Lesson video

Hello everyone.

Welcome back to another maths lesson with me, Mrs. Pochciol.

I can't wait for us to have lots of fun together and hopefully learn lots of new things.

So let's get started.

This lesson is called explain the patterns and relationships between the 5 and 10 times tables, and it comes from the unit doubling, halving, quotative, and partitive division By the end of this lesson, you should be able to explain the patterns and relationships between the 5 and 10 times tables.

(keyboard thumps) Let's have a look at this lesson's keywords.

Factor, product, double, and half.

Let's practise them.

My turn, factor.

Your turn.

My turn, product.

Your turn.

My turn, double.

Your turn.

My turn, half.

Your turn.

Fantastic.

Now that we've said them, let's get using them.

Let's have a look at this lesson's outline.

In the first part of our learning, we're going to be looking at when the factors are the same.

And in the second part of our learning, we're going to be looking when the products are the same.

So let's get started with exploring those factors when they are the same.

Jacob and Sofia are back to help us with our learning today.

Are you ready, guys? Let's get going.

Jacob and Sofia want to continue to explore why the multiples of 10 are double multiples of five.

So let's have a look at this array.

How many 10s are there? We can see that there are two groups of 10.

So we can record this as 2 X 10 = 20.

Two 10s are equal to 20.

Or we could record this as 10 X 2 = 20 because 10 two times is also equal to 20.

Well done, guys.

I really love how you are using those words to explain what your equation is showing.

That's wonderful.

Sofia decides to explore though what happens when they halve the array.

If we halve our array, we can also see that two times by 5 is equal to 10.

Here we can see two groups of 5 is equal to 10.

We can record this as 2 X 5 = 10.

Two fives are equal to 10 is how we could explain that one or we could record this as 5 X 2 = 10, where we could say 5 two times is equal to 10.

2 X 10 = 20, and 2 X 5 = 10.

We can see that when we multiply 5 or 10 by the same factor, the factor times by 10 is double the product of the factor times by five.

10 is double 5.

So two 10s is equal to double two fives.

Can you see? So we know that 2 X 10 = 20 because that's double, or we know that 5 is half of 10, so 2 X 5 will be half of 2 X 10.

2 X 10 = 20, but 2 X 5 = 10.

They now explore what this array shows.

How many tens can we see here? We can see that there are three groups of 10 here.

3 X 10 = 30, and we can explain this as three 10s are equal to 30, or we might record this as 10 X 3 = 30.

10 three times is equal to 30.

So let's have a look what we can create now.

Sofia and Jacob now halve their array again to see what they can see now.

If we half our array, we can see that we now have three groups of five, which is equal to 15.

So we can record that as 3 X 5 = 15, or explain this as three fives are equal to 15.

But we might also record this as 5 X 3 = 15 or five, three times is equal to 15.

We know that 10 is double 5, so three 10s will be equal to double three fives.

Three fives are equal to 15.

So three 10s will be double that, which is 30.

But we could also see this as 5 is half of 10.

So three fives will be equal to half of what three 10s is equal to.

So three 10s is equal to 30.

So three fives will be half of that, which is 15.

Well done for that, Jacob and Sofia.

Thank you.

I'm really starting to see how the halving and doubling work when we're looking at multiples of 5 and 10.

So when the factors are the same, we can use our doubling and halving knowledge to find the products.

Let's have a practise of this.

Let's have a go at this check.

Use this array to help you find the missing products.

So pause this video, use what you know, and find the missing products and come on back when you're ready to see how you've got on.

Welcome back.

Let's have a look then at what we can see.

We can see that there are five groups of five.

5 fives are equal to 25.

So five 10s is equal to double five fives, which we know double 25 is equal to 50.

Well done if you used that knowledge to help you.

Or you might've solved it this way.

We know that five 10s are equal to 50.

So because that factor is the same, times it by five, the product will be half of five 10s, which is 25! Well done, Jacob.

Well done, Sofia.

And well done to you.

Which strategy did you choose to use? Did you use the 5 times table to solve the 10? Or the 10 times table to solve the fives? I think I would've used Jacob's strategy there because I'm really confident with my 10 times table.

Let's have a look then at where this might be useful.

Sofia has eight 10-pence coins.

She has 80 pence all together.

Jacob has 8 five-pence coins.

So how much money does Jacob have altogether? They both have the same number of coins.

So does that mean they have the same amount? Hmm, I'm not sure.

What do you think? Do you think they have the same amount? They may have the same number of coins but they are worth different amounts, because Sofia has 10-pence coins but Jacob only has five-pence coins, so they can't possibly have the same amount, can they? So let's have a look then.

Sofia knows that she has eight 10-pence coins.

So 8 X 10 is equal to 80.

She has 80 pence.

Jacob has 8 five-pence coins.

Each coin is worth 5 one-pennies.

So eight times by five will give Jacob his total.

So let's have a look then.

We know that 5 is half of 10, so those five-pence coins are worth half of Sofia's 10-pence coins.

So eight fives will be equal to half of eight 10s.

Eight 10s are equal to 80, and we know that half of eight is equal to four.

So half of eight 10s is four 10s which is 40.

So that means that Jacob has 40 pence.

Look how we use that knowledge to help us to solve that problem.

Well done, guys.

So let's have a look then.

So although they had the same number of coins, Sofia has 10-pence coins, but Jacob only has five-pence coins.

They had the same factor, they have the same number of coins, but because 5 was half of 10, that means that Jacob will only have half the amount that Sofia has.

So that means he has 40 pence.

Half of 80 is 40.

So let's see if you can use this knowledge to help you to solve this problem.

Jacob has seven 10-pence coins.

He has 70 p.

Sofia has 7 five-pence coins.

So how much does Sofia have altogether? Remember, 5 is half of 10, so seven fives will be half of seven 10s.

So pause this video, see if you can work out how much does Sofia have and then come on back when you're ready to see how you've got on.

Welcome back.

Let's have a look then.

Come on then, Jacob, how did you work this out? We know that 70 is an odd multiple of 10, so we're going to partition it to help us to halve it.

Half of 60 is equal to 30 and half of 10 is equal to 5.

30 + 5 = 35.

So that means half of 70 is equal to 35, and that means that Sofia has 35 pence.

Well done to you if you got this correct.

I love how you remembered how to halve an odd multiple of 10 there, Jacob.

Well done.

Let's move on then to Task A so we can have a little bit more of a practise of this to really build your confidence.

Jacob and Sofia create a game to help them practise what they've learnt.

They've collected 12 counters.

Sofia's going to pick out a random number of counters and that's going to be the factor.

We can then multiply that factor by 5 and by 10.

Sofia picks out five counters, so five is going to be their factor.

Now they've got their factor.

They're going to record these as an equation.

So we're going to record 5 X 5 and 5 X 10, and they're going to practise finding the product.

So pause this video and have a go at Jacob and Sofia's game for yourselves.

Grab your 12 counters, randomly pick a selection of them, and that is going to be your factor.

Record these as your two equations, times by 5 and times by 10, and then work out the product using the knowledge that we've been practising in our learning.

Do this a few times and then come on back when you're ready to see how Jacob and Sofia have got on.

Welcome back.

I hope you enjoyed playing Sofia and Jacob's game there.

It was a really good game to help us to practise timesing by 5 and timesing by 10 when the factor is the same.

Using that knowledge that we've learnt today, let's have a look then at how Sofia and Jacob got on.

They picked out five counters for their first turn, so that means that they're finding the product of 5 and 5 and then 5 and 10.

We know that 5 X 1 = 5, so 5 times by one 10 is equal to five 10s, which is 50.

Because 5 is half of 10, we know that five fives will be half of five 10s, so half of 50 is equal to 25.

So 5 X 5 = 25.

Well done, Jacob and Sofia for using that knowledge, and well done to you for completing Task A.

Let's have a look at the second part of our learning, when the products are the same.

So in the first part of our learning, we were looking when the factors were the same.

We're now going to look at when our products are the same and how we can use this knowledge to help us.

Jacob picks up four counters and records it as an equation.

He has four counters that each represent five.

So this is four groups of five, and he records this as four times by five is equal to something.

Jacob decides to show this on a number line.

5, 10, 15, 20.

He knows that four fives are equal to 20.

So how many 10s will Sofia need to make the same product? We know that two 10s are equal to 20, or two times by 10 is equal to something.

We can also show this on a number line.

10, 20.

We know that two 10s are equal to 20.

So can you see the product is the same there, but what is different? Hmm.

Jacob and Sofia notice that when the products are the same, the factor times by 5 is double the factor times by 10.

We know that 2 X 10 = 20, but we know that one 10 is equal to two fives.

So if we double the factor, multiplied by five, the product will remain the same.

So if two 10s are equal to 20, we know that double the amount of fives will also be equal to 20, which is four! Four times by five is also equal to 20.

2 X 10 = 20, so double 2 times by 5 is also equal to 20.

Can you see how we can use that knowledge to help us here? When the products are the same, the factor times by 10 is half of the factor times by 5.

If we know that 4 X 5 = 20, how many 10s will be equal to 20? We know that two fives are equal to one 10.

So if we halve the factor, multiplied by 10, then the product will still remain the same.

Can we see? So we did have four fives, but now we need half that amount because they're now 10s.

So two 10s are equal to 20.

4 X 5 = 20, so half of four, which is two, times by 10 is equal to 20.

Can you see how we can use that knowledge of doubling and halving with 10 and 5 to help us to find those missing factors? Sofia and Jacob now check this knowledge.

Sofia knows that 5 X 10 = 50.

So what will be the missing factor here? We can see that we're not timesing by 10 anymore, we're timesing by five.

And we know that one 10 can be equal to two fives.

So we're going to have double the amount of fives to still equal 50.

5 X 10 = 50, so double five times by five will also be equal to 50.

Double 5 is equal to 10, so 10 times by 5 will also be equal to 50.

Can you see? For every one 10, we have two fives.

So we have to double the amount of fives to still have the same product.

So let's have a go at this one.

8 X 5 = 40.

So what will be the missing factor here? Now we can see that we're not using fives anymore, we're looking at tens.

So I know that I'm going to need half the amount to make the same product.

So if eight times by five is equal to 40, we know that half of 8 times by 10 will also be equal to 40.

So if I've got eight fives, how many 10s will that be? Half of eight is equal to four.

So we know that 4 times by 10 will be equal to 40.

You can see here how we can use our knowledge of fives and 10s to help us to find those missing factors.

Jacob and Sofia now explore the equations.

Here we can see that both 5 times by 10 and 10 times by 5 is equal to 50.

So we can show this as one equation because they're equal.

5 X 10 = 10 X 5.

That's a great way of showing that equation there, Sofia.

Well done.

Jacob's now going to have a go with his equation.

8 X 5 = 4 X 10 because we know that they both equal the same product.

So we can say that those expressions are equal.

Over to you then for a check.

So how many fives will be equal to six 10s? And then use this knowledge to complete the equations.

You can see that Jacob has also created one of those two expression equations that Sofia and Jacob have just created.

So pause this video, have a go at the check, and then come on back when you're ready to see how you've got on.

Welcome back.

Let's have a look then.

We know that one 10 is equal to two fives.

So the number of fives will be double the number of tens.

So if I've got six 10s, how many fives will I have? We're going to have double 6, which is equal to 12.

We're going to have 12 fives.

So 6 X 10 = 60, but we're going to need double the amount to still equal the same product when we're using fives.

So that's going to be 12 X 5 = 60.

Then we can show that these two expressions are equal by writing 6 X 10 = 12 X 5, because they both equal to 60.

Well done if you completed those equations, and well done for completing this check.

Let's move on then to some problems. Sam has 10 five-pence coins.

Lucas has some 10-pence coins.

They both have the same amount of money.

So how many 10-pence coins does Lucas have? Hmm, I wonder how we're going to use our knowledge to solve this one.

Sofia and Jacob both work out this problem.

So let's have a look.

Sofia is first going to work out how much Sam has, then she can then use that product to work out how many 10-pence coins Lucas has.

That's a good strategy, Sofia, I like it.

Let's have a look.

10 X five = 50, so we know that Sam has 50 p.

Now Lucas has 10-pence pieces, but we know that he has the same product, which is 50.

So we need to work out what that missing factor is.

We know that two fives are equal to one 10, but to have the same amount, Lucas will have half the amount of coins that Sam has.

So half of 10 is equal to 5.

So Lucas must have five 10-pence coins because 5 X 10 = 50 and 10 X 5 = 50.

Well done, Sofia.

Lucas does have five 10-pence coins.

I'm very impressed.

Well done! Jacob now shows us how he solved this problem.

Jacob knows that 10 is double 5, so half the number of 10-pence coins would be needed to make an equal amount.

He records the equation like this.

Ooh, I see.

So Jacob's not working out the product here.

He's just using his knowledge of the expressions to help him.

Half of 10 is equal to 5.

So we know that Lucas must have five 10-pence coins because 10 X 5 = 5 X 10.

Well done, Jacob.

That's a really efficient strategy there.

You didn't have to work out any products at all.

You just used the knowledge of the factors.

I'm really impressed.

That's a really efficient strategy.

Well done to you.

Let's see if we can use Sofia and Jacob's strategy to help you solve your own problem.

Laura has two 10-pence coins.

John has some five-pence coins.

They both have the same amount of money.

How many five pence coins does John have? So you can see that we have recorded Sofia and Jacob's strategy there to help you.

So pause this video, see if you can find out how many coins John has, and come on back when you're ready to find out how you've got on.

Welcome back.

I wonder if you preferred to use Sofia's or Jacob's strategy.

Let's see how they got on.

Sofia used her strategy first.

2 X 10 = 20, so we know that 20 is going to be the product for both of the piggy banks because they both have the same amount of money.

We can now work out how many five-pence coins because we know what the product has to be.

2 times by 10 is equal to double 2 times by 5, which is four.

So four times by five is also equal to 20 because it's double the amount of fives.

But you might've used Jacob's strategy where he just used the expressions to help him.

He knew that 2 times by 10 would be equal to four times by five.

He didn't need to work out the product, he just used the knowledge of the missing factors.

Well done for completing those equations, and from that, we can see that John will in fact have four five-pence coins.

Well done if you said that.

I wonder which strategy you preferred to use though, Sofia's or Jacob's? Let's continue to practise this in Task B.

So in Task B, Jacob and Sofia have now created another game to help them practise what they've learned.

This time, they're going to select a number card and that's going to be their product.

Then they're going to record how many fives and how many 10s they need to make that product.

Sofia has selected a card.

She's picked 30, so that's going to be their product.

Jacob now records these as equations.

Something times by 5 is equal to 30, and something times by 10 is equal to 30.

Can you see that their products are the same? And they've just left those missing factors for them to work out.

We'll come back to Sofia and Jacob to see how they completed their first turn after you've had a chance to play this game for yourselves.

So pick a card, record the equations, and find those missing factors.

Remember to use what we know to help us to solve them more efficiently, rather than using your times tables.

Pause this video and come on back once you've had a chance to play the game.

Welcome back.

As we said earlier, let's see how Jacob and Sofia got on with their first turn.

30, then.

We know that 3 times one 10 will be equal to three 10s, which is 30.

So we know that 3 X 10 = 30.

How can we now use this knowledge to help us find the missing factor times by five? Our product is the same, so we know that 5 is half of 10, so we're going to need double the amount of fives.

Double 3 times by 5 will also be equal to 30.

So double three is six, so 6 times by 5 will also be equal to 30.

Well done to Sofia and Jacob, and well done to you for completing Task B.

I hope you can now see how using our knowledge of the relationship between 5 and 10 can help us to find those missing products and those missing factors.

I hope you've enjoyed your learning today.

Let's have a look at everything we've covered.

When we multiply the same factor by 5 and 10, the product of the factor times by 10 is double the product of the factor times by five.

When we multiply the same factor by 5 and 10, the product of the factor times by five is half of the product of the factor times by 10.

When the products are the same, the factor times by 5 is double the factor times by 10.

And when the products are the same, the factor times by 10 is half of that factor times by 5.

Thank you for all of your hard work today.

I know you might've found some of this lesson a little bit tricky, but once you understand it, it really does help you to solve problems involving 5 and 10 more efficiently.

I can't wait to see you all again soon to continue our learning.

Goodbye!.