Lesson video

Welcome back to our next lesson on number addition and subtraction.

I'm Mrs. Furlong and I'm going to be taking your lesson today.

In the last lesson, Ms. Heaton left you with this question, Stephanie wants to learn to jump higher, she records her standing reach height, then the height she reaches when she jumps to calculate how high she can jump.

This is a really long question that you've been given with a lot of information.

So I'm just going to think about that first part.

So Stephanie wants to learn to jump higher, she records her standing reach height.

And then it talks about the height she reaches when she jumped, calculate how high she can go.

That's the different here between her standing reach and her jumping reach.

That would tell us how high she jumped.

In fact that arrow would be the same height here as it would be here, as long as long as her legs were still dangling, but we can't calculate it from our legs cause we might lift them and bend them when we do it.

Right.

So we understand that information now.

Stephanie tests herself each week for five weeks, Stephanie doesn't get any taller during the five weeks, so her standing reach doesn't change.

Her results are shown in the table, but some of the values are missing, fill in the missing information.

So at the moment, with all this information we are ringing in my head, I really need to unpack this question and think carefully because my brain's been basically pushed into overload.

So we know that she tests herself for five weeks, that's fine.

And it says that she doesn't get any taller.

So let's just have a think about the parts of our equation.

In a subtraction equation, we've got minuend, subtract the subtrahend equals the difference.

So what do you think the difference is, first of all, in this calculation? That's right.

We talked about it moments ago, I didn't we? And it's different between her standing reach and at dumping reach.

And that's written as her score over here, so that's the difference.

What would the minuend be, and what would the subtrahend be? Let's have a think about that.

So the minuend subtract the subtrahend, gives us the difference, right? Okay.

Well, let's look at the picture to help us.

It's going to be subtracted from what, Stephen? Yes, you're right, the jumping reach would have to be our minuend and then we would subtract the standing reach.

Let's spot this part of the sentence.

Stephanie doesn't get any taller during the five weeks.

So that means her standing reach doesn't change because she hasn't grown.

So we're going to fill that in.

And that is which part of our equation? Is it the minuend or is it the subtrahend? The number we're subtracting, or is it the number we're subtracting from? You're right.

It's the number we're subtracting, it's the subtrahend, so the standing reach is the subtrahend.

So therefore the minuend, the number that we're going to start with, the number we're going to subtract from to find that difference, is the jumping reach.

Right.

So now that we can understand that, we're ready to start to solve the equations.

Because I did a lot of talking on that last slide, I've just written out at the top here, last part, so you can understand.

So we've got the jumping reach, subtract the standing reach will give us our difference in reach, which is our score.

So we've got our minuend, subtrahend, and our difference.

I wonder where you started with the table, one , two, and filled in the 143s.

Which week did you look at? Well, I decided to begin with week one, because then I won't get confused, I won't miss any steps out.

And I notice that my differences are quite small, so my calculations will be quite straightforward.

So I decided I would start with week one, what does the number one mean in week one? We need to do anything with a value.

No, it's not, is it? You're right? It's just the name of the week, which just number one is the name, like the name of your house, it's not actually a value, right? Okay.

So let's have a think about week one then, we've got 143 as our subtrahend, we've got our difference as nine and we don't know our minuend, we don't know our jumping reach.

So what are we going to do to find that out? Yes, that's right, we need to use the inverse, don't we? And the inverse of subtraction is addition.

143 + 9 = 152.

So therefore our minuend must also be 152.

That means that her jumping reach in week one is 152, did we get that part right? I hope you did.

Okay.

So let's have a look at week two now.

So we already know week one, and in week two, we already know our subtrahend because remember, our subtrahend doesn't change.

She hasn't changed height over those weeks, she hasn't grown, so her standing reach is still 143.

So my hope, subtrahend is still 143.

And we know the difference, we know that she could jump eight centimetres in week two.

We can see that in the table, okay? We know that she can jump eight, so we know that difference.

We could use the inverse again, or we could use the information we've already got.

I wonder what you did.

I decided to use the information from week one, so we know 152 - 243 = 9.

And if you remember from last time, if the minuend is changed by an amount, and the subtrahend is kept the same, the difference changes by the same amount.

Well, this time we don't know the minuend, but we do know the difference.

So in other words, we could say, if the difference is changed by an amount and the subtrahend is kept the same, then the minuend is changed by the same amount.

So what's happened to our difference, from here to here, and from the nine week one, to the eight in week two it's reduced by one, hasn't it? Subtrahends are the same, so our minuend must also decrease by one, to give us 151.

Oh, and look, if you pay attention to week three as well, it's the same.

So we say 151.

Let's take a look at week four now.

Right, so we already know our subtrahend.

What is it? Brilliant.

The standing reach is 143.

The table in week four, score is what's the difference here.

And it's a brilliant last day, that she jumped from a standing reach, how much higher she could reach for her jumping reach.

So she was able to reach 10 centimetre, she's improved, hasn't she? So I'm going to use a previous week's example to help me with this, and I chose to go back to week one again.

Oh, and look, what's happened this time.

We can still say, our subtrahends are the same.

That generalised statement, if the minuend is changed by an amount, and the subtrahend is kept the same, the difference changes by the same amount.

Well, let's go to the difference, cause we know that one.

That's increased by one, subtrahends are the same, so our minuend must also increase by one from 152 to 153.

If you might've just number five, so actually I know those, because that number 10 is such an easy factor, isn't it.

And that's fine too, there's lots of ways to solve this.

We're not saying this is the way you have to do it, it's just a connection that you might've made from the previous lesson.

Let's take a look at week five now.

So this time we're given different information, we're given the standing reach and we're given the jumping reach.

Remember our minuend is that jumping reach, how high she could reach when she jumped.

Our subtrahend, how high she could reach when she was just standing with her arms high in the air.

When she jumped, this is how high she managed to get to.

So to find that score, we need to find that difference.

So what it is from here to here with that jump.

So we need to do 154 - 143.

You could calculate that as it is, or you might have decided to make a connection to a previous calculation, it's up to you.

So I'm going to make that connection using our generalised statement.

If the minuend is changed by an amount, let's just check what it's changed by.

Oh, look, it's increased by one, I decided to use week four's results.

Our subtrahend is kept the same, so difference must also increase by one so that then becomes 11.

I hope you did well with that, but lots of strategies you could have used, and this is just another string to your bow, another bit of your maths toolkit.

Okay.

Let's get started on today's session.

We're going to change our focus a little bit now, and we're going to look at a different aspect to our subtraction.

I wonder what happens when the Subtrahend changes, but the minuend stays the same? We're going to think about this in a reduction context.

Some of you might know this as takeaway, but I'm going to call it reduction because sometimes things can be removed, but that doesn't mean that they've gone away.

In the next clip where you see some examples with some juice, it will become more clear what I mean.

Okay.

So you can see here that I've got two glasses of juice and they've both got the same capacity in them, so they've got the same amount of juice in them.

From this first cup on the left-hand side, I'm going to pour some juice out into the smaller cup.

I'm going to pour a small amount out.

Okay.

So I've poured out a small amount, you can see that here.

All right.

And I'm left with quite a lot in the cup on side.

This time with my other cup of juice, I'm going to call some more out.

So you can see this time I've pulled out a lot more juice.

So in this one, the less juice I poured out, the more I was left with in my cup and on the right hand example, the more juice I poured out, the less I was left with in my cup.

Okay.

So I'd like to make a connection now between the juice example and the generalised statements that we're going to be using throughout today's lesson.

So let's take a look first of all, at the cups on the left-hand side and using this example, we can say that the less we subtract, so here's the amount we subtracted, the more we were left with.

So the less we subtract, the more we are left with.

And if we go to the right-hand example over here, we could say the opposite thing now, the more we subtract, this is what we subtracted from this cup, it came out of here and into here, didn't it? So the more we subtract, the less we are left with, and those sentences are going to feature a lot in today's session.

So I'd like you to read these generalised statements with me.

Let's have a go at the first one.

The more we subtract, the less we are left with.

Can you say it one more time? Brilliant.

Let's have a go with the second statement, The less we subtract, the more we are left with.

Can you repeat that? Fantastic.

Okay.

So these are going to form a key part of our lesson today.

I'd like you in a moment to pause the video, and I'd like you to see if you can come up with some other situations where these sentences might be true.

See if you can come up with three or four situations.

So pause the video here and have a go.

I wonder what you came up with.

I thought about strips of paper, I'll show you that now.

So I've got two strips of paper here, at equal length.

They are a bit bendy, so it's a bit hard to tell, but I promise you they are identical in length.

What I'm going to do is, I'm going to think about our generalised statements.

So the more we subtract, less we are left with.

So I'm going to start with my blue one, the more we subtract, so the less we all left with this.

This is the bit I'm subtracting, and this is the bit I am left with, okay? So now I'm going to think about that again with the green one, but this time, the less we subtract, the more we are left with.

So am I going to need to cut off a smaller or a larger piece if I am doing the less we subtract? That's right.

I'm going to have to cut off a smaller piece.

So I'm just going to subtract a small piece like this.

Okay.

The less we subtract, this is the bit I subtracted, the more I am left with.

Or, if we lay them out side by side like this, you can see, we start back with our original lengths.

Okay, and the more I subtract, the less I'm left with, the less I subtract, the more I am left with.

And you can see, because I took a smaller piece from the green, I've been left with more green.

So the less I subtract, the more I am less with.

Okay.

One thing that I haven't really focused on yet is, what is the same about both of my examples.

So I've got some photographs here at the start of both of my examples.

What do you notice in example one with the cups, and in example two with the strips of paper or the same about both examples? Have a think.

Yes, you're right.

So in both examples, I started with two items, so in the cups example, I had two cups of juice, and in the strips of paper example, I had two strips of paper.

In both the examples, I removed something, didn't I? All right, reduced the amount of paper, or the amount of juice.

So anything else that was same in our test? Yes, you're right.

The amount of juice in both cups was equal, the quantity was an equal amount or the amount of juice in each cook was the same.

And if we talk about the strips of paper, the lengths of the strips of paper were equal, or we can say they have the same length.

So in terms of subtraction, what does that mean? What is that quantity of juice or those strips of paper? Right.

So those quantities, are the amount that we started with, and we label this as the minuend, don't we? So the amount of the value that we begin with is that minuend.

So in both examples, both minuends were the same, so in my juice, for example, they had the same capacity.

So the menu ended in a cup on the left, on the cup and the right would be the same and the exact same idea with those strips of paper.

That's important part of today's session.

Okay.

Let's put some numbers on today's examples now.

You should have a really good understanding of those generalised statements now, from those juice examples and from the strips of paper examples, and perhaps from the ones that you came up with.

So you should have an idea that the more we subtract, the less we are left with and the less we subtract, the more we are left with.

So we're going to start with Sebastian on the left, he has 10 marbles.

We've also got Meghan on the right, she has 10 marbles.

Do you notice they start with the same amount? That's really important.

So Sebastian is going to give away four of his marbles.

Now I know you can tell me what 10 plus four is.

This isn't really about you being able to calculate 10, subtract four, but it's to do with using small numbers to help our brains to understand the structures.

If we use large numbers, sometimes our brain can literally go into overload and it becomes too difficult for it to understand.

So we start with small numbers and then we can apply it to any numbers that we want to.

Right.

So what's the question? Sebastian has 10 marbles, he gives away four of them.

Megan also has 10 marbles, she gives away five marbles.

And I want to know, will she be left with more or fewer marbles than Sebastian? So Megan gives away five marbles.

Let's see.

There we are, 10 subractt 5.

So who is left with the most marbles? That's right.

Sebastian's got more left because we can use our generalised statement, the more we subtract, the less we're left with.

So in this case, Megan subtracted more than Sebastian, so she is left with less.

Okay.

Let's have a look at one more example using 10.

So we've still got Sebastian, he had his 10 marbles and he's given four away.

This time, we've also got Peter, there is on the right.

Peter has 10 marbles too, he gives the away three marbles, will he be left with more or fewer marbles than Sebastian? So he gives away three.

There we go.

Was he left with more or fewer? Let's think about our generalised statements.

Our generalised statement is, the less we subtract, the more we are left with.

So if we have a look here, we subtracted less, so he is left with more.

So if we go down here, 10 - 3 > 10 - 4.

Or going the other way round, we can say that 10 - 4 < 10 - 3 because the less we subtract, the more we are left with.

Okay.

You've listened to me for long enough.

So I would like you to fill in the lesser than, greater than, or equals sign into these equations and decide where they go Thinking about those generalised statements.

Say them with me, the more we subtract, the less we are left with, or the less we subtract, the more we are left with.

Pause the video here and have a go.

Okay.

So let's have a look at this first one.

We should be able to see in a moment that I'm going to slide some of the numbers onto the scales.

And we've got our 10 - 2, and we've got 10 - 1.

So 10 -2 I'm going to put on the right-hand side and 10 - 1 is going to go on the left.

I wonder where those scales are flip.

We've just seen that they have, do you know why? Because if we think about 10 -1, the less we subtract, the more we are left with, so therefore 10- 1 is greater than 10- 2, And we probably know that anyway, cause you probably know that 10 -1 = 9, and 10 - 2 = 8 We can watch that video one more time.

So we're just going to see it one more time, so the 10 - 2 will go onto the right-hand side and then it's quite heavy, but once the 10 - 1 one goes on, that is greater.

So 10 - 1 > 10 - 2 Okay.

So let's take a look at this next one.

We've got 10 - 2, and we've got 10 - 2.

Let's see what happens when these go onto balance scales.

10 - 2 goes on one side, you put it on the other side, and they are equal.

We have balanced equations, don't we? And that's because a minuend depends if the same in both equations, and our subtrahend of two is equal in both equations.

So our result's going to be equal, therefore we have balanced equations.

Okay.

So let's take a look at this one now.

We have 10 - 3, and 10 - 2.

So 10 - 3 is going on the left-hand side of my scales.

What's going to happen when 10 -2 goes on? Oh, it's heavier.

Why is it heavier? Why is it have a bigger value? Ah, that's right, because if we think about 10 - 3, compared to 10 - 3, the less we subtract, the more we are left with.

So therefore 10 - 3 > 10 - 2, or we can say, 10 -3 < 10 -2.

I'll show you that to show you that one more time.

So here we are, 10 - 3 is.

It's gone on 10 - 3 < 10 - 2.

And here are all of those answers in order, just in case you missed them.

One of the thing that I wondered, whether you noticed is well, on the right-hand side here, what do you notice about all of those equations? That's right.

They're all identical.

What happens with the equations on the left-hand side? You spot a pattern? Yes, that's right.

So, and subtrahends are increasing by one each time, aren't they? And so at first, it starts off at 10 - 1 > 10 - 2.

Then 10 - 2 = 10 - 2, and now 10 - 3 < 10 - 2.

You might want to pause the video here and just make some observations linked to those generalised statements if you're still a little bit unsure.

Okay.

So now it's time for you to help me.

I asked my friends who use the 'greater than' sign, or the 'lesser than' sign or the equal sign to complete these following equations.

And I wonder if you could help me check whether they've got them right.

So if you could pause the video here and have a look and I'll be back with you in a moment.

Did you get chance to do that? Okay.

So let's have a look.

Did any of them jump out to you straight away? Ah, I agree.

I spotted this one first.

I spotted this 62 - 25 must be equal to 62 - 25 because the minuends and the subtrahends are the same on both sides.

So the 62 is the same, and the 25 is the same.

hat's about in that first equation, or first sets of equations.

The 62 - 25 and 62 - 35.

You're right.

We've got the same minuends, haven't we? Did you notice anything about the subtrahends? Ah, so the left-hand one, they subtracted 25, and on the right-hand one, they subtracted 35.

And if we think about our generalised statement relating to the 25, 25 is less than 35, so the less we subtract, the more we are left with.

So my friend made a mistake there, it should be, the less we subtract, the more we are left with, so 62 - 25 > 62 - 35.

Okay.

Let's have a look at this final set.

What did you notice? Yeah, you're right.

Our minuends are still the same, that's still 62.

What did you notice about the subtrahends? That's right.

On this equation, that's 25, and on this side, it's 15.

So what do we notice about those two subtrahends? Is 25 greater or less than 15? Is greater, it's more.

So now we need to think about this generalised statement over here, the more we subtract, the less they are left with.

So in this case, it's more, we are left with less.

So that for this must be less 62 - 25 < 65 - 15.

Thank you for helping me.

Okay.

So it's your turn to do some work independently now.

I've left you with these questions today, and in fact, one, you just need to insert the 'lesser than', 'greater than' or equal sign.

Make sure that you're always thinking about those generalised statements.

Think about the juice, think about what we did with the paper and have those visual images in your mind, help you.

And then in part two, I'd like you to spot the mistake over here and see if you can correct it and come up with a reason why, when you find the mistake.

And if you're ready for challenge, then I'd also like you see if you can insert the 'lesser than', 'greater than', or equal sign within these equations.

So I'm going to leave that with you.

Thanks for listening.

Take care.