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Hi, everyone.

My name is Ms. Ku.

And I'm really happy that you're joining me today.

Today, we're going to be looking at bounds.

Upper bounds, lower bounds, degrees of accuracy, error intervals, and more importantly, the real-life application of these.

I hope you enjoy the lesson.

I know I will.

Let's make a start.

Hi, everyone, and welcome to the lesson on upper and lower bounds under the unit Rounding, estimation, and bounds.

And by the end of the lesson, you'll be able to calculate the upper and lower bounds of values and express these as a is less than or equal to x, which is less than b.

Now we'll look at some keywords in this lesson today.

We'll be looking at the word upper bound, and the upper bound of a rounded number is the smallest value that will round up to the next rounded value.

For example, six has been rounded to the nearest integer.

So that means 6.

5 is the upper bound.

Another example would be 6.

3 has been rounded to one decimal place.

This means 6.

35 is the upper bound.

Let's have a look at the lower bound.

Well, the lower bound of a rounded number is the smallest value that the number could have taken prior to being rounded.

We'll look at these keywords in our lesson today.

We'll also be looking at the keyword error interval.

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

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

Today's lesson will be broken into two parts.

We'll be looking at the upper and lower bounds first, and then we'll be moving on to using upper and lower bounds with estimation.

So let's make a start, looking at upper and lower bounds.

Now notation in mathematics is so important.

And we use symbols and notation because they're just easier to read and understand.

They're concise and take up less space.

They can be used to represent complex concepts.

And they allow mathematical ideas to be communicated more effectively than words.

Now this can also be said when trying to find the range of unrounded values of a number.

We use an error interval.

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

And this notation allows us to concisely identify the unrounded values of a number.

Now Jacob says, here's the definition, but can you explain why a is less than or equal to x, which is less than b? What does it actually mean please? Andeep says, sure.

Andeep explains how to find the error interval when five has been rounded to the nearest integer.

Andeep says, the degree of accuracy is to the nearest integer.

So what integers are either side of the five? And Jacob replies, well, it's four and six.

So that means they need to look at the midpoint between four and five and the midpoint between five and six, which is 4.

5 and 5.

Now Andeep says, what are the range of values when rounded gives five to the nearest integer? And Jacob replies, well, I know the unrounded numbers have to be in this highlighted line.

Jacob then says, but I want the range of values to include 4.

5 and not include 5.

And Andeep says, exactly.

And that's what the error interval does.

When it says a is less than or equal to x, less than b, that's what it means.

So the error interval when five has been rounded to the nearest integer is 4.

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

And Jacob recognises this is because 4.

5 is included.

And Andeep says, and it's because 5.

5 is not included.

So this means when five has been rounded to the nearest integer, 5.

5 is the upper bound and 4.

5 is the lower bound.

So the error interval can really help us identify the upper and lower bounds.

Now let's have a look at a quick check.

Andeep and Jacob have drawn these number lines to help identify the error interval.

Can you fill in the bounds? See if you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

The error interval when nine has been rounded to one significant figure is 8.

5 less than or equal to x, which is less than 9.

That means the lower bound is 8.

5 and the upper bound is 9.

Let's have a look at B.

56 has been rounded to two significant figures.

So that means our error interval is 55.

5, less than or equal to x, which is less than 56.

So that means the lower bound is 55.

5 and the upper bound is 56.

Well done if you got this.

Jacob says, can we find the error interval when a number has been rounded to decimal places? And Andeep replies, of course, yes, we can find error intervals when any number has been given to a degree of accuracy.

So let's identify the upper and lower bounds when 3.

6 has been rounded to one decimal place.

Well, Jacob says, I'm going to draw a number line.

As the degree of accuracy is one decimal place, this means he adds 0.

1 to the 3.

6 and subtracts 0.

1 from the 3.

6, thus giving us this number line.

Jacob then recognises where we need to find the middle numbers.

What's in between 3.

5 and 3.

6? Well, it's 3.

55.

And what's in between 3.

6 and 3.

7? Well, it's 3.

65.

Identifying this highlighted region means we can see our error interval.

Our error interval is 3.

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

65.

Then from here, we can identify the upper and lower bounds.

The upper bound is 3.

65 and the lower bound is 3.

55.

Well done, Jacob.

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

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

A number line can help.

From here, we can use the error intervals to identify the upper bound, which I'm going to label in capitals as UB and then the lower bound, which I'm going to label in capitals as LB.

Let's have a look at another check question.

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

Can you identify the bounds? See if you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, for part A, 4.

8 has been rounded to one decimal place.

So you can see the diagram identifies plus 0.

1 to 4.

8, giving us 4.

9, and minus 0.

1 from our 4.

8 gives us 4.

Identifying the middle value in between 4.

7 and 4.

8 and 4.

9 gives us the error interval of 4.

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

85.

Identifying the lower bound is 4.

75 and the upper bound is 4.

85.

Well done if you got this.

42 has been rounded to three significant figures.

Once again, let's identify those midpoints.

From here, we can see the error interval.

The error interval will be 3.

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

425.

From our error interval, or our diagram, you can see our lower and upper bound.

Really well done if you got this.

Let's have a look at another check question.

I've taken away that number line, and you're very welcome to draw it if it helps.

30 has been rounded to the nearest integer.

Which of the following are the upper and lower bounds? So if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Firstly, let's have a look at our degree of accuracy.

Because it says to the nearest integer, then I know I'm adding and subtracting one either side of the 30, thus giving me this number line, 29, 30, and 31.

I know I'm talking about the midpoints, so I have my 29.

5 and my 30.

Now I can identify my lower and upper bound.

My lower bound is 29.

5 and my upper bound is 30.

So that means the answer is A.

Well done if you got this.

Let's have a look at another check.

30 has been rounded again but it's been rounded to the nearest 10.

I want you to have a little think.

Which of the following are the upper and lower bounds? If you want to draw a number line to help, feel free.

Well done.

Let's see how you got on.

Well, the degree of accuracy is to the nearest 10, so that means we plus or minus 10 either side of our 30, giving us this number line.

Identifying our midpoints, we know our lower bound would be 25 and our upper bound would be 35.

Well done if you got this.

There is a more efficient way to work out the bounds without drawing a number line.

So let's have a little think.

When we were using our number line, we plused or minused the degree of accuracy and then found the midpoint.

Now this is the same as plus or minus the degree of accuracy all divided by two.

So let's have a look at an example where we're not using a number line.

Here, we need to work out the bounds when eight has been rounded to the nearest integer.

So the lower bound would be eight subtract a half.

Now why is it a half? Well, it's one as our numerator because the nearest integer refers to the nearest one, so we had to do one.

And we're dividing it by two just like we did on a number line because we found the midpoint.

Thus giving us 7.

The upper bound will be eight add one half.

Same again, the degree of accuracy was the nearest integer, so that means it's the nearest one.

So then, just like we did before, we divided by two.

So that's why we're adding one half, giving me the upper bound to be 8.

Notice how we didn't use a number line to identify these bounds.

Let's have a look at another example.

Here we have 4.

6 and it's been rounded to one decimal place.

Well, to find the lower bound, it'd be 4.

6 subtract 0.

1 over two.

It's 0.

1 because the degree of accuracy says one decimal place and one decimal place means to the nearest 0.

Then, we divided by two just like we did before, thus giving us 4.

6 subtract our 0.

05, which gives our lower bound of 4.

55.

The upper bound is 4.

6 add 0.

1 over two.

Why is it 0.

1? Because the degree of accuracy says to one decimal place and this refers to the nearest 0.

So then we do our 0.

1 over two.

Working this out, we have 4.

6 add our 0.

05, which gives me the upper bound to be 4.

65.

In other words, the bounds were found by 4.

6 plus or minus our 0.

05.

Let's have a look at another example.

Here, we're asked to work out the bounds when 3.

82 has been rounded to three significant figures.

Well, the lower bound would be 3.

82 subtract our 0.

01, and we're dividing it by two.

Why is it 0.

01? Because three significant figures in this question refers to the nearest 0.

01.

Then we divide by two just like before, giving us the lower bound to be 3.

815.

The upper bound would be 3.

82 add 0.

01 and we're dividing it by two.

Why is it the 0.

01? Same again, because to three significant figures in the context of this question refers to the nearest 0.

01.

Working this out gives me the upper bound to be 3.

825.

In other words, the bounds were found by 3.

82 plus or minus 0.

005.

Well done if you got this.

Now let's have a look at a quick check.

See if you can work out the upper and lower bounds without a number line.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for A, it says 5.

8 rounded to two significant figures.

The second significant figure is 0.

So we divide this by two, which means to find the bounds, it's 5.

8 plus or minus 0.

05, which gives me the upper and lower bounds of 5.

75 and 5.

85.

For B, 70 rounded to the nearest 10.

The bounds would be found by 70 plus or minus five because the degree of accuracy of 10 divide by two is five, so it's 70 plus or minus five, which gives me the lower bound of 65 and the upper bound of 75.

45 rounded to the nearest five means the bounds can be found by 45 plus or minus our 2.

5 because the degree of accuracy of five divide by two gives us 2.

5, giving us the lower bound of 42.

5 and the upper bound of 47.

Well done if you got this.

Great work, everybody.

Now it's time for your task.

See if you give these a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

See if you can give these a go.

Press pause if you need more time.

Great work.

Moving on to question three.

Can you fill in the blanks? This is a little bit tricky.

Try your best.

Press pause if you need more time.

Well done.

Let's move on to these answers.

For question one, here are our bounds.

Press pause if you need more time to mark.

For question two, here are all our answers.

Press pause if you need more time to mark.

And for question three, this was a really tricky one.

Massive well done if you got any of these.

Great work, everybody.

Now it's time for the second part of our lesson where we're using upper and lower bounds with estimation.

What I want you to do is have a look at this error interval for a.

It's given as 4.

75 is less than or equal to a, which is less than 4.

85.

I want you to identify which of the following values of a are included in this error interval? See if you can give it a go.

Well done.

Well, hopefully you've spotted here are all our values that satisfy that error interval.

Now, I'm going to give you another one.

Let's have a look at b.

Here, we have an error interval of b.

I want you to have a little think.

Which of the following values of b are included in this error interval? See if you can give it a go.

Press pause if you need more time.

Well done.

Well, hopefully you spotted you should have got these answers.

So now we have an error interval of a.

Then we have an error interval of b.

We can recognise there's an infinite number of values that can satisfy a, given that error interval, and satisfy b, given that error interval.

And Andeep is asked to sum as many of the values of a and b as he can.

So he decides to put some possible values of a and b into a spreadsheet.

Then he sums them.

And what I want you to do is I want you to see if you can identify the lowest bound of the sum of a and b.

Have a little look and have a little think.

Well done.

Well, hopefully you spotted the lower bound is 13.

That is our lowest value in our little spreadsheet of the sum of a and b.

But can you see how we calculated the lower bound of the sum of a and b? Same again.

Have a little think.

Well, it would be the sum of the lower bounds of a and b.

Now what I want you to do is see if you can use Andeep's spreadsheet, and identify what would the upper bound of the sum of a and b work out as? What do you think? Well, it would be 14.

But same again, can you explain how we calculate the upper bound of the sum of a and b? Well, hopefully you spotted it's the sum of the upper bounds of a and b.

Really well done if you spotted this.

So now we have the error interval of the sum of a and b.

13.

8 is less than or equal to a plus b, which is less than 14.

And Andeep says, but I did not show all the possible values of a and b in his spreadsheet, and he's right.

But what I want you to do is explain why it's not important to show all the summations of a and b when identifying the error interval.

Why do you think? Well, it's because we only need the maximum and the minimum values.

This is because they're the upper and lower bounds.

Now it's time for a quick check.

Same again, we're using the error interval of a and the error interval of b.

But Andeep is asked to find the product of a and b.

So just like before, he put some possible values of a and b into a spreadsheet.

But what I want you to do is have a look at this spreadsheet and can you identify the upper and lower bounds of the product of a and b? Well, hopefully you spotted our lower bound would be 42.

9875 and our upper bound would be 44.

3775.

But now what I want you to do is, can you explain how the upper and lower bounds were calculated? See if you can give it a go.

The product of the lower bounds gives the minimum and the product of the upper bounds gives the maximum.

Now what I'm going to do is I'm going to show you two separate spreadsheets, one by Izzy and one by Andeep.

And using the same intervals of a and b as before, Izzy and Andeep were asked to work out the error interval when b is divided by a.

But their two spreadsheets are a little bit different.

And I want you to explain why their two spreadsheets show different maximum and minimum values for b divided by a? Have a good look at this and have a think.

Press pause, as you'll need more time.

Great work.

So let's have a little look.

Well, Izzy starts by dividing the smallest value of b by the smallest value of a.

Then she finishes her spreadsheet with the largest value of b being divided by the largest value of a.

But Andeep starts by dividing the largest value of b by the smallest value of a, and then finishes with the smallest value of b being divided by the largest value of a.

So from these results, from Izzy and Andeep, what are the bounds of b divided by a? Well, hopefully you can spot that the lower bound and the upper bound are seen in Andeep's spreadsheet because they are the lowest value achieved and the highest value achieved when dividing b by a.

The lower bound is 1.

86598 and the upper bound is 1.

92632.

Really well done if you spotted this.

Now it's time for your check.

Using the same values of a and b again, Izzy and Andeep are asked to work out the error interval when b is subtracted by a.

Here you can see we have two different spreadsheets.

So I want you to explain why their two spreadsheets show different maximum and minimum values for b subtract a? See if you can give it a go.

Well done.

Well, hopefully you spotted Izzy started by subtracting the smallest value of b by the smallest value of a.

Then she finishes with the largest value of b being subtracted by the largest value of a.

That gave us all the same answers.

But for Andeep, Andeep starts by subtracting the largest value of b by the smallest value of a, and then finishes with the smallest value of b being subtracted by the largest value of a.

So now what I want you to do is, can you identify what the bounds of b subtract a are? Well done.

Well, hopefully you spotted we're using Andeep's spreadsheet.

The lower bound is 4.

2 and the upper bound would be 4.

Well done.

Great work, everybody.

So now it's time for your task.

Here we have error intervals for x and y and I want you to fill in the table and work out the following, rounding to three significant figures where needed.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Question two has a completed table, but we are missing some information.

Have a little think and you can always go back to how you worked out question one to help you out.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on with these answers.

Well, you should have got these answers for the upper and lower bounds of these calculations.

Massive well done if you got any of these.

Press pause if you need more time to mark.

Great work.

Let's move on to question two.

This was a super hard question.

Really well done if you got this one.

Press pause if you need more time to mark.

Great work, everybody.

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, where a is less than or equal to x, which is less than b, where a is the lower bound and b is the upper bound.

Remember, any number that has been given to a degree of accuracy has an error interval.

And this is because of the rounding that has taken place.

The degree of accuracy is so important when identifying bounds.

Great work, everybody.

It was wonderful learning with you.

I've finished the video