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 today.

By the end, we want you to be able to say, "I can explain what each factor represents "in a multiplication equation." These are the key words that you are going to hear and that you'll need to understand in order to access the learning fully.

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

So I'll say the word, and then I'll say "Your turn," and you can repeat it back.

We'll start with my turn.

Factor.

Your turn.

My turn.

Product.

Your turn.

My turn.

Commutative.

Your turn.

Here's a definition for each of those keywords.

A factor is a whole number which divides exactly into another whole number.

A product is the result of two or more numbers being multiplied.

You can see an example equation at the bottom there.

5 multiplied by 3 is equal to 15, and in that equation, 5 and 3 were the factors, and the product was 15.

In addition and multiplication, numbers may be added or multiplied together in any order.

This is called commutativity.

And you can see some example equations below.

They've been paired.

We've got 5 multiplied by 3 is equal to 15, and using commutativity, the positions of the factors can swap to give 3 multiplied by 5 is also equal to 15.

We've got addition pairs on the right hand side there.

1 plus 2 is equal to 3, and 2 plus 1 is equal to 3.

This is the outline for today's lesson.

We're gonna start by thinking about what do factors represent? Then we'll move on to finding equivalent expressions.

These are two friends that we are gonna meet as well.

They're gonna help us by discussing some of the maths that we encounter during this lesson.

We've got Aisha and Laura.

Hi, you two.

Okay, then I hope everybody is ready to start.

It's time to begin.

Aisha and Laura are practising their four times table.

0 times 4 equals 0, 1 times 4 equals 4, 2 times 4 equals 8, 3 times 4 equals 12, 4 times 4 equals 16.

5 times 4 equals 20.

But what do the numbers represent? Aisha says the first factor represents the number of groups.

The second factor says Laura represents the group size.

The third number is the product and represents the total amount.

Aisha begins to say the four times table.

0 groups of 4 is equal to 0.

1 group of 4 is equal to 4.

2 groups of 4 is equal to 8.

3 groups of 4 is equal to 12.

Laura writes the factors the other way around.

There they are.

What do the numbers represent now? Hmm, I need to compare them both.

Aisha says four still represents the number in each group in all the equations.

So we can read it as 4, 0 times is equal to 0.

4, 1 time is equal to 4.

4, 2 times is equal to 8.

4, 3 times is equal to 12.

They say the four times tables both ways.

0 groups of 4 is equal to 0.

4, 0 times is equal to 0.

1 group of 4 is equal to 4.

4, 1 time is equal to 4.

2 groups of 4 is equal to 8.

4, 2 times is equal to 8.

3 groups of 4 is equal to 12.

4, 3 times is equal to 12.

Okay, let's check your understanding of what we've thought about so far.

You need to complete these speech bubbles by saying this times table fact both ways.

So you can see you've got some empty lines there, what would you put into that speech in order to make it complete? Pause the video here.

Have a go at that, and I'll be back to reveal the answers shortly.

Welcome back.

In the first one, you should have written 3 groups of 4 is equal to 12, and for Laura it was 4, 3 times is equal to 12.

Did you manage to get those? Okay, let's move on.

We can talk about these eight pairs of socks in different ways.

You can see them there.

We've got eight pairs.

Aisha says, we can say eight groups of two as we have eight pairs of socks with two in each pair.

Laura says, "Hmm, I think we can say, 2, 8 times because we have 2 socks 8 times over.

Two slightly different ways of thinking about the same thing.

Laura says, we can represent 2, 8 times as 2 times 8.

Okay, your turn to do something similar to check that you've understood that.

Describe the number of shoes using two different multiplication expressions.

You can see them at the bottom there, but they're incomplete.

We've got five groups of something and we've got two something times.

Pause the video, and see if you can complete those sentences.

Welcome back.

Here are the answers.

We've got five groups of two that's been written as an expression as well.

Then we've got 2, 5 times, so they were the two alternative ways of describing the number of shoes.

"Both ways represent five pairs of shoes," says Laura.

We've got an array here, and Aisha and Laura are gonna compare the multiplication equations that they would attach to this array.

Aisha says, "I can see 3 times 2 equals 6, the three represents the number of groups and two represents the number in each group.

Laura says, "I can see 2 times 3 equals 6.

2, 3 times.

Aisha looks at the array differently.

If we group horizontally.

The two now represents the number of groups and the three represents the group size.

So I can see 2 times 3 equals 6, whereas I can see 3 times 2 equals 6, because I think of it as 3, 2 times.

They both look at the arrays together, side by side.

And they're thinking about the questions, what's the same and what's different? Have a look yourselves.

What do you think is the same and what's different between those two representations and the accompanying equations? Well, Aisha says in both equations, three and two are factors and six is the product.

Laura says the factors can be written in a different order, but they will still represent either, the number of groups or the group size.

So factors can be swapped around, but the product will be the same.

I remember learning that before.

It's called commutativity says Laura.

Do you remember learning that before? You probably do, and we talked about it at the start as one of our key words as well.

Okay, then let's check your understanding.

True or false? Factors can be multiplied in any order.

Pause the video, and decide whether you think that's true or false.

Welcome back.

That was true.

But I'm gonna get you to try to justify the answer because as mathematicians, we need to be able to reason our thinking, not just choose.

You've got two justifications here that you need to select from.

A, you get to choose how you want to work things out or B, the product will still be the same because of commutativity.

For example, 3 multiplied by 2 is equal to 2 multiplied by 3, which is equal to 6.

Pause the video here, and choose which justification you think is best.

Welcome back.

B was the best justification here.

For A, you do kind of get to choose how you want to work things out, but there are laws, multiplicative laws that we have to follow and they can't be changed just by the order that you decide you wanna work things out with.

Okay, then let's look at the next part.

Aisha and Laura look at another array and write multiplication equations.

This array shows five groups of three.

I would write this as 5 times 3 is equal to 15.

I think it can be represented by 3 times 5 equals 15 because there is 3, 5 times.

This is one image represented by two equations, and they've been labelled there.

Can you see there's an equation on the left, an equation on the right, and there's an image in the middle.

Both of those equations describe that image.

Can we think of a real life context to replace the array? Could you think of a real life context where you might have 5 times 3 or 3 times 5 is equal to 15? Aisha and Laura consider a real life context.

My teacher once asked me to sort 15 pencils into 5 pots.

So you had three pencils per pot, must have looked like this.

Aha, a real life context, makes it easier to understand sometimes.

Again, one image represented by two equations.

So we've got one image in the middle there.

We've got five pots with three pencils in each pot, and we've got two equations that can be used to describe that.

What if we try and draw two images from one equation? That's an interesting thought, Laura.

I love challenging questions like that in maths.

This time they start with one equation and draw two images.

So the equation they've chosen is in the middle there, you can see it.

5 times 3 is equal to 15.

Aisha says, I have drawn five pots of three pencils again.

So there's that first image, and that's the one that we've just seen.

I wonder what the other image might look like? How will it be different, I wonder? Well, let's see what Laura came up with.

I have drawn three parts of five pencils.

Ah, of course, so the factors have actually swapped their roles in this one.

What was the number of groups has now become the group size? Both images are possible interpretations of one equation.

Yes, we have created two images from one equation.

Okay, then your turn to check your understanding.

Using the pencils in a pot context, draw two quick interpretations of the following equation.

And Laura gives some really good advice here.

Remember the drawings don't have to be perfect, they're just representations.

If I were you, I'd just draw a simple shape for a pot and just use stick lines to represent the pencils in them.

Okay, pause the video, and have a go at drawing two images from the equation.

2 times 3 is equal to 6.

Welcome back.

Now your drawings may not look exactly the same as this, but as long as they're representing the same number of pots and pencils, then you can consider yourself correct.

Here's one image.

We've got two pots with three pencils in each, and here's a second image, three pots with two pencils in each.

Now were your representation similar to that? I hope so.

It's time for your first practise task.

What you need to do is complete the stem sentences for each array.

Do take note of the groups in the array is represented by the purple boxes.

Some of them are vertical and some of them are horizontal.

So you've got to fill in those sentences with numbers.

For number two, for each of the images and expressions below, write a multiplication equation to represent the number of pencils and decide which factor represents the pots and which represents the pencils.

And Aisha says as well, what do you notice? For number three, using pencils and pots as a context, draw two different images for the two different interpretations of the equation below.

Then write a sentence describing the image and its factors.

So the equation to start with is 5 times 4 is equal to 20.

You've got to draw two different images for that, and then write a sentence describing each image and its factors.

Okay, pause the video here, and I'll be back shortly with some feedback.

Good luck.

Welcome back.

We're going to start by marking number one, so be ready.

4 times 3 is equal to 12 because there are 4 groups of 3 counters, or 3 times 4 is equal to 12 because there are 3 counters 4 times.

For the middle one, 4 times 5 is equal to 20, because there are 4 groups of 5 counters, or 5 times 4 is equal to 20 because there are 5 counters 4 times, and lastly, 5 times 4 is equal to 20 because there are 5 groups of 4 counters, or 4 times 5 is equal to 20 because there are 4 counters 5 times.

Pause the video here if you need some more time to make sure that you've marked accurately.

Let's do number two now.

So we had to fill in those blanks there by looking at the images closely.

On the first one, we had 3 times 4 as equal to 12.

There were three pots and four pencils, and for the second one, we had 4 times 3 is equal 12, four pots and three pencils.

Now, do remember that the factors could swap positions.

So it could be that you've written two identical equations or that your factors in the opposite way round for each of those.

Aisha said, "What do you notice?" The outcome is the same in both equations, 12 pencils.

However, the number of pots is different in each, so the factors are important in telling our story.

Time to mark number three.

Here's the first image that you might have drawn.

5 pots of 4 pencils is equal to 20 pencils.

There's the second image.

4 pots of 5 pencils is equal to 20 pencils.

Remember, your images didn't need to be entirely accurate as to how pencils and pots look in real life, as long as they were representative then that was okay.

It is time for the second part of the lesson now.

We're gonna look at finding equivalent expressions.

Aisha and Laura are asked to play true or false.

They've got a post-it note there with a statement on 4 times 3 and 3 times 4 have products that are of equal value.

What do you think? Think about what we've learned or revised so far.

Aisha says because of commutative, this is true.

Factors can be swapped, but the product will be the same.

Laura says, here's to arrays showing that.

Brilliant, Laura.

Making sure that you've got something to show as evidence to support what you think.

Again, that's what good mathematicians do.

The first showing four groups of three, the second showing three groups of four.

You can see the groupings there have been shown with the boxes.

In the first array, the boxes are vertical, so they're showing that there are four groups of three, and in the second array, you've got them horizontally showing three groups of four.

Here's another statement.

4 times 12 and 2 times 14 have products that are of equal value.

Again, what do you think? Look closely at those factors.

Examine them.

What do you think? Aisha said, "I'm not sure about this one." The digits are the same, but they aren't creating factors that are the same.

The digits are the same.

Let's have a look.

Oh yes, four, one, and two, and then in the second expression, two, one, and four, the digits are the same.

I think it's false.

I'll represent each.

Good.

Again, Laura is making sure that she can prove what she's thinking by using representations.

Laura sets out to prove that the statement is false.

I'll use base 10 this time.

4 times 12, so she's put 12 down 4 times.

4 groups of 12 is equal to 48.

We can see we've got 4 lots of 10 there.

That's 40.

And then we've got 4 lots of 2, that's 8.

Both of those combined give 48.

Now she's gonna do 2 times 14.

There's 1 lot 14, and there it is doubled, which is the same as multiplying it by two.

Again, you can see that she's got 2 lots of 10 and 2 lots of 4.

2 lots of 10, make 20.

2 lots of 4, make 8.

So combined.

It makes 28, 2 groups of 14 is equal to 28.

There is proof that the statement is false because those two products are not of equal value.

The products are different values, so the statement is false.

Well done, Laura.

Okay, it's your turn.

For each of the statements below, state whether they are true or false, and justify your answer with an explanation and a representation.

So you're doing the same thing that you have just seen the girls do.

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

Good luck.

Welcome back.

Here are the bits of proof for A and B.

Both of those were true.

2 times 3 and 3 times 2 have products that are of equal value, and 5 times 4 and 4 times 5 have products that are of equal value as well.

It says at the bottom, the factors are commutative.

The product is the same for each of these, and you can see the accompanying arrays as well.

The only difference between those two pairs of arrays is that sometimes the groups have been drawn horizontally and sometimes they've been drawn vertically, but that doesn't change the value of the product.

Here's C then.

This one was false.

5 times 5 and 5 times 6 have products that are of equal value.

One of the factors is different in the second expressions of the products are different.

You can see a comparison of their arrays there.

The groups have been drawn in the same direction because for each of them, there are five groups of the second factor.

The difficulty in the products being the same is that in the first expression, the second factor, or the size of the group is five, whereas in the second expression the size of the group is six, so the products are not of equal value.

Here's D.

Again, this was false.

12 times 3 and 13 times 2 have products that are of equal value.

No, they don't.

The digits within the expression are the same, but the factors are different.

It's a bit like the example we saw earlier, and base 10 has been used to show this.

On the left hand side, you can see 12 times 3.

We use base 10 to work that out.

You've got 3 lots of 10 added to 3 lots of 2.

3 lots of 10 are 30, 3 lots of 2 are 6, so we'd make 36.

Then on the right hand side, you can see 13 doubled.

You've got 2 lots of 10 and 2 lots of 3.

2 lots of 10 make 20, 2 lots of 3 make 6, so added together, that's 26.

The products are not of equal value.

Here's E, another false statement.

3 times 22 and 23 times 2 have products that are of equal value.

The digits within the expression are the same, but the factors are different.

You've got base 10 showing 22 times 3 on the left, and that gives you in total, 3 lots of 20 plus 3 lots of 2, 3 lots of 20 is 60, 3 lots of 2 is 6, so that's 66.

And on the right hand side, we've got 23 doubled.

2 lots of 20 of 40, 2 lots of 3 of 6, so that's 46.

That product is less than the first product.

So they are not of equal value.

Lastly, here's F.

This is correct, 3 times 4 and 6 times 2 have products that are of equal value.

Although the factors are different, the product is the same.

The factors tell us how the counters need to be arranged in terms of group size and number of groups.

So if you look at those two arrays, you can see that there's the same number of counters in each, but they've been arranged differently because the group size and the number of groups are different.

That's shown by the fact that the factors are different in the expressions.

Okay, I've really enjoyed learning that today.

Here's a summary of all the things that we thought about.

Factors can either represent the number of groups or the size of each group.

Factors are commutative so they can swap positions within a multiplication equation and the product remains the same.

When looking at multiplication expressions, the factors can be interpreted in two different ways to give two different images.

My name is Mr. Tazzyman, I hope you enjoyed that.

I know I did, and I hope to see you again in the future in another math lesson.

Bye for now.