Lesson video

Hello, my name is Mr. Tazzyman and I'm really looking forward to teaching you a lesson today from the unit based on understanding and representing multiplicative structures.

Get yourself sat comfortably and ready to learn, 'cause it's time to start.

Here's the outcome for the lesson then.

By the end, we want you to be able to say, I can explain where zero can be part of a multiplication or division expression and the impact it has.

These are the key words that you are gonna be hearing.

I'm gonna say them and I want you to repeat them back to me.

So I'll say my turn, say the word and then I'll say your turn and it'll be your turn to do that.

Okay, ready? My turn, dividend.

Your turn.

My turn, divisor.

Your turn.

My turn, quotient.

Your turn.

This is the meaning of each of those words and that's helpful to us before we get started.

The dividend is the number being divided.

The divisor is a number that will divide the dividend exactly.

The quotient is the result of the dividend being divided by the divisor.

And there's an equation at the bottom there which helps us in understanding these.

10 divided by two is equal to five and in that equation, 10 is the dividend, two is the divisor, and five is the quotient.

Okay, there's gonna be two parts to this lesson.

First of all, we're gonna look at zero as the dividend, then we're gonna look at zero as the divisor.

We'll start with zero as the dividend.

We've got Jun and Lucas with us as well.

They're gonna help us out.

They'll respond to some of the math prompts you'll see on screen, they'll discuss some of the learning and they might even give us some hints and tips as we go into the practise tasks.

Jun and Lucas are discussing zero.

Zero is a very strange number, isn't it? Have you ever thought that? Zero is a strange number, it's quite unlike the other numbers in lots of ways.

"Yes, but it is the easiest times table to learn," says Lucas.

You can see the times table there.

The product is always zero in the zero times table.

I suppose so, how does it work in multiplication and division? Well, why don't we explore that? Good thinking Lucas.

Jun and Lucas need to represent a number of pencils in the pots using equations.

I'd represent this as three times two equals six because there are three pots of two pencils, which is equal to six pencils in total.

Or two times three equals six because there are two pencils three times, multiplication is commutative.

So the related division fact is six divided by three equals two because the product becomes the dividend.

You can see there that the product of six starts our division equation, and that the start of a division equation is the dividend.

The divisor is three because that's the number of pots and the quotient is two, the number of pencils per pot.

Can you see that the representation has changed slightly? I wonder how they'll describe this one.

I'd represent this as three times one is equal to three because there are three pots with one pencil in each, which is equal to three pencils altogether.

Or one times three is equal to three because there is one pencil three times as there are three pots.

Multiplication is commutative.

So the related division fact is three divided by one equals three because the product becomes the dividend.

The divisor is three because that's the number of pots and the quotient is one, the number of pencils per pot.

Hmm, another change.

Can't see any pencils here.

Hmm, this is strange because there are absolutely no pencils.

How can we write an equation without any? Let's compare the equations we have written already to see if we can spot some patterns.

In each equation, the factor of three represents the number of pots or groups and you can see for the equations that we knew we've got the pots have been counted along one, two, three.

So the first equation is two pencils, three times.

The last image still has three pots, the same as the others.

So you can see they put three in as one of the factors on that final equation.

The other factor represents the number of pencils per pot.

In the top one you can see two pencils per pot, that gives a factor of two.

Then it's one pencil per pot that gives a factor of one.

But there aren't any pencils in the last image, so the other factor will be zero.

There it is zero multiplied by three.

The products give the number of pencils altogether.

So there were six pencils in that first image, three pencils in that second image.

Again, there aren't any pencils in the last image, the product is also zero then, so it's zero in the last image.

The equation could be zero three times or three times zero.

Time to check your understanding then.

You are gonna fill in the missing numbers for the multiplication equation representing the number of pencils in the pots.

You've got one at the top there that says one times four is equal to four or four times one is equal to four.

For the second row, we've got some missing numbers.

You need to work out what that equation would be and you might have two because of commutativity.

Pause the video and give that a go.

Welcome back.

Let's reveal the answer then.

It could have been four times zero is equal to zero or zero times four is equal to zero.

Did you get that? I hope so.

Alright, let's move on.

What about division? Good thinking Jun.

Let's use the related division equations, two times three equals six has a related division equation of six divided by three equals two.

One times three equals three has a related division equation of three divided by three is equal to one.

Zero times three equals zero has a related division equation, but what is it? The product becomes the dividend.

You can see that in the example equations we've already got.

So on the last one, zero was the product.

So our unknown dividend will be zero.

There it is.

The number of pots becomes the divisor.

So our unknown divisor will be three because there are three pots.

The quotient tells us how many pencils per pot, it's the group size.

So our unknown quotient will be zero because there are zero pencils in each pot.

Jun and Lucas writes some more related division and multiplication equations for the same context.

Zero times four is equal to zero, zero divided by four is equal to zero, zero times five is equal to zero, zero divided by five is equal to zero, zero times six is equal to zero, zero divided by six is equal to zero.

No pencils in sight.

Can we generalise by looking at these? What is true for all of them? What do you think? What can we say is true of all of these? When the dividend is zero, the quotient is zero.

Okay, let's check your understanding.

Which of the following images is a representation of the division equation below? And the division equation reads zero divided by three is equal to zero.

Is it A, B, or C? Pause the video and decide.

Welcome back.

The answer was A, zero divided by three because there are three pots is equal to zero.

Dividend is the number of pencils altogether, divisor is the number of pots and the quotient was the number of pencils per pot.

Okay, it's your first practise task now.

I want you to match the multiplication equations to their related division fact by drawing a line between them.

So you've got a column of multiplication on the left and a column of related division facts on the right, but they're not all on the same row as their related fact.

So you need to draw lines between them to match them up.

For number two, using the number cards below how many ways can you complete the equation, something multiplied by something equals zero? Here's number three, similar to number two but a slightly different equation.

Using the number cards below, how many ways can you complete the equation, zero divided by something is equal to something? And Lucas says, "What do you notice in comparison to question two?" So when you finish question three, compare it with question two and think about what you notice.

Okay, pause the video here and give those a really good go.

Try your best.

Good luck.

I'll be back shortly with some feedback.

Welcome back.

We'll start with number one.

The first equation, 24 equals four times six, matched 24 divided by six equals four.

Three times six equals 18 matched with 18 divided by six equals three.

Zero times seven equals zero matched up with zero divided by seven equals zero.

22 equals two times 11 matched with 11 equals 22 divided by two.

Zero times eight equals zero matched with zero divided by eight equals zero.

Five equals five multiplied by one matched up with five divided by one equals five.

And lastly, two multiplied by nine equals 18 matched with 18 divided by nine equals two.

Pause the video here for extra time to mark accurately.

Here's number two then.

Jun's gonna help us to explain.

"Because the product is zero, one of the factors must always be zero." So that meant you had these combinations, one times zero, two times zero, three times zero, four times zero, five times zero, six times zero, seven times zero, eight times zero and nine times zero.

Factors are commutative, so there are these as well.

And they're the same except the factors have swapped positions.

These are the different options that you could have had.

There were 18 ways in total.

Let's look at number three then.

Jun's gonna explain again.

"The quotient will always be zero because the dividend is zero." That meant you had these combinations, zero divided by one, zero divided by two, zero divided by three, zero divided by four, zero divided by five, zero divided by six, zero divided by seven, zero divided by eight and zero divided by nine.

There were nine ways.

And Jun, in response to what Lucas has asked says, "I noticed there are half the number of combinations because division isn't commutative whereas multiplication is." So because the factors can be swapped with commutativity in multiplication equations, that meant they would double the number of combinations available.

Okay, let's move on to the second part of the lesson now, zero as the divisor.

Jun and Lucas look at some related division and multiplication equations and generalise.

Four times six is equal to 24, relates to 24 divided by six equals four.

Three times six equals 18 relates to 18 divided by six equals three.

Two times six equals 12 relates to 12 divided by six equals two.

One times six equals six relates to six divided by six equals one.

Zero times six equals zero relates to zero divided by six equals zero.

The product of the multiplication becomes the dividend of the related division equation.

The divisor is the second factor of each multiplication here.

The numbers appear in reverse order for the division equation.

It's an example of the inverse.

So you can see that if we look at that top pair, 24 is almost mirrored into that second division equation.

It's the product in the first multiplication, but it also starts the related division equation as the dividend.

Jun and Lucas explore something new.

Three divided by zero is equal to something.

What if zero was the divisor, not the dividend, which is what we have used it as so far.

Interesting.

I wonder what the quotient would be.

So they're looking at that equation in the middle, which they've helpfully labelled for us and this time we are considering zero as the divisor, not the dividend.

Let's write out the related multiplication equation.

The product of the multiplication becomes the dividend of the related division equation.

So the dividend will become the product of the related multiplication equation.

Good thinking Jun.

So we know that the product is going to be three.

The divisor is the second factor in the multiplication.

There it is.

Oh dear, this can't be correct.

What has Jun noticed? Have a look at that multiplication equation with a missing factor.

What do we already know, the product of that multiplication equation should be given the factor of zero? Hmm.

If zero is a factor, the product will always be zero, but it's three here.

Let's write out some more to check.

So they start writing out some more and this is the one they've got to begin with, they know that it's incorrect so they put a cross next to it.

They try it with four, same thing.

Five, same thing.

Six, same thing.

We could definitely generalise here.

What can you generalise do you think? Look across all of those and what can we say is definite about all of them? We cannot write an equation where the divisor is zero.

Okay, your turn to check your understanding.

Explain why the following equation is incorrect by using a related multiplication.

So you're not being asked necessarily to explain in words, you're being asked to explain by finding a related multiplication fact that shows the equation must be incorrect.

Okay, pause the video here and give that a go.

Welcome back.

Here is the equation you might have written, three, which was the dividend became the product, which was three.

The dividend of the division becomes the product of the related multiplication equation.

And you can see that they put zero times zero and they put a funny symbol in there.

And that funny symbol actually means not equal to three, because zero multiplied by zero as we know is equal to zero, not three.

The divisor and quotient become the factors which can't be correct here because zero multiplied by zero equals zero and not three.

Jun and Lucas explores something else new.

What if we use zero for the dividend as well? Zero divided by zero is equal to something.

Let's write out the related multiplication equation.

The product of the multiplication becomes the dividend of the related division equation.

So the dividend will become the product of the related multiplication equation.

Our dividend here was zero, so our product is zero as well.

The divisor is the second factor in the multiplication.

So it's zero and it's zero there.

What do you notice? Hmm? Because one of the factors is zero, the other can have any value and the product will remain zero.

So that's also true of the quotient because that's the same value as the first factor.

So there are infinite answers.

I think I'd get this one right.

Very good, Lucas, you're absolutely spot on.

You could write any number in there.

I agree, but it's not necessarily a useful equation.

Okay, it's time for the second practise task.

You've got to tick the equations featuring zero that you can solve with one value, not an infinite number of values.

Explain your choices with a solution or a reason why they can't be solved.

For number two, using the number cards below, how many ways can you complete the equation, something divided by something is equal to zero? Pause the video here and have a go at those.

I'll be back in a little while with some feedback.

Welcome back, let's start with number one.

These were all equations that could be solved.

Zero divided by six is equal to zero.

Three multiplied by zero is equal to zero.

Zero multiplied by zero is equal to zero.

Zero multiplied by five is equal to zero.

Zero divided by seven is equal to zero For these two, zero can't be used as a divisor.

This can have infinite solutions.

Let's look at number two now then.

Jun's gonna explain to us again.

"Zero can't be the divisor, so it must be the dividend, to ensure the quotient is zero." So there are the combinations you could have had.

Each of your number cards could have been used once as the divisor.

That gave you nine ways in total.

Remember, division is not commutative, so you couldn't double up by swapping around the divisor and the dividend.

We've reached the end of the lesson.

Now it's time to summarise our learning.

If zero is a factor in a multiplication, the product is always zero.

In a division equation, when zero is the dividend, the quotient will always be zero.

The divisor can never be zero.

We will learn more about this in the future.