Lesson video

Hiya, my name is Ms. Lundell.

Really pleased that you've decided to pop along and do some maths with me today.

Welcome to today's lesson.

The title of today's lesson is "Problem Solving with Factors, Multiples, Squares, and Cubes", and it's within the unit on properties of numbers.

By the end of this lesson, you'll be able to use your knowledge of factors, multiples, squares, and cubes to solve problems. We've actually got some different problems here, some things that you might not have seen before.

So hopefully, you'll really enjoy those.

Key words that we will be using in today's lesson.

So it's worth just having a quick reminder of what those are.

So a square number is a product of two repeated integers.

A cube number is the product of three repeated integers.

A multiple is the product of a number and an integer.

And a numerical factor is a factor that is an integer.

So you should be really familiar with all of these words by now.

I'm going to split the lesson into two separate learning cycles, the first of which is going to be problem solving, where we're just going to be concentrating on squares and cubes.

And in the second part, we are going to be looking more at factors and multiples, but within that, we will be looking at things like prime numbers as well.

Let's start then, solve problems with squares and cubes.

Jun is arranging blocks so that they form a square.

He uses nine blocks.

I'd like to pause the video and draw a representation of what you think Jun has drawn.

So pause the video, come back when you are ready.

This is what Jun has drawn.

Have you drawn the same or a similar representation? I hope so.

What I want us to think about now is what's the largest square that he could make if we had 150 blocks? So we probably wouldn't want to get count out 150 blocks, going to use the knowledge that we've developed over the last few lessons to solve this problem.

We need to remember that a square number is the product of two repeated integers.

So remember, that's really important.

It's the product of two repeated integers.

Pause the video and have a go.

What is the biggest square you think you can make with 150 blocks? Pause the video, come back when you're ready.

Well done, Jun decides to start with 10.

So Jun decides to start with 10, and he makes a 10-by-10 square, and that gives us, he's used 100 blocks.

He knows though that he might be able to make a larger one 'cause he's still got 50 blocks left.

So he decides to do 13 multiplied by 13.

So a square of 13 by 13, that's 169.

Has he got enough blocks? No, he's only got 150.

He now knows that it's between 10 and 13.

So has another go, and this time tries 12.

And the product of 12 and 12 is 144.

So we knew he didn't have enough to make 13 squared, but he has got enough to make 12 squared.

The largest square that he can make then is 12 by 12.

Did you get as well? I hope you did, well done if you did.

Now he's going to stack the blocks to make a large cube.

He's used 27 blocks.

So this time, we're actually stacking the blocks to create a cube.

What will his cube look like? Pause the video, draw a representation of what you think his cube might look like, and then come back when you're ready.

Here's his cubes, we stack the blocks to make a large cube and we can see that it's a three-by-three-by-three cube.

We've still got 150 blocks.

I want to know what's the biggest cube now.

So moved on from square.

What's the biggest cube that you can make with those 150 blocks? So a reminder that a cube number is the product of three repeated integers because it's three dimensional.

He decides to start with four.

I wonder if you would've decided to start with four.

So let's start with four.

You're gonna pause the video and come back when you've got an answer.

Well done, so you started with four and we found the repeated integer for three times.

So four multiplied by four multiplied by four is 64.

Right, we've still got quite a lot of blocks left.

So he knows he might be able to make a larger one, only might, so we're gonna give that a go.

Five multiplied by five multiplied by five is 125.

Yeah, we've got enough blocks for that.

But do we have enough for six? So I think really to be certain, we do have to try six, don't we? Six multiplied by six multiplied by six is 216.

He does not have enough to make a six-by-six-by-six cube.

So the largest cube he can make is five by five by five.

True or false, 10 is a square number.

Is that true or false? And then I want you to choose your justification.

The justification is so important.

It cannot be written as a product of two repeated integers.

It can be written as a product of two repeated integers.

Pause the video, come up with your answer and that justification, come back when you're ready.

It's false.

It cannot be written as a product of two repeated integers.

Often, people get confused because you could do 10 squared.

People think that 10 is a square number, okay, but remember the hundred is the square number, not the 10.

We're now going to look at this cuboid, and it has a volume of 60 cubic centimetres.

60 centimetres cubed.

If the dimensions are all integers, what could they be? So we know that we can write 60 as a product of its prime factors and that is two multiplied by 30, then we can break the 30 down into three multiplied by 10, then we can break the 10 down into two multiplied by five.

And then remember, we're going to tidy that up and put it into index form.

Here's Sam's table, he's decided to draw a table out, length, width, and height.

They decide to firstly use the three separate factors.

So four, three, and five.

But actually, we could rearrange some of those.

So we could split the twos into two separate columns, okay? And pair the three with one of them, which would give us a dimension of six, two, and five.

We could do the same with that we did with the three, with the two, given us dimensions of 10, 2, and 3.

Or we could pair up the three and the five and separate out the two squared to give us 2, 2, and 15.

They decide that the table shows all of the possible solutions.

Do you agree that Sam is right? Do you agree that they have found all of the possible solutions to the dimensions of this cuboid? Sam's not right.

Let's take a look why.

Here's our product of prime factors, and he noticed that he could use combinations of those prime factors, that was absolutely brilliant.

So there was the table.

We need to fill in the rest of the table.

What dimension has Sam missed? What dimension did they miss out? They've missed out the fact that one centimetre could be a dimension because we've used the product of prime factors and we know that one is not a prime number.

So we need to add to our list of possible dimensions, 1, 1 and 60, 1, 2 and 30, 1, 3 and 20, 1, 4, and 15, 1, 5 and 12, and 1, 6 and 10.

Remember, they can be in any order.

We can put them in any order 'cause it doesn't matter which way up the cuboid is.

We're now ready to do task A.

So question number one is very similar to the question that we've just been doing with Jun.

So pause the video, come back when you've got your answers.

Good luck.

And the second question, so this is like the second part of it that Sam was doing.

So very similar, so remember to think about that dimension of one and then make sure you include that in your answer.

Pause the video, good luck with this.

Come back when you're ready.

We're now ready to take a look at some answers.

One, A, what is the largest square you can make with 80 blocks? And it's eight by eight 'cause that's 64.

To make a nine, we'd need 81 blocks and we've only got 80.

B, what is the largest square you can make with 200 blocks? When that's 14 by 14, that's 196.

To make the next square, we would need 225.

We don't have enough.

C, what is the largest cube you can make with 200 blocks? So that's a five by five by five, which is 125 because to make the next size up, which would be six by six by six, we would need 216 blocks.

We don't have quite enough.

D, what is the largest cube you can make from 350 blocks? Here, our answer is a seven by seven by seven, that's 343.

I think it's pretty obvious we haven't got enough blocks to make the next one, but we should always do a check.

And if we wanted to do an eight-by-eight-by-eight cube, we would need 512 and we certainly don't have enough.

E, justify why eight is not a square number.

There are not two repeated integers whose product is eight.

We've got two multiplied by two, gives us four.

So four is a square number, and three multiplied by three is nine.

So that is a square number.

So eight is not a square number.

Two, the volume of the cuboid was 90 centimetres.

If the dimensions are all integers, what could they be? I'm not going to read all of these out.

What I'm going to do is say pause the video, mark your answers, and then come back when you're ready.

Superb work on that, well done.

So nice to look at some problems in a slightly different way.

Now we're gonna move on to looking at factors and multiples.

So like I said, we were gonna do some slightly different things today.

Some more problem solving in a slightly different way.

So the following row contains consecutive numbers which satisfy the condition given in the box.

No number is greater than 20.

Consecutive, can you remember what consecutive means? So consecutive means numbers that follow on in order.

So the first box has to be the sum of the digits of the number is seven, followed by a consecutive odd, consecutive even, and then prime.

So I'm going to start with this first box.

So remember, I'm only considering numbers between 1 and 20.

We're going to start here.

We need to remember that no number is greater than 20.

So we're only using the number from 1 to 20.

The sum of the digits is seven.

So what numbers between 1 and 20 has a sum of the digits of seven? Well, that's 7 and 16.

Could we put seven in the box? If we put seven in the box, we need the consecutive number, which is eight, but that needs to be odd and eight is not odd.

We cannot put that in, so it must be therefore be 16.

So we can then complete the rest of the boxes, but we'll do a quick double check.

So 17, is it odd? Yes.

18, is it even? Yes.

And 19, is that prime? Yes, it is.

Let's have a look at another one together.

So the following row contains consecutive numbers which satisfy the condition given.

Again, so exactly the same problem, but this time, the conditions are different.

So we've got odd, cube, square, product of digits is zero.

Now it wouldn't be very sensible to start here at odd because there are lots of odd numbers between 1 and 20.

So I'm actually going to start with the final box, the last one, the product of the digits is zero.

Hmm, that must mean for it to have a product of zero, then one of the digits must be zero.

What numbers between 1 and 20 have a zero in the ones column? It can only be two, sorry, it can only be 10 or 20.

Now we need to think about the number preceding that.

It needs to be a square number.

So 19 is not a square number, but nine is.

So we now know that 10 goes in here.

Let's check, that's nine, that's a square.

Is eight a cube number? Yes, because two multiplied by two multiplied by two is eight, and then seven is that odd number.

One more to have a go at together.

It's exactly the same scenario.

This time, I've listed my square numbers up to 20 and my cube numbers.

We need to find a consecutive cube and square number.

So look at our list, can you find two that are consecutive? Yep, they would be eight and nine.

So we're gonna put eight in here, nine.

Then we're gonna check, this is 10.

Has 10 got four factors? 1 and 10, 2 and 5.

Yes, it does, and 11 is odd.

Now let's look at a different problem.

So here, we're going to be using our mathematical knowledge of prime factors, but also we're going to need to use our logic skills to place those into the grid.

These are written as a product of their prime factors.

So we're just looking at the last line in each of them.

Now we need to be careful here about where we start.

So I've decided to start with seven because it only appears in two of the numbers.

So seven is common to 42 and 231.

So I now know that I can put that in that box.

I'm now gonna do that with the number five 'cause again, it only appears in two numbers.

So I've got it appears in 110 and it appears in 30.

So we can put the five here.

Now I've spotted that there's 11, and that's in 110 and 231.

Now I've got one missing factor from 110 and one missing factor from 231.

So I can now put those into the grid.

So the missing one for 210 is two.

Let's just check that row.

Has it got a 2, 5, and 11? Yes.

And then the missing one from 231 is three.

So I'm gonna put that in there, let's just check.

I've got a 3, 7, and 11 in that column I have.

We're then just left with twos and threes.

So 30 and 42 share a common factor of two and three, and 18 and 12 share a common factor of two and three.

I'm gonna put those into the grid, and then I'm just going to double check 42, two, three, and seven, yes.

18, two, three, and three, yes.

11, sorry, 110, 2, 5, and 11.

231, 3, 7, 11.

30, two, three, five.

And 12, two, two, and three.

So like I said, we are using knowledge of prime factors, but we're also using our logic skills there.

Right, we're ready now to have a go at this one together, and then you'll have a go at the one on the right-hand side yourself.

Now it is challenging, but you have all of the skills you need.

Here, I've noticed that 11 is a prime factor of the top two numbers, but as they're both in rows, that can't be common to both of them in the grid.

So therefore, I know that the two 11s have to go here because of the 11s in 242.

Now there's only one prime factor missing from 242, which is two.

And now I've already placed this two into the grid for 42.

Now I'm going to look for something that's common again.

So this time, I've got five and five.

So 110 and 45 share a common factor of five.

So that needs to go where those two boxes meet.

Now we can see that there is only one missing prime factor from 110, which is two, so it has to go here.

Now I'm going to look at this two because I've already used the two now for the 28 by putting it in that column, and now I'm looking at the sevens.

So seven is in 42 and 28, so it needs to go here.

Now I can use the fact that there are only missing numbers, one missing number in some of these.

So I've decided to place that three, it's the only one missing for 42.

And then we know then that three has been used in 45.

So now I know that there is a three missing in the 45 column, and then there is the two missing in the 66.

And then like I said, just always double check.

So 66, we should have 2, 3, and 11.

110, 2, 5, and 11.

42, two, three, seven.

242, 2, 11, and 11.

45, three, three, and five.

And 28, two, two, and seven.

And we've got that.

Have a go at this one by yourself.

So exactly the same thing.

You need to place those prime factors into the grid.

Pause the video and come back when you're ready.

Well done, hope you didn't find that too much of a challenge.

Here are the correct answers.

So we should have five, seven, seven on the top row, 11, 5, and 2 on the middle row, and then 3, 2, and 11 on the next row, bottom row.

You're now gonna have a go at some of those problems independently.

So this is exactly the same as the first examples we've just been through.

So remember, no number is greater than 20.

They need to satisfy the conditions that are in the boxes and they can contain consecutive numbers.

When you're done, you can come back, and we'll move on to question number two.

Good luck with that.

Super work, well done.

Now we can move on to question number two.

So we did some examples together like this, so you should be okay to do these questions.

So what you are going to do is you are going to fit the prime factors into the grid so that each of the rows and the columns is right for the numbers in the green boxes.

When you get to the second one, what I'd like you to do is to practise using your calculator to write those products as prime factors of those numbers, remembering to use the prime factor button on your calculator, and then go ahead and complete the grid.

Good luck with that, pause the video, and come back when you are ready.

Well done, here are our answers then.

So A, you should have nine.

Three factors are one, three, and nine.

10, the sum of the digits is one because one, but add zero is one.

11 is odd, and 12 is a multiple of three.

B, 14, the sum of the digits is five.

Yep, the sum of one and four is five.

15, the sum of the digits is six.

That's correct.

16 has five factors, they are 1 and 16, two and eight and four.

And 17 is a prime number.

And then C, we should have six that has four factors, one and six, two and three.

Seven is prime, eight has four factors, one and eight, two and four.

And then nine has three factors, one, three, and nine.

Well done if you've got all of those right.

Here are your answers now to the product of prime factor grids.

So A should be in this order for 500 and what, sorry, 154, 2, 7, and 11.

For 30, two, five, and three.

And for 20, five, two, and two.

And then part B, for 275, we should have 5, 5, and 11.

99 is 3, 11 and 3.

And then 70 is seven, two, and five.

Give those a mark.

Well done if you've got all of those right.

Like I said, fairly challenging.

We're now going to summarise the learning that we've done today.

So what we've been able to do is actually use our knowledge of squares and cubes to solve real-world problems. Also, factors, multiples, and primes allow us to solve problems too.

Worked really, really hard on those problem-solving tasks today, so well done.

Thank you for joining me.