Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson from our unit on using equivalences to calculate.

So are you ready to use all your knowledge of the different operations and think about how the numbers within them relate to each other, how we can maintain equivalence whilst changing the way an equation looks? If you're ready, let's make a start.

So in this lesson, we're going to be explaining the effect on the quotient when scaling the dividend and the divisor by the same amount.

We've got 3 key words in this lesson, dividend, divisor and quotient.

You may know them, but let's just rehearse them and then think about what they mean as well.

So are you ready? I'll take my turn, then it'll be yours.

My turn.

Dividend.

Your turn.

My turn.

Divisor.

Your turn.

My turn.

Quotient.

Your turn.

Okay, let's explore what those words mean, 'cause they're gonna be really useful to us today when we are thinking about our division equations.

So the dividend is the amount that you want to divide.

The divisor is the number we divide by, and the quotient is the result after division has taken place and it's the whole number part.

So if we imagine 45 divided by 9 is equal to 5, the dividend is 45, the number that we want to divide.

The divisor, the number we divide by in this case was 9, 45 divided by 9, and it's equal to 5.

That's the quotient, the result after the division has taken place.

So there are two parts in our lesson today.

In the first part, we're going to be using multiplication to scale, and in the second, we're going to be using division.

So let's make a start on part one.

And we've got Andeep and Laura helping us with our learning today.

There are 3 jelly beans.

You might have come across Laura and Andeep and their jelly beans before, I wonder.

Laura says, "I take all 3 of the jelly beans." Oh, slightly selfish, Laura.

Let's hope Andeep will get some later.

So there's Laura and her 3 jelly beans.

Andeep says, "We could say that 3 jelly beans shared between 1 person is equal to 3 beans," and we could record that with a division equation.

3 divided by 1 is equal to 3.

Laura says, "What about if there are 6 jelly beans, and the jelly beans are equally shared between 2 children?" Oh, what do you notice? Andeep says, "There are double the number of beans, but there are also double the number of children." So we've doubled the number of beans, and we've doubled the number of children, but they each get the same number.

6 divided by 2 is also equal to 3.

What's happened? Think about those words we were thinking about at the beginning of the lesson, dividend, divisor, quotient.

What's happened? 6 jelly beans divided between 2 children is equal to 3 jelly beans each.

Ah, if the dividend and the divisor are both doubled, the quotient stays the same.

Can you see 3 times 2 is equal to 6, and 1 times 2 is equal to 2? So the dividend and the divisor have both been doubled, but the quotient, the number of jelly beans each child gets remains the same.

There are 9 jelly beans now, and the jelly beans are shared equally between 3 children.

What do you think is gonna happen this time? They're getting 3 each, aren't they? There are 3 times as many beans, but there are also 3 times as many children.

So if we think back to our 3 divided by 1 is equal to 1 when Laura ate all the jelly beans, we've now got 3 times as many children, but we've also got 3 times as many jelly beans.

3 times 3 is equal to 9, 1 times 3 is equal to 3.

9 divided by 3 is also equal to 3.

9 jelly beans divided between 3 children is equal to 3 jelly bean each.

If the dividend and the divisor are both scaled by the same amount, the quotient will remain the same.

Can you see how we've changed that generalisation? First of all, we were just thinking about doubling, but now we're thinking is this true if we scale by any amount? What about if there are 12 jelly beans and they're shared equally between 4 children? What can you see now? They're still getting 3 each, aren't they? Andeep says, "There are 4 times as many beans, but there are also 4 times as many children." 3 times 4 is equal to 12, 1 times 4 is equal to 4.

12 jelly beans divided between 4 children is equal to 3 jelly beans each again.

So our statement holds true again, doesn't it? If the dividend and the divisor are both scaled by the same amount, the quotient will remain the same, and we've now tried that out for doubling, multiplying by 3 and multiplying by 4.

Okay, can you predict what's gonna happen then with these 15 jelly beans shared equally between 5 children? That's right, 3 each again.

There are 5 times as many beans, but there are also 5 times as many children.

So we've scaled up our dividend and our divisor this time by 5.

15 jelly beans divided between 5 children is equal to 3 jelly beans each, and our statement is still holding true.

If the dividend and divisor are both scaled by the same amount, the quotient will remain the same.

Oh, big jump now.

45 jelly beans are shared between 15 children.

How many jelly beans do each of the children get? And Andeep says, "I can't think easily about 45 divided into 15 groups.

I wonder if I can scale the equation to make it more manageable." He says, "I know if 15 jelly beans are divided equally between 5 children, it's equal to 3 jelly beans each." He says, "There are 3 times as many beans, but there are also 3 times as many children." 15 times 3 is equal to 45.

5 times 3 is equal to 15.

So because both the dividend and the divisor have been scaled by the same amount, the quotient remains the same.

Each of the children still gets 3 jelly beans.

If the dividend and the divisor are both scaled by the same amount, the quotient will remain the same.

So now we've got 180 jelly beans shared between 60 children, how many jelly beans do each of the children get? Well, Andeep says, "I know that 45 jelly beans divided equally between 15 children is equal to 3 jelly beans each," so can we use that as a starting point? There are 4 times as many beans, but there are also 4 times as many children.

45 multiplied by 4 is equal to 180, and 15 multiplied by 4 is equal to 60.

Each of the children still gets 3 jelly beans, because we've scaled the dividend and the divisor by the same amount, 4 in this case.

Time to check your understanding.

75 jelly beans are shared between some children.

Each of the children gets 3 jelly beans each.

How many children are there? So this time, we've got a missing divisor.

And Andeep says as a starting point, "We know that 15 jelly beans divided between 5 children is equal to 3 jelly beans each." So can we use that to work out the missing divisor? How can you use scaling to help you solve this problem? Pause the video, have a go, and when you're ready for some feedback, press play.

Well, Laura spotted that there are 5 times as many beans, and the children still get the same number each.

The quotient hasn't changed, so there must be 5 times as many children as well.

5 times 5 is equal to 25.

So 75 jelly beans shared between 25 children is equal to 3 jelly beans each.

We'd scaled the dividend and the divisor by the same amount.

So they're going to use their scaling to solve some other problems. Laura's spotted in this first one, 12 multiplied by 3 is equal to 36.

The quotient in both equations is equal to 4, so she says, "I have to multiply 3 by 3 as well." She's got to scale the dividend and the divisor by the same amount.

3 times 3 is equal to 9.

So 12 divided by 3 is equal to 4, and 36 divided by 9 is also equal to 4.

Andeep spotted something in the second pair.

He says, "36 multiplied by 5 is equal to 180.

The quotient in both equations is 12, so I have to multiply 3 by 5 too.

3 times 5 is equal to 15." So 36 divided by 3 is equal to 12, and 180 divided by 15 will also be equal to 12.

Time to check your understanding.

Can you use scaling to find the missing number? Laura says, "Find the missing dividend." And Andeep's hint is, "8 multiplied by what number is equal to 32?" And can we then use that to help us? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Andeep says, "8 multiplied by 4 is equal to 32 and the quotient in both equations is 5, so you have to multiply the 40 by 4 too." We must scale the dividend and the divisor by the same amount.

40 multiplied by 4 is equal to 160, so the missing dividend is 160.

I hope you were successful with that.

Andeep and Laura find the missing dividend.

Andeep says, "Both sides of the equation are equal." Laura says she knows that 14 divided by 2 is equal to 7, and she spotted that 2 multiplied by 4 is equal to 8, so our divisor has been scaled up by a factor of 4.

And Andeep says 14 also needs to be multiplied by 4.

We must scale the dividend and the divisor by the same factor.

So, "To multiply by 4, I can double and double again," says Laura.

14 times 2 is equal to 28.

28 times 2 is equal to 56.

So the missing dividend is 56.

And if you know your times tables, it's my favourite times table fact, 56 divided by 8 is equal to 7, and 14 divided by 2 is also equal to 7.

Time to check your understanding.

Can you find the missing dividend in this equation? "How can you use scaling to help you?" says Andeep.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, Laura says she knows that 72 divided by 12 is equal to 6.

It's in her times table knowledge.

And she's also spotted that this divisor has been multiplied by 3, 12 has been multiplied by 3 to equal 36.

So Andeep says 72 also needs to be multiplied by 3 because we must scale the dividend and the divisor by the same amount for the quotient to remain the same, and we know that these 2 expressions are equal.

Laura says I'll partition 72 into 70 and 2.

70 multiplied by 3 is 210.

2 multiplied by 3 is 6, and 210 plus 6 is equal to 216.

So 72 multiplied by 3 is 216, and that's our missing dividend.

Well done if you got that right.

Time for you to do some practise now.

So for question one, can you complete the equations? Andeep says, "If the dividend and the divisor are both scaled by the same amounts, the quotient will remain the same." And Laura's reminding you, find the missing dividends and divisors.

And for question two, can you complete the equations? Andeep says the missing number needs to balance the equation.

And Laura's asking you to spot, "Has the divisor or the dividend been multiplied by 2, 3, 4 or 5?" Pause the video, have a go at the two questions, and when you're ready for some feedback, press play.

How did you get on? So here are the answers to question one.

And Andeep's saying, "If the dividend and the divisor are both scaled by the same amount, the quotient will remain the same." Laura's focused in on B.

She says, "Both the dividend and the divisor have been multiplied by 5.

24 multiplied by 5 is equal to 120." 6 multiplied by 5 is equal to 30, so 120 divided by 30 is equal to 4.

And here are the answers to question two.

Andeep's focused on A.

He says, "20 has been multiplied by 4 to equal 80, so 5 also needs to be multiplied by 4." He hasn't worked out the quotient this time.

He just knows that we've got dividends and divisors, and if they are scaled by the same amounts then they will still be equivalent.

And Laura's focused on C.

12 has been multiplied by 2 to equal 24, so 48 also needs to be doubled, and double 48 is 96.

So 48 divided by 12 will be equal to 96 divided by 24.

I hope you were successful with those.

And onto part 2 of our lesson, using division to scale.

So the dividend and the divisor can also be scaled down.

We've been scaling up in the first part of our lesson, but we can also scale them down.

If they're divided by the same amount, the quotient remains the same.

And Andeep says this can be really useful for helping you to solve problems. And Andeep says, "Let's have a look at some problems to help us understand how useful this is." Laura's gone back to the jelly beans.

"15 jelly beans divided between 5 children is equal to 3 jelly beans each," and there's our image and our equation as well.

What happens if there is just Laura and 3 beans? How many would she get? Do you remember this from part one? Well, 15 divided by 5 is equal to 3, but when it was just Laura, she just had 3 jelly beans and 1 person.

There are 1/5 the number of beans, but there's also 1/5 the number of children.

We need to divide the dividend and the divisor by 5.

15 divided by 5 is equal to 3, and 5 divided by 5 is equal to 1.

Laura would get 3 jelly beans.

There's 1/5 times the number of jelly beans and 1/5 times the number of children, and to find 1/5, we divide by 5.

So we can say that if the dividend and divisor are both scaled by the same amount, the quotient will remain the same, and that works for scaling up and scaling down.

So let's see if we can use scaling down to help us.

300 jelly beans are shared between 12 children.

How many jelly beans does each child get? Andeep says, "To make the calculation easier, I could divide both the dividend and the divisor by 6." 300 divided by 6 is equal to 50 and 12 divided by 6 is equal to 2.

There are 1/6 the number of beans, but there are also 1/6 the number of children.

To find 1/6, we divide by 6.

Andeep says, "I know 50 divided by 2 is equal to 25." So Laura says, "300 divided by 12 must also be equal to 25." We've scaled the dividend and the divisor down by the same amount, so the quotient must remain the same.

Andeep and Laura are looking for the missing dividend here.

We've got 240 divided by 60 is equal to something divided by 15.

Laura says, "We can use scaling to find the missing dividend." Andeep says, "I know that 60 divided by 4 is equal to 15, so we also have to divide 240 by 4." To divide by 4, we can halve and halve again.

Half of 240 is 120, half of 120 is 60.

So the missing dividend is 60.

240 divided by 60 is equal to 60 divided by 15, and we know that they are equivalent, so therefore their quotients would be the same.

Time to check your understanding.

Can you find the missing dividend? Use scaling to find that missing dividend.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, Andeep spotted 18 divided by 3 is equal to 6, so our divisor has been scaled down by 3.

To find the missing dividend, we must divide 450 by 3, and 450 divided by 3 is equal to 150, so the missing dividend is 150.

450 divided by 18 is equivalent to 150 divided by 6.

We've scaled the dividend and the divisor down by the same amount.

So Andeep and Laura write expressions equal to 240 divided by 48.

Laura says, "I can halve 240 and I must also halve 48, so 240 divided by 48 is equal to 120 divided by 24." Andeep says, "I can divide 240 by 3 and I can also divide 48 by 3.

240 divided by 48 is equal to 80 divided by 16." Time to check your understanding.

Can you write another expression equal to 240 divided by 48? Laura says, "Divide 240 and 48 by the same amount and find another expression equal to 240 divided by 48." Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Laura says, "You could divide 240 by 4 and you would also have to divide 48 by 4.

240 divided by 48 is equal to 60 divided by 12." And Andeep says, "You could divide 240 by 6 and you'd also need to divide 48 by 6.

240 divided by 48 is equal to 40 divided by 8." Did you find any other possibilities? I hope you enjoyed playing around with those equivalent expressions.

And it's time for you to do some practise.

In question one, you're going to complete these equations.

And Andeep's saying, "What has the dividend or divisor been divided by?" And Laura says, "Make sure both dividend and divisor are both divided by the same amount." So you're looking to find some missing dividends and divisors to make these equations equivalent.

And in question two, you're going to find different ways to complete each equation.

Andeep says, "Use division to scale the dividend and the divisor." And Laura says, "Work out the missing quotient." So pause the video, have a go at questions one and two, and when you're ready for some feedback, press play.

How did you get on? So here are the answers for question one.

Andeep's focusing on A.

He says, "In A, 240 has been divided by 3, so 12 must be divided by 3 to make the equation correct." 240 divided by 12 is equal to 80 divided by 4.

And Laura's looked at B.

She says, "In B, 300 has been divided by 5, so 25 must be divided by 5 to make the equation correct." 300 divided by 5 is equal to 60.

25 divided by 5 is equal to 5.

So 300 divided by 25 is equal to 60 divided by 5.

And in question two, here are some possible answers.

So Andeep says, "For A, 360 and 60 could both have been divided by 5.

360 divided by 60 is equal to 72 divided by 12.

Which one of those gave you the easiest way into the answer? I think I'd have taken 120 divided by 20, divided both my dividend and divisor by 10 and made 12 divided by 2.

And I know 12 divided by 2 is equal to 6, so if that equation is equal to 6, all the others must have a quotient of 6 as well.

What about B? 1,800 and 120 could both be divided by 6 to make an equivalent equation of 300 divided by 20.

And again, you could have scaled that one down by 10 and had 30 divided by 2, which is equal to 15.

And I think that might have been my way to get my quotient of 15.

And if all of those equations have the dividend and the divisor scaled down by the same amount, then all of them will have a quotient of 15.

I hope you enjoyed playing around with those numbers.

And we've come to the end of our lesson.

We've been explaining the effect on the quotient when scaling the dividend and the divisor by the same amount.

So what have we learned about? Well, we've learned that if the dividend and divisor are both scaled by the same amount, the quotient will remain the same.

We've also seen that unitizing helps to explain the effect on the quotient when scaling the dividend and the divisor by the same amount.

So an example, if 6 divided by 3 is equal to 2, then 6 twos divided by 3 twos is equal to 2.

Thank you for all your hard work in this lesson.

I hope you've enjoyed learning about scaling up and down dividends and divisors, and realising that the quotient stays the same.

Thank you for your hard work and your mathematical thinking, and I hope I get to work with you again soon.

Bye-bye.