at working this out or maybe you could do it both ways just to check.

Anyway, you are going to have a go at this on your own.

So pause the video and find the remaining missing value.

How did you get on? I wonder which way you did.

Did you use the column or did you use the row to work it out? Let's see what Izzy did.

Ah, Izzy did the row.

She likes that whole number I think, doesn't she? I wonder if you did the same.

So there's our bar model again, five is our whole five is the sum of the three numbers.

And we know that the parts this time are two and three quarters.

Oh, two and three quarters.

Oh, that's a bit of adding up we've done already.

But I wonder, did you subtract the whole number first? Yeah, Izzy did that.

She subtracted each parts this time again.

So five subtract two is equal to three and then three subtract three quarters are only dipping into one of those wholes so we're left with two and a quarter.

So our missing part is two and a quarter.

Well done Izzy.

We can put that in the bar model and add it into our puzzle as well.

And just to check, we could add up that final column, couldn't we? One and a quarter plus two and a quarter plus one and a half.

Let's add the whole numbers one plus two plus one, that's four.

And then we've got a half and two quarters and that gives us a whole, so that gives us five, doesn't it? So that worked in that column as well.

Well done if you've got that right.

Time for you to have a go at some puzzles as well now.

So for this first puzzle, the number in each circle is the sum of the three numbers in the row below it.

And your job is to fill in the missing numbers.

So let's just look at this puzzle.

So you can see at the top we've got a thousand and a thousand is the sum of the three numbers in the row below it.

We know two of those numbers, one of them is missing.

And then on the bottom row, we've got five numbers and the three of them at the bottom left, they sum to make 425, the three middle ones, sum to make that missing number in the middle row and the three on the right hand side, sum to make 250.

And we know two of those numbers as well.

So your first task is to work out the missing numbers.

I wonder where you'll start.

And for the second part of your task, you've got a magic square to complete.

So in this magic square, each row sums to 13.

6, each column sums to 13.

6, and the long diagonal, the diagonal made from the four squares, the diagonal sum to 13.

6 as well.

And your challenge is to work out the missing values.

I wonder if you could use a bar model to help you to identify the whole and the parts you know, and to find out what to do to work out those missing parts.

But wherever you start, however you decide to represent the problems, pause the video now to have a go and we'll come back for some feedback and answers later.

How did you get on? So in this puzzle, remember the number in each circle is the sum of the three numbers in the row below it.

So I wonder where we started here.

Where did you start? Yes, I started at the top there with a thousand.

So I know that 425 plus 250 is equal to 675, and 675 plus 325 is equal to 1000.

675 plus 300 is 975 plus another 25 to get me to 1000.

Now where can I go? Well I can see that there's a gap there, one of the numbers that sums to 250.

Now I wonder what you did this time.

Did you add them together and subtract from the 250 or did you subtract each one? Well I can see there that 250 subtract 50 would give me 200, and 200 subtract 75 would leave me with 125.

So that's quite a good strategy.

Now let's have a look.

I still haven't got enough information for the 425, but I know two of the numbers that sum to 325 now.

And again I'm thinking I might do some subtraction from the whole here because I can see 325 subtract 125, which will lead me with 200, and 200 subtract 50 will lead me with 150.

So my final number to sum to 325 must be 150.

And now I can fill in that final gap.

Now this time I can see that 150 plus 50 is equal to 200 and that's easy then to subtract from 425.

So this time I am going to sum my two parts and subtract them from the whole.

So 425 subtract 200, leaves me with 225 and I've completed my puzzle.

Did you start in the same place as I did? Did you use similar strategies, I wonder? I hope whatever you did, it was successful.

Let's look at question two.

So question two is our magic square and this time we were looking for a decimal value 13.

6 as the sum of each row and column.

You may have used some column additions and subtractions to help you here or you may have just used your mental methods.

So let's have a think.

We've got two places we could start here.

I think I'm going to start on the right hand column and I can see that 0.

4 and 5.

6, well I've got a number bond to 10 there in my 10th, haven't I? So they must equal six.

And then another one will give me 7.

Ah, and I want 13.

So 7.

6 plus six is equal to 13.

So I know that that missing number is six.

So there it was easier to use my number bonds to add the information I had and then work out the difference up to 13.

6 or maybe subtract from 13.

Which square are we going to look at now? I think we've now got to look at that bottom square.

So I've got 3.

6 plus 1.

6 plus two, so 5.

6, 6.

6, 7.

So I'm looking at 6.

4 there to give me my sum for the row of 13.

Now I'm gonna have to look at the diagonal now, aren't I? So that diagonal from the top right to the bottom left.

So I've got 0.

4, a gap, four and 6.

Well six and four is equal to 10.

So I've got 10.

4 plus the other 0.

4, 10.

Ooh, if I add three, that will be 13.

8, that's 0.

2 too many isn't it? So 2.

8 I think is my missing number.

Yes, I'm right.

8 is missing there.

And now I can either work out the row with 0.

8, 2.

8 5.

6 or the column, let's see.

Let's go along that row and I'm sure using either written methods or mental methods, you worked out that the missing square there was 4.

4 and then we can work out the one above it is 3.

The top left is 5.

So we can either then do the diagonal first or the column first.

We worked out the column first and our missing number there was 1.

And then finally to complete the diagonal, the final row and column, we had a missing value of 2.

So I wonder which way you worked around the magic square, especially towards the end.

There were lots of different options weren't there, but I hope you were successful in finding those missing numbers.

And Izzy's got a challenge.

Can you create an example for a friend to solve? Could you make them another magic square with a different sum for your columns, rows, and possibly diagonals as well? Good luck with that.

Let's move on to the second part of our lesson.

So we're going to be thinking about solving problems again, but in different contexts, so not just in number puzzles this time.

So Sofia's garden is currently being redesigned, lucky Sofia, it looks like fun.

So she knows that the whole garden has an area of 652.

3 metres squared.

She's got a pond which has an area of 95 metres squared, and she's got a decking area which has an area of 215.

2 metres squared.

And around that you can see the green area, which I think is gonna be grass.

Ah, Sofia says, dad says we need at least 360 metres squared of turf or grass for the rest of the garden.

Is he correct? So can we help Sofia to work out whether dad is right in saying that they need 360 metres squared at least to put grass in the rest of their garden? I wonder what we can do to help.

Sofia says, we know the area of the garden, the whole garden, and we know the area of two parts of the garden.

So we know the whole area and we know the area of the pond and the area of the decking.

Is this reminding you of something? Let's represent this as a bar model.

It may be a garden with pond and decking and grass, but it's still a problem where we know the whole and we know two parts and we need to find out the other part.

So let's draw that bar model.

So here we are.

So the bar at the top represents our whole garden, which is 652.

3 metres squared.

And the two of the parts represent the pond and the decking at 95 metres squared and 215 metres squared.

How are we going to work out the final part, which is the turf or the grass we need to put down the lawn.

Sofia says it might be easier to sum the known areas and then subtract from the whole area.

So we're going to add together the areas we know about and then subtract it from the whole area.

And Sofia says that 95 metres squared plus 215.

2 metre squared is equal to 310.

2 metres squared.

She worked that out very quickly.

Let's have a look at those numbers.

I spotted that 95 is quite close to a hundred, isn't it? So if we added a hundred on to 215.

2, we'd have 315.

2, but 95 is five less than a hundred.

So we need to subtract five from our 315.

2, which gives us 310.

2 metres squared.

So well done Sofia.

So now we know the total of the two known areas and we need to subtract them from the whole.

So 652.

3 metre squared, subtract 310.

2 metres squared.

They look like big numbers, but if we look, it's actually quite straightforward because all of the numbers in the value that we're subtracting are smaller than the digits in our whole.

So we can do a fairly simple subtraction and work out that the area that's going to be grass is 342.

1 metre squared.

Now what did dad say? At least 360 metres squared.

Sofia says, well we only need 342.

1 metre squared of turf.

So 360 metres squared of turf is more than enough and they might have a bit left over.

I wonder what they'll do with that.

Now they've redesigned their garden, they need some fence panels to go around four sides of the garden.

They're not going to put it around the little bit at the edge of the decking.

So they've got four sides of the garden that need fence panels.

And each side needs a set number of panels.

So you can see there the number of panels that are needed for each side, all the three of the size.

We've got one unknown, haven't we? Hmm, wouldn't it be nice if we knew how many panels were needed in total? Let's see if that's what we're going to find out.

Oh no.

So we know that each panel costs nine pounds and they buy 279 pounds worth of panels.

Oh, we've gotta do some work here to find out how many panels are on that fourth side, haven't we? I wonder how we're going to work that out.

You might want to have a think about that before we move on.

Let's see what Sofia thinks we should do.

So we need to know how many panels are needed for the longest fence.

The one that goes right along the long side of the grass.

Sofia says, we know the total cost of the panels is 279 pounds and we have four sides.

We can't just divide it by four, can we? Because the sides aren't the same length.

So we've got four parts, four sides, and we know something about the whole.

So we know that the whole cost was 279 pounds.

We also know that each panel costs nine pounds, so we can work out the cost of the three sides that we know.

So if we multiply the number of panels by nine pounds, we can work out how much each one cost.

So let's start with that short side.

So we know there are seven fence panels and they each cost nine pounds.

So we've got seven lots of nine pounds, that's one of our parts.

Then along the top we've got eight lots of nine pounds.

Down the other side we've got six lots of nine pounds.

And the final side is we don't know how many lots of nine pounds, but we know that all of those nine pounds must sum to 279 pounds.

Sofia says, so now we can add up all the parts we know and subtract them from the whole.

So I think we're gonna have to use our nine times table here.

Let's see how good's your nine times table.

So seven times nine is 63 pounds, eight times nine pounds is 72 pounds, six times nine pounds is 54 pounds.

And then we've got our unknown number of nine pounds.

So Sofia says, now we can add up all the parts we know and subtract them from the whole.

So 63 plus 72 plus 54 is equal to 189 pounds.

And we know that the total was 279 pounds.

So if we subtract 189 from 279, we end up with 90 pounds.

So Sofia says that means the remaining fence panels cost 90 pounds, which is 10 lots of nine pounds.

So the longest fence needed 10 panels.

10 lots of nine is equal to the 90 pounds that we needed.

So there must be 10 panels along the long side of the garden.

Well done if you did that and good nine times table work if you managed to work it out.

Okay, so time to check your understanding.

So this pie chart shows the percentage of the budget spent on the garden over the four week period it took to improve it.

And it says what percentage of the budget was spent in week four? Gosh, there's a lot of words to unpick there.

Let's just have a think about it.

So a pie chart just shows us how much of the whole was spent in each of those weeks.

And we can see it's labelled week one, week two, week three, and week four.

And so however much of the pie we've got for each week tells us how much was spent in that week.

And we know this as a percentage.

So percentages are parts out of a hundred.

So we know that in week 1, 21.

8% of the budget, that's the money they had was spent.

We know how much was spent in week two and we know how much was spent in week three, but we don't know how much was spent in week four.

So there's lots of words in there, but we are thinking about the sum to a hundred of our percentages.

So pause the video, have a go and see if you can work out what percentage of the budget was spent in week four.

How did you get on? Did you draw a bar model perhaps, I wonder? So what we needed to do though was work out how much of our budget was spent in week four.

And we know the percentages, it's like a proportion that was spent in each week, which part? So if we add up the three parts we know and subtract them from a hundred, we'll find out how much was spent in week four or what part of the budget was spent in week four.

So 21.

8 plus 32.

7 plus 10.

9 is equal to 65.

4%.

So we can add those together.

And in weeks one, two and three, 65.

4% of the budget was spent.

So now we have to find out how much is left from a hundred percent.

So a hundred subtract 65.

4 is equal to 34.

6%.

So that's the percentage of the budget that was spent in week four.

There were a lots of big words in there and lots of ideas that are quite complicated.

But what we really had to do again was work out what our whole was, a hundred percent, add up the percentages we knew, and then subtract them from a hundred to find out what was missing.

And that told us the percentage of the budget that was left to spend in week four.

Now we've got some money.

So if the budget, so the money they spent in total was 15,000 pound, how much money was spent in week four? Again, we've got a pie chart there and we've got the totals for week one, week two, and week three.

So pause the video and work out how much of the 15,000 pounds was spent in week four.

How did you get on? I hope you spotted again that this is another problem about parts and wholes.

Our whole is 15,000, our three known parts were the amounts spent in weeks one, two, and three.

So if we added those together and subtracted them from 15,000, we'd work out how much money was spent in week four.

So the total money spent in weeks one, two, and three was 9,810 pounds.

And 15,000 pounds subtract 9,810 pounds is equal to 5,190 pounds.

So week four was equal to 5,190 pounds of the budget spent in total.

I wonder how you calculated that, did you use a written method to help you, I wonder? Anyway, well done if you got that right.

But remember, sometimes problems can look quite complicated, but if we know what the whole is and we know what the parts are, we can work out those missing parts.

Time for you to have a go.

So the school is getting ready for sports day and there are 342 pupils in the school and they're divided into four houses.

There's Ash, Elm, Fir, and Oak, and we know the number of children in Ash, Elm and Fir, but we don't don't know how many pupils are in Oak house.

So how many pupils are in Oak house? And for part two, the caretaker is marking out the lines for a game of capture where the pupils have to capture items and return them to their base.

And the pitch is composed of a square centre and four equilateral triangles, hmm.

And your question is, how big is angle A? Can you use the information there thinking about parts and wholes and work out how big angle A is? And then at sports day, the children take part in different events.

So they take part in the 60 metre dash, the hurdles, the 400 metres and the long jump.

So we can see how many children took part in each event all together.

And we can see how many children there are all together in Ash, Elm, Fir, and Oak houses.

So can you use what you've got there to fill in the missing number of pupils for each section? So how many children in Oak did the 60 metre dash? And carry on through the table like that.

Pause the video now, have a go, and we'll come back for some answers.

How did you get on then? So this was about the houses, 342 pupils in the school in four houses, we didn't know how many were in oak.

So our whole was 342, our known parts are 87, 93 and 75.

So we can add those together and subtract them from the whole, or we could subtract each one in turn.

We've added them together.

So the number of pupils we know about is 255.

And so we need to subtract 255 from 342, and 342 subtract 255 is 87, so there must be 87 children in Oak House.

Now this one was about shapes, wasn't it? And we had to work out the size of angle A knowing that our capture pitch is made of a square and four equilateral triangles.

So there's something about the angles that all meet where the square, the two equilateral triangles and the outside of the pitch meet, and those angles around that point will sum to 360 degrees.

So our whole is 360 degrees.

We know that the angles in a right angle triangle are each worth 60 degrees.

So this angle must be 60 degrees and this angle must be 60 degrees, and the angles in a square are each 90 degrees.

So now we know three of our four parts, and we know that our whole is 360 and can add up our known parts and subtract them from the whole.

So 60 plus 60 plus 90 is equal to 210, and 360 subtract 210 is equal to 150 degrees.

So angle A is equal to 150 degrees.

And then part three, you are having to work out the number of children taking part in each event and fill in those gaps.

So for oak in the 60 metre dash, there were 27 children.

The total number of children in the long jump, and we can work this out because we know there's 340 children in total and we know the totals for the other three events.

So there must have been 88 children doing the long jump.

There were 19 children doing the long jump for Oak.

That gave us the total number of children in Oak at 87.

There were 23 children doing the long jump for Ash.

We could work that out either from the long jump total or from the Ash house total.

There was 17 children from Elm doing the 400 metres.

Again, two different ways to work that out depending on whether your whole was the 62 children doing the 400 metres or the 93 children in Elm.

And then finally, 16 children from Fir did the hurdles.

And again, there were two different ways to work that out.

But working out what the whole is and what the sum of the parts that you know is allows you then to work out the missing part.

Wow, we've done a lot in this lesson.

Lots of different contexts, lots of thinking and different types of numbers we've looked at.

Larger numbers, we've looked at whole numbers, we've looked at fractions, and we've looked at decimals.

You've worked really hard and I hope you've done lots of mental and written calculations alongside to help you to work out those missing parts.

So at the end of this lesson, hopefully you can use your knowledge of part-part-whole structures to solve additive problems in a range of contexts.

And you know that drawing a bar model to represent a problem can help you identify the calculations required to solve the problem.

Thank you so much for your hard work and I hope I'll get to work with you again.

Bye-bye.

I've finished the video