Lesson video

Hi everyone.

My name is Ms. Koo and I'm really excited to be learning with you today.

We're going to have lots of fun and look at some interesting new keywords and some keywords you may have come across before.

We're going to work hard, but I'm here to help.

In today's lesson from the unit properties of number, factors, squares and cubes.

We'll be checking your understanding of factors, multiples, squares, and cubes.

By the end of the lesson, you'll be able to state the factors and multiples of different numbers and identify square and cube numbers.

Let's go through our keywords.

We have five keywords in our lesson today.

The first is a product, and a product is the result of two or more numbers multiplied together.

The next keyword is a multiple, and a multiple is the product of a number and an integer.

We're also looking at the words numerical factor, and a numerical factor is a factor that is an integer.

We'll be looking at a square number and a square number is the product of two repeated integers.

Finally, we'll be looking at the word cube number, and a cube number is the product of three repeated integers.

Now don't worry if there's plenty of keywords here.

We will go through them all in our lesson.

So let's start our lesson.

Firstly looking at multiples.

So let's have a look at some multiples of three.

I'm going to start by using dots first.

Three is a multiple of three as we have three dots.

Six is a multiple of three as we've simply added another three dots onto our original three dots.

Nine is a multiple of three as we have simply added another three dots, so three, six, and nine are on multiples of three.

Another way to identify multiples is to use calculations.

For example, let's look at the multiples of five.

Five times one is five, so five is a multiple of five.

Five times two is 10, so 10 is a multiple of five.

Three times five is 15, so 15 is a multiple of five.

We are multiplying an integer by five, thus giving us a multiple of five.

Lastly, let's have a look at some multiples of four.

Here I'm simply gonna randomly pick a integer and multiply it by four.

So I've just chose four times seven, so that means 28 is a multiple of four.

Next, I've chose 10 times four, which is 40, so I know 40 is a multiple of four.

Lastly, I've chose 11 times four, which is 44, so I know 44 is a multiple of four, so a multiple is simply the product of two integers.

So let's have a look at a check question.

Here we have three students and each student makes a statement and we need to identify who is correct and explain why.

Aisha says eight multiplied by two is 16, so 16 is a multiple of eight and a multiple of two.

Sam says 10 multiplied by 1.

5 is 15, so 15 is a multiple of 10 and 1.

5 And Andeep says you're both correct.

See if you can give this a go and press pause if you need.

Well, hopefully you've figured out that only Aisha is correct because 16 is the second multiple of eight or the eighth multiple of two.

Sam and Andeep are incorrect because although 15 is a multiple of 1.

5, it's not a multiple of 10.

Well done if you got this one correct.

Let's check your understanding a little further.

Lucas writes five random multiples of five, and Sam writes five random multiples of two.

You are asked to put these 10 numbers in a Venn diagram.

Press pause for you to draw your Venn diagram and fill in these numbers.

Well done, now don't worry if your Venn diagram doesn't look exactly like mine as long as the key points are in here.

I'm going to identify the left part of our Venn diagram to be multiples of two.

And the right part of my diagram to be multiples of five.

You may have got it the other way round.

It doesn't make a difference.

Now, before I start inserting numbers into my Venn diagram, I'm going to identify which numbers are multiples of five and multiples of two.

Hopefully we can see we have 20, 30 in both of our list of multiples.

So I'm going to put the 20 and 30 in the intersecting area of our Venn diagram.

Then I'm going to simply put in the rest of our numbers, so hopefully you can see how our numbers are distributed in our Venn diagram.

So now let's do another check.

Here the questions want us to identify multiples of four from the list of 13, 4.

1, 12, 40 and four eights.

Part B, you want us to identify multiples of five from our list of 50, four, five, 5.

55 and 50,005.

Part C wants us to identify the multiples of seven from our list of note 0.

07, 0.

7, seven, 70 and 700 and part D wants us to identify multiples of both four and five from our list of nine, one, 20, 10, and 100.

Remember, the definition of a multiple.

A multiple is a product of two integers.

For part D you might find it useful to draw a Venn diagram, see if you can give it a go and press pause if you need.

Well done.

So let's start identifying our multiples of four.

Hopefully you've identified it's 12 and 40 because three times four is 12 and we also know four times 10 is 40, so that means 12 and 40 are multiples of four.

Our multiples of five are 50, five, and 50,005.

This was a tough one, so well done if you got this one right.

Well, the reason why 50 is a multiple of five is because five times 10 is 50, five is a multiple of five because five times one is five and 50,005 is a multiple of five because 10,001 multiplied by five is 50,005.

That was a tough one.

Next one, multiples of seven, hopefully you identified our multiples of seven are seven, 70 and 700.

The reason why it's seven is because seven times one is seven.

The reason why 70 is a multiple of seven, seven times 10 is 70, and the reason why 700 is a multiple of seven is because seven times 100 is 700.

Lastly, a tough one here, multiples of both four and five, drawing a Venn diagram may have helped you out.

Hopefully you've identified there are only two answers.

20 and 100, four is a multiple of 20 because four times five is equal to 20.

A 100 is a multiple of four and five because four times 25 is 100, and five times 20 is 100.

This was a great question to check your understanding on multiples.

Now let's move on to our task.

The task shows a grid, for part A it wants us to draw a circle next to all of the multiples of two.

For part B, it wants us to draw a square next to all the multiples of three.

For part C, it wants us to draw a triangle next to all the multiples of five, and part D wants us to identify what do the numbers with a circle and a square represent? And E, well, it asks what do the numbers with a circle, square and triangle represent? Press pause to start and resume when you're ready.

Well done.

This was a great question.

So let's have a look at our answers.

Drawing a circle next to all the multiples of two gives us all of these answers.

How did you get on? Now for part B, drawing a square next to all the multiples of three gives us all of these answers.

And for part C, drawing a triangle next to all the multiples of five gives us these answers.

This was a tough one, so well done if you got those right.

Now part D asks us what do the numbers with a circle and a square represent? So you can see them on our grid.

Well, the ones with a circle and a square for example is six and we also have 12, and we also have 18, so on and so forth.

But what do these numbers represent? Well, they represent multiples of both two and three.

So what do you think the numbers with a circle, square, and triangle represent? Hopefully you've spotted it.

They represent multiples of two, three, and five.

Well done.

This was a tough question.

So now let's move on to factors.

Well, a numerical factor is an integer which divides exactly into another integer.

Let's have a look at some examples using factors of 12, I'm going to use arrays here.

I'm simply going to list my 12 dots in one row.

So 12 divided by one is 12, so this means I know one and 12 are both factors of 12.

12 fits into 12 once and one fits into 12, 12 times.

Now I'm going to draw another array here.

I've done my 12 dots but arrange them in two rows.

So 12 divided by two is six.

This means I know that two and six are factors of 12 because six goes into 12 twice and two goes into 12 six times.

I'm going to look at another array still having 12 dots.

I have three rows of four, so I'm going to divide 12 by three, giving me four, that means I know three and four of factors of 12.

Now let's have a look at some numerical calculations.

Looking at 20 divided by two equals 10.

Using the definition we can see two fits exactly into 20 10 times.

So that means there is no remainder and two and 10 are factors of 20.

Now factors are multiples are inverse concepts.

So what does that mean? Well, multiples is the product of two integers.

So let's rewrite our calculation using multiplication and be rewritten as 10 times two is equal to 20.

So this means we know 20 is a multiple of both two and 10, but it also identifies that 10 and two are factors of 20.

So recognising factors and multiples are inverse concepts is a very important key point.

So now let's check our understanding with this question.

Here the question asks us to identify which of the following is a factor of 20 and explain.

It does say we can select more than one.

See if you can give it a go and press pause if you need.

Well done.

This was tough.

Hopefully you've identified the only factors of 20 are four and 10.

Four is a factor of 20 because 20 divided by four equals five and there's no remainder.

Or you may have understood it as four times five equals 20.

10 is a factor of 20 because 20 divided by 10 is two, or you may have understood it as 10 times two is 20.

Either one of those explanations are fine to justify why four and 10 are factors of 20, well done.

You may have extended your explanation by looking at 40 and 200 and spotted that 40 is a multiple of 20 and 200 is a multiple of 20.

This was a tough one if you spotted it.

So well done.

Let's have a look at another check for understanding question.

Lucas writes the following, one times 15 is equal to 15, two times 7.

5 is equal to 15 and three times five is equal to 15.

Lucas then says all of these pairs have a product of 15, so this means they all must be factors of 15.

Is Lucas correct? and you need to explain.

See if you can give it a go and press pause if you need more time.

Lucas is incorrect because the factors must be integers.

So the only factors of 15 are one, 15, three and five.

So let's move on to our task.

Here the question shows some cards, two, three, five, seven, 11, 12, 13, 14, 15, and 22, and the question wants us to use the cards once and fill in the gaps.

Part A, B and C are the same.

They say something is a multiple of something else.

Part D wants us to identify what number is a factor of another number.

And part E states that a certain number only has a certain number of factors.

See if you can give it a go and press pause if you need.

Well done.

So let's move on to the second part of the task.

In this part it gives us a table and from the table we are asked to identify if it's always true, sometimes true or never true.

When the answer is sometimes or never true.

We're asked to give an example.

So the first part says, given A, B, and C are integers, A multiplied by B is equal to C, so therefore C is a multiple of A and B, and A and B are factors of C.

Do you think this is always true, sometimes true or never true? The second part states a multiple of a number is bigger than its largest factor.

Do you think this is always true, sometimes true or never true? The next statement is a product of two integers is always an integer.

Is it always true, sometimes true or never true? And lastly, the last statement states one is a factor of every integer.

Do you think that's always true, sometimes true or never true? See if you can give it a go and press pause if you need.

Well done and let's go through our answers.

Starting with question one.

This is a great question and really does embed the understanding of a multiple and a factor.

The first three examples, any pairing is absolutely fine.

So 14 is a multiple of seven, 12 is a multiple of three, 15 is a multiple of five.

Then I've given an example of a factor.

11 is a factor of 22, and finally 13 only has two factors.

Now you may have had a different set of answers, so I'm going to show you another example.

You can see I've changed the pairing between the factors.

So I still have the pairing of 12 and three.

12 is a multiple of three.

I still have the pairing of 15 and five, 15 is a multiple of five, but I've swapped over 22 and 11.

Showing 22 is a multiple of 11, and I've also swapped over the seven and 14 to show that seven is a factor of 14.

I've done this a couple of times so you can see the relationship between the pairs.

Well done if you've got any of these right.

Let's have a look at question two.

Well, for question two, the first statement was always true, and we know this because we know factors and multiples are inverse concepts.

The second statement is sometimes true as five is a multiple of five, but five is also a factor of five.

So that means a multiple of a number isn't always bigger than the largest factor.

Next, we know a product of two integers is always an integer, and we also know one is a factor of every number.

So a huge well done if you've got that one right.

Great work so far.

So let's move on to the last part of our lesson.

Square and cube numbers.

Remember, a square number is the product of two repeated integers.

So let's start identifying our square numbers.

Starting with multiplying one by one.

Now remember, we multiplying because it's the product and the repeated integers are one.

So one multiplied by one is one.

Next, let's have a look at the product of two and two is four.

The answer to three multiplied by three is nine.

And four multiplied by four is 16.

Do you think you can list the remaining square numbers? Press pause and see if you can do it without me.

Well done.

So let's have a look.

Well five multiplied by five is 25, so we have that as a square number.

Six times six is 36, so 36 is a square number.

Seven times seven is 49.

So 49 is our square number.

Eight times eight is 64.

So 64 is our square number nine times nine is 81.

So 81 is our square number, and 10 times 10 is a 100.

So it's another square number.

Continuing on, we have 11 times.

11 is 121, which is another square number.

Remember, there is an infinite number of square numbers.

I'm not gonna go through them all in this lesson, but hopefully you can identify that a square number is the product of two repeated integers.

Let's move on to cube numbers.

Well, a cube number is the product of three repeated integers.

So let's list our cube numbers starting with listing a repeated number of one.

Well, the product of one multiplied by one, multiplied by one is one.

Now let's have a look at the repeated number two.

Well two multiplied by two, multiplied by two gives us eight.

I'm gonna help you out here.

Two multiplied by two is four, multiplied by two is eight.

Sometimes breaking it up into those parts can help.

Next, let's have a look at three.

Three multiplied by three, multiplied by three, well three times three is nine multiplied by three is 27.

Do you think you can work out the next two cube numbers? Press pause if you need.

Well done.

So let's have a look.

Four times four times four, well four times four is 16, multiplied by four is 64.

Next five times five times five, well five times five is 25.

25 multiplied by five is 125.

So well done if you've got those numbers.

Great work so far.

So let's check our understanding.

Which of the statements is true? And we need to justify our answer.

A states every number is a square number and cube number.

B states there are no numbers, which are both square and cube numbers.

C states 64 is a square and a cube number, and D states a 100 is a square and a cube number.

See if you can give it a go and press pause if you need.

Great work, well done.

So let's see how you did.

Hopefully you've identified 64 is a square number and a cube number A 64 is the product of eight times eight, and 64 is the product of four times four times four, really well done.

Let's move on to the task.

Well, question one wants us to identify the unknown number given the clues.

Part A says, I'm a number less than 100, I'm greater than 50, I'm a square number and I'm odd.

Another number in part B says I'm a cube number.

The sum of my digits is nine.

I'm a multiple of nine, and I'm also less than 100.

Part C is another number, which is a factor of 24, is also a multiple of two.

The number is less than 10 and is also a square number.

And part D is another number, which is a cube number, also a square number, and a factor of every number.

See if you can give it a go and press pause if you need.

Really well done.

So let's move on to the second question.

The next question wants us to complete a Venn diagram showing all the square and cube numbers up to 200.

See if you can give it a go and press pause if you need.

Really well done, so let's go through our answers.

For question one.

Hopefully you spotted for part A.

The answer is 81 for part B, the answer's 27.

For part C, the answer is four.

And for part D, the answer is one.

For question two, hopefully you've got all of these wonderful square and cube numbers, which are all our square and cube numbers, up to 200, and you spotted the only numbers which are in that intersecting area of both a square number and a cube number is one and 64.

A huge well done.

So in summary, we've done a lot today.

We've looked at the product of a number, which is the result of two or more numbers multiplied together.

We've also looked at multiplying or dividing integers.

And we can identify factors and multiples.

We know the product of two repeated integers is called a square number, and we know the product of three repeated integers is called a cube number.

A huge well done, and it was great learning with you.