Lesson video

Hi, everybody.

My name is Mr. East, and I am going to take you through today's lesson, and we're going to start off by looking at the practise activity from the previous lesson.

Now, this question says Dorota decides to use a mental strategy to calculate 342 subtract 96.

She starts from 342.

Which of these methods could she use? Tick those that are correct.

Now, I hope you had a go at this.

If you didn't have time to, pause the video now and have a quick go before I go through the solutions.

So, the first method says add four, then subtract 90.

You can see on the screen I've drawn my number line and I'm going to show you those steps on the number line, always looking back to that calculation, which has shown 342 subtract 96.

So, I have started my number line with 342.

The first step that Dorota makes is to add four.

After that, she subtracts 90, and that's going to help her find her unknown value.

Now, we need to think about what the difference is between that unknown value and 342, and think, is that equal to 96? So, in blue on my number line, I've shown that amount.

That is the difference between that question mark and 342.

Is that 96? If I look carefully, I've added on four, and then I subtracted 90, so that difference shown in blue is not going to be 96.

Actually, it is 86.

So, the first method that Dorota introduced or suggests is incorrect.

Let's think about the second method.

Again, it starts on 342.

The first step is to subtract 100, and then from this, to add four.

Again, we need to be thinking about what is the difference between that question mark, the unknown, and the starting number, 342? If I subtract 100, but then I add another four, that actually means I've only subtracted 96.

That's the same as the question, so the second method is correct.

For the third method, again, starting with 342.

First step is to subtract six, and then to subtract another 90 to find our unknown value.

And what has been subtracted in total? What's the difference between that question mark and 342? Subtracting six, so subtracting another 90.

That will be 96, and that's what it says in the question, so a third solution or a third method is also correct.

Finally, the fourth method.

So again, starting with 342.

First, subtracting four, then subtracting 100, to find our unknown value.

How much has been subtracted altogether? 104 is how much has been subtracted.

That's not the same as the question.

The question says subtract 96, so the fourth method is incorrect.

Another practise question that you were asked to consider was this.

All these calculations are equivalent.

Is this true or false? You are also then asked to explain your answer.

Now, I want to highlight two calculations.

I'm going to start with these two calculations to prove why I think this is false.

Did anybody else try to do the same thing? Let me show you on a number line why I think these two calculations show that this is false.

I'm going to use my number line again.

Both of them, the minuend is 2,000, but the top calculation, the subtrahend is 1,999.

The second question, the second calculation that I've highlighted is subtrahend is 1,997.

So, our two question marks are going to be at slightly different points.

So, if the minuend is the same, and the two subtrahends are different, then the calculations cannot be equivalent.

Let me show you that in a slightly different way.

I really want to focus on this same difference which you've been thinking about before.

So, on the top number line, if I show the difference between 2,000 and 1,999, we can all see the difference is one.

And if I do the same for my second part, 2,000 subtract 1,997, or the difference between those two numbers, we can see that they are not equivalent to each other, so we cannot use the equals sign.

What sign could we use, though, to compare 2,000 subtract 1,999, and 2,000 subtract 1,997? If they're not the same size that means one is greater than the other.

Is 2,000 subtract 1,999 greater? No, it's smaller, so we can use the less than sign.

So, we can say 2,000 subtract 1,999 is less than 2,000 subtract 1,997.

We're still sticking with these four calculations, but I want us now to think about the solutions to each to help us know which ones are equivalent.

If we think in our heads like we did in the previous example of counting up, hopefully we can see the difference between these two pairs of numbers can be calculated mentally.

I've shown you the difference between these two numbers is three.

The difference between 2,000 and 1,999 is one.

This one is three, and the bottom one is three.

So, three of those are equivalent to each other, it's just the second one, which is not equivalent.

Now, here is an image which you have seen in previous lessons.

This image of a seesaw helps us think about equivalent calculations.

We know that if it's level, that means they are equal to each other.

That means they are equivalent.

So, let's choose two of these calculations which we know are equivalent to each other.

And they're going to introduce you to a stem sentence, which we're going to use in this lesson.

I'm going to start actually with the blanks.

If I add to the minuend and add to the subtrahend, that difference stays the same.

So, let's look at this calculation.

2,000 has changed 2,001, so I've added one.

Great, so the first part of my stem sentence, I can put that information.

Let's think about the subtrahend.

How has the subtrahend changed? That's also increased by one, so I can add that information to my stem sentence.

Let's say the sentence together.

If I add one to the minuend and add one to the subtrahend, the difference stays the same.

Let me show you that on a number line.

At the moment, the difference is shown between 2,000 and 1,997, and we can see that, that difference is the same when we look at the other two numbers because it's the same size, we just moved its location on our number line.

Okay, watch carefully.

I'm going to change something.

Can you spot what I change? Did you see? Exactly the same calculations, but I've changed their order on our seesaw.

So, does anything change? Can we still use the same stem sentence? Let's have a look.

Has our minuend increased, if you think about the left to the right? No, it hasn't.

It's decreased, so I'm going to change my sentence stem slightly.

Rather than saying add, this time I'm going to say subtract, because we've subtracted one from our minuend.

So, if I subtract one from the minuend, and subtract one from the subtrahend, the difference stays the same.

Can you say that sentence with me? If I subtract one from the minuend and subtract one from the subtrahend, the difference stays the same.

Great.

Now, I've used two of the other calculations that are shown above.

Which stem sentence would I use this time? Would I use my addition stem sentence or my subtraction sentence? Let's look at the minuend first.

Has my minuend end increased or decreased? It's decreased, right? So, we're going to have to use our subtraction stem sentence because it's decreased by two.

So, let's say this together.

If I subtract two from the minuend, and subtract two from the subtrahend, the difference stays the same.

What generalisation can we make, then? What can we say about what we've learned? We've used different examples of it increasing by one or decreasing by one.

In this case, decreasing by two.

What generalisation can we make? What do you think? Here's my attempt at a generalisation.

If we change the minuend and subtrahend by the same amount, the difference stays the same.

That means we could change it by using addition, or in the case shown above, change it with subtraction.

But if we change both of them by the same amount, the difference will stay the same.

Okay, same example, different numbers.

The numbers are a bit bigger this time, aren't they? Which one are we going to use? Are we going to use our subtraction stem sentence, or our addition stem sentence? Let's look at the two minuends.

Remember, those are our first numbers.

We've got 324,000, and then on other side we've got 334,000.

Has that minuend increased or decreased? It's increased, hasn't it? How much has it increased by? Think carefully about your place value.

It's increased by 10,000, hasn't it? So, let's say the stem sentence together.

Remember, it's the addition one.

If I add 10,000 to the minuend, and add 10,000 to the subtrahend, let's just check.

290,000 add 10,000.

Yup, that's 300,000.

Sorry, let's go back to the sentence.

If I add 10,000 to the minuend and add 10,000 to the subtrahend, the difference stays the same.

What about this example? Can you notice something? Same as before, I've swapped those two around, and we'd noticed before that we would change our stem sentence from addition to subtraction.

So, this time let's see if we can use our subtraction stem sentence together.

If I subtract 10,000 from the minuend, and subtract 10,000 from the subtrahend, the difference stays the same.

So, is our generalisation still true? If we change the minuend and subtrahend by the same amount, the difference stays the same.

That is true, isn't it? Our generalisation is still correct.

Okay, still a seesaw image, but if you look carefully you can see I've now got a missing value.

We can use our understanding of same difference to solve missing boxes in our calculations.

I'm going to take the information from the seesaw and write it as a linear equation.

So, 10 subtract three is equal to 45 subtract something.

Pause the video and see if you can use the same difference principle to work out the value of the missing box.

Okay, let's see if we can do this together.

So, let's look at our minuend.

10 has increased to 45.

How much has it increased by? It's increased by 35, hasn't it? So, if we go to our subtrahend, that needs to increase by the same amount.

So, three add 35, that is going to be 38.

So, we know that the missing value in our calculation is 38.

So, the whole calculation should read 10 subtract three is equal to 45 subtract 38.

Now, some of you might have done that a different way.

Some of you might have looked at that and said, "Oh, I know that 10 subtract three is seven, so that means the difference between 45 and the unknown number must be seven, but I want us to concentrate on the relationship between the minuends and the subtrahends for the next few questions.

You'll see why in a second.

Here's another example for us to look at.

And if you look closely, some of you will have noticed there's something the same.

10 subtract three is the same, but now, there's an unknown minuend.

So, let's take that and write that as a linear equation.

10 subtract three is equal to something subtract 47.

Now, do we know what's happened to the minuend? We don't, do we? Because that is what the missing box is, let's start this time with the subtrahend.

how much is the subtrahend increased or decreased by? It's increased, hasn't it? And it's increased by 44.

I know that three add 44 is equal to 47.

So, if these are equal to each other, that means the minuend must also increase by the same amount.

That must also increase by 44.

10 add 44 is 54.

So, our missing value in our top equation is also 54.

This is what I was talking about before.

I don't want to work out the difference between 642 and 387, and then try and work out what I do with that difference 630.

Instead, I want to think about the relationship between the minuends that I'm shown, and use that to work out the unknown subtrahend.

So, let me write that equation as I've done before in a linear way.

642 subtract 387 is equal to 630 subtract something.

I can see we know both our minuends.

Has our menu add increased or decreased in size? It's decreased in size, and it's decreased by 12.

So, that means if I subtract 12 from my minuend, I need to also subtract 12 from my subtrahend.

387 subtract 12 is 375.

Then I can put that information into the top to complete the unknown box.

This is a question I'd like you to think about.

Something subtract 100 is equal to 185 subtract 98.

Pause the video and use same difference to see if you can work out the missing box.

Okay, now, do we know what's happened to the minuend? We don't, do we? We do though, know about the subtrahend, so let's look at the subtrahends.

We've got 100 in the first one and 98 in the second.

So, will we use our subtraction or addition stem sentence? We'll use our subtraction one, so let's say that together.

If I subtract two from the minuend and subtract two from the subtrahend, the difference stays the same.

So, we're going to use the information we know about the subtrahend to help us work out the missing minuend.

Now, be careful.

We don't do 185 subtract two.

We actually think of it as something subtract two is 185.

What would that be? It would be 187, wouldn't it? Now, that can be a bit confusing, so let me show you that in a slightly different way.

Exactly the same calculation, but we can think about it in this way.

We can now use our addition stem sentence to say if I add two to the subtrahend, and if I add two to the minuend, the difference will stay the same.

Then we can think of it as 185 add two is 187, and we get exactly the same answer.

So, let's think back to our generalisation.

Let's bring that up.

If we change the minuend and subtrahend by the same amount, the difference stays the same.

In this one, like we said before, the numbers are exactly the same, but depending on how we look at it, we can think about changing the minuend and subtrahend using addition or subtraction.

But as long as you do the same to both, the difference will stay the same.

Okay this one involves some decimals, but let's think about it in exactly the same way.

113.

5 subtract 87.

3 is equal to 114 subtract something.

So, which one will we use? Can you see we're going to use the addition stem sentence because our minuend has become larger? How much larger is 114 than 113.

5? It's 0.

5 larger, isn't it? So, if we go back to our stem sentence and our generalisation, that means the subtrahend also needs to increase by 0.

5.

And what is 87.

3 add 0.

5? It's 87.

8.

That is a lot easier than working out the difference between 113.

5 and 87.

3, and then working out what to do with that to find our answer.

So, our same difference model can be much simpler.

Okay, so your practise activities.

I've given you three expressions, and I want you to use the greater than, less than, or equal to symbols.

I want you to use same difference to help you make your decision.

Think about those stem sentences that we've used.

Our subtraction stem sentence, and our addition stem sentence.

Then I want you for the second question to fill in the missing numbers like we've done in this lesson.

So, 587 subtract 297 is equal to 590 subtract something, remembering, like I said before, we're still using the same difference to help us.

And then there's two more.

720,201 subtract 390,199 is equal to something subtract 390,200.

Oh, I think I can see a connection there.

And the final one with some decimals.

Something subtract 0.

57 is equal to 0.

87 subtract 0.

61.

And a challenge for you.

Here is a kind of question which I normally get in a pickle with.

Jon was born in 2007, and his dad was born in 1982.

Jon says that there will always be an odd difference between their ages.

Explain whether he is right or not.

I wonder if you could try and work that out, and I wonder if using the same difference might help us find out the answer.

Have a go.

Good luck.

See you again soon.