Lesson video

Hi, everyone, let's start today's lesson by going through the practise activities for the lesson before.

So you were asked to complete the following expressions using the greater than, less than or equal to symbols.

You were also asked to use the same difference principle to help you make your decision.

So let's look at the first question together.

We've got 15 subtract five and we've got 14 subtract four.

We've got the STEM sentences that were used in the previous lesson.

I can see that the minuends have decreased in size.

We've got 15 and we've got 14.

So, let's use the sentence on the left.

If I subtract one from the minuend, and subtract one from the subtrahend, the different stays the same.

That is true so they are equal to each other.

The next one, let's see, 123 subtract 87, 124 subtract 86.

Let's look at the minuends again, 123 is our first minuend, 124 is our second minuend.

The minuend has increased.

So let's use the sentence scaffold on the right.

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

Let's look at the subtrahend, 87 to 86.

Oh no, the subtrahend has decreased by one, so they're not equal to each other, but let's think if the minuend has increased, but the subtrahend has decreased, that must mean that the second expression is greater than the first expression.

So 123 subtract 87 is less than 124 subtract 86.

Final one, let's look at the minuends again.

We've got 13576.

5 and 13,577.

My minuend has increased, what's it increased by? It's being increased by 0.

5.

Let's see, let's look at the subtrahend, has that increased by the same amount? We've gone from 297.

4 to 297.

9.

Yep, that's a 0.

5 increase.

If I look in my 10th's column, it has increased by five one tenths.

So they are equal to each other.

The second question you are asked to fill in the missing numbers.

And the first one there's arrows to help you.

So let's think, the minuends, 587 to 590.

Has that increased or decreased? Increased, by how much? By three.

And it says is equal to, that means our subtrahend needs to increase by three also.

So three more than 297 would be 300.

I hope you got that one right.

Let's look at the second one.

720,201 subtract 390,199.

Now we don't know the minuend of the second expression, but we do know the subtrahend, so let's compare them.

We've got 390,200.

My subtrahend has increased by one.

Or if I think about right to left has decreased by one which is going to be more useful? Thinking about left or right, because we know that minuend on the left.

So if the subtrahend increases by one, that means my minuends needs to increase by one.

So the answer should be 720,202.

Yes, I am right.

Okay final one on here.

This time we don't know the minuend in our first expression, but we do know the two subtrahend.

So let's look at the subtrahends.

It has 0.

57 and 0.

61.

I'm going to think about comparing the subtrahend on the right to the subtrahend on the left.

So 0.

61 to 0.

57.

The subtrahend's decreased.

It's decreased, sorry, by 0.

04.

So my minuends needs to do the same thing.

So what's four hundredths less than 0.

87.

It would be 0.

83.

Well done if you've got those questions correct.

Okay now the challenge was quite challenging.

Let me read it to you.

John was born in 2007 and his dad was born in 1982.

John says that there will always be an odd difference between the ages.

Explain whether he is right or not.

So I have decided to use a number line for this question.

I'm going to write those two years on my number line.

I'm going to put 1982 on the left and 2007 on the right.

The question asks us about the difference between their ages.

We should be really good at knowing that difference now means that I can subtract the smaller number from the larger number to work out the gap in between these two numbers, the difference between them.

And my subtraction will be 2007 subtract 1982.

Now we don't always need to work out the number that is the difference but for this question we do, because it explicitly asks us if the difference is odd.

Now the way I would do this, I was thinking about counting up in my mind from 1982 or 1,982 to 2007, I would do that in two jumps.

I'm going to think in my mind of going from 1982 to 2000, that's going to be 18 more.

And then 2000 to 2007 would be seven more, 18 add seven is equal to 25.

Is 25 an odd number or an even number? It's an odd number, isn't it? So certainly at this point in the year when they were born, they had a difference of 25 years.

So John's dad must've been 25 years old when he had John.

Now let's think, one whole year later, how would this change? 1983 will be one year later than 1982, 2008 would be one year later than 2007.

So one whole year after John was born, is the different still the same? Yeah, it's exactly the same.

There's still 25 years between their ages.

So I can write the expression.

2007 subtract 1982 or subtract 1,982 is equal to 2008 subtract 1,983.

The year after that would be 1984 and 2009.

So what do you think? Do you think the difference still stays the same? Yeah, it must do because we can see when we move that difference along, the difference hasn't changed.

It's just now on two different years.

Now on 1984, which is the same as 1,984 and the year 2009 is still a difference of 25, which is still an odd number.

So I can continue my expression and say, this is also equal to the other expressions.

Let me read the whole thing.

2007 subtract 1,982 is equal to 2008 subtract 1,983, which is also equal to 2009 subtract 1,984.

The difference has always been the same, it's always been 25.

Does anybody think that this wasn't true? Let me show you something.

If we move on to part way through the year, maybe it's a July, it could be that John's dad has had his birthday but John hasn't had his birthday.

So it might be that John's dad's age has increased by one, but John's age hasn't.

But the difference between the ages as we can see is still the same because that's moved up our number line, but it would depend on whether we're talking about days, months, or years, and those would have different odd or even values.

So kind of a trick question, but it's important to remember what we learned that if both the minuend and the subtrahend increased by the same amount, the difference will stay the same.

Now in the last few lessons we've been thinking about same difference.

So when we change our minuends and subtrahend by the same amount, the difference stays the same, but we're going to change things slightly in this lesson.

And I'm going to use a new context to help us understand that.

So John has 10p in his money box.

I'm going to show that using a tens frame.

You might have seen a tens frame before.

If you haven't, here it is and it's full because there's 10 parts and I'm representing John's 10 pence.

Misha has three p in her money box.

I'm going to use a different tens frame initially.

And I've showed that by filling three of the parts up with a red counter.

It asks us, John has more than Misha.

So we need to work out the difference between 10 and three.

Now I know you will know the answer to that, but I'm going to show it to you in a certain way.

I'm going to put Misha's counters on top of John's counters.

And the difference between these two numbers is the number of counters that haven't overlapped.

So if I look on here, I can see that there is seven counters that haven't overlapped.

There's five down the left, two on the right, five and two is seven.

So the difference is seven.

That means John has seven p more than Misha.

Now, there's a new part to our question.

John adds 10p to his money box.

How'd you think I can show that on our representation? I can use another tens frame, can't I? So here I've now showed John's money.

Does he still have 10p? No, he's got 10p more.

And if we think about the section that would all be blue, John now has 20p.

How much more money does he now have than Misha's? Has Misha's amount of money changed? No, it's still three, it's still shown with the red counters on our tens frame.

Has the difference changed? Yeah, because we've added more blue counters for John's money, that means the difference has changed.

What's the difference now, can you see? Now hopefully no one's adding up in ones and counting or the counters, the tens frame works.

I know there's 10 at the top, I know there's seven blue counters in the bottom, 10 and seven is 17.

He now has 17 more p than Misha.

Okay final part of our question.

Now John adds another 10p to his money box.

Picture in your head how that will now look, what am I going to do? Is that what you thought? I've added another tens frame to represent the 10 more p that he's put in his money box.

It's now 30p because there's three lots of 10.

Has the difference changed? It must've done, Misha's amount hasn't changed, there's still three counters.

So if I've changed it from 20 to 30, there's a difference there of 17? No the difference increases, let's change that.

The difference now is 27.

He now has 27p more than Misha.

Now I want to show you that sequence again.

So we started off with ten subtract three is seven.

See if you can notice any patterns.

We then added 10 more and changed it to 20 subtract three is 17.

We then added 10 more again and changed it to 30 subtract three is 27.

Did you notice the pattern? Let me show you them all together.

So what's the same and what's different in our three representations? Think about the minuends, think about the subtrahends, think about the difference.

Pause the video, have a think.

So what did you notice? Let's start with the minuend, does the minuend change each time? Definitely does.

We had 10 as our minuend first, then 20, then 30.

So how much has the minuend increased by each time? It's increased by 10.

And we can see on our representation where that 10 is shown.

It's shown here on the middle one and then our next extra 10 is then shown by our next extra 10 frames.

What about the subtrahend? Does the subtrahend change? In this question the subtrahend with the amount of money that Misha had, does that change? No, the subtrahend stays the same.

And what do we say about the difference? Does the difference change? Is the difference different? We can see in our three equations, we have a difference of seven, a difference of 17 and a difference of 27, it's changing.

What's it changing by? It's changing by 10 as well.

So do you notice that if we increase the minuend by 10 and the subtrahend stays the same, the difference will change by the same amount that we change the minuend by.

Let me show you that in a slightly different way.

So I've given us a sentence STEM that we can use to describe what's happening.

It says I've added to the minuend and kept the subtrahend the same.

So I must add to the difference.

Can you remember what our next equation was? We added 10 to the minuend, we added 10 to the difference, the subtrahend stayed the same.

Our next equation was 20 subtract three is equal to 17.

So in that first example, I've added 10 to the minuend and kept the subtrahend the same.

So I must add 10 to the difference, can you see that? Great, okay, the next equation, what did we do? We added 10 to the minuend, we kept the subtrahend the same.

So I must add 10 to the difference.

We had our final equation, 30 subtract three is equal to 27.

So let's compare that third equation to the second equation.

I've added 10 to the minuend and kept the subtrahend the same.

So I must add ten to the difference, great.

I'm going to show you the same context with a different representation.

I'm going to show you the context on a bar chart.

So here I've got the amount of pence on the left and they're going to show the different occasions on the bottom.

So to start with, I got John's money shown in blue and I got Misha's money shown in red.

John's money goes up to 10 because at the beginning he had 10 pence.

Misha's money goes part of the way up to where three would be on our axis.

Why would the difference be shown? The difference is the part that is in between the top of Misha's bar and the top of John's bar.

What was our equation for this? 10 subtract three is equal to seven.

What happened next, what did we do? What did we change? Did we change Misha's money? Did we change John's money? Did we change the difference? We changed two of those, didn't we? We changed John's money and the difference therefore changed.

So my next bar looks like this.

John's money has increased to 20 pence.

Misha's money has stayed the same.

So John's money was our minuend, Misha's money was our subtrahend, and the difference must also have changed.

Seven was still there, but we have a new increase to our difference has increased by 10.

So what would our equation be now? How much money is it John had, that's our minuend? John had 20 pence.

How much money does Misha have? That's still three pence.

And what's our difference? Is increased, what was it? Great, 20 subtract three is equal to 17.

Now some of those numbers I can see if I look carefully.

Blue bar goes up to the line that shows 20, that's our minuend.

Misha's bar stayed the same, that's still three pence.

So it's still subtract three.

17, can you see 17 there? Ah, no but is 10 and seven because the difference before was seven, we've added on 10 more to the minuend so our difference has also increased by 10.

What is the next bar going to look like? Pause the video, see if you can explain if someone's there or tell yourself what it's going to look like and what was the equation? So, the next one was this.

John had 30 pence, Misha's money stayed the same.

What did you think? What lines are going to be on my bar this time? Seven is still the same, but this time is 20 pence.

Why'd you think 20, sorry, is shown in purple? What are we comparing it to? We're comparing this to the first two bars.

We're saying, the red bar, Misha's money, stayed the same.

The difference between what it was in the first bar between the red and the blue bar has stayed as seven.

But the amount of money here, let me show you, the amount of money that John has increased by from 10 has gone up by 20.

So this 20 is comparing this bar to this bar.

What was the equation? How much money does John have? It shows here that John has 30 pence, that's going to be our minuend.

This bar is still the same height it's going to be three is our subtrahend.

And what's the difference? What's 20 and 17? 20 and seven, sorry? The difference is 27.

So the next expression, the next calculation is 30 subtract three is equal to 27.

Now I'm going to put one more to continue our pattern.

What do you think the next set of bars is going to look like? Is that what you thought? Now, how much money has John got? John's got 40 pence.

Has the amount that Misha's got changed? No, that stayed at three pence.

So what's going to happen to our difference? Is it going to stay at 27? No, it's going to increase, isn't it? So how much bigger is it than the first bar? Remember the first difference was seven.

So how are we going to show the difference here? We're going to say it's still seven up to this line.

How much bigger is this? This difference will be three lots of 10, which is 30.

This is our calculation.

40 subtract three is equal to 37.

Okay, I've got a new context for you, still money, but two different children.

We have Evie and we have Hari.

Evie has 100 pence in her money box and Hari has two pence.

I've shown that with 100 here on the right hand side of my number line.

And Hari's amount of two pence here on the left.

If I want to know how much more Evie has than Hari, I'm going to do 100 subtract two.

What is 100 subtract two? It's 98.

So the size of the difference is 98.

So I can complete my calculation with is equal to 98.

And I can say, what do you think I can say if I'm comparing Evie and Hari's money? Evie has 98 pence more than Hari.

Now I'm going to change something.

On Monday, Evie spends 10 pence.

How's that going to change my representation? Which number's going to change? Is my two going to change? Is my 100 going to change? It's my 100, isn't it? 100 represents the amount of money that Evie has, but Evie spent 10p.

So is our minuend going to get larger or smaller? It's going to get smaller.

Let me show you that on our number line first.

Is going to go from 100 to 90 because 90 is 10 fewer than 100.

What's our calculation going to be? 90 subtract two is equal to 88.

So to complete our calculation, 90 subtract two is equal to 88.

Evie has 88 pence more than Hari.

I'm going to do it one more time.

On Tuesday, Evie spends another 10 pence.

What's going to happen this time? Can you picture it in your mind, what? The amount that Evie has has decreased by 10.

So what would our calculation be now? 80 subtract two is equal to, what's the difference? 78.

So I can complete my calculation.

80 subtract two is equal to 78.

She now has 78p more than Hari.

Okay let's look at that like we did before.

Here's our first calculation, 100 subtract two is equal to 98.

Did our minuend change? It did, didn't it? How much did it change by? What about our difference, how much did that change by? They both changed by the same amount, which was 10.

So let's describe what's happened with our STEM sentence.

I've subtracted 10 from the minuend and kept the subtrahend the same.

So I must subtract 10 from the difference.

What was our next calculation? What is 10 less than 100? It's 90.

So 90 subtract two is equal to 88.

Nine tens has become eight tens because we've subtracted one 10 and our ones have stayed the same.

What happened next in our problem? Evie spent more money, didn't she? How much did she spend this time? She spent another 10p.

So her minuend or the minuend in our calculation decreases by 10.

What happened to our subtrahend? It stayed the same.

And what happened to our difference? Did it stay the same? No, it decreased.

So the next line would be the same as before that we're subtracting 10 from our minuend.

We've kept our diff, sorry, our subtrahend the same.

So I must subtract 10 from my difference.

The next calculation was 80 subtract two is equal to 78.

So let's look at those side by side.

On the left we've got John's money and Misha's money and on the right we've got Evie and Hari's money.

What's the same and what's different about both of these? Let's see if we can come up with a generalisation from what we've learned so far in today's lesson.

So let's look at the minuend increased by 10 that's when John put more money in his money box.

So the difference increased by 10.

Then that happened again, increased by 10 more.

So the difference increased by 10 more.

That, what happened? Oh yeah, the subtrahend was the same all the time.

What about in our second problem with Evie and Hari? It decreased by 10, didn't it? She spent some money.

So the minuend became 10 fewer, which meant the difference became 10 fewer.

And that happened again.

We subtracted, didn't we? From the minuend so we had to subtract from the difference.

So let's look at our generalisation.

If the minuend is changed by an amount and the subtrahend is kept the same, the difference changes, what could we say? The minuend is changed when the first on the left is changed by adding and on the right it's changed by subtracting.

Then you'll notice that both those examples actually are by 10.

So the difference changes, for these, by 10, but we can say by the same amount.

Do you think it always has to be by 10, what'd you think? Would it work for other numbers, for other amounts of change? Let's have a look.

Let's look at this calculation.

20 subtract six is equal to 14.

I'm going to show you another calculation.

32 subtract six is equal to 26.

So what's changed? How much has the minuend increased or decreased by? It's increased by 12.

Has our subtrahend changed? No, has our difference changed? Yes, by how much? It's also increased by 12.

This time 44 subtract six is equal to 38, what's happened? Our minuend has increased by 12, our subtrahend has stayed the same, our difference has also increased by 12.

So that time both the minuend and the difference increased by the same amount each time.

Let me show you a different calculation.

99 subtract nine is equal to 90, what happens this time? What happens when I change my calculation? 96 subtract nine is equal to 87.

Has my minuend changed? Yeah, has it increased or decreased? It's decreased, it's decreased by three.

What about my subtrahend, has that changed? No, stayed the same.

What about my difference, has that changed? Yeah, that's decreased by three as well.

One more, 93 subtract nine is equal to 84.

What's happened to my minuend? Say it with me, it's decreased by three.

My subtrahend has stayed the same.

My difference has also decreased by three.

So let's bring up that generalisation we have before.

If the minuend has changed by an amount and the subtrahend is kept the same, the difference changes by the same amount.

That works for all values.

We've seen 10 for increase and decrease, 12, three, that will always be the case.

So for your practise activities, I'd like you to look at these two sets of calculations.

I want you to use the STEM sentences to describe what's happening in each pair of calculations.

I've given you all the minuends, all the subtrahends and all the differences, you have to just work out what goes in those blue boxes to describe the change.

Can you fill in those boxes? And can you complete the STEM sentence for each pair of calculations? Then I'd like you to still think about the same pairs of calculations, but I want you to create a math story to try and choose an appropriate context for the calculations.

So today so far, our context has been money.

I want you to think of your own.

Maybe you could use children in our class, think about how our class might be made up in different ways and what could change and what could stay the same thinking still about difference.

And maybe what about our length of ribbon? Could you use any of those numbers to describe what our length of ribbon could be and how that could compare to something else? And again, the minuend would change, the subtrahend would stay the same, but the difference would also change.

So get your thinking caps on right Mr. Math Stories.

And we'll go through that in the next lesson.