Lesson video

Hi everyone, my name is Miss Kew, 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 learn together.

In today's lesson, from the unit properties of number, factors, multiples, squares, and cubes, we'll be looking at divisibility tests for 6 and 9.

And by the end of the lesson, you'll be able to identify and explain whether or not a number is a multiple of 6 or 9.

Now, you might be thinking, why is 6 and 9 so important? Well, we know 9 is a square number.

We also know 9 is the highest single integer digit.

But what is so important about 6? Have a little think.

What makes 6 so important? Can you think of some examples of why 6 is so important? Well, firstly, it's the first non-square and non-prime integer value.

In other words, 1 is a square number, 2 is a prime number, 3 is a prime number, 4 is a square number, and 5 is a prime number.

So 6 is the first non-prime and non-square number.

Now, you might also notice 6 is the only number that is both the sum and the product of three consecutive positive integers, 1, 2, and 3.

So let's sum 1, 2, and 3.

1 add 2 add 3 is 6.

Now let's multiply.

1 times 2 times 3 is 6.

6 is a special number.

Also, 6 is the first product of two primes.

So the reason why we're looking at divisibility tests for 6 and 9 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 5 because we know 45 divide by 5 is 9.

A non-example is the fact that 8 is not divisible by 5 because 8 divide by 5 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 6.

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 2.

Well, a number is divisible by 2 if the digit in the ones column is always a 0, 2, 4, 6, or 8.

For example, we know 2,466 is a multiple of 2.

Now let's have a look at the divisibility test for 3.

Well, to identify if a number is divisible by 3, we simply sum the digits, and if the sum of the digits is a multiple of 3, then we know the number is divisible by 3.

For example, 2,466.

Summing those individual digits gives us 18.

We know 18 is a multiple of 3, so that means we know 2,466 is a multiple of 3.

I'm going to list some other examples here.

34 is a multiple of 2.

18 is a multiple of 2.

And 12 is a multiple of 2, and we also know 30 is a multiple of 2.

Now I'm gonna list some multiples of 3.

33 is a multiple of 3.

18 is a multiple of 3, 12 is a multiple of 3, and 30 is a multiple of 3.

So all I've done here is list some multiples of 2 and some multiples of 3.

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 2 and 3.

We can also see 12 is a multiple of 2 and 3, and 30 is a multiple of 2 and 3.

And 18 is a multiple of 2 and 3.

Now let's insert the rest of our numbers.

Well, we know 34 is only a multiple of 2, 33 is only a multiple of 3.

From our Venn diagram, what do you think all the numbers that are multiples of 2 and 3 represent? Well done if you spotted it.

All numbers which are multiples of 2 and 3 mean they are divisible by 6.

And this is our divisibility test for multiples of 6.

So let's check our understanding with this question.

Here it asks us to identify which of the following numbers are divisible by 6.

Now remember, if a number is divisible by 6, it means it's divisible by 2, and it must be divisible by 3.

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 2.

Remember, if a number is divisible by 2, the digit in the ones column must be a 0, 2, 4, 6, or 8.

Using this fact, that means we know the following numbers are not divisible by 2 so therefore they are not divisible by 6, leaving us with three numbers.

From these three numbers, let's find out if they are divisible by 3.

I'm gonna start off with 9,388.

Well, summing those individual digits gives us 28, and 28 is not a multiple of 3.

So that means we know 9,388 is not divisible by 3, so therefore is not divisible by 6.

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

Summing these individual digits we have 27.

27 is a multiple of 3.

So that means we know 19,386 is divisible by 2 and divisible by 3.

Thus we know it's divisible by 6.

Lastly, we have 521,238.

Summing these individual digits, we have 21.

21 is divisible by 3, so therefore we know 521,238 is divisible by 2 and divisible by 3, which means it's divisible by 6.

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 6.

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 4, 44, 444, 4,444, and 444,444.

You can see why we use divisibility tests, as we don't want to be wasting time dividing these numbers by 2 and 3 or 6.

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 2 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 6 points.

Lucas says, "I have 918 points." Jacob says, "I have 1,114 points." Sam says, "I have 1,195 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 1.

We needed to circle all the numbers which are divisible by 6.

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

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

Well the 2 and the 3 add the 2 make 7.

So that means it's not a multiple of 3.

The 8 add the 9 add the 0 add the 3 add the 8 add the 2 makes 30, which is a multiple of 3.

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 3.

Summing the individual digits of 8, 9, 9, and 2 gives 28.

This is not a multiple of 3.

Summing the digits 9, 3, 7, 4 gives me 23, which is not a multiple of 3.

Summing the digits 3 and 8 give me 11, which is not a multiple of 3.

Summing the digits 3, 4, 8, 3, and 0 gives me 18, which is a multiple of 3.

For part c, they're all even numbers.

So let's sum their individual digits.

Immediately you'll know 4 is not a multiple of 3, so I'm going to cross this one out.

But I'm gonna sum the individual digits for each of these numbers.

4 add 4 is 8, not a multiple of 3.

4 add 4 add 4 is 12.

This is a multiple of 3.

4 add 4 add 4 add 4 gives me 16.

This is not a multiple of 3.

Lastly, summing all of these 4s gives us 24, and 24 is a multiple of 3.

Question 1 was a great question in identifying which numbers are divisible by 6.

Question 2, we needed to have a look at the individual point scores for each student and identify if they are divisible by 6.

Lucas is telling the truth because 918 is divisible by 6.

Jacob, unfortunately, is not telling the truth because 1,114 is not divisible by 6.

Sam is also not telling the truth, as 1,995 is odd and multiples of 6 must be even.

Sofia is telling the truth because 1,194 is divisible by 6.

Great work so far.

So let's move on to the second part of our lesson, which is divisibility test for multiples of 9.

Well, a divisibility test for multiples of 9 is similar to the divisibility test for multiples of 3.

Let's have a look at our multiples of 9.

I've just listed a few here, 9, 18, 27, 36, all the way up to 505,953.

So I'm just gonna randomly pick two multiples of 9, 45 and 63, and look at their individual digits, 4 and 5, and 6 and 3.

Now, if you look at these individual digits, do you think you can spot a method to identify if number is a multiple of 9? Remembering it's very similar to the divisibility test for multiples of 3.

And does this method work if you are looking at 189? And 505,953? Well, if you sum the individual digits, 4 add 5, that makes 9.

6 add 3, that makes 9.

1, 8, 9, well that makes 18.

And the 5 add the 0 add the 5 add the 9 add the 5 add the 3? This makes 27.

You might be able to spot, if you sum the digits, the sum is a multiple of 9, so this means the number is divisible by 9.

In short, the divisibility test for multiples of 9 is where you sum the digits, and if the sum is a multiple of 9, then we know the number is divisible by 9.

So it's very similar to the divisibility test for multiples of 3.

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 9.

And you could only step vertically or horizontally.

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

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 9.

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 6 will be on the left and the multiples of 9 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 1 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 9, let's identify all our multiples of 9 first as a starting strategy.

Here you can see all our multiples of 9.

Now we're going to pick an entrance.

Which entrance do you think would be a sensible entrance given the multiples of 9 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.

No 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 2, 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 6 or is it a multiple of 9 or both? Let's start with 849.

Hopefully you would've worked out it is neither a multiple of 6 or 9.

For 32,377, it is neither a multiple of 6 or 9.

For 8,676, it is a multiple of 6 and 9.

For this huge number, 7,498,342, it is neither a multiple of 6 or 9.

22 is neither a multiple 6 or 9.

222,222,222 is a multiple of 6 and a multiple of 9.

9,090 is a multiple of 6 and 9.

984 is only a multiple of 6.

117 is only a multiple of 9.

And 7,931 is neither a multiple of 6 or 9.

Well done if you 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 6, it means it's divisible by 2 and divisible by 3 and if a number is divisible by 9, we simply sum the digits, and if the sum of those digits is a multiple of 9, then we know it is divisible by 9.

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 6, we identify if it's divisible by 2 and divisible by 3.

I really do hope you like looking at the special number 6 and 9.

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 6.

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.