Lesson video

Hi, everyone.

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

It's going to 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 square and cube root, and by the end of the lesson, you'll be able to explain the concept of square root and cube root as well as be able to calculate them.

So let's have a look at some keywords first, starting with a perfect square.

Now a perfect square is a number that is the second exponent of an integer, and it's usually shortened to a square number.

So we know our square numbers, we're simply calling them perfect squares.

For example, one multiplied by one is one, but we're calling it also a perfect square.

Two multiplied by two is four, so that means four is a perfect square.

Three multiplied by three is nine, so nine is a perfect square.

Four multiplied by four is 16, so 16 is a perfect square.

We have lots of perfect squares, because a number that is the second exponent of an integer is called a perfect square, but we also know it to be called a square number.

Now let's have a look at a perfect cube.

A perfect cube is a number that is the third exponent of an integer, and it's usually shortened to a cube number.

For example, one multiplied by one multiplied by one is one.

So we know one is a perfect cube.

Two multiplied by two multiplied by two is eight.

So we know eight is a perfect cube.

Three multiplied by three multiplied by three is 27.

So we know 27 is a perfect cube.

We know our perfect cube numbers, because we know our cube numbers.

So today, we'll be looking at perfect square and perfect cube numbers.

Today's lesson will be focusing on square and cube roots, and it'd be broken down into those two parts, focusing on square roots first and then cube roots second.

So let's have a look at square roots first.

Here's a table to help us see how a perfect square number is formed.

I've broken it down into an integer, an array, a calculation, and a perfect square number.

So looking at the integer one, making an array of a one by one is the same as the calculation one multiplied by one, giving us the perfect square number of one.

Looking at the integer two, I can create a two by two array.

So the calculation is two multiplied by two, which gives me the perfect square number of four.

Looking at the integer three, I can create a three by three array.

So the calculation is three multiplied by three, giving us the perfect square number of nine.

Lastly, looking at the integer four.

I can create a four by four array, so the calculation is four times four, giving us the perfect square number of 16.

Now the next table I've drawn, I've given you some integers, a calculation, and some perfect square numbers, but you have to work out the missing information.

See if you can figure it out and press pause if you need more time.

Well done.

So let's see how you got on.

Well, if you've got an integer of six, that means our calculation is six multiplied by six, giving us the perfect square number of 36.

Now we have an integer of eight.

I don't know the calculation, but I know the perfect square number is 64.

So hopefully you've identified the calculation would be eight multiplied by eight.

The next question doesn't tell us the integer, doesn't tell us the calculation, but it does tell us the perfect square number.

It's 100.

So, what do you think the calculation would be? Well done if you spotted it's 10 multiplied by 10.

So the integer we're looking for is 10.

What about 144? Well, we have a perfect square number of 144.

We don't know the calculation and we don't know the integer.

Well, hopefully you've spotted it's 12 multiplied by 12, therefore our integer is 12.

So how does that help us with the square root? Well, the square root of a perfect square is an integer that has been multiplied by itself to give that perfect square number.

For example, the square root of 25 is five, because five multiplied by five gives us 25 or the square root of 36 is six, because six multiplied by six is 36.

The square root symbol is represented as this.

It's called a radical, and we use this symbol to represent the square root.

So rather than writing words square root of 25 is five, we write it as the radical or square root of 25 equals five.

Same as the square root of 36 is six.

Rewriting this correctly is the square root of 36 equals six.

So the radical or the symbol here represents the square root, and we'll be using this a lot in the lesson.

So let's check for understanding.

169 is a perfect square.

So what is the square root of 169? And make sure you justify your answer.

Press pause if you need more time.

Well done if you've got this one right, it's 13.

It's 13 because 13 multiplied by 13 is 169.

Really well done if you've got that one right.

Let's have a look at another check question.

Which of the following is the square root of the perfect square 121? And I'd like you to justify your answer.

Well done.

Well, hopefully you've spotted it's 11, because 11 multiply by 11 is 121.

Well done if you've got that one right.

Let's put it into a bit of context.

A shop only sells square-based tents with integer lengths.

So tent A is a two by two tent.

Tent B is a three by three tent.

Tent C is a four by four tent.

And tent D is a five by five tent.

A family campsite has a campsite area for each family of a maximum size of 20 metres squared.

So what's the biggest tent a family can pitch at the campsite? See if you can give it a go and press pause if you need.

Well done.

This was a good question.

Hopefully you've spotted 20 is not a perfect square, so we do not get an integer when we square root the 20, but the closest perfect square to 20 is 16.

So that means the biggest tent we can get is tent C, because four multiplied by four is 16 or the square root of 16 is four.

Really well done if you got that one right.

Now let's move on to your task.

Here, question one wants you to match the square root of the perfect square with the correct answer.

The square root of 49, the square root of 225, the square root of 36, the square root of four, and the square root of 81.

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

Well done.

So let's move on to our next question, question two.

Question two wants you to work out the missing numbers using your knowledge on perfect squares and square roots.

Five adds the square root of what is 15? 30 subtract the square root of what is 28? The square root of what and the square root of what equals 10? See if you can figure out what those unknown numbers are.

Press pause if you need more time.

Great work.

So let's move on to our next question.

Question three.

An outdoor party has a plot allocation of 100 metres squared for all the guests, and the party will have a kitchen dining area, a cake table, a square bouncy castle, and a car to power the bouncy castle.

Now you're asked to insert all the items onto the plot, but you need to identify what's the largest bouncy castle that you can put on your plot, so there's at least one square gap between each item? It's important to remember each square is one metre squared.

This is a great question.

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

Well done.

So let's move on to question four.

Question four wants you to insert perfect square numbers only and fill in the missing gaps.

The square root of what perfect square is the closest to the square root of 65? The square root of what perfect square is the closest to the square root of 90? The square root of what perfect square is the closest to the square root of 152? This is a good question.

See if you can give it a go.

Well done.

So let's go through our answers.

For question one, we needed to match the square root of the perfect square with the correct answer.

Hopefully you spotted the square root of 49 is seven.

The square root of 225 is 15.

The square root of 36 is six.

The square root of four is two.

And the square root of 81 is nine.

Well done.

Now let's move on to question two.

We're asked to work out the missing numbers using our knowledge on perfect squares and square roots.

Five and the square root of what is 15? We know the square root of 100 is 10.

So five add 10 is equal to 15.

For B, 30 subtract what is 28? Well, it had to be four, because the square root of four is two.

30 subtract two is 28.

For C, this was a tough one, well done if you got the square root of 16 add the square root of 36.

Or you may have got the square root of 36 add the square root of 16, which gives you four add six, which equals 10.

Another example you may have got is the square root of one, add the square of 81, because the square root of one is one.

And the square root of 81 is nine.

One add nine equals 10.

Well done if you've got that one right.

Let's have a look at question three.

There are lots of different ways in which you could have positioned your items. But calculating the largest bouncy castle could be done by by subtracting the area of each item.

So we know we have 100 metres squared.

If we subtract the area of the cake table, which is four metres squared, and subtract the area of the kitchen diner, which is 18 metres squared, and then subtract the area of the car, which is 10 metres squared, this means we have 68 metres squared left.

So in theory, the closest perfect square to 68 is 64.

So is the largest bouncy castle that we could put on our plot an eight by eight? Well, if we did that, it's not practical, because we're not able to move around each item.

So what we should have done is work out the most practical bouncy castle with at least one square gap between the items, and this would be a six by six metre bouncy castle.

I've done an example like this, but there's lots of different ways.

Practically the biggest square bouncy castle you could get is a six by six.

Now let's have a look at question four.

We had to find the closest square root of a perfect square for each of these questions.

So the square root of what perfect square is the closest to the square root of 65? Well, it's 64.

For B, the square root of what's perfect square is the closest to the square root of 90? Well, it's 81.

For C, what is the closest perfect square to the square root of 152? Well, it's 144.

Great work so far.

So let's move on to the second part of our lesson, which is cube roots.

Well, a cube number is the product of three repeated integers, and we know a perfect cube is the same as a cube number.

For example, perfect cubes are one multiplied by one, multiplied by one, which is one.

Two multiplied by two multiplied by two, which is eight, so on and so forth.

So here are our perfect cubes.

Now what I'm gonna do is put these in a table just like before.

For example, the integer one, the calculation is one multiplied by one multiplied by one.

So the perfect cube number is one.

Looking at the integer two, we have two multiplied by two multiplied by two, giving me the perfect cube number of eight.

Can you fill in the rest of this table and work out the missing information? See if you can give it a go and press pause if you need.

Well done.

So let's go through our answers.

Well, if you are given an integer of three, this means the calculation would be three multiplied by three multiplied by three, which gives us a perfect cube number of 27.

Let's have a look at the integer four.

Well, four would give us a calculation of four multiplied by four multiplied by four, which gives us the perfect cube number of 64, but we're not given the integer or the calculation for the perfect cube number of 125.

So what do you think the calculation would be? Hopefully you've spotted it's five multiplied by five multiplied by five, which is 125.

So our integer is five.

What about the perfect cube number 1,000? Well, did you spot that the calculation is 10 multiplied by 10 multiplied by 10? So our integer is 10.

A huge well done if you got that one right.

Now let's have a look at notation.

Well, we know the cube root of a perfect cube is the in integer that has been multiplied by itself and then itself again to give that perfect cube number.

Focusing on the word cube root, the cube root of 125 is five, because five multiplied by five multiplied by five is 125, and the cube root of 1,000 is 10, because we know the integer was 10 multiplied by 10 multiplied by 10.

So let's use some correct notation.

We use that radical, but there is a little three there to indicate the cube root.

This symbol represents the cube root.

So let's change our sentence into a mathematical calculation.

The cube root of 125 is five is represented as this.

The cube root of 1,000 is 10 is represented as this, and we'll continue using the cube root sign in the rest of the lesson.

So now let's have a look at a quick check.

Here you need to match the cube root of the perfect cubes with the correct answer.

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

Well done.

So let's go through our answers.

Well, the cube root of 125 is five, because we know five multiplied by five multiplied by five is 125.

The cube root of one is one, because one multiplied by one multiplied by one is one.

The cube root of 216, that's six, because six multiplied by six multiplied by six is 216.

The cube root of eight is two, because two multiplied by two multiplied by two is eight.

Lastly, the cube root of 1,000 is 10, because 10 multiplied by 10 multiplied by 10 is 1,000.

Well done if you've got that one right.

Let's look at another check question.

What's the closest integer to the cube root of 90? Now, Alex says it's 10 and Sofia says it's five, Andeep says it's four.

Who's correct and can you explain? So let's start by looking at each integer.

Well, if it was 10, that means 10 multiplied by 10 multiplied by 10 is 1,000.

So is the cube root of 1,000 closest to the cube root of 90? Sofia says it's five.

So five multiplied by five multiplied by five is 125.

So is the cube root of 125 closest to the cube root of 90? Andeep says it's four.

So is four multiplied by four multiplied by four closer to the cube root of 90? Well, hopefully you can spot 64 is closer to 90, so therefore, Andeep is the closest.

This was a great question.

Well done if you got that one right.

Now let's have a look at your task.

Question one wants you to identify which of the following will give an integer answer.

Would the cube root of 1,000 give an integer, the cube root of 100 give an integer, the cube root of eight give an integer, the cube root of 50 give an integer, the cube root of 27 give an integer, the cube root of one give an integer, or the cube root of 64 give an integer? See if you can give this a go and press pause if you need.

Well done, so let's move on to question two.

Question two says, what's the closest integer to the cube root of 200? Laura says it's six, and Jun says it's five.

You need to identify who is correct and make sure you explain your answer.

Give it a go and press pause if you need more time.

Well done.

Let's move on to question three.

Question three says, we need to work out the answers to the following using our knowledge of square and cube roots of perfect squares and perfect cube numbers.

3a wants you to calculate the cube root of 125 subtract the cube root of one.

Question 3b wants you to calculate the cube root of 1,000 add eight.

Question C wants you to calculate the cube root of 1,000, add the cube root of eight.

And question D want you to calculate the root of 64, subtract the square root of 64.

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

Question four states that Izzy has a puzzle cube gift for Lucas, and the puzzle cube has a length of 4.

5 centimetres.

She needs to select the correct gift box, and the gift boxes are also cubes.

Which is the best choice for the puzzle cube? Explain your answer.

So we have a gift box with a volume of 1,000 centimetres cubed.

Remember, the gift box is also a cube.

We have another gift box with a volume of 512 centimetres cubed, and another gift box with a volume of 64 centimetres cubed.

So which gift box do you think is best for the puzzle cube for Lucas? See if you can give it a go and press pause if you need.

Well done.

So let's move on to our answers.

For question one, which of the following give an integer answer? Well, hopefully you spotted it's only these answers we have here.

The cube root of 1,000 gives the integer 10.

The cube root of eight gives the integer two.

The cube root of 27 gives the integer three.

The cube root of one gives the integer one.

And the cube root of 64 gives the integer four.

Well done if you got that one right.

Question two, what is the closest integer to the cube root of 200? Laura says it's six and Jun said it's five.

Who's correct? So let's look at Laura's answer of six.

Well, six multiplied by six multiplied by six is 216.

Is that closest to the cube root of 200? Jun says it's five.

So five multiplied by five multiplied by five is 125.

Is that the closest to the cube root of 200? Hopefully you've spotted 216 is closer to 200 than 125.

Therefore Laura is correct.

Well done if you got that one right.

Question three is a great question, as it really does check your understanding on perfect squares, perfect cubes, square roots, and cube roots.

For 3a, the cube root of 125 subtract the cube root of one.

Well, the cube root of 125 is five, subtract the cube root of one is one.

So our answer is four.

B, the cube root of 1,000 add eight.

Well, the cube root of 1,000 is 10, add eight is 18.

For C, the cube root of 1,000 add the cube root of eight.

Well, the cube root of 1,000 is 10, and the cube root of eight is two, so our answer is 12.

D, the cube root of 64 subtract the square root of 64.

Well, the cube root of 64 is four, subtract the square root of 64 is eight.

So our answer is negative four.

Well done if you got that one right.

So for question four, we had to identify the correct cube gift box for Izzy to put the puzzle cube gift for Lucas.

Well, hopefully you've identified because the gift box is a cube, all we need to do is cube root each of our volumes.

So the cube root of 1,000 is 10.

So the length of the gift box we have here is a 10 centimetre by 10 centimetre by 10 centimetre.

The cube root of 512 is eight.

So that means our gift box is an eight centimetre by eight centimetre by eight centimetre, and the cube root of 64 is four.

So that means we have a gift box of a four by four by four.

So which gift box is the most suitable gift box for the puzzle cube? Well, hopefully you've spotted it's the second one, because the four by four by four would be too small, and the 10 by 10 by 10 would be too big.

Really well done if you got this one right.

So in summary, the square root of the perfect square is the integer that has been multiplied by itself to give that perfect square number, and the cube root of the perfect cube is the integer that has been multiplied by itself and then itself again to give that perfect cube number.

Really well done.

It was great learning with you today.