Lesson video

Hello, I'm Miss Miah and I'm so excited to be a part of your learning journey today.

I hope you enjoy this lesson as much as I do.

In this lesson, you'll use unitizing and multiplying and dividing by 10 to scale division facts.

Your keywords are on the screen now and I'd like you to repeat them after me, scale.

Scaling.

Inverse.

Well done, let's keep going.

Now, scaling is when a given quantity is made hmm times the size.

In this lesson, scaling will involve making values 10 times the size.

So that's what we are focusing on.

Inverse means the opposite in effect, the reverse of.

So for example, the inverse of addition is subtraction.

The inverse of subtraction is addition.

I wonder what the inverse of multiplication would be.

Well, that's what we are going to figure out.

This lesson is all about scaling divisions derived from multiplication facts by 10.

So we've got two lesson cycles here, the first lesson cycle is about exploring the inverse of multiplication, and then our second lesson cycle is about scaling division.

Let's begin.

In this lesson, you'll meet Andeep and Izzy.

You can use your multiplication facts to also find division facts.

So Izzy says that, "If you know four times three is equal to 12." Andeep says that, "Then you can also calculate 12 divided by four, which is equal to three." So in today's lesson you'll be exploring how to use the inverse to solve calculations.

And if you're confused at this point, don't worry, I'm going to break it down and explain this to you.

Whoa, we've got a crate of eight watermelons here.

Now Izzy and Andeep are writing down multiplication facts.

Izzy says that she can see two groups of four, that's four times two, and four times two is equal to eight.

Andeep says he can see four groups of two.

That's two times four is equal to eight.

Izzy says she can see one eight times.

So that's one times eight is equal to eight.

And then Andeep says that he can see one group of eight and that's equal to eight times one, which is equal to eight.

Now using an array, you can actually also find division facts.

So we've still got the same group of watermelons altogether, we've got eight watermelons, so let's see how we can do that.

Now let's look at what Izzy says, "There are two equal parts, each with a value of four." So you can see two equal parts there, and within the two equal parts you've got four watermelons, which means there's a value of four, and the whole is eight.

So that is the same as saying eight divided by two is equal to four.

Now Andeep says that he knows that the whole is eight and the number of parts this time is four because you can see four parts there, and the value of each part is two, because there are two watermelons in each part.

So that is the same as saying eight divided by four is equal to two.

So can you see how we've derived two division equations from this one array, which represents eight.

Now you can derive division facts from multiplication facts without having to use an array.

So for example, if I know four times two is eight, and two times four is eight, then you also know that eight divided by two is four, and eight divided by four is two.

So this is the key part here, the inverse of multiplication is division.

The inverse of division is multiplication.

These are also sometimes known as fact families.

Over to you, which known facts match this array? Can you think of any other equations that represent this array equations here? So you've got two times four, eight divided by two and one times 10.

You can pause the video here and click play when you're ready to rejoin us.

So how did you do? Well, we know that there's one group of eight, so that's one times eight, or we could have had eight times one.

So that was another way you could have represented this equation.

What you should have ticked was two times four and eight divided by two.

And that's because we can see that in the array, we've got two groups of four, or four groups of two, and also could have derived the division equation eight divided by two, which would've given us four.

Now Izzy is finding fact families for the number 24.

She's writing down three times eight is equal to 24, eight times three is equal to 24.

And she's represented it in this image here.

So the product is 24, our factors are three and eight.

So that means three times eight is 24.

And the order in which we multiply three and eight does not matter 'cause we still get the same, this is known as the commutative law.

Now we've got our division facts.

So from this, Izzy has written down that you can have 24 divided by eight, which is three, so our dividend is 24, our divisors is eight in this case, and our quotient is three.

And then you can also have the division fact of 24 divided by three is equal to eight.

So the dividend has stayed the same, that's 24.

The divisor has changed.

So this time the divisor is three and the quotient is eight.

Back to you, can you think of two other multiplication facts for this array? So in this array, first I'd start off by counting what the total number is.

Off you go, you can pause the video here.

So what did you get? Well, you may have got quite a few equations here.

Let's start off with one group of 24.

So we know that one group of 24 or one times 24 is 24.

And then you could have also had 24 times one is 24.

Secondly, let's look at six times four and four times six.

Those are two other multiplication facts that you also could have had because they give you a product of 24.

And lastly, you could have had two times 12, which is 24 and 12 times two, which is 24.

Hmm, moving on, back to you.

Can you think of two other division facts for this array? So again, I'd start off with how many you have altogether.

You can pause the video here and then click play when you're ready to rejoin us.

Well, you can see that we needed to think of division facts.

Our dividend had to be the greater number.

So this is what you could have had, you could have had 24, so your dividend was 24 in each equation, but the divisors could have changed.

So you could have had 24 divided by six, which is four, 24 divided by four is equal to six, or you could have had 24 divided by two is equal to 12, 24 divided by 12 is equal to two.

Or you could have had 24 divided by one is equal to 24, and 24 divided by 24 is equal to one.

Don't forget, anything divided by itself is one.

Onto the main task for your lesson cycle.

For question one, you are going to describe the arrays with two multiplication facts and two division facts.

And you've got three arrays on the screen there, don't forget to count how many there are in total.

This will help you.

When you're ready, click pause to begin and click play when you're ready to rejoin us.

So how did you do? Let's look at the first array.

Now there were many ways you could have attempted this, but for this example, the total was 12.

So you could have had a multiplication factor of three times four or four times three.

And that your division facts would've been 12 divided by three, which is four or 12 divided by four, which is three.

You could have also had one times 12, which is 12, and 12 times one, which is 12.

And then your division facts would've been 12 divided by one, which is 12 and 12 divided by 12, which is one.

For B, you could have had this, so you could have either had two groups of eight or eight groups of two, and then your dividend would've been 16 and your divisors would've been eight and two, and your factors could have been four and four as well.

And lastly, for the third array, we've picked out five times four, which means another fact that we could have had was four times five, which is equal to 20, and our division facts would've been 20 divided by four, which is five, and 20 divided by five, which is four.

So that means you could have also had the factors one and 20 as well, and 10 and two.

Let's move on to our second lesson cycle, and this has all to do with scaling division.

You may have seen this before, this is known as a gattegno chart and it can help you to explore the relationship when scaling by 10.

So here you can see all the ones are in the bottom row.

Then you've got your 10s column, then you've got your 100s column.

We're just going to be focusing in on the first two rows.

So I want you to ignore the 100s row for now.

The first thing I'd like you to do is point to the number 10 and then drag your finger down to the one.

What do you think has happened there? Well, we have scaled down by 10.

In other words, we've divided 10 by 10 to get one.

Now I'd like you to point to the number 20 and drag your finger down.

You should have ended up at the number two.

What do you think has happened there? Well, we've divided by 10.

In other words, we've scaled down by 10.

So our equation is 20 divided by 10 is equal to two.

Now I want you to point to the number 30, and I want you to drag your finger down to three.

What do you think has happened there? Correct, we have scaled down by 10 or divided by 10 to get the number three.

Let's move on.

Andeep represents four divided by four is equal to one using place value counters.

The whole is four ones.

I can divide it into four equal groups.

There we are.

Each group has a value of one, so four divided by four is equal to one.

Andeep then represents 40 divided by four, which is equal to 10 using place value counters.

And you can see you've got place value counters which represent 10 now.

The whole is four 10s, I can divide it into four equal groups.

And we can see that that has been done there.

Now even though the whole is 10 times the size, I still have four counters.

So the number of counters has not changed, the value has.

So each group has a value of 10.

So 40 divided by four is equal to 10.

Izzy writes the two divisions into a diagram.

So we've got four divided by four, which is equal to one, and 40 divided by four, which is equal to 10.

She thinks she has noticed something.

What do you notice? Well, if you've said something along the lines of if the dividend is made 10 times the size, in this case the dividend here is four, and then it's made 10 times the size, which is 40, the quotient will also be 10 times the size.

We have scaled the equation by 10 and it is 10 times the size.

What do you notice here? If you use a known division fact, it can help you to solve problems like this.

So pretend that you can only see 72 divided by eight, which is equal to nine.

You would then be able to work out 720 divided by eight, which is equal to 90 because you have scaled the dividend by 10, which means your quotient will also have been scaled by 10.

Over to you, I'd like you to fill in the gaps.

So you've got 12 divided by four, which is equal to three, and 120 divided by four, which is equal to.

Think about what has been scaled.

You can pause the video here, off you go.

So how did you do? Well, 12 divided by four is equal to three.

So 12 10s divided by four is equal to three 10s.

60 donuts are packed into three large boxes.

If the donuts are divided equally between three boxes, how many are in each box? Well, Izzy says, "If there were six donuts divided between three boxes, there would be two donuts in each box", which is true? So she's using facts that she already knows.

Make sure you do that as well.

So six divided by three is two.

But in this case, there are 60 donuts divided between three boxes.

There are 10 times as many donuts altogether, which means there are 10 times as many donuts in each box.

So you know that three times 20 is equal to 60.

60 divided by two is equal to 30.

Over to you, which calculation will help you to solve the worded problem? 480 donuts are packed into eight large boxes.

If the donuts are divided equally between the eight boxes, how many are in each box? You can pause the video here and click play when you're ready.

So how did you do? You should have got C, 48 divided by eight.

So this is the division fact that you could have used because you know that 48 divided by eight is equal to six.

So 48 10s divided by eight is equal to six 10s.

Andeep is solving missing number problems. He says that he can derive facts from his four times tables to help him.

So we can see on the screen that we've got 80 multiplied by four, and that unknown number divided by four gives you 80.

I love questions like this.

So let's have a look.

Remember, if a factor scales by 10, then so does the product.

Now let's use the facts that we know, if we know, so let's ignore the fact that it's 80, let's think about eight, eight times four is equal to 32.

So that means 80 times four is equal to 320, 320 is 10 times the size of 32.

We can then use the inverse to check that we are correct.

So if we know that 32 divided by eight is four, so 320 divided by eight is 40.

Over to you.

You are going to find the missing number.

Remember, if a factor scales by 10, so does the product.

So you've got here 90 multiplied by eight gives you the unknown product, and then that unknown number divided by eight will give you 90.

Again, pause the video here, off you go, good luck.

So what did you get? You should have got 720.

And that's because if you know that eight times nine is 72, then you also know 80 times nine is equal to 720 because we've scaled one of the factors by 10, so our product also has been scaled by 10.

And then we can use the inverse to check this.

So we know that 72 divided by eight is nine.

So 720 divided by eight is 90, because as our dividend has scaled by 10, that means our product will also scale by 10.

Onto your main task for this lesson.

So for question one, you're going to use your knowledge of the twos, fours and eights, match up the pairs of calculations to solve the problems and the answers.

So you can use this calculation, so for example, 24 divided by two, 56 divided by eight, 40 divided by 10 and 72 divided by eight to then help you with the following calculations, so 240 divided by two, 400 divided by 10, 720 divided by eight, and 560 divided by eight.

Using that knowledge, you are then going to link it to the answer.

So for question two, you're going to find the missing numbers.

So for question two A, you're starting off with 20, then 60, then 70, and then 60 again.

And then for question two B, you've got 40 multiplied by four is equal to.

Something divided by four is equal to 40.

Then you've got 80 multiplied by something gives you 320.

320 divided by something is equal to 80.

And then you've got something multiplied by two is equal to 180, and 180 divided by 90 is equal to something.

Use the facts that you know to help you calculate these answers.

You could pause the video here, off you go, good luck, and click play when you're ready to rejoin us.

So how did you do? For question one, this is what you should have got, so in knowing that 24 divided by two is equal to 12, you then know that if 24 has been scaled by 10 to 240, the quotient would be 120.

56 divided by eight should have been linked to 560 divided by eight, and that's because 56 has been scaled up by 10.

The answer would've then been 70.

And that's because 560 divided by eight is 70.

Remember, if one of the factors has been scaled by 10, then so will your quotient.

40 divided by 10 should have been linked to 400 divided by 10, and that's because 40 has been scaled by 10, which means our answer would've been 40 because 400 divided by 10 is 40.

And lastly, 72 divided by eight is equal to 720 divided by eight because 72 has been scaled up by 10, which means that our quotient would've also been scaled up by 10.

So instead of nine, you should have got 90 as your answer.

Well done if you matched all of those correctly.

Now for two A, this is what you should have got, 20 multiplied by two is 40, and that's because you know two times two is four, so 20 multiplied by two is 40, we've scaled four by 10.

60 times four is 240.

The inverse of that, so 240 divided by four would've given you 60, 70 times eight is 560.

The inverse of that is 560 divided by eight is 70.

And we are going to look into more detail for the last question.

So 60 times eight would've given you 480.

And that's because if you know that 60 times eight is 48, then you know that 60 times eight must be 480 because we are scaling up the product by 10.

And then to check our answer, we would've used the inverse, so 480 divided by eight should have given you 60.

And for question B, this is what you should have got, so 40 times four is equal to 160.

So if you know that multiplication fact, you should have also derived that 160 divided by four is 40.

The multiplication fact that you could have used for this was four times four.

The multiplication fact that you could have used for 80 times four is eight times four.

So if you know that eight times four is 32, then you also know that 80 times four is 320.

320 divided by four is equal to 80.

And lastly, if you knew that nine times two is 18, then you also knew that 90 times two is 180.

180 divided by 90 is two.

Well done if you've got all of those correct, I'm very impressed.

We've made it to the end of the lesson, let's summarise our learning.

So in this lesson, you were scaling divisions derived from multiplication facts by 10.

You should now understand that division is the inverse of multiplication.

You also understand that if you know that three groups of four are equal to 12, then there are three groups of four in 12.

And lastly, you should also know that there are three groups of four 10s in 12 10s.

Again, thinking about scaling and the relationship of what happens when we multiply a factor by 10 to our product or to our quotient.

I really hope you enjoyed this lesson and that it's allowed you to unlock even more ways of calculating questions that you may not have been so sure of before.