Lesson video

Hi everyone.

My name is Miss Ku and I'm really happy and excited to be learning with you today.

It's going to be a fun and interesting lesson and some of our Oak pupils will be here to help as well.

It's gonna be tricky in part, but don't worry, we'll all be here to help and I really do hope that you'll enjoy the lesson and find it challenging too.

Really excited to be working with you.

So, let's make a start.

Welcome to this lesson on rounding errors, under the unit estimation and rounding.

And by the end of the lesson, you'll be able to appreciate the impact of rounding errors when using a calculator or other technology and the way that these can be compounded to result in large inaccuracies.

We'll be looking at significant figures as one of our key words.

And remember, significant figures are the digits in a number that contribute to the accuracy of the number.

The first significant figure is the first non-zero digit and number lines can be used to round to one significant figure.

For example, 6,523 is equal to 7,000 to one significant figure.

6,523 is equal to 6,500 to two significant figures and 6,523 is equal to 6,520 to three significant figures.

Today's lesson will be broken into two parts.

We'll be looking at number lines first and then the impact of rounding errors second.

So, let's make a start.

We round in everyday life to make calculations easier and sometimes we're given rounded values and must identify the range of values of the unrounded number.

For example, let's say 250 has been rounded to the nearest 10, what integer values could the unrounded number be? I want you to use this number line to help you if you need.

Well, let's see how you got on.

Well, Sam says 245 is one answer and Alex agrees.

Good, because rounding 245 to the nearest 10 does give us 250.

Now, Sam also says 247 is also an answer and Alex agrees.

Rounding 247 to the nearest 10 does give us 250.

Sam says 244 is one answer.

Now, this won't work because rounding 244 to the nearest 10 gives 240, so this won't work.

"253 is an answer," says Sam.

Well that would work, because rounding eight to the nearest 10 does give 250.

Next, Sam says 255.

Well, 255 won't work and this is because rounding 255 to the nearest 10, gives 260.

So, that means Sam says, "Okay, 254 is an answer." And Alex agrees and says, "Well done." Rounding 254 to the nearest 10 gives 250.

But can you see a way to identify all the unrounded integers that give 250 when rounded to the nearest 10? Let's see.

Well, Sam says, "I see identifying the middle on the number line segments "on the number line will help me see the range of values "for the unrounded number." So, you can see Sam's identified the middle number in between 240 and 250 and the middle number between 250 and 260.

And then from here Sam has identified 245, does round up to 250 to the nearest 10, and 254 does round down to 250 to the nearest 10.

So, all our integer values are 245 all the way up to 254.

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

What are the smallest and largest integers that when rounded to the nearest 50, give 300.

Let's see if Alex and Sam can do this.

Well, Sam suggests let's do a number line, but is unsure how to draw it.

So, Alex helps.

First of all, let's draw a line and put 300 in the middle.

From here as the degree of accuracy is 50, all we need to do is add and subtract 50 either side of that 300.

So, therefore we have our number line with 250, 300 and 350 drawn.

Now from here, do you remember what Sam suggested? Well, Sam suggested drawing that middle line, so identifying that middle line in between 250 and 300 and 300 and 350.

From here, he knows this has to be 275 as it's the midpoint between 250 and 300.

Now, Sam also knows this has to be 325, but if we round 325 to the nearest 50, it's 350.

So, that means Sam must use 324.

So, what is the largest and smallest integer values? Well, 275 is the smallest integer value and 324 is the largest integer value.

Remember, we couldn't use that 325 as rounded to the nearest 50 gives 350.

So, now let's move on to a check.

Here you have to work out the largest and smallest unrounded integers and we've drawn a number line so it helps.

See if you can identify what the largest and smallest integers are when 430 has been rounded to the nearest 10.

What's the largest and smallest integers when 6,800 has been rounded to the nearest a hundred? What's the smallest and largest integers when 160 has been rounded to the nearest 20? So, you can give it a go and press Pause if you need more time.

Well done, let's see how you got on.

All our number lines are drawn, so let's identify those middle values.

Well, you know the middle value here would be 425 and 435.

So, that means we know the smallest integer value is 425 and the largest integer value is 434.

For B, identifying that middle value, hopefully you can spot, we have 6,750 and 6,850.

So, that means we know the smallest integer value has to be 6,750 and 6,849.

For C, let's identify those middle values again.

We have 150 and 170, so that means we know 150 is the smallest integer and 169 is that largest integer.

Well done if you got this one right? For the next check question, you need to draw the number lines and work out the largest and smallest unrounded numbers.

So, remember what Sam did, you're very welcome to go back and have a look, but give these a go.

Press pause for more time.

Well done, let's see how you got on.

Well first of all, put 20 in the centre, because we are rounding to the nearest 10 as that is our degree of accuracy.

You plus 10 and minus 10, either side of that 20, thus giving goes 10 and 30.

Identifying those middle values.

We have 15 and 25, so that means the smallest integer value is 15 and the largest integer values 24.

Next one, 20 is rounded to the nearest five.

So, that means we put 20 in the centre with 15 and 25 either side.

Now, the centre is 17.

5 and 22.

5, but remember, we're only interested in the integers, so that means 18 is the smallest integer and 22 is the largest integer.

For C, 9,900 is rounded to the nearest 50.

So, that means we put 9,900 in the centre and draw 9,850 and 9,950.

From here you can see that the middle value, which we have 9,875 and 9,925.

So, that means the smallest integer value is 9,875 and the largest integer value is 9,924.

Great work everybody.

So, let's move on to your task.

Here I want you to work on the largest and smallest unrounded integers and you can use a number line if it helps.

See if you can give it a go and press Pause if you need more time, great work.

So, let's move on to the next one.

For the continuation of question one, same again, work out the largest and smallest unrounded integers and use a number line if it helps.

Well done.

Let's move on to question two.

Question two wants you to work out the unknown number and draw number lines if it helps.

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

Great work, let's move on to question three.

Question three says, in a school the head teacher says there are 480 pupils rounded to the nearest 10.

What is the largest number of pupils in the school? Question four says, an advert says 56,000 people use their product.

What is the least amount of people who use their product? See if you can give it a go and press pause for more time.

Fantastic work everybody.

So, let's go through these answers.

Well, for question one, hopefully you've spotted this is our number line, thus giving giving us 295 is the smallest integer and 304 is the largest integer.

For B with this number line, you can see 250 is the smallest integer and 349 is the largest integer.

For C, we should have this number line and identify 275 is the smallest integer and 324 is the largest integer.

Great work.

For D, you've got this number line with the smallest integer being 635 and the largest being 644.

For E, this number line shows we have the smallest integer is 2,950, and the largest integer is 3049.

And for F, the smallest integer is 25 and the largest integer is 74.

Great work.

Let's have a look at question two.

For 2A, you can see I've drawn my number line and I've identified 715 as the smallest integer.

Given that 724 is the largest integer, that means the mid has to be 725.

So, I can identify those missing numbers now, thus telling me 720 has been rounded to the nearest 10, looking at B, I can draw my number line.

Well, I know 130 is the smallest integer.

149 is the largest integer.

That means 150 must be the mid.

From the question I know 140 is the value.

So, what did we round to? Well, it had to be rounded to the nearest 20.

For C, using my number line, I know I've got my smallest integer values 875 and the largest integer values 924.

So, I'm drawing 925.

Thus I can fill in my values, identifying that 900 has been rounded to the nearest 50.

That was a great question.

Huge, well done if you got any of those right.

For question three, did you find out the largest number of pupils in the school? Well, it would be 484.

Did you find out out the least number of people who use the product? Well, there'd be 55,500.

Well done if you got this one right.

Great work, everybody.

So, now let's have a look at the impact of rounding errors.

Now, scientific calculators are fantastic and there are lots of different scientific calculators out there, but all have very similar functionality and notation.

They're all fantastic instruments and incredibly powerful.

Usually when there are errors with a calculation, it's not the fault of the calculator.

So, let's have a look.

For example, here's two calculations done in a calculator.

Can you explain why Laura and Jun get different answers? The question wants you to work out the volume of this cuboid to one decimal place.

Laura has an answer of 352.

5 metres cubed.

You can see the calculation she's inputted on her calculator.

Jun has the answer of 348.

8 metres cubed and you can see his calculation.

Can you explain why their answers are different? Hopefully you can see Laura has rounded too early.

It's important to keep the values as accurate as possible and round at the end of the calculation.

Now, Laura says, that's not fair.

The question says one decimal place and she rounded to one decimal place.

But Jun reminds her that the question wants the volume to one decimal place, so you keep the length as accurate as possible.

Laura says it takes too long to input the numbers, can't she just round? And Jun reminds her, if you round throughout the calculation, you do lose accuracy.

But Laura says, "Well, how else can I keep accuracy but be more efficient?" And a great way to do this is generally rounding to three significant figures is a good practise or using the Ans button on a calculator can help.

So, let's have a little look.

So, hopefully you remember that the Answer button on your calculator, recalls the previous answer.

So, let's try with a calculation.

Well, here we have a circular garden and it's 120 metres squared.

Now, our rectangular summer house has been placed in the garden, and what we have to do is work out the remaining area that is left to one decimal place.

I want you to try this question and round answers to the calculations to three significant figures and then finally round to one decimal place.

Well, let's have a look at that accuracy, 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, hopefully you've spotted to work out the area of that rectangular summer house, you're going to multiply 8.

24 by 7.

15 giving us 58.

916.

Now, rounding this to three significant figures, gives us 58.

9.

Then subtracting this from 120 gives us an area of 61.

1 metre squared.

Now, let's try using the Answer button on your calculator and then round the final answer.

Well, putting in the area of the rectangle, 8.

24, multiply by 7.

15 and then pressing Execute, does give us that 58.

916.

Then all we have to do from here is remember the calculator has saved this as our previous answer.

So, we simply press 120, subtract the Ans button, then pressing Execute, will give us exactly the same area, which is 61.

1 metres squared to one decimal place.

Laura is really happy with this, but she doesn't know how to show her working out when she's using the Ans button.

And Jun says, "Well, this is a good question, "but you can use the ellipsis to show "that you used an unrounded number." For example, instead of using the 58.

9 to three significant figures, you could show the 58.

916 with an ellipsis to show that you haven't rounded it.

Thus, 120, subtract the Ans, because it was the previous answer, gives you the answer to one decimal place.

Great work everybody.

So, let's have a look at a check.

Here which of the following working out shows the correct answer to the question and explain the errors.

The question says the area of a Pentagon is 14 metres squared and a triangle's being cut out.

We're asked to work out the remaining area to one decimal place.

See what you can do and press Pause if you need more time.

Well done.

Let's see how you got on.

Well, for the first piece of working out, they have rounded each measurement to one significant figure before any calculating.

This has meant a lot of loss of accuracy.

Now, the second working out, shows the calculation and working out is correct, and it's a really nice use of that ellipsis.

Now the third working out, well, hopefully you've spotted, although the answer's correct, there is an error in their working out.

They've incorrectly rounded to three significant figures and it should be 5.

27 to three significant figures, not 5.

28.

That's quite a costly mistake, so just be aware of that.

Well done everybody.

So, now let's have a look at your practise task.

For question one, here's a plot of land, and you have to explain if and where students, have lost accuracy and resulted in an incorrect answer when finding the area of this rectangle to two decimal places, see if you can give it a go and press Pause if you need more time.

Great work.

Let's have a look at question two.

Well, question two states, a circular sheet of metal has an area of 25 metres squared.

It gets two square patches with lengths, 2.

6574 metres cut from it using a laser cutter, which pupil has correctly worked out the unpatched area to two decimal places and explain the errors from the other pupils.

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

Fantastic works, and let's move on to these answers.

Well, hopefully you spotted, Aisha has incorrectly rounded her answer.

Jun has correctly rounded and Jun has used the accurate measurements given.

Andeep has rounded the measurements, before calculating the area and has resulted in a loss of accuracy.

And Laura has incorrectly written her values.

She's changed the place value of the digit, so her answer is far too big.

For question two, hopefully you can spot Aisha has rounded too early.

She rounded her first calculation to one decimal place, which meant she has struggled to give the correct accurate answer to two decimal places.

Lucas has answered the question correctly.

He did round early, but stated it to two decimal places which resulted in less error loss.

And finally, Laura has used the ellipsis to show she's using the Ans button on her calculator.

Great work, everybody.

So in summary, we round in everyday life to make calculations easier.

Sometimes we're given rounded values and must identify the range of values of the unrounded number.

Number lines can help us identify the smallest and largest in integer values of unrounded numbers.

And lastly, scientific calculators are fantastic instruments and incredibly powerful, but we have to be careful when to round and choosing our degree of accuracy.

Sometimes when there are error with a calculation, it's not the fault of the calculator.

Great work, everybody.

It was wonderful learning with you.