Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson.

And it comes from the unit all about the seven times table, odd and even patterns, square numbers and tests of divisibility, lots to be thinking about.

So if you're ready, let's make a start on today's lesson.

So in this lesson we're going to be thinking about identifying patterns of odd and even numbers in times tables.

So let's have a look at what's in our lesson today.

We've got two key words.

They're odds and even, I'm sure they're words that are very familiar to you, but we'll practise saying them and then think about what they mean.

So I'll take my turn and then it'll be your turn.

Are you ready? My turn, odd.

Your turn.

My turn, even.

Your turn.

I'm sure those are words you've been using for a very long time in your maths and in your thinking, but let's just remind ourselves of what they mean 'cause they are going to be very useful to us today.

So an odd number is any whole number that cannot be divided exactly by two.

And we can spot an odd number because it's ones digit is one, three, five, seven or nine.

And if you remember, it's only the ones digit that we have to look at because all multiples of 10 and a hundred and a thousand are going to be even numbers.

So it's just that ones digit that's important to us.

And any whole number that can be divided exactly by two is an even number.

And the ones digits in even numbers are zero, two, four, six, and eight.

If it's a zero in the ones, then we know we've got a multiple of 10 and those are even.

And two, four, six and eight are our even single digit numbers.

So that tells us that any number with a ones digit of zero, two, four, six or eight will be even.

So look out for that and that thinking in our lesson as we go on today.

There are two parts to our lesson.

In the first part we're going to be identifying patterns and in the second part we're going to be spotting patterns across multiplication charts.

So let's make a start identifying patterns and we've got Jacob and Sofia helping us in our lesson today.

Jacob and Sofia are looking for patterns in the times tables.

They investigate the products that are odd and even numbers.

So the products are the results of our multiplication, aren't they? Jacob says, "Remember an odd number cannot be exactly divided by two and the ones digit is one, three, five, seven or nine." And Sofia's reminding us that an even number can be divided exactly by two and the ones digit is zero, two, four, six or eight.

So let's remember that those were our keyword definitions, weren't they? So let's focus on those odd and even numbers for a moment.

We've got some numbers here on the screen.

Can you sort them into odd and even and explain how you know to a partner or somebody you are with at the moment? Pause the video, have a go, when you're ready for some feedback, press play.

How did you get on? Let's sort these numbers then.

There they go.

Is that how you sorted them? Did you for the ones digit? So our odd numbers all have ones digits of one, three, five, seven or nine.

In this case we had seven, which is an odd number.

It's only a single digit number.

19 has a ones digit of nine, so it must be odd.

And 23 has a ones digit of three, so it must be odd.

It's an even number of tens, but that doesn't matter when we're thinking about odd and even numbers, it's that ones digit that matters.

So our even numbers in that set were 188 and 242.

And in fact, all of those, well 88 and 242 were made from even numbers, weren't they? But in all of them the ones digit was a zero, a two, and in this case an eight.

And they tell us that those numbers are even.

So during the maths lesson, the children make some predictions about factors and products.

So the factors are the numbers that we multiply together and the product is the result of the multiplication, isn't it? Jacob says, "I think when both factors are odd, the product will be odd." And Sofia says, "I think when both the factors are even the product will be even." I wonder if you've thought about odd and even factors before.

You might want to pause and have a little think before Jacob and Sofia go on and share their thinking.

They're going to begin by investigating Jacob's prediction first.

So Jacob predicted that if both the factors were odd then the product would be odd.

So he's starting thinking about one times three is equal to three.

So one, lot of three, one group of three is equal to three.

What do you notice about the factors and the products? That's right, they're all odd.

One is odd, three is odd.

Those are our factors.

And our product of three is also odd and we can see it's an odd number can't we? Because it's not made up of pairs and we've got our number shape there showing us that it's one pair and one extra one and we know that multiplication is commutative so we can change the order of the factors and the product stays the same.

So we can also think about three times one is equal to three and again we've got odd factors and an odd product.

So the factors one and three are odd and the product three is also odd.

So can we say that if both the factors are odd, the product is odd, but we can for this one, let's have a look at another one.

So they're going to choose a different odd factor pair each.

So one of them chooses seven times five is equal to 35.

We've got two odd factors and our product is odd as well.

Seven and five are odd numbers and 35 is odd because five in the ones digit is an odd number and they also try three times nine is equal to 27.

The factors are both odd.

Three and nine are odd numbers and 27 is odd because it has a seven in the ones.

So it's an odd number.

So we can say that the factors seven and five, and three and nine are odd and the products 35 and 27 are also odd.

If both factors are odd, the product is odd and we are demonstrating that aren't we? With lots of different examples.

It's over to you now to check your understanding.

Pick two digits that are odd from the digit zero to nine and multiply them together.

Do you get an odd number as your product? Pause the video, have a go.

You might want to choose a few examples to have a go at and when you're ready for some feedback, press play.

I wonder which numbers you chose.

Say you may have got three times five is equal to 15.

So there's one group of five, two groups of five, three groups of five and three times five is equal to 15 and 15 is an odd number.

And we can see that with the number shapes as well.

Two groups of five went together to make 10 and our third group of five gives us our five in the ones in a way, doesn't it? And that's got that one extra one.

So 15 is an odd number.

We have two odd factors and an odd product.

So Jacob's prediction is correct in as far as we've tested it.

He was right.

He said that when both factors are odd, the product will be odd and we've demonstrated that with all the examples we've chosen so far.

So I think we can probably say that that's true.

So you can also say it like this, odd factor multiplied by odd factor is equal to an odd product.

Something to watch out for as we go on through this lesson.

Do you remember Sofia's prediction? She said I think when both the factors are even the product will be even.

So let's test her prediction now.

She started with four times six is equal to 24.

What can we see there? Well let's have a look.

Let's look at four groups of six.

There's one, two, three, four groups of six.

Does that look like an odd number or an even number in those number shapes? It looks even to me because it's all made from groups of two, isn't it? And we can also see that 24 has a four in the ones.

So we've got two even factors, four and six and we've got an even product of 24.

Let's think about it with our commutativity.

So we can also think about it as six groups of four, six times four, let's have a look.

One, two, three, four, five, six groups of four and that's also equal to 24.

We can see that those number shapes cover the same space and we can also see that that is very much an even number.

It's made from groups of two, six and four are even factors.

And 24 is an even product.

It has a four in the ones, so it's an even number.

So the factors four and six are both even.

And the product 24 is even.

So we can start to say that if both factors are even the product will be even.

So let's test with another factor pair.

They're going to choose a different factor pair each.

So we've got two times six is equal to 12.

Two and six are even numbers.

So we have even factors and 12 is an even number, it's an even product.

And we've also got four times 10 is equal to 40.

Four and 10 are even numbers, the factors, and 40 is an even product.

It's a multiple of 10 and all multiples of 10 are even numbers.

They have a zero in the ones.

So the factors two and six and four and 10 are even.

And the products 12 and 40 are also even.

So two more to add to Sofia's prediction that if both factors are even the product is even as well.

So we think that Sofia's prediction is correct as far as we've tested it.

So well done to Jacob and Sofia, you made predictions and they were both correct and we could also write Sofia's as an even factor multiplied by an even factor is equal to an even product.

So we can think about it in terms of the multiplication and the equation that we would write down.

Time just to check your understanding, take a pause and think true or false when one factor is even.

And the other factor is even the product will be odd.

Is it true or false? And why? Pause the video and have a go when you're ready for some feedback, press play.

What did you think? It's false isn't it? This is Sofia's prediction, isn't it? If one factor is even and the other factor is even the product will be even.

When two even numbers are multiplied together, the product will be even.

So eight times six is equal to 48.

48 is an even number and eight and six are all even factors.

And if you imagine those number shapes, if you've got an even number of even number shapes, you're going to have an even number as the product aren't you? Because we won't have any of those sort of extra ones sticking out.

So you can visualise the number shapes to help you think about this as well.

Jacob makes one last prediction.

He says, "I think when one factor is odd and the other is even the product will be even." Hmm, I wonder what you think about that.

I wonder how we could test this one as well.

Ah, there we go.

So he's chosen one even factor and one odd factor two multiplied by three is equal to six.

There's one group of three and two groups of three.

Ah did you see that the two sort of odd ones came together to make an even number.

So two is an even number, three is an odd number and those are our factors.

We've got one even factor and one odd factor and our product is six and that's an even number.

Let's look at it the other way round.

So we looked at two groups of three.

Let's look at three groups of two.

One, two, three groups of two.

And you can see we've got six.

We know it's going to be six because we know multiplication is commutative.

But again we've got one odd factor and one even factor and our product is even.

So the factor two is even and the factor three is odd.

And the product six is even.

If one factor is even and the other factor is odd, then the product is even.

If you think about all those little odd ones that we've got, if we've got an even number of them, they're going to make an even number.

So again, we can use our number shapes to visualise what's happening there.

Let's try it again.

We've got three times four is equal to 12.

So let's have a look.

Three groups of four One, two, three groups of four.

So we've got three even numbers, however many even numbers we have, we're always going to have an even number, aren't we? So three might be an odd factor but we are multiplying it by an even factor.

So we've got three groups of four, three groups of an even number and that will give us an even product.

And 12 is our even product and we know that because we can see it in the number shapes, but two is an even number and we have a two in our ones.

Let's look at it the other way around four groups of three.

This time we're gonna see that odd number.

So there's one group of three, two groups of three, three groups of three and four groups of three.

And could you see that all those odd ones linked together to make a new pair.

So four times three is equal to 12, four is an even factor and three is an odd factor.

But when we multiply them together we get an even product of 12.

So the factor four is even and three is odd, but the product 12 is even.

If one factor is odd and the other factor is even the product is even.

And we can think about those number shapes to help us to visualise what's happening there.

So it's your turn to check.

You are going to pick two digits, one odd and one even from zero to nine and multiply them together so you get an odd or an even product and you might want to try a few different examples to convince yourself and to describe what's happening.

Pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? I wonder which numbers you chose.

We chose four times seven.

Four times seven is equal to 28.

There's one group of seven, two groups of seven, three groups of seven, four groups of seven.

Did you see all those extra ones pairing up? And we get a product of 28 and 28 is an even number.

So an even factor multiplied by an odd factor gives us an even product.

Time for you to do some practise now.

You are going to choose some cards from one to 12 and make two factors.

So you're going to choose two numbers to be your factors and you're going to write the expression into the table without calculating the product.

Will the product be odd or even? Can you tell by the factors whether your product will be odd or even.

Think about those predictions that Sofia and Jacob were making.

So for example you might choose two and seven.

So two is even and seven is odd.

So Jacob says, "The product will be even because we've got one even factor and one odd factor." And Sofia says, "Yes, that's because odd factor multiplied by even factor gives us an even product.

So we'd write that expression into the even box.

So have a go at drawing two factors and deciding whether the product will be odd or even without calculating what the product is.

Pause the video, have a go and when you're ready for some feedback, press play.

How did you get on? What did you choose? I wonder you may have chosen a four and an eight.

So that's two even factors.

Four is even an eight is even.

So the product will be even, yes so Sofia.

That's because even factor multiplied by even factor is equal to an even product.

So four times eight would've been written in the even box.

You could also have had an even factor and an odd factor into the even box.

But to get an odd product you'd have needed an odd factor and an odd factor.

And time to move on into the second part of our lesson.

This time we're going to be spotting patterns across multiplication charts.

We're still gonna be thinking about those odd and even factors though.

So let's look at the two and five times tables.

Here they are.

Well some of them written out for you.

Sofia says, "I'll circle the odd products in blue and the even products in red." So let's have a go see what you notice as she does that.

So the odd products in blue and the evens in red.

So even, even, even even and even in the two times table, what about in the five times table? Can you see what's going to happen? Well there's two even products in red and then we've got some odd products in blue there, haven't we? What do you notice? Oh, I wonder if you spotted this because two is an even factor for all multiples of two.

The products will be even because remember if we've got one even factor then our product will be even.

So all the products in the two times table will be even.

And we sort of know that, don't we? Because even numbers can all be divided by two.

So they all must be multiples of two.

What about the five times table though? Ah, did you spot this? Because five is an odd factor for all multiples of five.

The products alternate between odd and even if we have two odd factors, we will have an odd product.

So if five is multiplied by an odd number, the product will be odd and if five is multiplied by an even number, the product will be even.

And we can see that one times five is equal to five.

two odd factors.

Two times five is equal to 10, one even and one odd factor.

So we get an even product and Sofia says the multiples of two have either an odd and an even factor or two even factors.

And that will always give us an even product.

The multiples of five have either an odd and an even factor or two odd factors.

So they will alternate between odd and even products.

Time to check your understanding.

Will the next product in the two and five times tables be odd or even? And how do you know? So can you use the pattern to help you predict, and can you explain it? Pause the video, have a go, and when you're ready for some feedback, press play.

What did you think? Well we got six times two.

Well we know something about the two times table, don't we? Six times two is 12, isn't it? So that's going to be an even product and we know that because all the products in the two times table are even.

But what about the next multiple of five? Well we have an alternating pattern, don't we? And this time it was six times five, so we had one even factor.

So our product was going to be even.

So both products will be even because at least one factor in both equations is an even number.

Six times two is equal to 12 and six times five is equal to 30.

Well done if you spotted that.

So let's look at the four and seven times tables.

Does our pattern still work? Can you predict the pattern of odd and even numbers in the multiples? You might want to have a think before Sofia and Jacob share their thoughts.

So what did you think? Well Sofia says, "I've circled the odd products in blue and the even products in red again." So what do you notice again? Well because four is an even factor for all multiples of four, the products will be even if we've got an even factor then the product will be even.

So it doesn't matter what we're multiplying four by, the product will always be even.

But what about the seven times table? Because seven is an odd number for all multiples of seven.

The products alternate between odd and even because sometimes we are multiplying seven by an odd number and then sometimes we're multiplying it by an even number and if we're going up in order through our times table, one times seven is seven, two odd factors.

Two times seven is 14.

We've got an even factor.

So our product will be even and that will alternate three times seven is odd, four times seven will be even, five times seven will be odd and we can predict that six times seven will have an even product because we are multiplying seven by an even number.

And Sofia's reminding us if one or both of the factors is even, then the product will be even.

And remember that both factors need to be odd for the product to be odd and that's why we'll never get an odd product in the four times table.

Time to check your understanding.

Jacob's making a prediction.

Do you agree and can you explain your thinking? Jacob says, "The nine times table will result in a pattern where the product alternate between odd and even and there's some of the nine times table for you to explore." Have a think, what do you think? Do you agree with Jacob? Pause the video now and when you're ready for some feedback press play.

What did you think? Well Jacob is correct, isn't he? If the times table is odd, so this is the nine times table, an odd times table.

The products will alternate between odd and even because we'll have nine multiplied by an odd number and then nine multiplied by an even number.

So our products will alternate.

Well done if you spotted that.

And it's time for your final task.

So question one says, "Use two different colours to mark odd and even products in the multiplication charts." And you've got those on a separate sheet.

What do you notice? Are there any patterns? Question two says, "Is it true that products in the eight times table will always be even because eight, one factor, is even?" And question three says, "Is it true that products in the seven times table will always be odd because seven, one factor, is odd?" So here are your times table charts to highlight and circle those odd and even products and then have a think about questions two and three.

And when you're ready for some feedback, press play.

How did you get on? So the first one you were going to look at those multiplication charts and you were going to use different colours to identify the odd and even products and what did you notice? So you might have noticed that if the times table is even, all the products will be even and are there any patterns? Well you might have said if the times table is odd, the products will alternate between odd and even depending on whether we are multiplying our times table factor by an odd number or an even number.

So this is what your charts may have looked like for the odd times tables we alternate odd, even, odd, even products.

And for the even times tables, the product is always even because we have always have one even factor.

So question two asked, is it true that products in the eight times table will always be even because eight, one factor, is even? Yes, that's true isn't it? If the times table is even, so this is the eight times table, then all the products will be even.

Eight is the factor and it's even so all the products will be even.

And for question three we asked is it true that products in the seven times table will always be odd because seven, one factor, is odd? Well, that's false, isn't it? To get an odd product we need two odd factors.

And in the seven times table the other factor will alternate one times seven, two times seven, three times seven.

So the factors will alternate between being odd and even.

Well done if you've got all those right.

And I hope you enjoyed thinking and reasoning about whether the products would be odd or even when you multiplied.

And we've come to the end of our lesson.

So what have we been learning about today? Well, hopefully we now understand that when both factors are odd, the product will be odd.

And we also understand that when one factor is odd and the other is even the product will be even and when both factors are even the product will be even as well.

Thank you for your hard work and your mathematical thinking and I hope I get to work with you again soon.

Bye-Bye.