Lesson video

Hello there.

How are you today? I hope you're having a really good day.

My name's Ms. Kay, and I'm really excited to be working with you on this unit, looking at the relationship between the three and the nine times table.

Now you've probably already done some learning about the three and the nine times table, and hopefully you're able to start to recall some of those key facts for those times tables.

Well, in this unit we're going to be really looking carefully about how those facts are connected and how we can use one to find the other.

By the end of this lesson today, you will be able to say that you can solve problems using the relationship between pairs of three and nine times table facts, including those with the same product.

We have one key word in this lesson today.

I'm going to say it and I'd like you to say it back to me.

Are you ready? My turn, product.

Your turn.

Great job.

Let's just check we know what that word means.

A product is the answer when two or more numbers are multiplied together.

So for example, five multiplied by three is equal to 15 is the equation, and 15 is the product.

We can say that 15 is the product of five and three.

In this lesson today we're going to be solving problems using the relationship between the three and nine times table facts and hopefully you're becoming more and more fluent with your three and nine times table.

Our lesson today has two cycles.

In the first cycle we're going to be solving problems with the three and nine times table, and then we're going to look at problems in different contexts.

So if you are ready, let's get started with the first learning cycle.

In this lesson today we're going to meet Andeep and Izzy.

Andeep and Izzy are going to be asking us some questions and helping us with our math learning along the way.

So let's start here.

What can you see in the counters? And we're asked to complete the table.

If you have some counters, you might want to recreate the array that we have created here.

How many counters are there all together? How many groups can you see? Let's see how we might complete this table.

Well, we can see that we have nine counters.

How many groups of nine do we have? Well, we have one group of nine.

There are nine counters altogether.

So that is one group of nine, and we can see from the colours that we have three groups of three.

We also know that if there is one group of nine, then there are three groups of three.

What can we see here, what's changed? Well, this time we have 18 counters.

How did I know that so quickly? Well, I can see that we now have two groups of nine.

I can see that we have two rows each with nine in the row.

I know that two multiplied by nine is equal to 18, so the total number of counters must be 18.

I also know that if there is one group of nine, then there are three groups of three.

There are three times as many groups of three as there are groups of nine.

So I can do two multiplied by three to find out how many groups of three there are.

So I can see that there are six groups of three.

I could also count to double check.

One, two, three, four, five, six.

I can see six groups of three.

What's happened now? That's right, we've added another row or group of nine counters.

I know that three multiplied by nine is 27, so I have 27 counters altogether.

We know there are three rows of nine, so that's three groups of nine and we know that there are three times as many groups of three as there are groups of nine.

Three times three is equal to nine, so there are nine groups of three.

I wonder if you can predict what's going to happen next.

That's right, we've added another group of nine.

We've added another row with nine counters in, so that's four groups of nine.

How many counters are there altogether? That's right, there are 36 counters altogether because there are four groups of nine which is equal to 36.

How many groups of three can we see? That's right, there are 12 groups of three.

You might be starting to see some patterns in this table.

What's going to happen for the last row of our table? Well, we have another group of nine.

So now we have five groups of nine, and five times nine is? That's right, it's 45.

So that means there are 45 counters altogether.

We know that there are five groups of nine, and we know that there are three times as many groups of three as there are groups of nine.

So that means there are 15 groups of three.

And remember, we could always count the groups of three if we needed to check.

Goodness me, that was a lot of hard thinking about groups of nine and groups of three.

Let's take a close look at that table.

Andeep is asking us, what do we notice? Can you see any patterns emerging from those numbers? Izzy says, that each group of nine is the same as three groups of three.

Remember that the groups of three are three times as many as the groups of nine.

Let's take a look at what she means by that.

If we look at the first row, we have one group of nine.

Each of those groups of nine is the same as three groups of three, so that means there are three groups of three.

If we look at the last row, there are five groups of nine.

If each group of nine the same as three groups of three, there are three times as many, which means there had to be 15 groups of three.

We can also say that there are three times as many groups of three, and hopefully you spotted that in these two columns, the groups of nine are groups of three, every time the number of groups of three was three times the size of the groups of nine, or we've times the number of groups of nine by three, to get the number of groups of three.

Let's think about some equations this time.

What do you notice about these equations? Here's the first one.

Five multiplied by three is equal to 15.

Think about what that means.

Let's just show that as an array.

Here I can see five groups, if we think about the horizontal rows and each group has a value of three.

There are 15 counters altogether.

The product is 15.

Five multiplied by three is equal to 15.

Let's look at equation B.

Five multiplied by nine is equal to 45.

What do you expect to see for an array? That's right, this time we still have five horizontal rows or groups, but this time they have a value of nine.

There are nine counters in each row or group.

This time there are 45 counters altogether, because five multiplied by nine is equal to 45.

What do you notice about those two equations? Can you spot any relationship between them? What's the same, what's different? Well, that's right, Andeep.

Five multiplied by nine is three times the size of five multiplied by three, and we can see that in the arrays.

We can see that the array for five multiplied by nine is three times the size of the array for five multiplied by three.

Let's think about this relationship using the array.

Here, I have highlighted five groups of three.

We can see that there are five rows with three in each row.

That's five groups of three.

And Izzy is pointing out that nine is equal to three groups of three.

So the product of B will be three times the product of A.

We can see that the second factor, three and nine, we can see that nine is three times the size of three.

We can think about nine as three times three, and we also notice that the product, the answer to our multiplication equation is three times the size.

15 multiplied by three is 45.

Let's take a look at these six equations.

What's are the missing products in these equations? So we have four multiplied by three, 10 multiplied by three, 100 multiplied by three, four multiplied by nine, 10 multiplied by nine, 100 multiplied by nine.

We are missing the product.

Remember, the product is the answer to the multiplication equation.

Hmm, how can you work out one product from the other? I can definitely see some patterns in some of these equations.

I wonder what you would do to find the products.

Well, thanks, Andeep.

Andeep is reminding us that the product when you multiply a number by nine is three times the product when you multiply the number by three.

So that should be able to help us out when we're thinking about the multiples of nine.

If we know the product of four and three, so if we know that four times three is equal to hmm, then we can find the product of four and nine by multiplying our original product by three and that will give us the answer.

And this is because nine is three times the size of three, so the product will also be three times the size.

So we can see here that three times three is equal to nine.

If one of our factors is three times the size, then the product needs to be three times the size, as well.

We know that four multiplied by three is equal to 12.

We know our four times table, we know our three times table, so we can do four multiplied by three is equal to 12, and we know that the product for four multiplied by nine is three times the size.

So we can do 12 multiplied by three, and 12 multiplied by three is equal to 36.

So four multiplied by nine is equal to 36.

Let's look at the second pair.

10 multiplied by three.

Well, 10 multiplied by three is 30.

10 multiplied by nine, well, nine is three times the size of three, so my product will be three times the size as well.

Hmm, 30 times three.

Ooh, I wonder if I can work that out.

Well, if 10 multiplied by three is 30, then 10 multiplied by nine is 90.

Three times three is equal to nine.

So three tens, or 30 times three, is equal to nine tens or 90.

What about 100 times three or 100 times nine? Well, 100 multiplied by three is equal to 300.

So 100 multiplied by nine is equal to 900.

The product for 100 multiplied by nine is three times the size of the product of 100 multiplied by three.

Time to check your understanding.

Fill in the missing products in these equations.

So we have 11 multiplied by three equals hmm.

11 multiplied by nine equals hmm.

And then we have six multiplied by three, and six multiplied by nine, seven multiplied by three, and seven multiplied by nine.

Take a moment to pause the video here, and have a think.

Welcome back.

How did you get on? Well, Andeep is helpfully reminding us that the product when you multiply a number by nine will be three times the product when you multiply by three.

So if we know that 11 multiplied by three is 33, then we know that the product for 11 multiplied by nine will be three times the size, or 33 times three.

And this is because nine is three times the size of three, so the product will also be three times the size.

That means if 11 multiplied by three is equal to 33, 11 multiplied by nine is equal to 99.

What's six times three? That's right, if six multiplied by three is equal to 18, then six multiplied by nine is equal to 54.

And finally, seven multiplied by three is equal to 21, and seven multiplied by nine is three times the size of that.

So, 21 multiplied by three is 63.

Well done if you got all of those.

Izzy and Andeep move on to thinking about using inequality symbols.

So remember the inequality symbols and the equal sign are symbols we can use to compare expressions or equations.

The first one you can see there is the greater than symbol so we can say hmm is greater than hmm.

The second one is the less than symbol, so we can say hmm is less than hmm.

And finally the last one is the equal symbol, so we can say hmm is equal to hmm.

I wonder can you see which symbols are missing between these expressions? Take a look.

Do you have to calculate each time, or are there some things you can notice about the expressions that will help us work more efficiently? Well, Andeep is reminding us that nine is three times the size of three, and I wonder if we can use that to help us.

Izzy is also saying that there must be three times as many threes as nines for the expression to be equal, and I think that's a really useful thing to think about as well.

If the number of groups of three is three times as many as the number of groups of nine, then we know that the expression is equal and we don't need to calculate.

Let's look at that first pair of expressions, two groups of three, one group of nine.

Can we say that two groups of three is greater than one group of nine? Can we say that two groups of three is less than one group of nine, or can we say that two groups of three is equal to one group of nine? Well, Andeep and Izzy have helpfully reminded us that there is a relationship between the groups of three and groups of nine.

And for them to be equal, the number of groups of three has to be three times as many as the groups of nine.

One times nine, well, that's one group of nine.

So I would need to have three groups of three for it to be equal.

And I can see that there are two groups of three.

Two groups of three must therefore be less than one group of nine.

And I can use the less than symbol to show this.

I could also have calculated two multiplied by three is six.

One multiplied by nine is nine.

I know that six is less than nine, but it's more efficient to reason about which one is greater and why using our knowledge of groups of three and groups of nine.

Let's look at the second pair.

One multiplied by nine or four multiplied by three.

Hmm, I wonder if you can predict which one of those is greater or whether they are equal.

Well, this time we have one group of nine, but we have four groups of three.

For them to be equal, we need three groups of three.

We have one more group of three than we need it to be equal.

So therefore, we can say that the four groups of three are greater than the one group of nine.

We can use the less than symbol between these two expressions.

One group of nine is less than four groups of three.

Again, we could have calculated, one times nine is nine, four times three is 12.

I know that 12 is greater than nine, so therefore nine is less than 12.

What about the next set of expressions? Six multiplied by three, two multiplied by nine.

Well, in this case I can see that the number of groups of three is three times the size of the number of groups of nine.

I have two groups of nine.

Two times three is six, and I have six groups of three.

So these expressions are equal.

Six multiplied by three is equal to two multiplied by nine.

What about the next set? 10 multiplied by three and three multiplied by nine.

Well again, we can look at the relationship here.

Three groups of nine.

If I have three groups of nine to make it equal, I would need nine groups of three.

I don't have nine groups of three.

I have 10 groups of three.

I have one additional group of three.

So that must mean that 10 groups of three is greater than three groups of nine.

So I can say 10 multiplied by three is greater than three multiplied by nine.

For the next pair, again, I can think about that relationship.

If I have four groups of nine, then I need four times three, which is 12.

I need 12 groups of three.

I have fewer than 12 groups of three.

So I can say that 10 multiplied by three is less than four multiplied by nine.

What about the last pair? Well again, I can see nine groups of three, and three groups of nine.

This one is easy.

I can use my understanding of commutativity to know that these two expressions are equal.

Nine and three are the factors, and it doesn't matter which way around they go in a multiplication equation.

These expressions are equal.

Time to check your understanding.

Which symbols are missing between these expressions? Explain how you know.

So we should try really hard not to calculate the products of these expressions.

Instead, can you use your reasoning about groups of three and groups of nine to decide which symbol to use? Pause the video here, and come back shortly for some feedback.

Welcome back.

How did you get on? Remember that nine is three times the size of three.

So there needs to be three times as many threes as nines for the expressions to be equal.

And if they're not, they're unequal.

So let's take a look.

Can we see any equal expressions? Well, yes, the middle pair are equal.

One group of nine is equal to three groups of three.

There are three times as many threes as there are nines.

What about above and below? Well, if I know that three groups of three is equal to one group of nine, then I must know that two groups of three is less than one group of nine.

It is smaller than nine.

And then I have four groups of three.

Well, if three groups of three is equal to one group of nine, four groups of three is another group of three.

So it must be greater than one group of nine.

Well done, if you placed those symbols correctly, but extra well done if you used your reasoning about groups of three and groups of nine to explain why.

Time for your first practise task.

For question one, I'd like you to fill in the missing numbers in these equations.

So you have A, five multiplied by hmm is equal to 15, and if you look at B, which is below it, five multiplied by hmm is equal to 45.

For question two, I'd like you to take a close look at the equations.

Are they correct? Explain how you know.

For question three, any that you've decided are incorrect in two, can you use the symbols less than or greater than to make those equations correct? So, if you think any are incorrect because they are not equal to one another, which symbol would you use instead of the equal sign? Good luck with those two tasks.

Pause the video here, and I'll see you shortly for some feedback.

Welcome back.

How did you get on with those two tasks? Let's take a look at the first one.

I hope you noticed the relationship between the products here and between the three and the nine times table.

Five multiplied by three is equal to 15.

I can see that the product 15 to 45, is three times the size, therefore the missing factor had to be three times the size as well.

Five multiplied by nine is equal to 45.

For C and D, four multiplied by three is equal to 12, and four multiplied by nine is equal to 36.

12 times three is equal to 36.

36 is three times the size of 12.

So the missing factor had to be three times the size of three.

For E and F, you may have noticed that the product was written first in the equation.

It doesn't matter, it doesn't matter which way round or which side of the equal sign the product or the expression goes.

27 is equal to three multiplied by nine.

27 is also equal to nine multiplied by three.

And finally, 18 is equal to two multiplied by nine.

So therefore, 18 is equal to six multiplied by three.

For question two, A and C are incorrect, but B and D are correct.

B is correct because five groups of three is equal to four groups of three plus three.

If you think about the second expression, four groups of three, well, we know that four groups of three plus another group of three is five groups of three, so they are equal.

For D, we know that four groups of nine is equal to 12 groups of three.

And if we know that, well, 11 groups of three plus three is the same as 12 groups of three, so we know they're correct.

Remember, I asked you to think about the incorrect ones, which we now know are A and C, and I asked you to think about whether you could change the equal sign for a greater than or less than sign to make these correct.

I wonder how you got on.

For A, we could say that five groups of three is less than four groups of three plus nine.

Five groups is equal to four groups of three, plus one more group of three.

So adding nine would be too much.

And for C, four groups of nine is less than 11 groups of three plus nine.

Four groups of nine is equal to 12 groups of three, which is the same as 11 groups of three, plus one more three.

But if we added nine, that would be too much.

Well done, if you identified that and changed those symbols.

Let's move on to the second part of our learning where we're solving problems in different contexts.

We're going to think about shapes and polygons.

A nonagon has three times as many sides as a triangle.

So a triangle is a polygon with three sides and a nonagon is a polygon with nine sides.

A nonagon has three times as many sides as the triangle.

Andeep and Izzy are using straws to create triangles and nonagons.

Each straw is one side of the shape.

Andeep is going to use nine straws to create a nonagon, and Izzy uses three straws to create a triangle.

Izzy makes six triangles.

How many nonagons must Andeep make to use the same number of straws? So remember for one triangle, Izzy has used three straws, because the triangle has three sides.

Andeep says that he knows that one nonagon will need three times the number of straws as one triangle, so he can make two nonagons with the same number of straws, because there are two groups of three triangles here.

We can say that six groups of three is equal to two groups of nine.

He can make two nonagons, if Izzy has made six triangles.

This time Andeep starts, and he makes four nonagons.

How many triangles must Izzy make to use the same number of straws? Hmm, I wonder.

Well, Izzy knows that one nonagon needs three times the number of straws as one triangle.

She can make three times as many triangles.

So how many nonagons are there? There are four.

So, if we can make three times as many triangles, what do we have to do to four? That's right, we have to multiply it by three, so she can make 12 triangles with the same number of straws.

There's a three from one nonagons, three from another, three from another, and three from another, three, six, nine, 12.

There are 12 triangles.

So we can say that four groups of nine is equal to 12 groups of three.

Time to check your understanding.

If Andeep makes three nonagons, how many triangles can Izzy make with the same number of straws? Take a moment to have a think.

Welcome back.

So, Izzy can make three times as many triangles, which is nine triangles.

If one nonagon can make three triangles, then three nonagons can make nine triangles.

Well done, if that's what you said.

Another quick check of your understanding.

If Izzy makes 15 triangles, how many nonagons can Andeep make with the same number of straws? Again, take a moment to have a think.

Welcome back.

What did you say? Well, Andeep can make one nonagon with the straws in three triangles.

There are five groups of three triangles.

So that means Andeep can make five nonagons.

There's one, two, three, four, five.

And we can say that 15 groups of three is equal to five groups of nine.

Well done, if that's what you said.

Time for your second practise task.

You're going to start with some digit cards.

I'd like you to turn one over, and you have that number of nonagons.

If you used all of the straws in the nonagons to make triangles, how many triangles could you make? Record your answers in the table.

What patterns do you notice? So for example, if you turned over the digit three, you had that number of nonagons, remember, there are nine sides to a nonagon.

If you use straws to make those nonagons, and you had three nonagons, how many triangles could you make? So you would fill out the number of straws, the number of nonagons, which for me would be three, and the number of triangles that you could make with the same number of straws.

Do that a few times, see what patterns you notice.

For question two, three, and four, we have a few worded problems. For question two, apples come in bags of three and in bags of nine.

If Andeep and Izzy, buy the same number of apples, Andeep has bought two bags, Izzy's bought six bags, how many apples did they each buy? How do you know? For question three, a baker baked 54 cupcakes altogether.

If he puts them into boxes of nine, how many boxes will he need? If he puts them into boxes of three, how many boxes will he need? And for question four, Izzy he makes three triangles using straws.

Andeep makes the same number of nonagons.

How many straws did Andeep use? Good luck with those tasks, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? Hopefully you will have had fun turning over digit cards, thinking about nonagons, and how many triangles you could make.

We've put ours in order, but remember you might not have chosen the digit cards in this order.

Did you spot any patterns, or did you notice anything about the relationship between the number of straws, the number of nonagons, and the number of triangles? Well, you might have noticed that you can always make three times as many triangles as nonagons.

Each extra nonagon means that you can make three more triangles and you may have linked that to the three and nine times table, because the products are all multiples of three and multiples of nine.

Remember that all multiples of nine are also multiples of three.

For question two, we know that Andeep bought two bags.

He bought fewer bags than Izzy, so he must have bought bags of nine.

We know that two multiplied by nine is equal to 18.

Izzy must have bought bags of three apples, because remember, the apples come in either bags of nine or bags of three, and we know that six multiplied by three is equal to 18.

There has to be three times as many bags for the product to still be 18.

For question three, the baker had 54 cupcakes.

If he put them into boxes of nine, how many boxes will he need? Well, we know that six groups of nine is equal to 54, so he needed six boxes of nine cupcakes.

If he had those 54 cupcakes and he put them into boxes of three, we would need three times as many boxes.

So we know therefore that if he needed six boxes of nine, he would need 18 boxes of three to use the same number of cupcakes.

And for question four, if Izzy made three triangles using straws, Andeep made three nonagons, he used three times as many straws.

So three groups of three is nine, and three groups of nine is 27.

He used three times as many straws as Izzy.

Well done, if you answered all of those word problems and thought really hard about the relationship between the three and nine times table.

We've come to the end of the lesson, and in this lesson we've been explaining the relationship between multiples of three and multiples of nine.

Let's summarise what we've learned.

We know that multiples of nine are also multiples of three.

In any product in the nine times table, there will always be three times the number of threes as there are nines.

And we've shown that we can use that knowledge to solve problems. Thank you so much for all your hard work today, and I look forward to seeing you in another math lesson soon.