Lesson video

Hello everyone, my name is Ms. 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 Oak pupils do.

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.

In today's lesson, under the unit estimation and rounding, we'll be looking at checking by estimation.

And by the end of the lesson, you'll be able to estimate and check if solutions to problems are of the correct magnitude.

We'll be looking at those key words, significant figures.

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

And the first significant figure, is the first non-zero digits.

Number lines can used to round to one significant figure.

So here's a few examples.

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

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

We'll also be looking at the word, estimate, and a quick estimate for a calculation is obtained from using approximate values, often rounded to one significant figure.

And we show estimation using this approximate symbol.

Two wiggled lines, if you like, to identify that the calculations are approximately the same but not equal.

And you can see it in this example, 892 multiplied by 176.

9, is approximately the 900 multiplied by 200.

Today's lesson will be broken into two parts.

First, we'll be looking at estimating answers, and then we'll be looking at checking answers.

So let's make a start with estimating answers.

So remember, quick estimate for a calculation is obtained from using approximate values often rounded to one significant figure.

In everyday life, it's important to estimate, so to know how to quickly calculate a rough answer or to check the accuracy of answers.

For example, here's a calculation and pupil responses.

272.

56 is multiplied by 9.

769.

Have a look at the pupil responses, and who do you agree with and why? Alex says, "The answer is 2,700.

It is maths, so there's only one answer." Jacob says, "The answer is 2,500, because 250 multiplied by 10." Sofia says, "The answer is 3,000, because 300 multiplied by 10 is 3,000." And Laura says, "You are all right except Alex's explanation." What do you think? Hopefully, you've spotted Laura is correct.

So let's have a little look.

Well, looking at Alex's answer only, as estimation gives an approximate answer, there is more than one estimate.

So when Alex says the answer is 2,700, Alex rounded 272.

56 to two significant figures, and 9.

769 to one significant figure, thus making the 2,700.

So Alex's approximation is correct.

Let's have a look at Jacob.

So Jacob has rounded 272.

56 to the nearest 50, and 9.

769 to one significant figure, thus making 2,500, Jacob is correct.

Sofia's answer is obtained by rounding 272.

56 to one significant figure, and 9.

769 to one significant figure, thus making 3,000.

So Laura is correct because they were all correct.

Estimation gives a rough idea, and the more significant figures we round to, the more accurate the estimation.

So all of our Oak pupils were correct here.

Knowing how to round so to estimate and knowing which sensible units to use is as equally important as knowing which operation to apply when given the context of the question.

For example, which operation do we tend to use when calculating the area of a 2D shape, the perimeter of a shape, difference between values, and how many times a number fits into another number, which operation do you think we use here? Well, hopefully, you spotted.

We tend to use multiplication with area, addition with perimeter, subtraction signifies the difference between values, and we use division to find out how many times one number fits into another.

So once we have this in our mind, let's see if we can apply it to a check.

Which of the following is a sensible measurement for the length of an average house? Then from this, can you calculate an estimate for the area of the floor of the house? See if you can give it a go, and press Pause if you need more time.

Well done.

So let's see how you got on.

Well, 844 centimetres by 596 centimetres, is a very sensible measurement for the front and side of our house, as is 8.

44 metres and 5.

96 metres.

Well done, if you chose those two.

Now, using this, let's see if we can estimate the area of the floor of our house.

Well, using centimetres, the area is approximately 800 by 600 which is 480,000 centimetres squared, but you may have actually chosen to use the metres as a measurement.

So the area of the floor, would be approximately eight multiplied by six, which is 48 metres squared.

Really good work if you've got this one right.

Let's have a look at another check question.

A bookshop needs to buy the latest maths puzzle books.

Each book costs six pounds 89.

Now, the bookshop only has 279 pounds and 45 pence to spend on these puzzle books.

Estimate how many books the bookshop can buy, and I want you to make sure you show you're working out.

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

Well done.

So let's see how you did.

You may have chosen to round 279.

45 to two significant figures.

It gives us 280, and you may have chosen to round 6.

89 to one significant figure giving us a seven.

So that means the approximate calculation, would be 280 divided by seven, which is 40 books.

This is absolutely fine.

But perhaps, you may have chosen to round to one significant figure.

So 279.

45 rounded to one significant figure would be 300, and maybe you chose around 6.

89 to the nearest 10.

So that means our calculation would be 300 divided by 10, thus giving us an approximate calculation of 30 books.

Estimation gives a rough answer, so both are absolutely fine, but the more significant figures we round to, the more accurate the estimation.

Massive well done if you got this one.

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

Lucas bought some items in a shop and paid with a 10 pound note.

Now, the receipt was damaged and the total is not seen, and he receives two pound 12 pence change.

He quickly knew there was an error.

How did he know there was an error without calculating the exact change? See if you can have a look at this and give it a go.

Well done.

So let's see what you did.

Well, hopefully, you spotted the first two items, round to around about three pounds in total.

The next two items are also rounded to three pounds in total.

The bill will be around about six pounds, so we should get close to four pounds in change.

And he only got two pounds 12 pence in change, so he knew something wasn't right there.

Well done, if you got this one right.

Now, it's time for your task.

A rectangle has the dimensions 145 metres by 18.

4 metres.

Which calculation would give you a suitable estimate for the area of the rectangle? And I want you to explain why you think it is a suitable estimation.

See if 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 shows three damaged receipts.

Each pupil pays with a 20 pound note.

Aisha gets 12 pound 23 change, Jacob gets six pound 89 change, and Sam gets nine pounds 11 change.

I only want you to use estimation, and identify which pupil went to which shop.

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.

Well, question three shows another damaged receipt.

John pays with a 50 pound note and receives around about three pounds in change back.

Estimate how much the board game was, and you must show your working out.

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

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

Well, for question one, you could have chose 150 multiplied by 18.

This is an absolute suitable choice because each number has been rounded to two significant figures.

You may have chose 150 multiplied by 20.

This is also a suitable choice because each number has been rounded to the nearest 10.

Perhaps you chose 100 multiply by 20, absolutely fine.

This is because each number has been rounded to one significant figure.

You may have also chose 18 multiplied by 145.

This is suitable because each number has been rounded to the nearest integer.

Great work, everybody.

You got any of those justifications right.

Let's have a look at question two.

Remember we had three receipts.

So let's have a look at shop A.

Looking at shop A, let's estimate.

If you sum all of these, I'm gonna simply round the nearest pound.

We have five pounds, at one pound, at two pound, at three pound, which is 11 pounds.

Given that each pupil paid with a 20 pound note, that means we expect around about nine pound change.

That means Sam went to shop A.

For shop B, we're going to round again to the nearest pound.

One pound, at eight pound, at three pound, at one pound, gives 13 pounds.

So I expect approximately seven pounds change.

So that means Jacob went to shop B.

For shop C, same again, I'm going to round summing these together, gives seven pound 80.

Given the fact that each pupil paid with a 20 pound note, I expect a round about 12 pound 20 change.

So that means Aisha went to shop C.

Really great work if you've got this one.

Now question three, we have to find out the price of that board game.

So I'm going to round appropriately.

I'm going to round to the nearest pound, giving me 30 pounds.

Given all the items sum to approximately 30 pounds, and Jun is given three pounds change, that means we expect the board game to be approximately 17 pounds.

Well done, if you got this one right.

Great work, everybody.

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

Now, sometimes we are given answers to calculations, and the answer just doesn't seem correct.

For example, three pupils give an estimate to a calculation.

Immediately, the Oak teacher knows who's correct.

I want you to have a look at these calculations and identify who is correct and how could it be seen so quickly.

See if you can give it a go, press Pause for more time.

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

Well, it was the middle calculation that was seen to be correct.

So let's have a look at each calculation.

We know it can't be 150, and this is because 150 is more than 121, and any number multiplied by a number less than one, will always give a smaller number.

So we knew 121 multiplied by four-seventh, cannot be approximately 150.

Now, for the middle calculation, four-seventh is slightly more than half.

So we expect the answer to be slightly more than half of 121.

Slightly more than half of 121 is around about 65.

Last one, four-seventh is slightly more than half.

So that means we expect the answer to be slightly more than 60.

So getting an answer of 45 just doesn't seem right.

Estimation is such a powerful tool, because it allows us to reflect on the calculation, and think, "Does this answer look right?" Let's have a look at a check.

Andeep has given this diagram to his friends.

Below are their responses.

Who do you agree with and why? So look at our triangle.

It's 1.

9 centimetres in length, 2.

89 centimetres of length, and 5.

7 centimetres in length.

Alex says, "Looks good, and I'll check with my ruler to see if it's right." Laura says, "Impossible.

I don't even need my ruler to know this is wrong." Who do you agree with and why? Well done, let's see how you got on.

Well, it's Laura.

And let's have a look at the explanation.

Well, if you summed the other two lengths, it must be longer than the third in order for them to connect.

So Laura's just simply estimated, and she said, "Two outta three is less than six," so she knows it's impossible.

Great work, and well done, Laura.

So we can use our common sense when looking at calculations and their answers, but we can also use estimation to check the answer as well as identify how the error has come about.

Let's have a look at another check.

Bookshop needs to buy the second edition of maths puzzle books, and each book costs four pounds 82.

The bookshop only has 371 pounds and two pence to spend on these puzzle books.

The three Oak pupils worked out an estimate number of books that the bookshop could buy.

So Izzy says, "80 books." Jun says, "2,000 books," and Alex says, "74 books." Who do you think is correct? And see if you can explain the mistake if you think there is one.

Let's see how you got on.

Well, Izzy and Alex are correct.

Izzy rounded 371.

02 to 400, and then she rounded four pounds 82 to five, thus making the calculation 400 divided by five, which is 80 books.

Jun has correctly rounded, but has incorrectly multiplied the numbers together, giving an answer of 2,000 books.

Alex? Well, Alex has rounded 371 pounds and two pence to two significant figures giving us 370 pounds, and divided four pound 82 rounded to one significant figure, which is five, thus giving us 370 divided by five, which is 74 books.

Really good work.

So reflecting on the answer, it is important to ask yourself, is the answer realistic? Looking how Jun calculated, 2,000 books is a lot of books for the amount of money that the bookshop has.

Now, it's time for your task.

For question one, I want you to use estimation only, and put a tick with the calculation which gives the approximate correct answer where the calculation is incorrect.

Identify the error.

See if 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 gives us a calculation, 7.

03 multiply by five over 11.

You have to identify, is it approximately six, 14, three, four, or it's not possible to estimate.

And I want you to ensure you show your reasoning or working out.

Well done, let's move on to question three.

Question three gives a floor plan of a one-bedroom apartment and the Oak teacher spots an error immediately with the measurement.

Can you find the error, and can you estimate what the correct measurement really should be? Then part B says, "Can you estimate the area of the missing room?" See if you can give it a go, and press Pause if you need more time.

Great work, let's see how you got on.

Well, for question one, our first two calculations were correct, buying 34 books each costing 20 pound 32, is approximately 700 pounds.

This is because you could have rounded 34 to 35, the nearest five, and you could have rounded 21 pounds 32 to 20, which is the nearest 10, thus giving us 700.

For the next calculation, a rectangle with lengths 3.

21 centimetres by 9.

56 centimetres has an area of 30 centimetres squared.

That's correct.

You may have rounded 3.

21 to the nearest integer, and you may have rounded 9.

56 to the nearest integer.

The skewing is 30.

So let's have a look at the other two calculations.

A square with length 8.

23 millimetres, has a perimeter of approximately 64 millimetres.

Well, let's round.

To work out the area, its length multiplied by length, so let's round that.

8.

23 to eight multiply by eight, which is 64, and the perimeter would be the sum of eight at eight at eight at eight, which is 32.

So the mistake in that calculation was, they worked at the numerical value for the area as opposed to the perimeter.

So the last part states, "There are approximately 45 50P pieces in 91 pounds." Now, you could have tackled this calculation a couple of different ways.

You may have thought, well, 45 multiply by 0.

5, gives me 22 pound 50.

So it's definitely not 91.

Or perhaps, you may have gone, "Well, I'm gonna round 90 pounds to 90, and then divide it by 0.

5 because we're trying to find how many 50 pence pieces are in 90." This is the same as 90 multiplied by two, which is 180.

So this answer is wrong for a couple of reasons.

45 50P pieces makes 22 pound 50.

Or if you wanted to find out how many 50 pence pieces make 91 pounds, well, it's 180 50 pence pieces.

Well done, if you got that one right.

So let's have a look at question two.

Well, question two, should give you the approximate answer of three.

Let's have a look at why.

Well, there are a few different ways to determine this answer.

For me, I'm looking at five-eleventh, and I think five-eleventh is slightly below a half.

So that means the question is asking you to work out a number which is slightly below a half of seven, and the number which is slightly below a half a seven is three.

Another example of rounding, perhaps, you may have chose seven multiplied by five-eleventh is equal to 35 over 11th, which is approximately the 33 over 11th, thus given us approximate answer of three.

Either one of those to justify the answer three, is absolutely fine.

Well done, if you got this one right.

Let's have a look at that third question.

Did you find the error in the measurement? Well, hopefully, you spotted it's going to be that 92.

4 metres.

It's far too big given the area stated.

So what should it have been? Well, it should have been nine metres, given the dimensions that are given on our floor plan.

Anything between eight and 12 metres inclusive, is absolutely fine.

So let's see if we can work out that missing area.

Well, if we use that nine metres, let's start rounding.

Nine multiplied by, I'm going to use seven metres to represent the 7.

03 metres, which is 63 metres squared in total.

So, all of the areas of the floor plan must add up to 63.

Subtracting everything, that means, I should have approximately 17 metres squared for that missing area there.

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

And remember, we have a range of solutions.

So anything between 50 and 20 metres squared inclusive, is absolutely fine.

Great work, everybody.

So let's summarise.

In everyday life, it's important to estimate, so to know how to quickly calculate a rough answer or to check the accuracy of answers.

Estimation is important because sometimes, we're given answers to calculations, and the answer just doesn't seem correct.

So when reflecting on an answer, it is important to ask yourself, is the answer realistic? And if not, it helps to identify where the error has occurred.

Great work, everybody, and well done.