Lesson video

Hi, welcome to today's lesson on accuracy of final answers.

By the end of today's lesson, you'll be able to appreciate the accuracy in leaving an answer without carrying out a final operation.

Here are some of the keywords that we'll be using today in our lesson.

What I'd like you to do is write down what you think each of these keywords means.

You will have seen some of these before and so it's important to recap these because your understanding of them now may have changed from when you first heard them and it's important that you understand what they are in this context.

Pause the video now and write down what you think these words mean.

Welcome back.

Let's see how you got on.

A recurring decimal is one that has a digit or sequence of digits that repeat endlessly.

An example might be 0.

3 recurring or in other words a third.

A terminating decimal is one that has a finite number of digits after the decimal point.

So for example, 0.

25, which we know is a quarter.

That's one example of a terminating decimal.

Now the words rational, irrational and surd are actually going to be coming up in our lesson and you'll see why very soon.

So hold onto the definitions of those if you have them for now and we'll come back to them.

Our lesson comprises three parts and we're going to start with part one, rational and irrational numbers.

The real number system is made up of groups of numbers.

You can see some of these groups here.

We have the natural numbers.

Now the natural numbers are any positive whole number which means there is one number you might think belongs there, but it doesn't and that's zero.

Zero is neither positive nor negative.

And because it is not a positive whole number, it cannot go into the natural numbers, so it sits in the whole number group.

But all whole numbers also include all the natural numbers.

Can you see how we're building out? Then we have the set of integers.

As you can see, integers include our negative numbers.

In this lesson we're gonna extend this diagram to have two more groups.

Now let's review our definition of what a rational number is.

A rational number is one that can be written in the form A over B, where A and B belong to that group of integers and B is not equal to zero.

And that's really important, isn't it? Because B is our denominator in our fraction.

If you're not sure what happens when you try to divide by zero, get out your calculator, type it in now.

You can Pick any number for the numerator and just see what happens.

Now let's see some examples of rational numbers.

Here we go, we have 3/4 or 2/9.

We can see that they're rational because we can write them as fractions and we know that we can do the same with any integer because we can write any integer as a fraction by dividing by one.

What we can tell is that a rational number when written as a decimal will be either a terminating decimal or a recurring decimal.

For example, that 3/4 of earlier can be written as 0.

75 and the 2/9 can be written as 0.

2 recurring.

What about irrational numbers? Well, an irrational number is one that cannot be written as a fraction.

In other words, we can't write it in the form A over B where their integers and B's not equal to zero.

Can you think of an example of an irrational number? You've actually met one of these before.

In fact, you've met a lot of them.

You just might not have realised.

Let's see some of those examples now.

There's Pi.

You've been using Pi since year eight when you were talking about circles and Pi's an example of an irrational number.

You've seen on your calculator that those digits go all the way to the end of the display and they haven't started a repeating pattern.

Actually, Pi goes on and on and on.

It's one of those examples of an irrational number, but we could also have the square root of two.

If you've not tried this before, type this into your calculator now and look at the display.

So to recap, an irrational number when written as a decimal is non-terminating and non-recurring.

Here's Pi written out.

Can you see the ellipsis at the end? It's to represent the fact there are still more digits.

There's route two.

Is that what you got on your calculator? Again, that actually carries on.

What I've done there is stopped 'cause that's as far as my display let me go, but I know there's actually more digits.

It's time for a quick check.

Identify which of the following are rational numbers.

Remember, rational numbers can be written as a fraction where both the numerator and the denominator are integers and the denominator is not equal to zero.

Pause the video now while you identify the rational numbers.

Welcome back.

How did you get on? Here's our rational numbers.

We know Pi's irrational.

We're not touching that one.

The second number there is written as a fraction so that's clearly fine.

The cube root of 27 is three, that's rational.

1/5 is written as a fraction, that's definitely fine.

Square root of three, definitely irrational, and you can check that on your calculator if you're not sure.

Now the square root of 0.

25 might have confused you except we know that a half times a half is a quarter.

So the square root of a quarter must be a half and a half is definitely a rational number.

And then we have the cube root of 36.

That doesn't give us a rational number.

And again, feel free to check on your calculator.

So now we know about the set of rational and irrational numbers.

Let's put these onto our diagram.

There's our rational numbers.

Both the numerator and the denominator are integers and the denominators aren't equal to zero.

So it makes sense that the rational numbers would include the integers, whole numbers and natural numbers, and there's some examples.

Now the irrational numbers, remember, there's no way that these two sets can overlap because either of that number is rational or irrational.

It can't be both at the same time.

And that's why we don't have any overlap here.

And in the irrational numbers set, there's some examples that we've seen earlier.

It's now time for your first task.

I'd like you to complete the following table.

You see that you have numbers on the left.

Put them into your calculator and write down what your calculator shows.

You then need to tell me if the number is rational or irrational.

For the bottom two, you get to choose what number is going to go there.

Remember in row five, you need to give me an irrational number and in row six, a rational one.

Pause the video now while you complete this task.

Welcome back.

Let's see how you got on.

So for the first four rows, I type those numbers into my calculator and I wrote down what they displayed.

I then filled in if they were rational or irrational.

So the top two are irrational and the bottom two are rational.

Remember, rational numbers as a decimal are either a repeating decimal or a terminating decimal.

And in the bottom two, there were lots of options you could have chosen.

All you need to do is make sure that as a decimal it either terminates or has a repeating pattern if it's to be rational and it doesn't if it's irrational.

It's now time for part two of our lesson.

And in this section, we're focusing on accuracy of our answers.

Now the answers to math questions can be given in a variety of forms. For example, money.

We tend to give money to two decimal places.

That makes sense, doesn't it? We've got pounds and pence.

We could give our answer as a fraction.

We could give our answer in standard form, and sometimes we round.

For example, with centimetres here, I've chosen to round to one decimal place.

Well, that makes sense because I have a number of centimetres and the first digit after the decimal point is actually representing how many millimetres I have.

Well, I can measure that accurately on a ruler, so it makes sense to go to that level of accuracy.

When dividing a restaurant bill, it makes sense to round to two decimal places.

Can you remember why that would make sense? That's right, we're dealing with money.

So for example, if I'm going to split a 90 pound bill between seven people, when I do that on my calculator, as you can see, I've got a decimal value here but no one can pay 12 pounds 0.

8571428 amount of money.

So rounding makes sense.

If everybody pays 12 pounds and 86 pence, A, that's an amount that's actually possible to create, and B, 'cause we've rounded up, we're guaranteed we're going to have enough money for that bill.

When reporting attendance at a large sporting event, it makes sense to round to three significant figures.

For example, if 784,231 people attended a large sporting event, that number doesn't have that much impact because trying to read that out loud or announcing it in the stadium, it's such a long number that people can't necessarily hold all of that in their head.

What's easier and what has more impact is if we report the number to three significant figures.

There are 784,000 people here today.

Because you're hearing that thousand so quickly, you understand this is a very large number and it gives you an idea of the scale of the people in the large sporting event.

So in other words, we've rounded for impact here.

Now sometimes we require our values to be exact.

Here's an example.

Alex wants to frame a circular portrait.

The portrait has a diameter of 50 centimetres.

Which frame should be bought? So we've got two circular frames here.

One has a circumference of 150 centimetres and the other 158.

Which one should Alex get? Now you know how to work out the circumference of a circle because you did this back in year eight.

So what I'd like you to do is very quickly work out what the circumference of the portrait is and write down which of the two frames you think Alex should buy.

Do this now.

Welcome back.

Have you decided what frame Alex should buy? Let's see if you're right.

Well, we know that in order to work out circumference, it's Pi times the diameter.

Since diameter is 50 multiplied by Pi, his portrait has a circumference of 157 and a little bit centimetres.

Well it's pretty clear then he needs the second of the frames 'cause he needs the larger one.

That's 158, it will definitely fit around his Picture.

The 150 centimetre frame, however, won't.

But what if instead of using the Pi button on my calculator I used a rounded value for Pi instead? Well we know Pi is 3.

1.

Well, okay, that rounds to three then.

So 50 times three, 150.

Alex should definitely buy the smaller frame.

Ah, but there's a problem isn't there? You know full well the smaller frame isn't going to fit.

And this is exactly the point we're talking about.

When we use a rounded value for future calculations, the answer we get is far less accurate than if we'd used the correct accurate value in the first place.

The more accurate we are with the values we use in our working, the more accurate our answer will be.

So when recording or using values, you need to decide what level of accuracy best fits the situation.

And if you're going to perform more calculations with a value, it's always best to be exact.

Quick check now.

I'd like you to match each time to the most appropriate degree of accuracy.

So in other words, if I want to record the 200 metre time in the OlymPics, what should I record it to? The nearest whole number, one decimal place, or two decimal places? Then we do the same for seconds to sing a song and the 200 metre time in a school sports day.

Pause the video now while you do this.

Welcome back.

Let's see what you said.

Well, let's start with seconds to sing a song.

If it's just singing a song, I don't need to be accurate to the millisecond, so it makes sense that I can go to the nearest second or in this case, nearest whole number.

Now of the two that are left, my 200 metre time in my school sports day and my 200 metre time in the Olympics.

Although school sports day is important, I'm going to venture to say it's not as important as the Olympics.

So my time in my school sports day to one decimal place seems pretty accurate.

But if I'm competing in the Olympics, then we want that time as accurate as possible because actually competing at such a high level, there are often tiny, tiny differences in runners' race times, so we need to be as precise as we possibly can be.

It's now time for your second task.

In this task, I'd like you to create two questions, one that is best left with an exact answer and one that is best left as a rounded answer.

Now, remember you've seen examples of where we need to be exact and examples of where we need to round.

Think about those as you complete this task.

Pause the video now.

Welcome back.

Now there are lots of different things you could have written.

The important bit is in your exact answer, you should have written down something maybe that required a future calculation or for example, where you talked about Alex's photo frame where it was really important that he'd been exact so he knew which one to buy.

Now the rounded answer can be anything where estimation is more practical, for example, with our money.

It's now time for the final part of today's lesson.

And in this we're gonna ask what is a surd? Remember surd was one of the key words at the start of today's lesson.

So by the end of this section, you'll be able to officially or formally have that definition.

On the screen, you can see some examples of surds.

Underneath, you can see examples that are not surds.

Using what you can see there, write down what you think the definition of a surd is.

Pause the video while you do this now.

Welcome back.

What did you put? Perhaps something like this.

I've gone for, "I think a surd has a square root sign "because I can see in the top line "all of those examples that are surds have square roots "and all the ones underneath don't." You may have written something similar.

Okay, I'm quite happy with that.

Oh, brilliant, okay, let's check to make sure that I understand what a surd is.

Identify which of these are surds.

Pause the video and make your choice now.

Welcome back.

Well, if a surd means it has a square root sign, then all of these must be surds.

Oh no, one of them isn't.

See in fact the square root of 25, that's not a surd? Hmm, better go back and update my definition.

All the ones we've just saw that were surds have gone into into the top group and square root of 25, which we know is five, has gone into the bottom group.

Bearing that in mind, I'd like you to refine, so go back and update, your definition of a surd.

Pause the video while you do this now.

Welcome back.

What have you got now? Well, I've said a surd needs a square root sign.

Ah, but it's an irrational number so it's a square root and it has an irrational number when I try to put it into my calculator.

Because the square root of 25 was five, it didn't have that non-repeating decimal.

And I know the square root of two and the square root of three do, and I can test the others on my calculator.

Right, pretty certain I've got this now.

All right, let's check this then.

Oh, all of these say cube root.

Hmm.

Which of these do you think are surds? Pause the video and make your choice.

Welcome back.

Are any of these surds? I mean, I guess not, because they don't have that square root sign and I said that's what a surd is.

Let's see then.

Oh no, the bottom two are surds but the top two aren't.

Ah, the top two evaluate to give me rational numbers.

The cube root of 64 is four and the cube root of eight is two.

But if I try the cube root of 16 or the cube root of 36 on my calculator, I get a non-repeating decimal.

So at the top I have my group of surds and at the bottom, my group of not surds.

Refine your definition of a surd once again.

Pause the video while you do this.

Welcome back.

What does your definition say now? Mine says that a surd has a root sign so I've no longer said it has to be a square root.

It can be any root, and it's definitely an irrational number left in an exact form.

One last check then.

Which of these are surds? Pause the video now and make your choice.

Welcome back.

So which of these are surds? That's right, it's A and C.

B evaluates to a half, half is definitely rational and D evaluates to a fifth, which is also rational.

You may want to refine your definition of a surd one last time based on the fact that under the square root sign now or under the root sign now, you have something that isn't a whole number.

Pause the video and do this now.

Welcome back.

I've updated my definition.

It now says that a surd has a root sign.

It's an irrational number left in an exact form and the value inside the root sign can be a fraction or a decimal.

So just showing it doesn't just have to be a whole number.

Here's our formal definition of a surd.

A surd is an irrational number expressed as the root of a rational number.

It is a way of leaving an answer exact with one operation remaining.

In this case, it's the finding of the Nth route.

Remember, it's Nth route because we could have a square root sign, cube root sign, et cetera.

Most commonly, surds happen to be the irrational square roots of numbers.

So it tends to be we only see the square root sign but it doesn't have to be.

True or false, the square root of A will always be a surd.

Do you think that's true or false? And remember to justify your answer by selecting either A or B.

Pause the video and do this now.

Welcome back.

What did you put? That's right, it's false.

And the reason it's false is because remember, A could represent a square number.

For example, A could be 16.

Well, the square of 16 is four.

That's not a surd, that's a rational number and therefore this statement's definitely false.

It's now time for our final task.

On the screen you can see 10 potential surds, and what I want you to do is select the ones that are in fact surds as supposed to rational numbers.

Pause the video and do this now.

Welcome back.

Let's see which ones you put.

The ones you can see circled are the surds.

A evaluates to two, E evaluates to five, G is equivalent to a half.

Now I might have been a little tricky.

I's actually 2/3 and I suspect you probably wanted to use a calculator for that one.

Well done if you've got these right.

Let's summarise what we've covered today.

Answers can be given to different degrees of accuracy.

A surd is a way of writing a number exactly with one operation remaining to be completed, i.

e.

, finding the Nth route.

Answers are left as surds when completing the square route or just route would give an irrational answer.

Well done.

You've worked really well today and I appreciate the effort you've put into your lesson.

I look forward to seeing you next time.