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 equivalence 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 solving problems using equivalence by scaling factors.

We've got four keywords today.

We've got equivalent, factor, product and factorise.

So I'll say them 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.

My turn, factorise, your turn.

Four words you might be familiar with, but always worth just reminding ourselves what they mean.

They're gonna be really useful to us in our lesson today.

So equivalent means having the same value.

A factor is a number that exactly divides another number so no remainders.

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

And factorise means to show a number as the product of its factors.

So for example, 45 can be factorised as 5 and 9.

So look out for those as we go through our lesson and perhaps you'll be using them as you discuss your work as well.

So there are two parts to our lesson today.

In the first part we're going to be using scaling to find a missing factor or product.

And in part 2 we're going to be using scaling to find two missing factors.

So let's make a start on part 1.

And we've got Andeep and Laura helping us in the lesson today.

So Andeep and Laura are solving missing number problems. Andeep says, "What is 3 multiplied by 84?" And Laura says.

"I can partition 84 into 80 and 4." And she's used a part-part-whole model to do that.

3 multiplied by 80 is equal to 240 and 3 multiplied by 4 is equal to 12 and 240 plus 12 is equal to 252.

So Andeep says, "3 multiplied by 84 is equal to 252." Andeep and Laura solve other missing number problems. So we had 3 multiplied by 84 is equal to 252.

Now we've got 30 multiplied by 84.

Andeep says, "30 is 10 times the size of 3.

So I can use 3 multiplied by 84 as equal to 252 to help me work out 30 multiplied by 84." He says, "30 multiplied by 84 is equivalent to 3 times 10 times 84.

It's also equivalent to 252 times 10." So he knew that 3 times 84 was equal to 252, but he also knew he had to then multiply by 10 and 252 multiplied by 10 is equal to 2,520.

So 30 multiplied by 84 is equal to 2,520.

2,520 then is equal to 3 times what? Hmm, Andeep says, "30 divided by 10 is equal to 3.

So to keep the product the same, I have to multiply 84 by 10." Ah, he's using his knowledge of equivalence when we multiply and divide the factors by the same number.

So he's looking at 30 multiplied by 84 is equal to 2,520 and he says that 30 divided by 10 is equal to 3.

So to keep the product the same, he must multiply 84 by 10.

So 3 multiplied by 840 must be equal to 2,520 as well.

What about 2,520 is equal to 6 multiplied by something then? Well Laura says, "3 doubled is equal to 6.

So to keep the product the same, I have to halve 840.

Half of 840 is equal to 420.

The product is the same as 3 multiplied by 840, but one factor is doubled and one factor is halved." And we know that if we double one factor and halve the other factor, the product stays the same.

So our missing factor there is 420.

Andeep and Laura solves some other missing number problems. 3 multiplied by 84 is 252 again, what about 3 multiplied by 85? Andeep says, "3 multiplied by 84 can be said as 84 groups of 3." Hmm, so we could say 3 groups of 84, but we could also say 84 groups of 3, because we know that multiplication is commutative.

So 3 times 85 can also be said as 85 groups of 3 or can you see we had 84 groups of 3.

Now we've got 85 groups of 3.

So 3 times 85 is one more group of 3 or 252 plus 3.

So 3 times 85 must be equal to 255.

If we think of 84 lots of 3 and 85 lots of 3.

What about 2,550 is equal to 3 times something then? We know that 3 times 85 is equal to 255.

Andeep says, "255 multiplied by 10 is equal to 2,550." He says, "The product is 10 times the size, so I need to multiply 85 by 10." The three is the same here.

So if our product is 10 times bigger, our other factor must be 10 times bigger.

So 2,550 must be equal to 3 multiplied by 850.

Oh gosh, what about 2,550 is equal to 15 times something? Can you think about what will happen to our factors? Remember our product is the same.

Well, Laura says, "3 multiplied by 5 is equal to 15." So what do we need to do to the 850? Ah, to keep the product the same, I have to divide 850 by 5.

She says, "I know that 85 divided by 5 is equal to 17.

So 850 is 10 times the size of 85.

So 850 divided by 5 would be equal to 170." So 2,550 is equal to 15 multiplied by 170.

What a lot of facts we can work out by working out our one known fact of 3 multiplied by 84 is equal to 252.

We've got all the way to 2,550 is equal to 15 multiplied by 170, all by thinking about scaling the factors to maintain the product, and by thinking about what happens when our product is scaled by a factor of 10.

Time to check your understanding, can you find the missing factor? 2,520 is equal to 12 multiplied by something and we've got our starting factor, remember that 3 multiplied by 84 is equal to 252.

Andeep says, "How can you use this fact to help you?" 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, "3 multiplied by 840 is equal to 2,520." So we can write this down and now we can think about where that twelve's come from.

Laura says, "3 is made 4 times the size," so 3 times 4 is equal to 12.

"To keep the product the same, I have to make 840 one quarter times the size." To make 840 one quarter times the size I can halve and halve again.

840 divided by 2 is 420 and 420 divided by 2 is 210.

So our missing factor is 210.

And 2,520 is equal to 12 multiplied by 210.

Well done if you worked that out.

Really good mathematical thinking going on.

So division facts can help complete multiplication equations as well.

We've got a division fact here, not one of our times table facts.

195 divided by 13 is equal to 15.

So what is 15 multiplied by 13? So Andeep says, "195 divided by 13 is equal to 15.

So 13 multiplied by 15 must be equal to 195." If 195 is our product, then if we divide by one factor we can find the other factor.

So those two must be our factors, 13 and 15.

So what multiplied by 13 must be equal to 1,950? Well, Andeep says, "195 multiplied by 10 is equal to 1,950.

The product is 10 times the size, so I need to multiply 15 by 10." Our 13 is the same, so our other factor has to be 10 times the size.

So it's 150.

15 times 10 is equal to 150.

Or what about 50 multiplied by something is equal to 1,950? I can't see 50 relating to 15 or 13.

What are we going to have to do? Let's look at that middle calculation to help us.

Laura says, "150 divided by 3 is equal to 50." Thanks Laura for helping with that.

So 150 divided by 3 is equal to 50.

So what must we do to the 13? She says, "To keep the product the same I have to multiply 13 by 3." To keep the product the same, we have to multiply one factor and divide the other factor by the same number.

13 times 3 is equal to 39.

So 50 multiplied by 39 must be equal to 1,950.

Again, what a long way we've got from 195 divided by 13 is equal to 15.

Fantastic how our mathematical thinking can help us.

Time to check your understanding.

Can you find the missing factors? Andeep says, "Use 50 multiplied by 39 is equal to 1,950 to help you to complete these equations." And Laura says, "To remember when one factor is scaled up and the other is scaled down by the same amount, the product remains the same." Pause the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? We were trying to find out what the missing factor was with 5 multiplied by something is equal to 1,950.

So did you spot that we could divide the 50 by 10 to equal 5? But because 50 has been made one 10th times the size, 39 must be made 10 times the size.

39 multiplied by 10 is 390.

So our missing factor there is 390.

What about the next one? Well, let's go back to that 50 again.

50 halved is equal to 25, because the product hasn't changed, 39 must be multiplied by 2, and 39 times 2 is equal to 78.

So 25 multiplied by 78 must be equal to 1,950.

Time for you to do some practise.

So in question 1, you're going to use the equation we've given you to work out the missing numbers.

So we've given you 6 multiplied by 48 is equal to 288.

And Andeep says, "Remember when one factor is scaled up and the other is scaled down by the same amount, the product remains the same." And in question 2, you're going to use this equation to work out the missing numbers.

This time you've got 560 divided by 35 is equal to 16.

And again, remember when one factor is scaled up and the other is scaled down by the same amount, the product remains the same.

And Laura says, "Start by writing 560 divided by 35 equals 16 as a multiplication equation." That might be a good start.

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

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

I'll pause for a moment so you can have a look and check and then we'll have a look at a couple in detail.

So Andeep says, for a, "2,880 is equal to 288 multiplied by 10." So 2,880 must be equal to 6 times 10 times 48.

So for a, the missing factor was 60, 10 times 6, because our product was 10 times the size.

And Laura says, "2,880 is equal to 48 times 10 times 6.

So 2,880 must be equal to 480 times 6." So that was b.

And the similar thinking, thinking about what those factors were and what things had been scaled up and scaled down by would help you to work out the other answers as well.

And here are the answers for question 2.

Andeep says, "560 is equal to 16 multiplied by 35." That was Laura's advice, wasn't it? To rewrite it as a multiplication equation.

So Laura says, well, "5,600 must be equal to 560 multiplied by 10." So all our answers were 5,600, weren't they in our questions 8, a, b, c, d, e and f.

So knowing that 560 had been multiplied by 10 was gonna help us to work out our missing factors.

So she says, "5,600 is equal to 16 times 10 times 35." So we could use that understanding and then do some doublings and halvings and multiplying by fours and things to help us to work out what those missing factors were.

So knowing that if the product is 10 times bigger, one of the factors must be 10 times bigger, and knowing that if we multiply and divide the factors by the same number, we will keep the product the same, help us to reason our way through those answers.

Well done for all your mathematical thinking in that task.

And on into part 2, we're going to use scaling to find two missing factors.

Andeep and Laura, look at this problem.

Andeep says, "We need to factorised 1,100 using two 2-digit numbers." Do remember factorise meant to express a number as the product of its factors.

Well Laura says, "I know that 110 multiplied by 10 is equal to 1,100." Will that work? Andeep says, "No, 110 is a 3-digit number.

I'm not sure how that helps." Ah, I wonder if Laura can find a way for that to help.

Laura says, "We can use multiply one factor and divide the other factor by the same number." Ah, that's right, and the product stays the same.

Well remembered Laura.

Andeep says we could halve 110 and double 10.

So 110 divided by 2 is equal to 55 and 10 multiplied by 2 is equal to 20.

Now we've got two 2-digit numbers.

Andeep says, "So 1,110 can be factorised as 55 multiplied by 20." So there's one possible solution.

They're going to look for another one now.

Andeep says, "We could divide 110 by 5 and multiply 10 by 5." Laura says, "100 divided by 5 is equal to 20.

So 110 divided by 5 must be equal to 22 and 10 multiplied by 5 is equal to 50." So Andeep says, "1,100 can be factorised as 22 multiplied by 50." So there's another solution.

Can they find another solution? Andeep says, "Well, we could double 22 and halve 50." 22 times 2 is equal to 44 and 50 divided by 2 is equal to 25.

So Andeep says, "1,100 can be factorised as 44 multiplied by 25." There's another possible solution.

They can start with a different product now, they've got 960.

So Andeep says straight away, "The factors could be 96 and 10." 96 multiplied by 10 is 960.

And those are both 2-digit numbers, so there's one solution.

960 is the product of 96 and 10.

They're gonna find a different solution.

96 is a multiple of four.

So Andeep, how does he know that? Well, we've partitioned it into 80 and 16.

And they're both multiples of four, aren't they? So yeah, 96 is a multiple of four.

Andeep says, "We could divide 96 by 4 and multiply 10 by 4." And Laura says, "To divide 96 by 4, I can halve and then halve my answer.

96 divided by 2 is equal to 48 and 48 divided by 2 is equal to 24." So 96 divided by 4 is equal to 24.

So then we've got to multiply 10 by 4 and that gives us 40.

So there's another solution, 24 multiplied by 40.

can they find another solution? 96 is also a multiple of three.

Ah, can you remember to test for divisibility by three add the digits 9 plus 6 is equal to 15.

And if that digit sum is a multiple of three, then the number will be a multiple of three.

We can also see that 90 is a multiple of three and six is a multiple of three in this case.

So we could divide 96 by 3 and multiply 10 by 3.

90 divided by 3 is 30.

So 96 divided by 3 must be 32 and 10 multiplied by 3 is equal to 30.

So 32 multiplied by 30 is equal to 960.

There's another solution.

Time to check your understanding.

Can you find a different way to factorise 960? Find two other 2-digit numbers that have a product of 960 and you could use these equations to help you.

These are the ones they found already.

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

So here are some other solutions.

So you might have had 16 multiplied by 60, 64 multiplied by 15 or 12 multiplied by 80.

So the 24 multiplied by 40, well, we could halve 12 and double 40.

For the 64 multiplied by 15, we could double 32 and halve 15.

And for the 16 multiplied by 60.

Well, we could have multiplied 10 by 6 and then divided 96 by 6 as well, 'cause 96 divided by 6 is equal to 16.

So lots of different ways that you could have come up with.

I hope you had fun playing around with those.

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

Can you find different ways to factor 1,600? Andeep says, "Both numbers have to have two digits." And Laura says, "To remember that scaling one factor up and the other factor down by the same amount maintains equivalence." And for question 2, can you find different ways to factorise 1,440? And again, both numbers have to have two digits.

And Laura says, "What number multiplied by 10 is equal to 1,440? You could start with that," and then work from there to find your solutions.

Pause the video, have a go at playing around with factorising, and when you're ready for some feedback, press Play.

How did you get on? Here are some answers.

Andeep says, "You could start with 160 times 10 is equal to 1,600." It's not two 2-digit numbers, but you could start from there.

And Laura says, "Then you could have halved 160 and doubled 10 to get 80 multiplied by 20 is equal to 1,600." And then from there you could halve the 80 and double the 20 to get 40 times 40.

Or you could have started with 16 multiplied by 100 is equal to 1,600.

And then you could have halved the 100 to get 50 and double the 16 to get 32, halve the 50 to get 25 and double the 32 to get 64.

So you could have started with multiplying by 10 or multiplying by 100 to get to your different solutions.

I wonder where you started.

And how about question 2? So you could have started with 144 multiplied by 10 is equal to 1,440.

And Laura says, "Then you could have halved the 144 and doubled the 10 to get 72 multiplied by 20." Then you could have doubled the 20, halve the 72 to get 36 times 40, and you could double the 40 and halve the 36 to get 80 multiplied by 80.

24 multiplied by 60.

Oh, we could have said 3 times 20 is equal to 60 and 72 divided by 3 is equal to 24.

Lots of different ways in and lots of different ways to work around those equations.

But I hope you've had lots of fun playing around with factorising larger numbers into 2-digit numbers by starting with a known fact and working from there.

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

So we've been using knowledge of equivalence when scaling factors to solve problems. What have we learned about? 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.

Scaling one factor up and the other down by the same amount, maintains equivalence.

Those equations or expressions that we create are equivalent and they have the same product.

And factorising means describing a product using factors that multiply together to give that product.

And we've explored different ways of factorising, drawing on our understanding of scaling up and down factors to maintain equivalence.

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

Bye-bye.