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 are 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 how adjusting the subtrahend affects the difference in a reduction structure.

These are the key words that you'll hear during the lesson.

I'm gonna say them and I want you to repeat them back to me, so I'll say my turn, say the words, 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, adjust.

Your turn.

Here's the meanings of those words just to make sure that everyone is 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.

We've got an equation at the bottom there, a subtraction equation.

Seven subtract three is equal to four.

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

When you adjust, you make a change to a number.

This is done to make a calculation easier to solve mentally.

You might have come across adjusting previously, it's a really useful method.

Here's the outline then for today's lesson on explaining how adjusting the subtrahend affects the difference with a reduction structure.

First, we're gonna start by adjusting the subtrahend in a reduction structure, then we're gonna move on to comparing expressions with adjusted subtrahends.

Hi Alex, hi Aisha.

These two are gonna be here today to help us with their thinking, their explanations and some hints and some prompts.

Okay, ready to learn? Let's do this.

A full box of eggs contains 12 eggs.

First there were nine eggs in the box, then four eggs were used to make pancake batter, now there are five eggs left over in the box.

This is a type of subtraction question.

The whole has been reduced to create a new amount.

This type of subtraction is called reduction.

Alex and Aisha both have a box of a dozen eggs for different meals.

Aisha says, "I will use three eggs for an omelette." 12 eggs subtract three eggs is equal to nine eggs.

Alex says, "I'll use five eggs to make scrambled egg for my family." 12 eggs subtract five eggs is equal to seven eggs.

What's the same and what's different? The minuend is the same, 12 eggs.

The subtrahend is greater in my equation, five eggs compared to three eggs.

My difference is greater.

Yes, you have nine eggs left over, but I only have seven eggs.

Alex and Aisha make a table of how many eggs are needed for different meals from a box of a dozen.

The headings are meal, number of eggs needed, eggs left over.

We've got omelette which needed three eggs and there were nine eggs left over from a box of a dozen.

Then we've got pancakes which needed four eggs, there were eight eggs left over.

Scrambled egg, which needed five eggs, seven eggs were left over.

And quiche, which needed six eggs, so there were six eggs left over.

Aisha says, "The minuend is the 12 eggs to start with." The subtrahend is the number of eggs needed.

The difference is the eggs left over.

What do you notice? Have a look at each of those rows.

Can you explain what's happening? Maybe you can use some of the key words like subtrahend, minuend and difference.

Well, Alex says, "The more we subtract, the fewer we are left with." As the subtrahend increases, the difference decreases.

"I wonder if that is always true," says Alex.

Okay, let's check your understanding so far.

Describe what is happening to the difference as the subtrahend increases in the table below and remember, the minuend has remained the same.

Pause the video and make sure you have a go at that explanation.

Welcome back.

Here's what Alex said, "The more we subtract, the less we are left with." Alex and Aisha explore by pouring out a drink into a beaker.

The bottle of Juice Blast contains one litre.

We've got beakers poured out and left in.

So you can see there that when there's one beaker, 200 millilitres was poured out and 800 millilitres was left in.

Two beakers, 400 millilitres was poured out and 600 millilitres left in.

For three beakers, 600 millilitres was poured out and 400 millilitres was left in.

For four beakers, it was 800 millilitres and 200 millilitres left in.

And for five beakers it was 1000 millilitres and zero millilitres left in.

Well, what do you notice then? Have a look at each row.

Aisha says, "It's the same again.

As the subtrahend increases, the difference decreases." Again, the more we subtract, the less we are left with.

Okay, let's check your understanding again.

If we add any number to the subtrahend, then the difference will decrease.

Is that always true, sometimes true or never true? Pause the video and decide what you think.

Remember to give reasoning as well.

Welcome back.

Well Aisha says, "This is sometimes true, it would always be true if the numbers were greater than zero." Ready to move on? Okay.

So far we have used examples in which the subtrahend has increased.

What if it decreased? "I think this table shows that as well." What does Aisha mean? Look from the bottom to the top, the subtrahend decreases.

Aha, so if you look at each row going up rather than down, you can see the subtrahend is decreasing, what's happening to the difference? You are right.

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

Alright, let's check your understanding again, true or false? The more we subtract the greater the difference.

Pause the video and decide whether you think that's true or false.

Welcome back.

That one was false.

But let's look at the reasons why.

Here's two justifications.

A, the more that is subtracted, the less the difference as you are subtracting more from the minuend, or B, the difference is always greater than the subtrahend and minuend.

Which of those two do you think is the best justification? Pause the video and decide.

Welcome back.

A, was the best justification here.

The more that is subtracted, the less the difference as you are subtracting more from the minuend.

Okay, let's move on.

Aisha says, "I've left the deliberate mistake in the workings above." She's got 95 subtract 37 is equal to 58, so if she adjusts the subtrahend by adding one, 95 minus 38 is equal to 59.

Can you spot the mistake in those workings? Alex says, "I know that if the subtrahend increases, then the difference should decrease by the same amount.

The difference in the second equation should be 57." Time for your first practise task then.

For number one, I'd like you to fill in the stem sentences below.

You can see that they are all stem sentences that we have come across during our learning.

For number two, you've got to complete the table and explain the pattern.

For number three, you've got to explain the mistakes in the pairs of equations below, just like the example Aisha gave us just now.

Okay, pause the video here and I'll be back in a little while with some feedback.

Good luck.

Welcome back.

Let's start with number one, then.

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

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

The greater the subtrahend, the smaller the difference, the smaller the subtrahend, the greater the difference.

We should probably add that for those last two sentences, we are also assuming that the minuend has remained constant.

Here's number two then, we had our tables to complete.

I'm gonna read out the numbers from top to bottom that are missing.

505.

50, 45, 520 and 525.

Now how can we explain that table? Well, the subtrahend is reducing by five each row, so the difference is increasing by five each time.

Let's move on to B then.

These were the missing numbers going from top to bottom, 88, 261, 104, 112, and 237.

What was the explanation? The subtrahend is increasing by eight each row, so the difference is decreasing by eight each time.

Here's number three then, explain the mistakes in the pairs of equation below.

Well, in the first one, the subtrahend has increased, so the difference should reduce.

You can see that the subtrahend in the first equation was 34, and in the second one it was 37, that's an increase of three.

That means that the difference should have reduced by three, which means the difference should have been 28.

There we go.

In B, again, the subtrahend has decreased, so the difference should increase.

You can see that it was 56, the subtrahend in the first equation and 51 in the second, that's a reduction of five.

That means that the difference should have increased by five.

It should have been 112.

Let's look at C then.

The difference was incorrect again, because the subtrahend has increased, so the difference should reduce.

The subtrahend went from 127 to 227, that means that the difference should have reduced by 100.

It should have been 19,315.

Okay, that's the first part of the lesson completed.

Let's look at the second part now.

Comparing expressions with adjusted subtrahends.

Alex and Aisha compare a sequence of expressions.

What symbols would you place between each? So have a look at them, and although it's quite tricky with some of the simple equations you can see, is there a way of working out without calculating the value? "Do we need to work each of these out or can we use reasoning instead," says Alex.

The subtrahend for the expressions on the left is adjusting by one more each time.

If the subtrahend increases, then the difference decreases, so the expressions on the left are decreasing in value.

The middle row of expressions are the same, so they are equal to each other.

The greater subtrahends gives smaller values.

So we've got a greater than in the first slot and a less than in the last slot.

We did use reasoning instead of calculating the value.

Alex and Aisha compare a new sequence of expressions.

What symbols would you place between each? Same idea again, but the numbers this time are two digit numbers.

What do you notice? What can you use to help you here? We can do this without calculating the value of the expressions, I think.

I agree, you can see that the value of the expressions on the right are decreasing because the subtrahend is increasing.

Whereas the expressions on the left are the same, so equal value.

So 66 subtract 33 is less than 66 subtract 23.

66 subtract 23 has a smaller subtrahend, which means that the difference is greater.

In the middle, it's equal, and in the last one, 66 subtract 33 is greater than 66 subtract 43.

That's because the subtrahend of 43 is greater than the sub of 33.

A greater subtrahend means a smaller difference.

Okay, it's your turn to check understanding so far.

Without calculating the values of the expressions, place the correct symbols between each.

Pause the video and give it a go.

Welcome back.

Here were the symbols, 75 subtract 45 is greater than 75 subtract 55 because 55 is greater than 45 and that means that the difference was smaller.

In the middle row you can see both expressions are the same, so they are of equal value, so there's an equal symbol.

And in the bottom row, 75 subtract 45 is less than 75 subtract 35.

The subtrahend on the second expression is smaller, which means the difference is greater.

The expressions on the right are increasing in value because the subtrahend is decreasing.

This time Aisha and Alex have to sort the expressions into the correct places, so this is a little bit of a puzzle.

"Let's start with the equation.

These expressions are the same," so when she says that she means the middle row because that has an equal symbol, so it becomes an equation.

86 subtract 15 is equal to 86 subtract 15.

The value is greater when the subtrahend is less.

So 86 subtract 17 is the least and 86 subtract 13 is the greatest, so they put in 86 subtract 15 is greater than 86 subtract 17.

And 86 subtract 15 is less than 86 subtract 13.

Okay, it is time for you to do your second practise task.

For each of the following sequences of expressions, without calculating their value, insert the correct symbol between them.

So you've got to use your understanding of the generalisation that if the minuend stays the same and the subtrahend increases, then the difference will have to decrease by the same amount and vice versa.

For number two, you've got to sort the expressions into the correct places without calculating their values.

We've got A and B.

Pause the video here and have a go at those.

Enjoy.

Welcome back.

We'll start with number one and we'll talk through each of these.

123 subtract 110 is the same as 123 subtract 110, so those two have equal value.

For the second one, 123 subtract 101 is greater than 123 subtract 110.

The subtrahend in that first expression is less than the subtrahend in the second, which means that the difference is greater.

In the last one on A, we've got 123 subtract 111 is less than 123 subtract 110.

Again, compare those two subtrahends.

In the first expression, the subtrahend is greater, meaning that the difference is smaller.

Let's move on to B then.

For the first one, 7,541 subtract 110 was less than 7,541 subtract 101.

That's because again, the subtrahend was greater, meaning the difference was smaller.

For the second one, we had a subtrahend of 1.

1 compared to a subtrahend of 101.

The greater the subtrahend, the smaller the difference, so that meant that there should have been a greater than symbol in between.

For the last one, it was greater than, because a subrahend of 11 is smaller than a subtrahend of 101, and that means that the difference is greater.

I didn't mention the minuend of the last few there because that remained constant at 7,541.

Here's number two then, an interesting puzzle.

We'll start with 4.

9 subtract 2.

8 that's greater than 4.

9 subtract 2.

9.

The subtrahend in that first expression is smaller, meaning that the difference is greater.

For the middle ones, well, both of those expressions were the same, so they were equivalent that's why there's an equal sign.

And lastly, 4.

9 subtract 2.

8 is less than 4.

9 subtract 2.

1, and that's because 2.

8 is greater as a subtrahend, meaning that the difference is smaller.

The minuend is the same for each expression.

When the subtrahend is greater, the difference is smaller.

Okay, let's look at B then.

I've uncovered all of them there and I'll explain.

For the first one, we've got 892,005 as the subtrahend, and for the second expression in that top row it's 892,015.

That's a greater subtrahend leading to a smaller difference.

In the middle row, the subtrahend is 891,999, and in the second expression it's 892,005.

That subtrahend is greater.

That means that the difference is smaller.

In the bottom row, the subtrahends were 892,005 compared to 892,002.

The one ending in two was smaller, meaning that the difference was greater.

Again, I didn't mention the minuend there because that remained constant at 923,678.

This was my solution, but there are others.

Alex is reminding us that you can actually chop and change between some of these provided that you look at the subtrahends carefully and follow our generalisation.

Again, the greater the subtrahend, the smaller the difference.

Thanks for reminding us, Aisha.

Okay, we've reached the end of the lesson, and here's a summary of all the things that we've learned about.

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 comparing expressions featuring subtraction, and selecting a correct inequality to complete inequality statements.

My name is Mr. Tazzyman, I enjoyed that lesson.

Hope you did too and I hope I'll see you again soon.

Bye for now.