Lesson video

Hello, my name is 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 poised 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 looking at representing an equation in a part-part-whole model correctly.

You've probably used part-part-whole models before, so this is an opportunity to practise that perhaps in some different contexts.

We've got one key word in our lesson today, and that's represent.

So my turn, represent, your turn.

Excellent.

I'm sure it's a word you've used before, but let's just check what it means.

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

So we're going to be taking information from a problem and representing it in a way that helps us to make sense of the problem and helps us to solve it.

So let's make a start.

So our lesson today has got two parts.

We're going to be thinking about how we can connect models together, so connecting different representations.

And then we're going to look at calculating strategies to help us to solve those problems. And we've got Sam and Jun helping us in our lesson today.

So let's make a start.

How can you describe the information provided in this equation? So just have a look at the equation first.

So what do you notice? We've got 987 is equal to 354 plus something plus 252.

Hmm, well Sam says she knows that the whole is 987.

So there is our whole.

And Jun says the whole consists of three parts, so there are the three parts.

He says, "Though we only know two of the parts, we have one unknown part." So we have two known parts, 354 and 252, and one unknown part.

Sam says, "We can represent this as a bar model." So the first representation is as an equation, but now we're using the bar model to help us to make a bit of sense of what that equation is telling us.

So we've got our whole there, which is 987, and our three parts, two of which we know about.

And then the one which we've shown again in the middle is unknown.

So the bar model shows the whole.

There it is.

And the three parts are equal to the whole.

That's really important.

It's what the equal sign means in the equation.

The three parts are equal to the whole.

They are worth the same.

What's the same about this equation? You see anything with the numbers? What do you notice? Well, Sam says, "This equation still has a whole and three parts." A whole and three parts.

And Jun says, "There is still also one unknown part." Sam's spotted something else.

"It also uses numbers with the same digits." So we had 987, now we've got 9.

87.

We had 354, now we've got 3.

54.

We had 252, now we've got 2.

52.

Hmm, what do you notice? Ah, Sam spotted it.

The numbers are all 1/100 times the size of the previous numbers.

We've divided them all by 100.

They are 1/100 times the size.

But Jun says the structure of the problem is still the same.

We still have a whole, three parts, and one unknown part.

And we can still represent this in a bar model.

And our bar model looks very similar, doesn't it? We've got some decimal points in there to show us that those numbers are 1/100 the size, but the bar model is very similar.

What's the same and what's different about the equation this time? Have a look.

Hmm.

Well, Sam says, "Once again we have a whole which is equivalent to three parts." A whole and three parts.

And the numbers used this time are also the same.

So what is different? Ah, the position of the parts has changed.

That's all that's changed, isn't it? We still have two known parts, but the bar model could look like this now.

So all we've done is shifted that unknown part from the middle to the left-hand side.

And in the equation it's shifted from the middle to just after the equal sign on the left of the parts.

So have a look what's the same and what's different.

"Both of these equations and their bar models represent the same thing.

In fact, we can represent it one more way as well," says Sam.

Can you think of what the other way would be? That's right, we could have the unknown part at the end, couldn't we? Exactly the same equation.

And because we know that we can add our add ends in any order, because addition is commutative, it doesn't matter where that missing number is.

We can still add the parts we know, and use that information to work out the part that we don't know.

So can you tick the bar model that represents this equation? Have a careful look, pause the video, and then we'll get back together for some feedback.

Which one did you reckon it was? That's right.

It was A, wasn't it? The whole is 658, the two known parts are 204 and 329, and then we've got a missing part.

We've shown the missing part in the middle again.

The other two bar models don't have the parts and wholes in the correct place.

Each of the others shows a part as the whole, and that's not going to represent the same equation.

What about this time? Tick the bar model that represents this equation.

So have a good look, pause the video, and we'll get together for some feedback.

What did you think this time? Ha-ha, that's right, there were two of them.

We talked about the fact that the order doesn't matter.

What matters is that our whole is 658, and our parts are 204 and 329, and an unknown part.

Doesn't matter where that unknown part appears.

So what was wrong with the one on the far side? Oh, no, 604 isn't a part in our equation, is it? So that had the wrong value as one of the parts.

Okay, time for you to have some practise.

You're going to draw a bar model for this equation.

So represent this equation using a bar model, and then it says and then draw another and another.

So can you draw three bar models that all represent this equation? Pause the video, have a go, and then we'll have a look together.

How did you get on? So our whole is always going to be 27.

4.

Our known parts are 20.

1 and 3.

44.

But what we can do is change the order of those parts, can't we? So here we've got the missing part in the middle, here we've got it on the right-hand side, and here we've got it on the left-hand side.

All of those bar models represent our equation.

Okay, so let's have a look then at how we can bring all this together, and use some calculating strategies to help us to solve our problems, 'cause if you noticed, in the first part of our lesson, we drew lots of representations and talked about the problems, but we didn't actually solve them.

Let's have a look at some calculating strategies we can bring in to help us solve these as well.

Okay, do these numbers look familiar? This is our problem from the first part, wasn't it? So how can you find the value of the missing part? So Sam says, "We could subtract the two known parts from the whole.

So let's have a think about that.

So she says, "987 subtract 354." She says, "I can do this mentally." Yet the numbers are quite friendly, aren't they? If we wrote it out as a column subtraction, there'd be no exchanging, no regrouping to do, so we could just use a mental method here.

So we can see that if we subtract all those digits, we get 633.

So 987 - 354 = 633.

633 - 252.

Woo, Sam says she might use a written method for this.

Can you see why? Now the ones are all right, but then in the tens we've got three tens and we're subtracting five tens.

So we will have to do a bit of regrouping here won't we? So let's see what that will look like.

So she set it out, remembering to line up the digits in the columns.

So we've got the ones, the tens, and the hundreds all lined up nicely.

Where do we start? We gonna start with the ones, aren't we? Because then if we do need to do any regrouping, we are moving in the right direction.

So three ones, subtract two ones is equal to one ones.

So then let's look at the tens.

We've got three tens, subtract five tens.

So we're going to need to do something there.

We haven't got enough tens in the number that we're subtracting from.

So we're going to exchange a hundred for 10 tens.

So we've now got five 100s left and we've now got 13 tens.

So 13 tens, subtract 5 tens is equal to 8 tens, and then 500 subtract 200 is equal to 300.

So our subtraction has given us an answer of 381.

So we'd already subtracted our 354 from 987, which gave us our 633, now we've subtracted the 252.

So our missing number is 381.

There it is.

Jun says, "We could add the two known parts together and then subtract them from the whole." So we could do that, couldn't we? Let's have a look at that then.

"354 plus 252," he says, "is equal to 606." So we've got 300 plus the 200, 550, plus 50 is another 100, so that's 600, 4 plus 2 is 6, 606.

And he says that's easy to subtract from 987 now.

You're right, it is, isn't it? And it is of course 381.

Sam says, "I preferred that method for this example." It all depends on the numbers that you're working with.

Sometimes one method will be the best way to go about it, and sometimes another will be.

It's always good to look carefully at the numbers involved before deciding on the strategy that you're going to use.

Jun says it's also the same as counting on from the value of the known parts when added together.

Ah, so there's yet another strategy we could use.

So we knew that our known parts added together were 606.

So now we can count on from 606 up to 987.

So 606 plus 100 takes us to 706, another 100 to 806, another 100 to 906.

And then we're going to add on 81.

So again, 381.

606 + 381 = 987.

"So we counted on 381 as we expected," said Jun, "'cause we knew the answer was 381, but another way of thinking about it." Sam says, "I quite like the idea of using a combination of these methods." Absolutely right, Sam.

Different methods will work at different times.

So let's have a look at another example.

"I could add the hundreds from the known parts first and then subtract these from the whole." Gosh, Sam's got another way of looking at it.

Let's have a look at Sam's way.

So she's just taken the hundreds from the known parts.

So 300 from 354, 200 from 252, and that equals 500.

She's taken that away from 987, which is 487.

Now Jun's gonna add the tens from the known parts, and the ones, and subtract them too.

So we've therefore got 54 plus 52, which is 106, 487 - 106 = 381.

Of course we know it's going to be 381.

Yet another different way to look at it.

You've got to be careful there to keep track of what you have subtracted and what you still need to subtract from the whole for that method.

Jun says, "I like that strategy too." It was a good one.

It was full of calculations that were easy to do mentally, wasn't it? So I think what Sam and Jun are trying to help us with is that we don't always have to reach for a pencil and paper method, but sometimes that might be the best way to think about it as well.

Time to check your understanding.

So we've got some decimals here.

So we've got a whole and three parts and one unknown part.

And your job is to calculate the missing value.

I wonder how you're going to do it.

I wonder if you could do it in more than one way, or maybe you could use a combination of those strategies that Sam and Jun were talking about.

So pause the video, calculate the missing value, and then we'll get together for some feedback.

I wonder how you worked it out.

Sam says, "I added the known parts together first mentally." So yes, there was some friendly numbers to add together.

So 13.

2 + 3.

1 = 16.

3.

Then she says, "I subtracted the value of those two parts from the whole." 18.

4 - 16.

3, again, quite friendly numbers to work with there, is equal to 2.

1.

No written strategies needed there, and I don't think too much thinking around other strategies.

That was definitely the way to go with this particular calculation.

I hope you found that out too.

So our missing part is 2.

1.

Time for you to have some practise now.

We've got some sets of calculations for you to have a look at, with some missing values.

So can you find the missing value in each equation? So you've got a set of additions there for question one.

Question two, you've got some additions that are equal to some multiplication.

And then in set three, some more additions.

And have a look at each set carefully.

Think about the calculations you're going to use, but also what do you notice about these sets, and what do you notice about your answers when you've completed them? So have a go.

Three sets to have a go at.

Pause the video, and we'll come back for some feedback together.

How did you get on? I wonder how you calculated the missing part in this first set of calculations.

Did you add the known parts and subtract them from the whole, or did you subtract one part at a time? However you did it, and whatever strategy you used, I hope you got that the missing value was 21.

9.

I wonder if you then spotted anything in the other two.

Did you spot that actually the values were all the same.

The numbers were identical in the other two parts.

We just moved that missing part.

So all the missing parts were 21.

9.

Did you spot that? Let's have a look at part two.

Now, this time we had to think about a multiplication that was equal to an addition with a missing part.

So in the first one, 3 multiplied by 200, so 600, is equal to 250 plus 150 plus something.

So 250 plus 150 is equal to 400.

So our missing part was 200.

Now I wonder if you spotted anything with the next one.

We had 4 multiplied by 200 this time, so an extra 200, but our parts were the same.

So our whole must have been 200 more, so it must have been 400.

Did you calculate the last one or did you spot that pattern? The missing part was 600.

So in the first set, the missing part was the same, because the parts and wholes were the same in each question.

And in the second set, the missing part was 200 more in each equation, because there was one more group of 200 in each multiplication, 3 times 200, 4 times 205, and 5 times 200.

But the two known parts stayed the same.

What's about set three? This time, all the equations were equal to 700.

So what did we notice about the parts this time? So in the first one, we had 124 plus 343 plus something equals 700.

So you might have added those two parts and subtracted them from 700, or you might have subtracted each one in turn from 700.

But whatever you did, you found that the missing part was 233.

Now in the second equation we still had 124, but this time we were adding 346.

So three more than we added last time.

Our whole is still the same.

So if one part has got bigger by three, another part must have been smaller by three.

The first known part was the same.

So our missing number had to be three fewer than in the first equation, 230.

Did you calculate that or did you work that out? I wonder if you can use some reasoning to work out what the missing number in the third equation is.

So, our whole is the same, 700.

Our first part is the same, 124.

This time, our second known part is 347, one more than the value in the previous equation.

So our missing part must be one less, 229.

And then in our final part, we've got 124 and we've got 347.

So two of the same parts as we had in the previous equation, but we've also got another 100.

So our missing part must be 100 fewer than what we calculated before, so it must be 129.

So by really thinking about the numbers involved, we avoided doing any calculation more difficult than adding and subtracting three, one or 100.

As one addend increases, or we add another addend, the missing value decreases by the same amount.

And we've come to the end of our lesson.

In this lesson, we've looked at the fact that missing part equations can be represented using a bar model.

Missing part problems can be solved using a range of strategies, including counting on, subtracting each part from the whole, summing the known parts, and subtracting from the whole, but also that it's really useful to look carefully at the numbers involved, 'cause you never know when there might be a really easy route to the answer.

Thank you very much for your hard work today, and I hope I get to work with you again.

Bye-Bye.