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 scaling factors up and down by the same amount and explaining the effect.

So are you ready to explore some multiplication magic? Let's make a start.

We've got three key words in our lesson today, they are equivalent, factor and product.

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

Are you ready? My turn, equivalent.

Your turn.

My turn, factor.

Your turn.

My turn, product.

Your turn.

Fantastic.

I'm sure you are very familiar with those words, but let's just double check what they mean.

They're gonna be really useful to us today, so it's important that we know exactly what we're talking about.

So equivalent means having the same value, exactly the same value, even when things look a bit different.

A factor is a number which exactly divides another whole number, no remainders, whole number quotient.

And a product is the result of two or more numbers multiplied together.

So keep those in mind, we're gonna be using them a lot in this lesson.

There are two parts to our lesson.

In the first part we're going to be scaling up and down by three, four, and five.

And in the second part we're going to be using scaling to solve problems efficiently.

And we've got Andeep and Laura working with us today.

Andeep and Laura each have a chocolate bar, lucky them.

Can you see? They look a bit different, don't they? Andeep says, "I've got a very large bar of chocolate.

I'm going to eat one row each day." That's very restrained of you, Andeep.

I wonder how many of us could just eat one row of the chocolate bar each day.

Laura says, "My bar is even bigger than Andeep's." Hmm.

Is she correct, I wonder? We'll come back to this later, keep it in mind.

Andeep and Laura compare multiplication equations.

Andeep says, "If we double one factor, we must halve the other factor for the product to stay the same." You might have heard about that before.

So let's have a look.

We've got six multiplied by four is equal to 24, and we've got it arranged as an array of dots or counters.

Here's that same array, we're going to do something to it.

Oh, did you see what happened? Let's just look at that again.

So there's that same array and half of it has moved.

We've now got an array of the same number of counters, but there is three multiplied by eight.

We've halved the six and we've doubled the four.

So we've doubled one factor and halve the other one and the product stays the same.

So six multiplied by four is equivalent to three multiplied by eight.

Laura says though, "What happens if we multiply one factor by three and divide the other factor by three?" We've learned about doubling and halving, what about multiplying and dividing by three? She says, "Does the product stay the same?" Andeep represents two multiplied by nine using counters.

He says, "There are 18 counters altogether." There we are two rows of nine counters.

He says, "I can rearrange the counters." Oh, he's divided them into three equal parts this time and stacked the three parts on top of each other.

So now we've got an array that is six multiplied by three.

Laura says, "If I multiply one factor by three, I must divide the other factor by three for the product to stay the same." Ah, let's have a look at this with the equations.

Two multiply by three is equal to six and nine divided by three is equal to three.

And we can see from our arrays that the product is the same, they're both equal to 18.

Two multiplied by nine is equivalent to six multiplied by three.

Andeep wonders about multiplying and dividing by four, "So what happens if we multiply one factor by four and divide the other one by four? Does the product stay the same?" What do you think? Laura says, "Let's represent 40 multiply by six using an area model." So there we go.

Too many to put counters in this time, so we're going to imagine them.

We've got 40 multiply by six or six multiply by 40.

She says, "Let's divide 40 by four and multiply six by four." So we're dividing our 40 by four and we are multiplying our six by four.

So we've got four lots of six and one lot of 10 now.

So now we've got 24 multiplied by 10.

So six multiplied by four was equal to 246.

Four of 24 multiplied by 10 is 240.

But we've multiplied the six by four to give us 24 and we've divided the four by 10 to give us 10.

And we've got 24 multiplied by 10, which was part of the calculation we did working out the first one, wasn't it? And 24 multiplied by 10 is indeed 240.

So Andeep says, "We've got a new generalisation if we multiply one factor by four, we must divide the other factor by four for the product to stay the same." So we found out that it's true for doubling and halving, multiplying and dividing by three, and now multiplying and dividing are factors by four as well Andeep and Laura look at the equation eight multiplied by nine.

And we know that the answer is 72 to this, it's part of our times table knowledge.

But it's always useful to take something we know in order to work out something new.

Andeep says, "I can multiply or divide the factors by three." Eight multiplied by three is equal to 24 and nine divided by three is equal to three.

So eight multiplied by nine is equivalent to 24 multiplied by three, the product will still be 72.

And Laura says, "I can multiply or divide the factors by four." Eight divided by four is equal to two and nine multiplied by four is equal to 36.

The product will still be 72.

And now we've got just a doubling to do, two multiplied by 36 is equal to 72.

So if you multiply one factor and divide the other factor by the same number, the product stays the same.

Ah, so we've developed our generalisation a bit.

We don't have to be specific about the number as long as the number that we multiply one factor by is the same as the number that we divide the other factor by, the product will stay the same.

These factors have been scaled.

Eight is made three times the size and nine is made one third times the size.

The two equations are equivalent because the factors have been scaled up and down by the same factor.

And Laura says in her equations, "Eight is made one quarter times the size and nine is made four times the size." So again, they've been scaled by the same factor, one up and one down.

The two equations are equivalent.

Time to check your understanding.

Andeep and Laura find equations equivalent to 15 multiplied by eight.

Andeep says, "I think five multiplied by 24 is equivalent to 15 multiplied by eight." And Laura says, "I think 60 multiplied by two is equivalent to 15 multiplied by eight." Who is correct? Pause the video, have a think about it and when you are ready we'll get together for some feedback.

So what did you think? Well Andeep says, "I'm going to divide 15 by three and multiply eight by three." So his 15 multiplied by eight becomes five multiplied by 24, which is what he had.

And remember Laura is trying to prove that it's equivalent to 60 multiplied by two.

Let's see what she does.

She says, "I'm going to multiply 15 by four and divide eight by four." So that's her starting equation.

She's multiplied 15 by 40 equals 60, and divided eight by four to equal two.

So she has got 60 multiplied by two.

So they're both correct.

Five multiplied by 24 and 60 multiplied by two are both equivalent to 15 multiplied by eight.

And I think I know which equation of those I'd like to solve to find out the product.

Andeep wonders about multiplying and dividing by five.

Well we've tested it out on two, three and four, haven't we? Do you think it's gonna work for five? Let's see.

He says, "Does the product stay the same? And so Laura says, "Let's look at 15 times eight.

15 multiplied by eight is equal to 120." I wonder if they use 60 times two from your check for understanding slide, that's the one I'd have used.

If 15 is divided by five it is equal to three, and eight multiplied by five is equal to 40.

So we've created an equivalent equation, three multiplied by 40.

And Laura says, "I know that three multiplied by four is equal to 12.

So three multiplied by 40 must be equal to 120, 10 times bigger.

So 15 multiplied by eight is equivalent to three multiplied by 40." So it does work, doesn't it? And it's another way of demonstrating our generalisation.

If you multiply one factor and divide the other factor by the same number, the product stays the same.

And we've now tested that by multiplying and dividing our factors by two, three, four and five.

Let's go back to those chocolate bars and check your understanding.

Remember Laura thinks her chocolate bar is larger than Andeep's.

Andeep's bar has nine rows with four chunks in each row.

And my bar she says, "Has three chunks in each row and there are 12 rows." So is Laura correct in thinking that her chocolate bar is larger than Andeep's? Pause the video, have a go.

And when you've come to a decision, press play for some feedback.

What did you think? Well Laura says, "You can divide nine by three and multiply four by three." Nine divided by three is equal to three and four multiplied by three is equal to 12.

"Oh no," she says, "Nine times four is equal to three times 12.

The chocolate bars are the same size." And it just proves that idea of things being equivalent they don't have to look exactly the same.

I dunno quite whose chocolate bar I thought looked bigger, but it's clear now from doing the maths that we've proved that they are the same size.

Although if they only at one row a day, Laura's would last longer, wouldn't it? Okay, Laura finds the missing number in the equations.

What do you notice about the equations? Have you had a think? Let's see what Laura thinks.

She spotted that eight divided by four is equal to two.

So we've divided a one factor by four, so I need to multiply six by four, which is equal to 24.

So now we know that six multiply by eight will be equal to 24 multiplied by two.

And what about this one? Well, 20 multiplied by five is equal to a hundred.

So I need to divide 25 by five, which is equal to five.

And again, by using those factors and scaling one factor up and the other factor down by the same number, we've created two easier equations to solve.

And another demonstration, if you multiply one factor and divide the other factor by the same number, the product stays the same.

Time to check your understanding again.

Can you find the missing number in the equation? Three multiplied by 16 is equal to something multiplied by four.

Andeep says, "Look carefully at how 16 has been scaled down.

What's the missing factor?" Pause the video, have a go.

And when you're ready for some feedback, press play.

How did you get on? Well, Laura says, "If you multiply one factor and divide the other factor by the same number, the product stays the same." So 16 divided by four is equal to four.

So you need to multiply three by four, which is equal to 12.

So three multiplied by 16 is equal to 12 multiplied by four.

Time for you to do some practise, find the missing numbers in the equations.

Scale one factor up and the other factor down, but remember by the same number, that way we keep the products the same.

In question two, we're going to find the missing numbers in these equations.

And Andeep says, "Has one of the factors been multiplied or divided by three, four, or five?" And Laura says to remember, "If you multiply one factor, you need to divide the other factor by the same number so that the product remains the same." So 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.

You might want to have a quick look.

Andeep's focused in on A, he says, "Seven is scaled up and 40 is scaled down.

Seven multiplied by four is equal to 28 and 40 divided by four is equal to 10." So we've created a calculation where we just have to multiply by 10.

So 28 multiplied by 10 is equal to 280.

In B, we scaled one factor up by three and the other factor down by three.

So one was made three times the size, and 15 was made one third the size by dividing by three.

And in C, Laura says, "30 divided by five is equal to six and 12 multiplied by five is equal to 60." So we had made the 30, one-fifth times the size and the 12 five times the size.

But in both cases the product is 360.

And I suppose it's up to you to decide which is the easier way to find that product, isn't it? So here are the answers to the parts of question two.

Andeep says, "In A, nine divided by three is equal to three and 20 multiplied by three is equal to 60." So we'd had that factor of three multiplied one by three and divided the other by three so that the product remained the same.

20 multiplied by nine is equal to 60 multiplied by three.

And in B, Laura spotted that, four multiplied by four is equal to 16 and 40 divided by four is equal to 10.

So 40 multiplied by four is the same as 10 multiplied by 16.

And hopefully in the others you noticed how the numbers had been multiplied and divided to keep the products the same, to maintain that equivalence.

And on into part two of our lesson, we're going to use scaling to solve problems efficiently.

Andeep and Laura use scaling to answer 36 multiplied by four.

How we do use scaling to find an efficient way to solve that? Well, Andeep says, "I could divide 36 by three and multiply four by 3." 36 divided by three is equal to 12 and four multiplied by three is equal to 12.

So 12 multiplied by 12 is equal to 36 multiplied by four.

And we know that 12 times 12 is 144, Andeep does as well.

36 multiplied by four is equivalent to 12 multiplied by 12, so it's also equal to 144.

I wonder what Laura's going to do.

She says, "I could multiply 36 by four and divide four by four.

To multiply 36 by four, I could double 36 to get 72 and then double it again to get 144." Oh and four divided by four is one.

Oh and she says, "I know 144 multiplied by one is equal to 144." So 36 multiplied by four is equivalent to 144 multiplied by one, so it's also equal to 144.

Whose strategy was most efficient do you think? Well, they both work, don't they? But Andeep's strategy may be quicker.

36 multipli by four is what we were originally trying to solve.

So actually Laura took more steps to get back to the product, whereas Andeep managed to change his equation to an equivalent equation that gave him an easier calculation to do.

They're both using scaling to answer 15 multiplied by 30.

Andeep says, "I could divide 15 by three and multiply 30 by three." So 15 divided by three is equal to five, 30 multiplied by three is equal to 90.

So he's got five multiplied by 90.

And he says, "I know five times nine is 45, so five times 90 must be equal to 450." And because those equations are equivalent, 15 multiplied by 30 is also equal to 450.

I wonder what Laura is going to see.

She says, "I could multiply 15 by three and divide 30 by three." Ah, so 15 multiply by three is equal to 45 and 30 divided by three is equal to 10.

Ah, she's created and multiplied by 10, hasn't she? She says, "I know that 45 multiplied by 10 is 450 and because they're equivalent then 15 times 30 is also equivalent to 450." Whose strategy was most efficient this time do you think? I think Laura's was this time, wasn't it? Because the final calculation she had to do was slightly easier than the one that Andeep had to do.

Time to check your understanding.

Can you use scaling to answer nine multiplied by 33? Andeep says, "Multiply one factor and divide the other factor by the same number to maintain that equivalence." And Laura says, find a calculation that's more efficient to work out." Find one that you can do mentally rather than having to do a written calculation.

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

How did you get on? Andeep says, "You might have divided nine by three and multiplied 33 by three." So you might have ended up with three multiplied by 99.

And Laura says, "Well, three times a hundred is equal to 300, and 3 times 99 must be equal to 300 subtract three.

So three times 99 is equal to 297, and nine multiplied by 33 is equivalent to three multiplied by 99.

So it is also equal to 297." Not the easiest calculation, but possibly easier than nine multiplied by 33.

Andeep and Laura use scaling to answer five multiplied by 37.

Andeep says, "I'm not sure what to do about this calculation." Can you spot something with these calculations? Laura says, "five and 37 are both prime numbers.

We could divide five by five." So five divided by five is equal to one, but 37 multiplied by five, well that's the equation we're trying to solve anyway.

He says, "That means we still have to multiply 37 by five." So that's not gonna help us is it? So sometimes scaling is not a strategy that can be used.

You might have looked at that and thought, well, I could double the five to make 10, but then I've got to half 37 again and 37 is an odd number, so it's not going to give us a whole number.

So sometimes scaling is just not a strategy that can be used.

Time to check your understanding, which expressions could be calculated more efficiently using scaling.

Andeep says, "Can you multiply one factor and divide the other factor by the same number easily?" And Laura says, "Are both of the factors prime numbers?" Pause the video, have a look at those expressions, and when you're ready for some feedback, press play.

What did you think? Well, for A, both of these numbers are prime.

So scaling isn't useful here.

We'd end up with fractions or decimals to work with, and that's not going to make it any easier, is it? What about B? Well, eight multiply by 13 is equivalent to two multiply by 52.

We can divide the eight by four and we can multiply the 13 by four.

So that might give us an easier way in to calculating the answer.

And what about C? Well, 25 multiplied by six is equivalent to 75 multiplied by two.

So that might be an easier one to look at, mightn't it? Six divided by three is equal to two, and 25 multiplied by three is equal to 75.

So yes, we could possibly change those equations to equivalent ones to make them easier to calculate mentally.

But it's always worth checking and thinking.

Sometimes it's a really good strategy, but sometimes it might not be the most efficient one.

Time for you to do some practise.

So can you find the product efficiently using scaling? So you've got 36 multiplied by 3, 45 multiplied by 20, and we've given you a starting point of where you might go with the scaling.

And Andeep says, "Multiply one factor and divide the other factor by the same number." And Laura says, "Look carefully for where both factors are prime numbers." And you've got some other examples there, and we haven't given you any starting points for those ones.

So pause the video, have a go at these equations, and when you're ready for some feedback, press play.

How did you get on? Andeep says, "In A, three multiplied by three is equal to nine and 36 divided by three is equal to 12." So Laura says, "36 multiplied by three is equivalent to 12 multiplied by nine, so it's also equal to 108." We know 12 multiplied by nine is 108 from our times tables.

And what about B? 45 divided by five is equal to nine and 20 multiplied by five is equal to a hundred.

So that gives us a much easier equation to solve.

Nine multiplied by a hundred is equal to 900.

So 45 multiplied by 20 must also be equal to 900.

So here are some possible answers for C, D, E and F.

We didn't give you any starting points this time.

So you may have come up with different equivalent equations to solve.

In C, 25 divided by five is equal to five and 200 multiplied by five is equal to a thousand.

So 25 multiplied by 200 is equivalent to five multiplied by 1000, which is 5,000.

In D, 110 multiplied by three is equal to 330 and nine divided by three is equal to three.

So we've got two equations there, both of which are relatively easy to solve mentally, aren't they? But we can see that they are equivalent, 110 multiplied by nine is 990 and 330 multiplied by three is equal to 990 because those equations are equivalent.

What about E? Oh, well both these numbers are prime numbers, so scaling isn't useful here.

Prime numbers only have factors of one of themselves.

So not going to help us to think about halving, dividing by three, dividing by four, dividing by five with 13 and 17.

I think I'd use a partitioning method here, maybe even a grid method to help me, an area model.

And what about F? Well, 150 multiplied by four is equal to 600, and 16 divided by four is equal to four.

Six times four is 24, so 600 times four must be 2,400.

So by using multiplying and dividing our factors by four here, we have created an easier equation for us to solve mentally.

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

We've been explaining the effect on the product when scaling the factors up and down by the same amount.

What have we learned then? Well, we've learned that when one factor is scaled up and the other is scaled down by the same amount, the product remains the same.

And we've tested that with lots of different amounts, with two, three, four and five.

And you may even have tried others in that final activity.

We've seen that arrays can represent the product staying the same when one factor is scaled up and the other down by the same amount, think about the chocolate bars.

And we've also seen that scaling one factor up and the other down by the same amount maintains equivalence.

The product remains the same because the expressions that we create are equivalent.

They have exactly the same value.

Thank you for all your hard work and your mathematical thinking.

I hope you've enjoyed playing around with factors and I look forward to seeing you in another lesson soon.

Bye-Bye.