Lesson video

Hi, everyone.

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

It's gonna be an interesting and fun lesson.

We'll be building on some previous knowledge, as well as looking at some keywords that you may or may not know.

It might be easy or hard in some places, but I'll be here to help.

I'm really excited to be learning with you, so we can land together.

In today's lesson, from the unit properties of number, factors, multiple, squares, and cubes, we'll be looking at divisibility tests for six and nine, and by the end of the lesson, you'll be able to identify and explain whether or not a number is a multiple of six or nine.

Now, you might be thinking, "Why is six and nine so important?" Well, we know nine is a square number.

We also know nine is the highest single inter digit.

But what is so important about six? Have a little think.

What makes six so important? Can you think of some examples of why six is so important? Well, firstly, it's the first non-square and non-prime integer value.

In other words, one is a square number, two is a prime number, three is a prime number, four is a square number, and five is a prime number.

So six is the first non-prime and non-square number.

Now, you might also notice six is the only number that is both the sum and the product of three consecutive positive integers.

One, two, and three.

So let's sum one, two, and three.

One add two add three is six.

Now, let's multiply one times two times three is six.

Six is a special number.

Also, six is the first product of two primes.

So the reason why we're looking at divisibility tests for six and nine is because they are such important numbers.

So let's start by looking at some keywords.

Divisibility is a mathematical concept that refers to one number's ability to be exactly divided by another number leaving no remainder.

Here, I have an example and a non-example.

45 is divisible by five, because we know 45 divided by five is nine.

A non-example is the fact that eight is not divisible by five, because eight divided by five is 1.

6.

We'll be using this word divisibility a lot throughout the lesson.

The lesson will be broken up into two parts, the first being a divisibility test for the multiples of six.

So let's make a start.

Remember, a divisibility test checks if a number can be divided into another number without a remainder.

Divisibility tests are quick processes to identify if a number is divisible and we use divisibility tests instead of working out as it saves lots of time.

Let's have a quick recap on the divisibility test for two.

Well, a number is divisible by two if the digit in the ones column is always a zero, two, four, six, or eight, for example, we know 2,466 is a multiple of two.

Now let's have a look at the divisibility test for three.

Well, to identify if a number is divisible by three, we simply sum the digits, and if the sum of the digits is a multiple of three, then we know the number is divisible by three.

For example, 2,466.

Summing those individual digits gives us 18.

We know 18 is a multiple of three, so that means we know 2,466 is a multiple of three.

I'm going to list some other examples here.

34 is a multiple of two, 18 is a multiple of two, and 12 is a multiple of two, and we also know 30 is a multiple of two.

Now I'm gonna list some multiples of three.

33 is a multiple of three.

18 is a multiple of three, 12 is a multiple of three, and 30 is a multiple of three.

So all I've done here is list some multiples of two and some multiples of three.

Now, I'm going to put them in a Venn diagram.

Where do you think each of these numbers go? You can probably see from the list, 2,466 is both a multiple of two and three.

We can also see 12 is a multiple of two and three, and 30 is a multiple of two and three, and 18 is a multiple of two and three.

Now, let's insert the rest of our numbers.

Well, we know 34 is only a multiple of two, 33 is only a multiple of three.

From our Venn diagram, what do you think all the numbers that are multiples and two and three represent? Well done if you spotted it.

All numbers which are multiples of two and three mean they are divisible by six.

And this is our divisibility test for multiples of six.

So let's check our understanding with this question.

Here, it asks us to identify which of the following numbers are divisible by six.

Now, remember, if a number is divisible by six, it means it's divisible by two and it must be divisible by three.

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

Okay, so let's give this question a go.

Well, first of all, let's identify which of these numbers is divisible by two.

Remember, if a number is divisible by two, the digit in the ones column must be a zero, two, four, six, or eight.

Using this fact, that means we know the following numbers are not divisible by two.

So therefore, they are not divisible by six, leaving us with three numbers.

From these three numbers, let's find out if they are divisible by three.

I'm gonna start off with 9,388.

Well, summing those individual digits gives us 28, and 28 is not a multiple of three.

So that means we know 9,388 is not divisible by three, so therefore is not divisible by six.

Next, let's have a look at 19,386.

Summing these individual digits, we have 27.

27 is a multiple of three.

So that means we know 19,386 is divisible by two and divisible by three.

Thus, we know it's divisible by six.

Lastly, we have 521,238.

Summing these individual digits, we have 21.

21 is divisible by three, so therefore we know 521,238 is divisible by two and divisible by three, which means it's divisible by six.

Well done if you got this one right.

Now let's move on to your task question.

Here, the question wants you to circle the number or numbers from the list that are divisible by six.

1a gives us 389, 232, 890,382, and 348,997.

b gives us 8,992, 9,374, 380,000, 34,830, and C gives us four, 44, 444, 4,444, and 444,444.

You can see why we use divisibility tests as we don't wanna be wasting time dividing these numbers by two and three or six.

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

Well done.

So let's move on to our second question.

Question two states we have four students and they play a computer game.

Now, the game has 200 levels and every time a student moves onto the next level, they score six points.

Lucas says, "I have 918 points." Jacob says, "I have 1,114 points." Sam says, "I have 1,284 points," and Sofia says, "I have 1,194 points." Now who's telling the truth, and can you justify your answer for each person? See if you can give it a go and press pause if you need.

Great work.

So let's go through these answers, starting with question one.

We needed to circle all the numbers which are divisible by six.

Firstly, 389 is not even, and 348,997 is not even.

Looking at our remaining numbers, let's sum their digits.

Well, the two add the three add the two makes seven.

So that means it's not a multiple of three, the eight add the nine add the zero add the three add the eight add the two makes 30, which is a multiple of three.

So that means we only have one answer to part a.

Well done if you got that one right.

Moving on to b, you can see we have all even numbers.

So let's identify which ones are divisible by three.

Summing the individual digits of eight, nine, nine, and two gives 28.

This is not a multiple of three.

Summing the digits, nine, three, seven, four, gives me 23, which is not a multiple of three.

Summing the digits three and eight give me 11, which is not a multiple of three.

Summing the digits three, four, eight, three, and zero gives me 18, which is a multiple of three.

For part c, they're all even numbers.

So let's sum their individual digits.

Immediately you'll know four is not a multiple of three.

So I'm going to cross this one out, but I'm gonna sum the individual digits for each of these numbers.

Four add four is eight, not a multiple of three.

Four add four add four is 12.

This is a multiple of three.

Four add four add four add four gives me 16.

This is not a multiple of three.

Lastly, summing all of these fours gives us 24, and 24 is a multiple of three.

Question one was a great question in identifying which numbers are divisible by six.

Question two, we needed to have a look at the individual point scores for each student and identify if they are divisible by six.

Lucas is telling the truth because 918 is divisible by six.

Jacob unfortunately is not telling the truth because 1,114 is not divisible by six.

Sam is also not telling the truth as 1,995 is odd and multiples of six must be even.

Sofia is telling the truth because 1,194 is divisible by six.

Great work so far.

So let's move on to the second part of our lesson, which is divisibility tests for multiples of nine.

Well, a divisibility test for multiple of nine is similar to the divisibility test for multiples of three.

Let's have a look at on multiples of nine, I've just listed a few here, nine, 18, 27, 36, all the way up to 505,953.

So I'm just gonna randomly pick two multiples of nine, 45 and 63, and look at their individual digits, four and five, and six and three.

Now, if you look at these individual digits, do you think you can spot a method to identify if number is a multiple of nine? Remembering, it's very similar to the divisibility test for multiples of three.

And does this method work if you are looking at 189 and 505,953? Well, if you sum the individual digits four and five, that makes nine, six add three, that makes nine.

189, well, that makes 18, and the five add the zero add the five add the nine add the five add three, this makes 27.

You might be able to spot, if you sum the digits, the sum is a multiple of nine, so this means the number is divisible by nine.

In short, the divisibility test for multiples of nine is where you sum the digits, and if the sum is a multiple of nine, then we know the number is divisible by nine.

So it's very similar to the divisibility test for multiples of three.

Now let's jump into our tasks.

Here, you need to enter a maze through entrance A, B, C, D, or E, and you're only allowed to step on the numbers that are multiples of nine, and you could only step vertically or horizontally.

Now you must exit through either F, G, H, I, or J.

For question 1a, which entrance would you use? For b, can you draw your path? And c, what exit do you use? This is a great question and needs you to use your knowledge on multiples of nine.

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

Well done.

So let's move on to the second question.

The second question gives us a Venn diagram.

In our Venn diagram, the multiples of six will be on the left and the multiples of nine will be on the right.

We have 10 numbers to insert in our Venn diagram.

See if you can fill in the Venn diagram using these 10 numbers.

Press pause if you need more time.

Well done.

So let's go through these answers.

Question one is a great question and requires you to think of a strategy before attempting this question.

So knowing you are only allowed to step on numbers that are multiples of nine, let's identify all our multiples of nine first as our starting strategy.

Here you can see all our multiples of nine.

Now we're going to pick an entrance.

Which entrance do you think would be a sensible entrance given the multiples of nine we have on our screen? Well, if we chose A, we couldn't move horizontally or vertically from that point.

So that means entrance A would be no good, nor would entrance B, C, or D.

So the only entrance we have is E.

So for part b, we need to draw our path.

Now, remember, we can only move horizontally and vertically.

We can't use diagonally.

So that means this is the only path that we can use.

This identifies our entrance must be E and our exit must be F.

Well done if you got that one right.

For question two, we needed to use the numbers below and fill them in our Venn diagram.

So let's have a look first.

For each of the numbers, you needed to work out, is it a multiple of six, or is it a multiple of nine, or both? Let's start with 849.

Hopefully you would've worked out it is neither a multiple of six or nine.

For 32,377, it is neither a multiple of six or nine.

For 8,676, it is a multiple of six and nine.

For this huge number, 7,498,342, it is neither a multiple of six or nine.

22 is neither a multiple of six or nine.

2,222,222 is a multiple of six and a multiple of nine.

9,090 is a multiple of six and nine.

984 is only a multiple of six.

117 is only a multiple of nine, and 7,931 is neither a multiple of six or nine.

Well done if you've got this one correct.

So in summary, divisibility tests are quick processes to identify if a number is divisible by another number.

If a number is divisible by six, it means it's divisible by two and divisible by three, and if a number is divisible by nine, we simply sum the digits, and if the sum of those digits is a multiple of nine, then we know it is divisible by nine.

Hopefully you've enjoyed this lesson and you can see why divisibility tests are so important.

They are quick processes and much better than dividing huge numbers using short division or long division.

To simply identify if a number is divisible by six, we identify if it's divisible by two and divisible by three.

I really do hope you like looking at the special numbers six and nine, and have a look if you see anything in the real world which is packed into groups of six or nine.

A nice little investigation for you is to have a look when you go shopping, for example.

Look at how supermarkets or food producers group their product.

How have they grouped them? You may see groups of nines or sixes.

You may have even heard the expression half a dozen.

This means half of 12, which is six.

Have you ever wondered why food producers or supermarkets group them according to that number and not another number? I really do hope you found this lesson very interesting and I've really enjoyed learning with you today.

A huge well done.