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 are going to be explaining the effect on the quotient when scaling dividend and the divisor by 10.

Let's explore those words.

Yes, those are our three keywords today.

We've got dividend, divisors, and quotient.

So I'll take my turn, and then it'll be your turn.

Are you ready? My turn, dividend, your turn.

My turn, divisor, your turn.

My turn, quotient, your turn.

Let's explore and remind ourselves what those words mean.

So the dividend in a division equation is the amount that you want to divide.

The divisor is the number that we divide by.

And the quotient is the result after division has taken place.

It's the whole number part.

So if we had 45 divided by 5 is equal to 9, our dividend is 45, our divisor, the number we're dividing by, is five, and our quotient is nine.

There are two parts to our lesson today.

In the first part, we're going to be scaling up by 10, and in the second part we're going to be scaling down by 10.

So let's make a start on part one.

And we've got Andeep and Laura helping with our learning in the lesson today.

So Andeep and Laura have some jelly beans, lucky them.

Andeep says, "I have four beans.

I eat one each day." Laura says, "I have 40 beans.

I eat 10 each day." Lucky Laura.

Well, if you're a fan of jelly beans, then lucky Laura.

How many days do Andeep's and Laura's jelly beans last for? So they eat their jelly beans.

And Andeep says, "I eat one jelly bean each day." And Laura says, "I eat 10 jelly beans each day." So on day one, Andeep eats one, and Laura eats 10.

Same again on day two, and on day three, and on day four.

What do you notice? Andeep says, "My jelly beans lasted for four days." Laura says, "My jelly beans also lasted for four days." So both their sets of jelly beans lasted them for four days.

Andeep had four and ate one each day.

Laura had 40 and ate 10 each day.

Andeep and Laura think about why this happened, and they have recorded them as equations.

Andeep says, "Four divided by one is equal to 40 divided by 10." For both children, the jelly beans lasted for four days.

So the quotient, the result after they divided, was four.

So four divided by one is equal to 40 divided by 10.

And Laura spots something.

The dividend in 40 divided by 10 is 10 times the size of the dividend in four divided by one.

4 multiplied by 10 is equal to 40.

Andeep's also spotted that the divisor in 40 divided by 10 is 10 times the size of the divisors in four divided by one is equal to four.

The divisor in four divided by one is equal to four is one, and that multiplied by 10 is equal to 10, which is the divisors in 40 divided by 10 is equal to 4.

And Andeep says, "The quotient is the same in both equations." The results when we've divided is four in both cases.

So they're going to suggest something here.

If the dividend and divisor are both multiplied by 10, the quotient will remain the same.

And we can see that in these two equations.

We've multiplied the dividend and the divisors by 10, but the quotient has remained the same.

So why does the quotient remain the same? Let's explore it in a different representation.

Four is divided into equal parts of one.

There are four equal parts and the quotient is four.

Four tens is divided into equal parts of one 10.

So can you see we've used our unitizing language, 40 is the same as four tens.

So rather than four ones being divided into equal parts of one, this time we've got four tens being divided into equal parts of one 10.

And there are four of those one tens.

There are four equal parts, so the quotient is four again.

And we can see again in this representation, if the dividend and the divisors are both scaled up by 10, the quotient will remain the same.

Our dividend was four, and our divisors was one in Andeep's case, and we've made four equal parts, so the quotient is four.

And in Laura's case the dividend was 40, and the divisors was 10.

But again, we've got four equal parts, so our quotient is still four.

Andeep and Laura are going to compare some other equations.

So we've got six divided by two is equal to three, a fact we know from our times tables, but let's just represent it, and think about what's happening here.

There are three groups of two ones in six.

We're dividing by two, we're trying to find out how many groups of two ones there are in six.

And there are three groups of two ones in six.

What about 60 divided by 20? Well, Laura says, "There are three groups of two tens in six tens, or 60." So 60 divided by 20 is also equal to three.

There are three groups of two tens, or 20, in 60.

There are three groups of two, or two ones, in six.

And we can see that in the bar model.

And again, we can see the dividend has been multiplied by 10, 6 times 10 is equal to 60, and the divisors has been multiplied by 10, 2 times 10 is equal to 20.

The quotient, or the number of groups, is the same in both equations.

We made three groups of two ones from six, and we made three groups of two tens from 60.

So they carry on, and compare some other equations.

We've got 40 divided by 8 is equal to 5.

Again, we know that fact from our times table knowledge.

We know that there are five groups of eight ones in 40.

So if we take 40 and we make groups of eight ones, we will make five of those groups.

What about 400? Can we think about that? And Laura says, "There are five groups of eight tens in 40 tens, or 400." So we can still see that 40 and that 8, but this time we are thinking about tens.

There are 40 tens, and we're dividing them into groups of eight tens.

And we can see again in both that we make five equal groups.

The dividend has been multiplied by 10, 40 multiplied by 10 is 400.

And the divisors has been multiplied by 10, 8 multiplied by 10 is equal to 80.

But the quotient, or the number of groups, is the same in both equations.

We had five groups of eight ones in 40, and five groups of eight tens in 400.

What about 36 divided by 9? We know it's equal to four from our times table knowledge, and there is our bar model to show it.

There are four groups of nine ones in 36.

What about 360 divided by 90? Can you think about what the sentence is going to read for that? That's right, there are four groups of nine tens in 36 tens, or 360.

So we could also say there are four groups of 90 in 360.

The dividend has been multiplied by 10, 36 multiplied by 10 is 360.

And the divisor has been multiplied by 10, 9 times 10 is equal to 90.

But the quotient, or the number of groups, is the same in both equations.

And by using our knowledge of unitizing, thinking of dividing into groups of ones and groups of tens, we can see that that quotient of four is the same in both equations.

They're going to use what they know this time to see if they can find the missing divisor.

So nine divided by three is equal to 90 divided by something.

And Andeep says, "There are three groups of three ones in nine." Now we can see it in the bar model.

So Laura says, "There are three groups of three tens in nine tens, or 90." Or 9 times 10 is equal to 90.

3 times 10 must be equal to 30.

So our missing divisor is 30.

Over to you to test your understanding.

Can you find the missing divisor in this pair of equivalent expressions? 15 divided by 5 is equal to 150 divided by something.

So pauses the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? Andeep says, "There are three groups of five ones in 15." And there's the bar model.

And Laura says, "So there will be three groups of five tens in 15 tens, or 150." And there's the bar model.

What's going to be the size of that group? Well, Andeep says, "The dividend has been multiplied by 10." 15 multiplied by 10 is 150.

So the divisors also needs to be multiplied by 10 for the quotient to remain the same.

50 multiplied by 10 must be equal to 50.

So our missing divisor is 50.

And if we think about 150 divided into groups of five tens, we will have three of them.

150 divided by 50 must be equal to three.

Time for you to do some practise.

So in question one, you're going to complete the equations.

And we've given you some scaling hints there.

And Andeep's reminding you, "If the dividend and divisors are both multiplied by 10, the quotient will remain the same." And Laura says, "You could use bar models to help you represent each equation." And in question two, you're going to complete the equations again.

And Andeep says, "The missing number needs to balance the equation." And again, Laura is saying, "You could use bar models to help you to represent each equation." We've not given you any scaling hints this time, so it's over to you to think about applying that scaling up by 10 to find these missing values.

So pause the video, have a go at your two questions, and when you're ready for some feedback, press Play.

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

So you might want to pause and have a little look, and then we'll look at a couple in detail.

Okay, so Andeep says, "In A, the dividend has been multiplied by 10, so the divisor also needs to be multiplied by 10." 30 multiplied by 10 is equal to 300.

And 3 multiplied by 10 is equal to 30.

30 divided by 3 is equal to 10.

There are 10 groups of three ones in 30.

And 300 divided by 30 must be equal to 10.

There must be 10 groups of three tens in 30 tens.

300 divided by 30 is equal to 10.

Laura says, "In B, the divisor has been multiplied by 10, so the dividend also needs to be multiplied by 10." We knew that 24 divided by 6 is equal to 4.

So something divided by 60 was equal to 4.

Well, if we're thinking about four groups of six ones in 24, there are four groups of six tens in 24 tens, 240.

And you can see the same thinking in C as well.

And here are the answers to two.

Again, you might want to pause, and just check against your working, and then we'll focus in on a couple.

Andeep's looked at A.

"20 has been multiplied by 10 to equal 200, so 4 also needs to be multiplied by 10 to equal 40." So 20 divided by 4 must be equal to 200 divided by 40.

This time we haven't actually worked out the quotient, we haven't evaluated those expressions, but we know from the work we've done that if we scale up the dividend by a factor of 10, we must also scale up the divisors by a factor of 10 for the expressions to be equivalent.

Laura's picked on C.

She says, "In C, five has been multiplied by 10 to equal 50, so 50 also needs to be multiplied by 10." So we knew 50 divided by 5 was equal to something divided by 50.

So we can see that our divisor has been scaled up by 10, so our dividend also needs to be scaled up by 10.

50 times 10 is equal to 500.

So 50 divided by 5 must be equal to 500 divided by 50.

Well done if you've got those correct.

And time for the second part of our lesson.

This time we're going to look at scaling down by 10.

So Andeep wonders what happens when the divisor and dividend are scaled down by 10.

So he knows that 210 divided by 70 is equal to 3.

Andeep says, "There are three groups of seven tens in 21 tens, or 210." Three groups of 70 in 210, we could also say.

So now they're looking at 21 divided by 7, which they know is equal to three from their times table knowledge.

And you can see the similarity there in the bar model.

And Laura says, "There are three groups of seven ones in 21." So those are equivalent equations again.

The dividend has been divided by 10 this time.

210 divided by 10 is equal to 21.

And the divisors has been divided by 10.

70 divided by 10 is equal to 7.

So if the dividend and the divisors are both divided by 10, the quotient will remain the same.

They're going to look at another example.

320 divided by 80 is equal to 4.

You might see a times table fact you know in there that helps you to work that one out.

But we can see it there represented in a bar model.

There are four groups of eight tens in 32 tens, or 320.

Ah, was that the times table fact you were using? There are four groups of eight ones in 32.

So again, we can see the dividend has been divided by 10.

320 divided by 10 is equal to 32.

And the divisors has been divided by 10.

80 divided by 10 is equal to 8.

So if the dividend and the divisors are scaled down by 10, the quotient will remain the same.

And you can see that clearly in the bar models as well.

Over to you to check your understanding.

What is the missing dividend? Something divided by six is equal to five.

And we've got another equation there.

300 divided by 6 is equal to 5.

So pause the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? Well, Andeep says, "There are five groups of six tens in 30 tens, or 300." 300 divided by 60 is equal to 5.

So Laura says, "The missing dividend is equal to five groups of six ones." And we can also think about how the divisors and the dividend have been divided by 10.

So the divisors has been divided by 10.

60 divided by 10 is equal to 6.

And the dividend also has to be divided by 10.

300 divided by 10 is equal to 30.

So we know that our missing dividend is 30.

But Laura's way of thinking about the missing dividend must be five groups of six is an interesting way of thinking about it as well.

And we can also see that five groups of 60 are equal to 300.

They're gonna think about some decimal numbers now.

So we've got 180 divided by 45 is equal to 4.

180 is divided into four equal parts of 45, so there are four lots of 45 in 180.

What about if we divide the divisor and the dividend by 10 again? 18 is divided into four equal parts of 4.

5.

There are four lots of 4.

5 in 18.

The dividend has been divided by 10.

Or 180 divided by 10 is equal to 18.

And the divisors has been divided by 10.

45 divided by 10 is equal to 4.

So we've shown here that with decimals as well, if the dividend and the divisors are both divided by 10, the quotient will remain the same.

And that will happen whatever the scale of the number.

Time to check your understanding now.

What's the missing divisor in this equation? So we've got six divided by something is equal to five.

And we've also got 60 divided by 12 is equal to 5.

So pause the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? Andeep says, "We can see that 60 is divided into five equal parts of 12." And we've shown that in the bar model.

So Laura says, "Well, we know that six is divided into five equal parts, and we can work out the size of the parts." Remember it's our divisors that's missing this time.

So Andeep says, "The dividend has been divided by 10." 60 divided by 10 is equal to 6.

So, "The divisor also has to be divided by 10." And 12 divided by 10, we think about our place value knowledge, we're going to move all the digits one place to the right, 12 divided by 10 is equal to 1.

2.

So 6 divided by 1.

2 is equal to 5.

The missing divisor is 1.

2.

There are five groups of 1.

2 in 6.

Well done if you work that out.

It's useful to know that these strategies work not just for whole numbers, but for decimals as well.

So Andeep and Laura find the missing quotients here.

460 divided by 60 is equal to something.

64 divided by 80 is equal to something.

Andeep says, "We can divide the dividend and the divisor by 10 and work out the quotient." So 420 divided by 10 is equal to 42.

And 60 divided by 10 is equal to 6.

So by scaling down the dividend and the divisor, he's created an equation that he couldn't solve himself using his times table knowledge.

He says, "I know that 42 divided by 6 is equal to 7, so 420 divided by 60 must also be equal to 7." And Laura says, "Well, 640 divided by 10 is equal to 64, and 80 divided by 10 is equal to 8." So again, she's created an equation within her times table knowledge.

She says, "I know that 64 divided by 80 is equal to 8.

So 640 divided by 80 must also be equal to 8.

So we can use this strategy to help us to simplify equations, and that's really useful.

Ah, so they're gonna find the missing quotient here.

4,400 divided by 400.

And Andeep says, "We can divide the dividend and the divisor by 10 to work out the quotient.

4,400 divided by 10 is equal to 440.

And 400 divided by 10 is equal to 40." Hmm.

Can you spot something there? Laura spotted it.

"We can divide the dividend and the divisor by 10 again", she says.

"440 divided by 10 is equal to 44.

And 40 divided by 10 is equal to 4." So now again, we've got a fact from within our times tables.

She says, "I know that 44 divided by 4 is equal to 11.

So 440 divided by 40 is equal to 11.

And 4,400 divided by 400 is also equal to 11." All of those equations have a quotient of 11.

Time to check your understanding.

Can you find the missing quotient here? Andeep says, "Divide the dividend and the divisor by 10 to work out the quotient." Pause the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? Well, Laura says, "350 divided by 10 is equal to 35.

And 70 divided by 10 is equal to 7.

I know that 35 divided by 7 is equal to 5.

So 350 divided by 70 must also be equal to 5." And Andeep says, "If the dividend and divisor are both divided by 10, the quotient will remain the same." Well done if you spotted that.

Time for you to do some practise.

Question one says, complete the equations.

And this time, again, we've given you some scaling tips to work with.

And Andeep says, "If the dividend and divisors are both divided by 10, the quotient will remain the same." And Laura says, "Find the missing quotient, dividends and divisors." And for question two, can you use scaling to help work out the missing numbers? This time you've got equivalent expressions, and you've got to work out the missing dividend or divisors in each one.

And for question three, can you use scaling to work out each quotient? Can you use what you know about scaling down by 10 to simplify these equations into ones within your times table knowledge? And again, "Divide the dividend and the divisors by 10 to help you work out the quotient." And, "You might need to divide both the dividend and the divisor by 10 more than once in some cases." Pause the video, have a go at your tasks, and when you're ready for some feedback, press Play.

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

Andeep's looking at A.

He says, "25 divided by 5 is equal to 5.

So 250 divided by 50 must also be equal to 5 as well." Laura's looked at B.

"49 divided by 7 is equal to 7.

So 490 divided by 70 must also be equal to 7." The dividends and the advisors have both been scaled down by 10.

And you could use the same to work out in C that 81 divided by 9 is equal to 9.

So 810 divided by 90 must be equal to 9.

Here are the answers for B.

Andeep is looking at A.

He says, "40 divided by 8 is equal to 5.

So 400 divided by 80 must also be equal to 5." And we can see that 400 had been divided by 10, so 80 must also be divided by 10 for the expressions to be equivalent.

And Laura's looked at B.

"600 had been divided by 10 to give a dividend of 60, so the divisors of 120 must also be scaled down by 10 to give a divisor of 12.

60 divided by 12 is equal to 5.

So 600 divided by 120 is also equal to 5." And that thinking would help you to work out the missing dividend or divisor in the remaining expressions.

And here are the answers to three.

This time we were looking for that missing quotient.

Andeep looked at A.

"96 divided by 8 is equal to 12.

So 960 divided by 80 is also equal to 12." He scaled down both the dividend and the divisor by 10.

And Laura looked at B.

"108 divided by 12 is equal to 9." So if we scaled down both the dividend and the divisor, we could create that equation.

So we also know that, "1,080 divided by 120 must also be equal to 9".

And did you spot that for D, you could scale down by 10 twice, and we could create 48 by 3, which we might know, or is much easier to work out that our quotient is 16.

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 divisors by 10.

So what have we learned about? Well, we've learned that if the dividend and the divisor are both scaled by 10, the quotient will remain the same.

And that's whether we scale up by 10 or we scale down by 10.

We've also seen that place value and unitizing can help to explain the effect on the quotient when scaling the dividend and the divisor by 10.

If six divided by three is equal to two, then six tens divided by three tens is equal to two.

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

Bye-bye.