Lesson video

Hello, I'm Mr. Taziman.

And today, I'm going to teach you this lesson from the unit that's all about equivalence, compensation, and subtraction.

So sit back, be ready to listen, and engage.

Here's the outcome for today's lesson.

By the end, we want you to be able to say, I can explain how adjusting the subtrahend affects the difference in a partitioning structure.

These are the key words that you might hear today.

I'm gonna say them and I want you to repeat 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, clear, ready? Okay, my turn, minuend, your turn.

My turn, subtrahend, your turn.

My turn, difference, your turn.

Okay, here's what each of those words means, just so that we can be really clear.

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 example subtraction equation at the bottom.

Seven subtract three is equal to four.

In that equation, the minuend is seven, the subtrahend is three, and the difference is four.

Here's the outline of today's lesson on explaining how adjusting the subtrahend affects the difference in partitioning structures.

To begin with, we're gonna think about increasing the subtrahend.

Then we're gonna move on to looking at decreasing the subtrahend.

We'll start with increasing the subtrahend, and I think we need some help.

Here's Jacob and Sam.

They can be our help.

They're going to explain some of their thinking, give us some hints and prompts.

They look like they're ready to learn.

Let's hope you are because we're about to start.

There are five yellow and red counters.

Two of them are red.

How many are yellow? This is a subtraction question.

We've got the minuend, the difference, and the subtrahend.

One part of the whole amount has been partitioned from the whole group.

This is a type of subtraction.

Now I know that that might seem like a really simple form of the subtraction, but actually, it's important for us to understand the structure beneath it.

That's why we're starting with a really simple version.

One counter is flipped over.

How has this changed the solution? How many yellow counters are there now? The subtrahend has increased.

The subtrahend has increased, so the difference has decreased.

And there are the labels again.

The difference is now two, where it was three.

Here's a worded problem then that uses some of that concept.

Jacob and Sam have collected tokens for a voucher for 20 pounds off football equipment for their club.

They purchase 88 pounds of new equipment and use their voucher.

How much will they pay overall? I'll use a number line to model this.

The minuend is 88.

The subtrahend is 20 because that's the voucher, so the difference is 68 pounds, which is how much we pay.

Well done, Sam, good calculations.

Jacob looks closely at the voucher and realises it was 30 pounds off instead of 20 pounds.

That's a really good result, isn't it? He remodels the question on the number line.

The minuend is unchanged, so it's still 88 pounds.

The subtrahend is now 30 pounds, which is a 10 pound increase, so the difference has decreased by 10 pounds to 58 pounds.

What made the difference smaller? The minuend remained unchanged.

When the subtrahend increased, the difference decreased.

That's because the minuend and subtrahend got closer to each other.

You can see, at the bottom of the number line there, the minuend of 88 and the subtrahend of 30 are much closer together than they were in the one on top where they were 88 and 20 pounds.

In fact, they were 10 pounds closer together.

Okay, true or false to check your understanding so far.

If the minuend stays the same and I increase the subtrahend, the minuend and subtrahend will be closer together on a number line.

Is that true or is that false? Read it carefully and have a think.

Pause the video here.

Welcome back, that was true, but why? Let's look at two possible justifications and choose which one we think is best.

A.

The minuend and subtrahend are constant so they always remain in the same place on a number line.

Or B.

The greater the subtrahend, the smaller the difference so the minuend and subtrahend will be closer.

Which of those do you think justifies the answer of true? Pause the video and have a think.

Welcome back, B was actually the correct justification here, the greater the subtrahend, the smaller the difference, and the minuend and subtrahend will be closer.

Ready to move on? Let's go for it.

Sam challenges Jacob.

"Can you show this as two bar models instead?" "I like a challenge!" says Jacob.

Remember, it's not just about getting the correct answer here.

It's about representing things in different ways to show a strong understanding of the structure underneath.

That's what good mathematicians can do.

Here's what we thought at the start and it's written before next to it.

We've got the minuend of 88.

We've got the subtrahend of 20, and a difference of 68.

Remember, 88 was what the equipment cost, 20 pounds was what they thought the voucher was to begin with.

We know it changed.

And 68 pounds was the difference, which was what they needed to pay.

Here's the change in the subtrahend.

So now, Jacob has taken the subtrahend and he's added 10 onto it, which means he's also had to write in subtract 10 from the difference because the minuend was staying the same.

Here's the difference of 58 pounds.

He's completed those small jottings now.

And you can see, we've got a minuend of 88, a subtrahend of 30, which was the actual value of the voucher, and then the difference of 58 before the change and after.

Jacob challenges Sam, "Can you turn these into jottings?" Aha, another challenge based on representations.

"Definitely!" says Sam.

"Good, great confidence." Here's the starting equation.

88 subtract 20 pounds is equal to 68 pounds.

So there, 88 was the cost of all the equipment, 20 pounds was the original value of the voucher that they thought anyway, and then that meant they would have to pay 68 pounds.

Here's the change in the subtrahend and difference.

So they added 10 pounds onto the subtrahend, meaning they had to remove 10 pounds from the difference.

Here's the final equation showing the adjusted subtrahend and difference.

88 pounds subtract 30 pounds is equal to 58 pounds.

We kept the minuend the same and added 10 pounds to the subtrahend, so we had to subtract 10 pounds from the difference.

Let's check your understanding again.

Complete the stem sentence to describe the change in the subtrahend and the change in the difference.

We've got 27 subtract 15 is equal to 12.

15 has been increased by adding eight, so then underneath, we've got 27, the minuend which remains constant, subtract 23 now is equal to four because the difference has been decreased by eight.

Every time you increase subtrahend, you have to decrease the difference by the same amount provided the minuend stays the same.

Can you complete that sentence at the bottom? Pause the video and have a go.

Welcome back, I've kept the minuend the same and added eight to the subtrahend, so I had to subtract eight from the difference.

Did you get that? Hopefully.

Here's your first practise task then.

For number one, you need to fill in the blanks on the number line and use it to complete the stem sentence describing the change in the subtrahend and difference.

For number two, you need to fill in the blanks on the bar models and use them to complete the stem sentence describing the change in the subtrahend and difference.

For number three, use the description below to draw three bar models that represent the change in the subtrahend and difference.

So you can see, we've got three empty bar models there ready to be drawn upon.

And the sentence at the bottom reads, "I kept the minuend of 200 the same "and added 30 to the subtrahend of 50, "so I had to subtract 30 from the difference." Here's number four, fill in the blanks in the jottings and stem sentences below.

So we've got some jottings with some empty boxes there that need completing and we've also got some sentences with some gaps.

You should, with the information you've got, be able to work all of those out.

At least give it a go anyhow.

All right, pause the video here and enjoy.

Welcome back, let's do the first one to begin with.

On the number line, the missing numbers were 62 at the top, 52 for the difference on the second number line, and 75 was the minuend on the second number line because that remains constant, so the sentence should have read as follows: "I've kept the minuend the same "and added 10 pounds of the subtrahend, "so I had to subtract 10 pounds from the difference." That's number one.

Let's look at number two then.

Fill in the blanks on the bar models.

So let's look at those to begin with.

On the first bar model, the difference was 85 pounds.

On the second bar model, the difference was 85 pounds subtract eight, so the missing number was still 85 pounds.

And on the last bar model, we had adjusted those so we ended up with 33 and 77 pounds.

The sentence should have read as follows: "I kept the minuend the same "and added eight pounds to the subtrahend, "so I had to subtract eight pounds from the difference." Let's look at number three then.

You had to draw bar models on this occasion.

The first bar model was 200 as the minuend, 50 as the subtrahend, and 150 as the difference.

The change was adding 30 to the subtrahend and subtracting 30 from the difference, so it should have had 200 at the top as the minuend, 50 plus 30 as the subtrahend, and a 150 subtract 30 as the difference.

Lastly then, we needed to calculate the value of those expressions.

We end up with 200 as the minuend, 80 as the subtrahend, and a 120 as the difference.

Here's number four then.

For A, these were the missing parts in the jottings.

We had add 35 for the arrow between the subtrahends, and subtract 35 for the arrow between the differences.

That meant that the new value of the difference was a 155 pounds.

The sentence should have read as follows: "I kept the minuend of 250 pounds the same "and added 35 pounds to the subtrahend, "so I had to subtract 35 pounds from the difference." Let's look at B.

The missing numbers were as follows.

The difference on that first equation was 268 pounds, 40 pounds was subtracted from that difference to give 228 pounds as the difference at the bottom, and 40 pounds was added to the subtrahend to give a 112 pounds as the second subtrahend.

The sentence read as follows: "I kept the minuend of 340 pounds the same "and added 40 pounds to the subtrahend, "so I had to subtract 40 pounds from the difference." Okay, that's the first part of the lesson completed.

Let's move on to the next part, decreasing the subtrahend.

Jacob has saved 200 pounds and wants to buy a new games console which costs 380 pounds.

How much more will he need to save? I'll use a number line to model this.

The minuend is 380 pounds.

The subtrahend is 200 pounds because that's my savings total, so the difference is 180 pounds, which is how much I still need to save.

Jacob goes bowling with his friends and spends 30 pounds of his savings.

How much more will he need to save now? The minuend is unchanged.

It's still 380 pounds for a games console.

My savings have decreased by 30 pounds, so the subtrahend is now 170 pounds, so the difference has increased by 30 pounds to 210 pounds.

What made the difference greater? "Well, the minuend remained unchanged." says Jacob.

"When the subtrahend decreased, "the difference increased." "That's because the minuend and subtrahend "got further away from each other." And you can see that on the number line there if you compare the calculation at the top with the calculation at the bottom.

The minuend and subtrahend are further away from each other in the bottom calculation and the difference is greater.

Okay, let's check your understanding then.

True or false, if the minuend stays the same and I decrease the subtrahend, the minuend and subtrahend will be closer together on a number line.

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

Welcome back, that statement was false, but I wonder why.

Well, here's two justifications for you to select from.

A.

The minuend and subtrahend are constant so they always remain in the same place on a number line.

B.

The smaller the subtrahend, the greater the difference, so the minuend and subtrahend will be further apart.

Which of those two explains why that statement was false? Pause the video, have a think, and I'll give you the feedback in a moment.

Welcome back, the correct justification was B.

The smaller the subtrahend, the greater the difference, so the minuend and subtrahend will be further apart.

Okay, let's move on to the next bit.

Jacob challenges Sam, "This time, "can you show this as two bar models instead?" "Nice idea!" says Sam.

Here's what you had before going bowling.

We had a minuend of 380.

We had a subtrahend of 200, and we had a difference of 180, and let's just remind ourselves, 380 was the cost of the games console.

200 pounds was what Jacob had already saved, which meant that he needed to save a 180 pounds more.

"Here's the change in the subtrahend." says Sam.

Off he went bowling.

It cost him 30 pounds.

So you can see, the minuend is still the same.

The price of the games console didn't change, but the subtrahend changed by subtracting 30 pounds because that's what he spent at bowling.

Consequently, the difference was increased by 30 pounds.

Here's the difference of 210 pounds.

And you can see, the subtrahend is a 170 there because that's what remains of Jacob's savings.

"Now, your turn to transform these into jottings." "Sure!" Says Jacob.

Here's the starting equation.

380 pounds subtract 200 pounds is equal to 180 pounds.

Here's the change in the subtrahend and difference.

So the subtrahend has subtracted by 30 and the difference has added 30.

Here's the final equation showing the adjusted subtrahend and difference.

380 pounds subtract 170 pounds is equal to 210 pounds.

We kept the minuend the same and subtracted 30 pounds on the subtrahend, so we had to add 30 pounds to the difference.

Okay, let's check your understanding with this.

Complete the stem sentence to describe the change in the subtrahend and the change in the difference.

Can you complete the sentence at the bottom that will describe that by filling in the blanks? Pause the video and give it a go.

Welcome back, I've kept the minuend the same and subtracted 14 from the subtrahend, so I had to add 14 to the difference.

Did you get that? Hope so.

Jacob and Sam compare two examples of adjusting the subtrahend.

What's the same? What's different? So have a look at those two different examples.

What do you notice about them? Where are the numbers the same? Where are they different? And what about the operations? Can you see any changes in those? Well, Jacob says, the minuend is the same in all the equations.

So the minuend is remain in constant.

The subtrahend has been increased in the first pair, but decreased in the second pair.

These two sets feature the same equations, but the order has been reversed.

This shows how we have to use the inverse to adjust the difference if the subtrahend changes.

Okay, time for your second practise task.

Fill in the blanks on the number line and use it to complete the stem sentence describing the change in the subtrahend and difference.

This is very similar to what you did in the first task.

For number two, you've got to fill in the blanks on the bar models and use them to complete the stem sentence describing the change in the subtrahend and difference.

Same sort of task, but this time, the representation is a bar model.

For number three, you'll turn to draw the bar models.

Use the description below to draw three bar models that represent the change in the subtrahend and difference.

And for number four, fill in the blanks in the jottings and stem sentences below.

Pause the video, enjoy the questions, and I'll be back in a little while for some feedback.

Welcome back, let's start with number one.

The missing numbers in the number line were as follows: 400 was the difference on the top calculation.

490 was the difference on the bottom one, and the subtrahend, which was missing on the bottom calculation was 750 because that remains constant, so the sentence should have read as follows: "I've kept the minuend the same "and subtracted 90 from the subtrahend, "so I had to add 90 to the difference." Okay then, number two, we'll start with the bar model and looking at the blanks there.

On the top, the difference was 850.

In the middle, the missing number was 850 because, of course, that was about to have 200 added to it since 200 had been removed from the subtrahend.

After calculating those new values then, we had 950 as the subtrahend and 1,050 as the difference.

The sentence should have read as follows: "I've kept the minuend the same "and subtracted 200 from the subtrahend, "so I had to add 200 to the difference." For number three, drawing the bar models, they might have looked like this.

In the top one, you might have had a minuend of 400, the subtrahend of 250, and the difference of 150.

And just quickly a word on this, don't worry too much about the proportionality of your different parts making up the whole.

As long as your 150 is smaller than your 250, then that's okay.

For the second one, 400 was the minuend because that remains constant.

The subtrahend should have had 250 subtract 90 written into it, and the difference should have had a 150 add 90.

When you calculate those, you end up with a bar model that looks like this.

400 is the minuend still, 160 is the subtrahend, and 240 as the difference.

For number four then, let's start with the blanks in the jottings.

45 was taken away from the subtrahend, 45 was added to the difference, and the missing difference was 795 pounds.

The sentence should have read as follows: "I kept the minuend of 900 pounds the same "and subtracted 45 pounds from the subtrahend, "so I had to add 45 to the difference." Let's look at B, the missing parts.

Well, the difference in that first equation was 407 pounds.

21 pounds needed to be added to that giving a difference of 428 pounds in the second equation, and the subtrahend had had 21 pounds removed from it, meaning that that was 25 pounds.

The sentence should have read as follows: "I kept the minuend of 453 pounds the same "and subtracted 21 pounds from the subtrahend, "so I had to add 21 pounds to the difference." Okay, we've reached the end of the lesson.

Here's a summary of our learning today.

A greater subtrahend means that more is subtracted from the minuend and the difference is reduced.

A smaller subtrahend means that less is subtracted from the minuend and the difference is increased.

This concept is very useful when considering problems involving a partitioning structure.

My name is Mr. Taziman.

I hope you enjoyed that today.

I certainly did, and maybe I'll see you again in another maths lesson.

Bye for now.