Lesson video

Hi there.

My name is Mr. Tazzyman and I'm really looking forward to teaching you the lesson today from the unit that's all about equivalence, compensation, and subtraction.

If you're ready, we can get started.

Here's the outcome for today's lesson then.

By the end, we want you to be able to say, I can explain and represent constant difference for subtraction.

These are the keywords that you're going to hear during the lesson and it's important that you know how to say them and that you understand them.

We'll start with you repeating them back to me.

I'll say my turn, say the word, and then I'll say your turn and you can say it back.

Ready? My turn, minuend.

Your turn.

My turn, subtrahend.

Your turn.

My turn, difference.

Your turn.

My turn, constant.

Your turn.

Well done.

Now let's see what each of these words means.

The minuend is the number being subtracted from.

A subtrahend is a number subtracted from another.

The difference is the result after subtracting one number from another.

You can see an equation written down at the bottom there.

Seven takeaway three is equal to four.

In this equation, seven was the minuend, three was the subtrahend, and four was the difference.

Finally, a constant is a quantity that has a fixed value that does not change or vary, such as a number.

This is the outline for the lesson then.

It's a lesson about explaining and representing constant difference for subtraction, and we're going to start with thinking about the concept of constant difference.

Then we're going to put constant difference in context.

So we'll start with just learning about constant difference.

And to do that, we're going to meet a couple of friends.

We've got Sam and Jacob, and they're going to help us by discussing some of the maths prompts, giving us some hints and tips and sometimes even revealing some answers.

Hi, Sam.

Hi, Jacob.

Okay, then, everyone, ready to start? Let's go for it.

What's the same? And what's different? Look at each of these representations.

What do you notice? Jacob says, the Base 10 blocks are the same in each number line.

There are two.

Their position is different on the number line.

They are moving along two.

Okay, let's look at these ones.

What's the same? And what's different? It's a similar activity to last time.

Jacob says, the number rod remains constant.

It is representing 30.

So you can see in all three of those representations that the number rod is exactly the same.

That's constant.

It doesn't change its value at all.

The position on the number line moves right by 30 each time.

So the starting position for the number rod is actually increasing by 30 each time.

Although the number rod remains constant, its starting position changes.

Here are another three representations to compare.

This time, we've got three rulers.

Take note of what the unit is on each ruler.

What's the same and what's different here? The arrow is 25 millimetres in each of these.

The arrow has been moved on by 10 millimetres in each of them.

So you can see in the top one there, the arrow started on zero and finished on 25.

That's probably the easiest way to work out what the length of the arrow is.

But it remains constant, and we know that because in the second one, it starts on 10 and finishes at 35 millimetres, and on the bottom one, it starts on 20 and finishes at 45 millimetres.

The length of the arrow doesn't change.

It remains constant.

Jacob turns the representations into equations.

So these are the sequences of equations that were represented by what we've just seen.

What's the same? What's different? Have a look at each of those.

Can you find something that's the same? Can you find something that's different? The difference remains constant in each sequence.

The subtrahend and minuend increase by the same amount.

What can we generalise from this? If the minuend and subtrahend are increased by the same amount, the difference stays the same.

That's a really good generalisation.

That's going to be really important in this lesson today.

Okay, it's your turn.

Let's check your understanding.

I'd like you to try and fill in the missing numbers.

And then, crucially, you need to think about what you notice and explain it.

This lesson, a lot of the time is not going to be just about finding the answers.

It's going to be about explaining some of the patterns that you might see.

Okay, have a good go at that, pause the video, and I'll be back to reveal the answers shortly.

Welcome back.

Let's have a look at some answers then.

The difference was 20 in that first equation and in the second and in the third.

So what do you notice and can you explain it? Well, Jacob says, the minuend and subtrahend are increasing by five each time.

So what he means by that is that the minuend went from 30 to 35 to 40 and the subtrahend went from 10 to 15 to 20.

The increase matches on each of those.

And Sam says, the difference remains constant at 20, because the difference for all three equations was 20.

Okay, so more representations now.

What's the same and what's different here? Look closely at these representations, the cubes that they've used.

What do you notice? Can you compare the top layer of cubes with the bottom layer of cubes? What can you see? The difference between the two sets of cubes is always two.

The number of each colour cube set is decreasing by one each time.

The top layer, you can see, starts at six, and on the second one, it's five, and on the bottom one, it's four.

Similarly, the bottom layer, which is our subtrahend, starts at four.

In the middle, it's three, and in the bottom, it's two.

Okay, let's look at these number lines this time.

What's the same and what's different here? The purple arc is the same in each number line.

It's three.

The position of the purple arc is decreasing by three each time.

So you can see in the top one, it starts on 10.

In the middle, it starts on seven.

And in the bottom one, it starts on four.

It's moving along each time, but it's decreasing in the value of its starting position.

Okay, the rulers are back.

Let's look at these one.

What's the same and what's different? The arrow is 25 millimetres in each of these.

The arrow has been moved 10 millimetres closer to zero in each of them.

So in the first one, you can see it starts on 70, then it goes to 60, and then it goes to 50.

But each arrow is still the same length.

It's still constant.

It's 25 millimetres.

Sam turns the representations into equations.

What's the same? What's different? The difference remains constant in each sequence.

The subtrahend and minuend decrease by the same amount, and we saw that in the representations.

What can we generalise then? Hmm, so what can we say that is true of all three of these sets and indeed probably other sets of calculations if we were to put some together? If the minuend and subtrahend are decreased by the same amount, the difference stays the same.

And another way of saying that the difference stays the same is that the difference stays constant.

That's the part of the lesson that we're in, learning about constant difference.

If the minuend and subtrahend are changed by the same amount, the difference stays the same.

So Sam has slightly amended the generalisation there and Sam's done that because they noticed that actually, if it's changed by an increase or a decrease, the difference still stays the same, provided that the minuend and subtrahend are changed by the same amount.

Well done, Sam.

Okay, your turn.

True or false? If the minuend and subtrahend are changed by the same amount, the difference remains constant.

Okay, spend some time thinking about that now and decide whether you think that's true or false.

Pause the video here.

Welcome back.

That was true.

But it's not good enough just to say whether it was true or false.

Let's think about justification as well.

Here are two, and you're going to need to choose between them.

A said, the minuend and subtrahend are commutative so they can be changed.

B said, increasing or decreasing the minuend and subtrahend by the same doesn't change the difference.

Which of those do you think justifies the truth in that statement? Pause the video, have a discussion if you need to, have a think, and I'll be back in a moment to reveal the answer.

Welcome back.

B was the justification there.

Increasing or decreasing the minuend and subtrahend by the same doesn't change the difference.

Okay, it's time for your first task then.

Use the number cards to complete the subtraction equations with constant difference.

So you can see that there are some number cards at the bottom and then there are three equations for you to insert the number cards into, giving a constant difference of three each time.

When you've finished A and B, Jacob says, what do you notice? Make sure that you explain it really well.

Try and use some of the language that we've already learned about in the lesson so far.

Number two says using number rods, how many ways can you create a difference of three? An example is shown below.

If you've not got any number rods, you can always draw out some squares in your book if needed.

For number three, it says using a ruler and number rods, can you find five different ways of showing a difference of 50 millimetres? Write each as an equation.

An example is shown below, and they've written 90 millimetres subtract 40 millimetres is equal to 50 millimetres.

Okay.

Good luck with those.

Pause the video here and I'll be back to give you some feedback in a little while.

Enjoy.

Welcome back.

Let's start with number one.

A might have looked something like this.

Six subtract three is equal to three.

Five subtract two is equal to three.

Four subtract one is equal to three.

Consequently, B might have looked like this.

96 subtract 93 is equal to three.

95 subtract 92 is equal to three.

And 94 subtract 91 is equal to three.

Jacob said, what do you notice? Set B has the same ones digit as Set A, which is what creates the difference.

Okay, here's number two.

Using number rods, how many ways can you create a difference of three? Well, here are the differences of three written as equations.

You will of course have used number rods if you could.

10 subtract seven equals three.

Nine subtract six equals three.

Eight subtract five equals three.

Seven subtract four equals three.

Six subtract three equals three.

Five subtract two equals three.

Four subtract one equals three.

And three subtract zero equals three.

That last one is one that sometimes people forget, but we must still include it because it still shows constant difference.

Okay, let's move on to number three.

Using a ruler or number rods, can you find five different ways of showing a difference of 50 millimetres? Well, here are Jacob's, and as the question asks, he's written them as an equation.

90 millimetres subtract 40 millimetres equals 50 millimetres.

70 millimetres subtract 20 millimetres equals 50 millimetres.

60 millimetres subtract 10 millimetres equals 50 millimetres.

55 millimetres subtract five millimetres equals 50 millimetres.

And 51 millimetres subtract one millimetre equals 50 millimetres.

Well done, Jacob.

You of course might have had a different range of equations, but that's okay.

Provided that the difference was 50 millimetres in each, then you've got it spot on.

Okay, then, ready to move on? Let's go for it.

We're now going to look at constant difference in context.

Jacob and Sam are playing a football skills game.

Jacob scores 83 points and Sam scores 92 points.

They have a second go and both score another 56 points.

How many more points does Sam have now? Jacob says, I'll write this as an equation.

92 points subtract 83 points is equal to nine points.

I don't need to recalculate the new scores.

Why not? Why is it that Jacob thinks he doesn't need to do any recalculation? Well, because of constant difference, we know that nine points will remain the difference between the two scores.

When they had their second go, they both got the same number of points.

So we could add on 56 to the minuend and the subtrahend, but the question isn't asking us what the minuend and subtrahend are.

It's asking how many more points does Sam have? And that's the difference.

Both the minuend and subtrahend have been adjusted by the same amount, so the difference remains constant.

It's still nine points.

Time to check your understanding then.

Another true or false? If you calculate the difference and the minuend and subtrahend increase by the same amount, you don't need to recalculate the difference.

Is that true or false? Pause the video and have a think.

Welcome back.

That was true.

But let's think about justifications.

Here are two.

You've got to choose which one you think is the correct justification.

Is it A, it takes too long to recalculate so you can just leave it.

Or B, if the minuend and subtrahend are changed by the same amount, the difference remains constant.

Pause the video and think which one is best.

Welcome back.

B was the correct justification here.

If the minuend and subtrahend are changed by the same amount, the difference remains constant.

Okay, let's move on.

Sam collects battle robot stickers.

Sam has 45 more flying robots than aqua robots.

Sam gives away 17 flying robots and 17 aqua robots.

How many more flying robots than aqua robots does Sam have now? I'll write this as an equation, says Sam.

Flying robots subtract aqua robots equals 45.

You can see that written in the question.

But you don't know the subtrahend and minuend, says Jacob.

I don't need to know them, says Sam.

Why not? Have a think.

Why doesn't Sam need to know them? What is it about the question and the things that we've learned so far that tell Sam they don't need to know it? Well, actually, if you look at the question more closely, you can see that right at the start, it says, Sam has 45 more flying robots than aqua robots.

Then Sam gives away the same number of each type.

Well, that means that the minuend and subtrahend have both decreased by the same amount, so the difference remains constant.

It's still 45.

Right, then, let's check your understanding with another true or false.

If you calculate the difference and the minuend and subtrahend decrease by the same amount, you don't need to recalculate the difference.

Is that true or false? Pause the video and decide.

Welcome back.

That was true.

Let's look at the justification possibilities for this.

If you're going to justify your answer, would it be A? If the minuend and subtrahend are changed by the same amount, the difference remains constant.

Or B, the minuend and subtrahend don't affect the difference.

Pause the video and decide which one you think is the correct justification.

Welcome back.

A was the correct justification here.

If the minuend and subtrahend are changed by the same amount, the difference remains constant.

The other justification was certainly not true.

The minuend and subtrahend don't affect the difference.

Well, we use the minuend and subtrahend to find the difference, so that can't be true.

Okay, let's look at some more situations in which constant difference is really useful.

This says, how can we use constant difference to find missing numbers in this sequence of equations? Have a look at that sequence of equations and what do you notice? Sam says, we should calculate the difference for the first equation.

It's four.

Eight subtract four is equal to four.

In the next expression, the minuend and subtrahend both increased by two but the difference remained at four.

Can you see that? There's been a change of adding two to both the minuend and subtrahend, so the difference remains the same.

Now we can use that pattern to finish the sequence.

So we know that that's going to be eight because the difference has remained at four, the minuend has increased by two from the equation above it, so the subtrahend also needs to increase by two because the difference has remained constant.

In fact, the difference is constant for all of the rest of the equations, so that can help to fill in those missing numbers.

10.

16.

18.

Okay, it is time for you to have a go at your second task.

Use your understanding of constant difference to solve the worded problems below.

We've got A and we've got B.

A, Jacob collects seashells.

He has 27 more slipper shells than turret shells.

He uses eight slipper shells and eight turret shells to make a necklace and gives it away as a gift.

How many more slipper shells does he have now? And for B, Jacob and Sam have a go at archery.

Jacob scores 32 points and Sam scores 25 points.

They have a second go and both score 20 more.

Who has won and by how many points? And for number two, you've got to use your understanding of constant difference to find the missing numbers below.

Remember to use the equation above to help you with the next one.

Identify what has the change been in the subtrahend, the minuend, or the difference? Good luck with those, pause the video, and I'll be back shortly with some feedback.

Enjoy.

Welcome back.

Let's look at one A first.

He still has 27 more slipper shells because the minuend and subtrahend have both been changed by the same amount, so the difference remains constant.

Now let's look at B.

32 points subtract 25 points is equal to seven points.

That was the difference between Jacob and Sam after their first go.

There's no need to recalculate the difference after the second go because both the minuend and subtrahend have changed by 20 points, so the difference will be the same.

The difference remains constant.

Okay, let's look at number two.

I'm going to go through these and read them out, but at the end, I'll invite you to pause the video again in case you need to catch up with marking.

Let's start with the first set then.

We've got 10 minus three is equal to seven.

20 subtract 13 is equal to seven.

30 subtract 23 is equal to seven.

40 subtract 33 is equal to seven.

50 subtract 43 is equal to seven.

60 subtract 53 is equal to seven.

Now for the second set.

10,000 subtract 9,750 is equal to 250.

8,500 subtract 8,250 is equal to 250.

7,000 subtract 6,750 is equal to 250.

5,500 subtract 5,250 is equal to 250.

4,000 subtract 3,750 is equal to 250.

2,500 subtract 2,250 is equal to 250.

That was the second set.

Let's do the final set.

0.

84 subtract 0.

42 is equal to 0.

42.

2.

34 subtract 1.

92 is equal to 0.

42.

3.

84 subtract 3.

42 is equal to 0.

42.

5.

34 subtract 4.

92 is equal to 0.

42.

6.

84 subtract 6.

42 is equal to 0.

42.

8.

34 subtract 7.

92 is equal to 0.

42.

Phew! They're all done.

But of course, you might need more time to catch up with that marking.

So pause the video here if that's what you need.

We've reached the end of the lesson then and here's a summary of all of our learning.

If the minuend and subtrahend are changed by the same amount, and remember that means either increasing or decreasing, the difference stays the same or remains constant.

We can represent this using physical resources such as number rods and visual representation such as number lines.

This concept can help with solving equations and finding missing numbers efficiently.

My name is Mr. Tazzyman and I've enjoyed today's lesson.

I hope you have as well.

Maybe I'll see you again on the next one.

Thank you very much.

Bye-bye for now.