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Hello, everyone.

My name is Miss Ku, and I hope you enjoy today's lesson, as I'm really happy you've chosen to be learning with me today.

In today's lesson, it's gonna be easy and perhaps hard in some parts, but don't worry, I'm here to help, as well as some of our other pupils, too.

Now, you might come across some keywords that you may or may not know, but we'll go through those in the lesson, so don't worry.

I really hope you enjoy today's lesson.

So let's make a start.

Hi everyone, and welcome to this lesson on inequality notation to express error.

It's under the unit "Estimation and Rounding," and by the end of the lesson, you'll be able to calculate possible errors expressed using inequality notation, where a is less than or equal to x, which is less than b.

Now, we're going to look at a keyword called "error interval." An error interval for a number x shows the range of possible values of x, and it can be written as a is less than or equal to x, which is less than b.

Another keyword that we'll be looking at is "range," a range of values where every value in the range estimates to the given value.

For example, rounding an integer is 300 to the nearest 100, the range of values for the unrounded number is 250 to 349.

9 recurring.

Rounding 23 to two significant figures, so the range of values for the unrounded number is 22.

5 to 23.

49 recurring.

We'll look at this in our lesson, so don't worry.

Today's lesson will be broken into two parts.

The first is where we'll look at the range of values on a number line, and the next is where we'll be looking at those error intervals.

So let's make a start.

We use decimals and integers in everyday life, and sometimes rounded numbers are presented to us; and it's important we are able to calculate the largest and the smallest unrounded numbers.

For example, what's the largest and smallest integer when 430 is rounded to the nearest 10? A number line might help you out.

Well, let's see if we can draw a number line together.

Here's my number line, and we're going to put 430 in the centre.

Now, given the degree of accuracy is 10, this means we're going to plus 10 and minus 10 either side of that 430, thus giving us 420 and 440.

From here, we're going to do those middle lines.

But what do those middle values represent? Hopefully, you can spot it's 425 and 435.

From here, 425 is the smallest integer when rounded to the nearest 10, as this does give us 430.

434 is the largest integer when rounded to the nearest 10, as this does give us 430.

But why do you think we didn't use 435 as the largest integer? Well, hopefully, you can spot it's because 435 rounded to the nearest 10 gives us 440.

So let's extend this by continuing to look at our number lines and look at the conversation Jacob and Sofia are having.

Well, Jacob says, "Are the largest and smallest values of an unrounded number always an integer?" And Sofia says, "Good question.

Let's find out." Let's have a look at an example where five has been rounded to the nearest integer.

What are the smallest and largest unrounded values? So let's draw a number line.

Here, with our number line, we're going to put five in the middle.

Now, given the fact that the degree of accuracy stated by Sofia said "the nearest integer," this means we put four and six on either side of the five, as they're the nearest integers.

From here, we're going to identify those middle values, so the middle line in between four and five and the middle line in between five and six.

What do you think those values are? Well, it'll have to be 4.

5 and 5.

Sofia says, "Good.

So what do you think are the largest and smallest unrounded values?" Jacob recognises 4.

5 is the smallest.

Sofia says, "Yes, because when you round 4.

5 to the nearest integer, you are correct, you do get five.

But is 5.

5 the largest?" Now let's think about it.

If we round 5.

5 to the nearest integer, what do we get? Well, we get six.

So that means we know it can't be 5.

Now, a nice way to visualise where the largest number would be is to zoom in.

So let's zoom in to see if we can find that largest number that, when rounded, gives five.

So now we've zoomed in, we're looking at the values in between five and 5.

5, and we want to find that biggest value that, when rounded, we get five.

Let's put it on our intervals.

We have 5.

1, 5.

2, 5.

3, and 5.

Jacob looks at this number line and says, "Is the largest number when rounded to five to the nearest integer 5.

4?" Sofia recognises there is a larger number than 5.

4, when rounded, it gives five.

So let's see if we can zoom in again.

Now we've zoomed in.

We have our lowest number, 5.

4, and our largest number, 5.

Let's identify these intervals again: 5.

42, 5.

44, 5.

46, 5.

48.

Can you see the largest number? Jacob says, "Is the largest number 5.

49?" Now, Sofia says, "There's a larger number than 5.

49." So let's see if we can zoom in again.

Now our interval goes between 5.

49 and 5.

So, Lucas is identifying these intervals and says, "Right, is the largest number 5.

499?" Sofia says, "There is a larger number than 5.

499, which, when rounded, gives five." Jacob says, "Is it 5.

4999?" Sofia then again says, "There is a larger number than 5.

4999, which, when rounding, gives five." And now Jacob sees.

So Jacob needs to answer the question: What are the largest and smallest unrounded values? The range goes from 4.

5 to 5.

49 recurring.

That dot above the nine means there are an infinite number of nines there.

Really well done if you spotted this.

We could zoom in an infinite number of times to see that largest unrounded number.

We're recognising that we can't use 5.

5, but we have to use a number which is infinitely smaller than this, which is actually 5.

49 recurring, and that's our largest number.

Thus, the range of values for the unrounded number is 4.

5 to 5.

49 recurring.

Really well done if you got this.

Now let's move on to a check.

Here, we're asked to identify the range of unrounded values when seven has been rounded to the nearest integer, 12 has been rounded to the nearest integer, and 100 has been rounded to the nearest integer.

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

Well done.

Let's see how you got on.

Well, using a number line, you can see I've used seven in the middle and six and eight either side, as we're rounding to the nearest integer.

Identifying those mid-values, can you spot the largest and smallest numbers? Well, it's gotta be 7.

49 recurring as our largest and 6.

5 is our smallest.

Twelve has been rounded to the nearest integer.

So here's my number line.

There are those middle values.

The largest value has to be 12.

49 recurring, and the smallest is 11.

100 has been rounded to the nearest integer.

So let's identify that number line, those middle values.

That means our largest value is 100.

49 recurring, and our smallest is 99.

Well done if you got this one right.

Now, let's have a look at another check.

Here's a question and answers from Andeep and Laura.

Who's correct, and I want you to explain why? The question says: Work out the range of unrounded numbers when 40 has been rounded to two significant figures.

Andeep says the smallest is 39.

5 and the largest is 40.

49 recurring.

Laura says the smallest is 35 and the largest is 44.

9 recurring.

So, who do you think is correct? And I want you to explain why.

Well done.

Let's see how you got on.

Well, Andeep is correct.

The second significant figure is in the one's column of the place value table, so the question is asking to round 40 to the nearest integer.

Laura has rounded, but she rounded to the nearest 10.

Well done if you got this one right.

Now remember, when trying to work out the range of unrounded numbers with units, it's important to focus on which units are given and compare to the degree of accuracy.

For example, 2.

13 metres, which digit represents the integer centimetres? Well, let's put it in our table, and we know 2.

13 metres is actually this in our table.

So that means the three represents the integer centimetres.

Therefore, we know 2.

13 metres is 213 centimetres.

Now, if we got a question stating that 2.

13 metres has been rounded to the nearest centimetre and we're asked what are the smallest and largest unrounded values, we can use this conversion to help us identify the range.

So, what do you think the largest and smallest unrounded values are? Let's put it in our number line again.

We know we're using centimetres now, so putting 213 in the centre, one centimetre either side, as that's our degree of accuracy.

Identifying those middle values will help us identify the smallest unrounded number to be 212.

5 centimetres, and the largest unrounded number to be 213.

49 recurring centimetres.

Great work if you got that one right.

Great work, everybody.

So let's have a look at another check.

126 kilometres has been rounded to the nearest metre.

What is the range of unrounded values? And 21.

3 centimetres has been rounded to the nearest millimetre.

What is the range of unrounded values? See if you can give it a go, and press pause if you need more time.

Great work, everybody.

So let's see how you got on.

I'm gonna use that table again to identify 5.

126 kilometres is 5,126 metres.

So rounding to the nearest metre means we're looking at that integer six.

So that means we're adding one integer either side of the six to give me 5,125 and 5,127.

Identifying those mid-values, we have this, thus giving us the smallest value to be 5,125.

5 metres and 5,126.

49 recurring metres.

Well done if you got this one right.

What about 21.

3 centimetres rounded to the nearest millimetre? Using our table again, well, that means we can spot 21.

3 centimetres is the same as 213 millimetres.

So rounding to the nearest millimetre means we need to focus on the three, which is in the units column.

So, therefore, using our number line again, we have 213 in the centre, 212 to the left, and 214 to the right, giving us our centre values and our smallest value to be 212.

5 millimetres and 213.

49 recurring millimetres.

Great work if you got this one right.

Well done, everybody.

Now it's time for your practise questions.

See if you can identify the range of unrounded values for these questions.

Press pause if you need more time.

Well done.

Let's move on to question two.

Question two wants us to identify the range of these unrounded values given these degrees of accuracy.

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

Well done.

Let's move on to question three.

Here's some missing numbers.

Can you work them out? See if you can give it a go and press pause for more time.

Let's look at our answers.

Well, for question one, you should have these answers to represent the range of unrounded values.

Well done if you got any of these.

For question two, you should have these answers to represent the range of these unrounded values.

For question three, let's see how you got on.

The missing values are here.

Massive well done if you got this one right.

Great work, everybody.

Now, let's move on to the second part of our lesson: error intervals.

Now, notation in mathematics is so important.

We use symbols and notation because they're easy to read and understand.

They are concise and take less space.

They can be used to represent complex concepts, and it allows mathematical ideas to be communicated more effectively than words.

And this can also be said when we're trying to find the range of unrounded values of a number.

We use an error interval.

And an error interval for a number x shows the range of possible values of x.

And it's written as an inequality: a is less than or equal to x, which is less than b.

The notation we use allows us to concisely identify unrounded values of a number.

Jacob says, "Can you explain what 'less than or equal to x, less than b' means given our definition of what an error interval is?" And Andeep says, "Sure." So remember, we did this question at the start of the lesson.

Five has been rounded to the nearest integer, and we needed to identify what the largest and smallest unrounded values were.

Jacob remembers, and he remembers zooming in.

He also remembers that the smallest value had to be 4.

5, and the largest value had to be 5.

49 recurring.

So what we're gonna do now is we're going to change the question a little bit.

What we're gonna do is we'll identify what are the range of values of the unrounded numbers.

Jacob recognises that he needs to look at the smallest and largest possible values and then highlight them on the line.

So, if we're looking at all of these numbers on the highlighted number line, that means we can identify our range, but our range has to include 4.

5 and not include 5.

And Andeep agrees and says that is exactly what the error interval does.

The range of values of the rounded number is given as: 4.

5 is less than or equal to x, which is less than 5.

In other words, 4.

5 is included in the range of values, but 5.

5 is not included.

You may notice the difference in notation there.

So let's see how you get on with a check.

Here, Andeep and Jacob have drawn these number lines to help, and what we have to do is identify that error interval.

Do you think you can fill in those missing limits? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Nine has been rounded to one significant figure.

You can see that highlighted range of numbers.

So what does that look like on our error interval? Well, it looks like this: 8.

5 is less than or equal to x because we're including that 8.

5, and x is less than 9.

5; we're not including that 9.

For b: 16.

5 is less than or equal to x because we're including that 16.

5, and x has got to be less than 17.

5 because we're not including it.

Lastly, 55.

5 is less than or equal to x, as we're including that 55.

5, but x is less than 56.

5, as we're not including that 56.

Well done if you got this one.

Here's another check.

Here's another number line we're using to help identify the error interval.

I want you to identify whose answer is correct.

Here, the question says, "Work out the error interval given 43 has been rounded to the nearest integer." Sam says the error notation is written as this.

Sofia says it's written as this.

And Izzy says it's written as that.

Who do you think is correct, and explain? Well done.

Let's see how you got on.

Well, hopefully you can spot Sam is incorrect because Sam is including 42.

5, and he's including 43.

5 in the range of values.

43.

5 would round to 44, so Sam has incorrectly written the error interval.

Now, Sofia is saying that the range of values are greater than or equal to 42.

5 and greater than 43.

That does not fit the diagram that we've drawn here.

Izzy is correct.

She's used the correct notation to show the lowest value is 42.

5 and includes the 42.

5, and the largest value is 43.

5, not including 43.

Great work if you got this one.

Jacob now asks, "Can we find error intervals when a number has been rounded to decimal places?" And Andeep says, "Yes, we can find error intervals when any number has been given to a degree of accuracy." So let's say we can find the error interval when 3.

6 has been rounded to one decimal place.

First of all, drawing a number line always helps.

So Jacob says he's gonna draw the number line and identify the degree of accuracy to be one decimal place.

Thus, we're adding 0.

1 and subtracting nought point one from our 3.

So our number line looks like this.

From here, we're gonna identify those middle values.

So we have 3.

55 and 3.

65.

Then you can see our highlighted range of values.

So, from here, it's quite obvious that the error interval can be seen.

It is 3.

55 is less than or equal to x, which is less than 3.

65.

Really well done if you spotted this as well.

So what's important to remember is any number that has been given to a degree of accuracy has an error interval, and this is because of the rounding that's taken place.

It's important to recognise what degree of accuracy is used, and then the error interval can be used.

A number line can help.

Moving on to another check: Andeep and Jacob have drawn these number lines to help identify the error interval.

Can you identify the error interval of each of these questions? See if you can give it a go and press pause if you need more time.

Well done.

Let's see how you got on.

While using the number line, you can see this would be the range of values identifying all the unrounded numbers which give 4.

8 when rounded to one decimal place.

Therefore, the error interval would be 4.

75 less than or equal to x, less than 4.

85.

For B, hopefully, you can spot here are our range of unrounded values when 1.

3 has been rounded to two significant figures.

So that means our error interval is 1.

25 less than or equal to x, less than 1.

35.

For C, 3.

42 has been rounded to three significant figures.

Well, here is our range of unrounded values that, when 3.

42 is rounded to three significant figures, thus our error interval is 3.

415 less than or equal to x, less than 3.

425.

Amazing work if you got this one right.

Now, let's have a look at another check.

But here you'll notice we haven't drawn the number line.

See if you can draw a number line, if it helps you identify the error interval of these questions.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

8 rounded to one decimal place is 4.

75 less than or equal to x, less than 4.

85.

7 rounded to two significant figures is 7.

65 less than or equal to x, less than 7.

75.

18.

49 rounded to two decimal places is 18.

485 less than or equal to x, less than 18.

495.

Great work if you got those.

Now let's move on to your task.

Question one has given you the number lines, and you need to identify the error interval.

So you can give it a go and press pause if you need more time.

Well done.

Let's move on to question two.

Question two hasn't given you a number line, but you're very welcome to draw one if it helps.

Identify the error interval of the following: Well done.

Let's move on to question three.

Question three wants you to pair up the question with the correct error interval.

See if you can give it a go.

Press pause if you need more time.

Great work, everybody.

Let's move on to the answers.

Well, for question one, I've highlighted the range of unrounded numbers that, when rounded to one significant figure, give two.

So that means our error interval is 1.

5 less than or equal to x, less than 2.

33 rounded to two significant figures is 32.

5 less than or equal to x, less than 33.

10 rounded to two significant figures is 9.

5 less than or equal to x, less than 10.

For question two, let's see how you got on.

See if you can mark these.

Well done if you got any of these correct.

For question three, we had to pair them up.

Well, hopefully you've got these pairs.

Amazing work, everybody.

Excellent work.

So, in summary, an error interval for a number x shows the range of possible values of x, and it's written as an inequality: a is less than or equal to x, less than b.

Any number that has been given to a degree of accuracy has an error interval, and this is because of the rounding that's taken place.

It's so important to recognise what degree of accuracy is used, and then the error interval can be used.

A number line can help.

Great work, everybody.

It was tough today.

Well done.

I've finished the video