Lesson video

Hi, everyone.

My name is Ms. 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 parts, 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.

In today's lesson under the unit Estimation and Rounding, we'll be estimating numerical calculations.

And by the end of the lesson, you'll be able to estimate numerical calculations.

And today's lesson will be broken into two parts.

The first we'll be using estimating using significant figures, and the second we'll be using estimation in context.

So let's make a start.

Now, just remember those keywords: significant figures.

Now, 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 are so helpful as they can be used to round to one significant figure.

For example, if you are rounding 6,523 to one significant figure, that's 7,000.

Perhaps you wanted to round 6,523 to two significant figures.

Well, that would be 6,500.

Or maybe you wanted to round 6,523 to three significant figures, where the answer would be 6,520.

So significant figures are really important when looking at estimation.

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

When we show estimations, we use these two squiggly lines, and this represents when calculations on numbers are approximately the same but not equal.

For example, if we were looking at 892 multiplied by 176.

9, what I'm gonna do is I'm going to round 892 to one significant figure, giving us 900.

And then I'm going to round 176.

9 to the nearest hundred, giving us 200.

Thus my approximate calculation would be 900 multiplied by 200.

So now we know a quick estimation for a calculation is obtained using approximate values, often rounded to significant figures.

When we estimate, it's not really required to be exact, but why do you think we estimate? This is because it makes it easier and quicker to do calculations where the answer is very close but not accurate to the answer.

So let's have a look at this example that I showed you before.

Remember, we're using this approximate symbol to identify that the calculations are approximately the same but not equal.

We looked at 892 multiplied by 176.

9, and we know it's approximately 900 multiplied by 200.

It could also be approximately the same as 890 multiplied by 180.

It could also be approximately the same as 890 multiplied by 200.

All of these calculations will give an approximate answer to the calculation of 192 multiplied by 176.

9.

So let's have a look at an example.

Here, a typical car in the UK measures around about 4.

399 metres long.

Assuming that the cars are bumper to bumper, what would the calculation be to estimate the length of this traffic jam on the screen? See if you can give it a go.

Let's see how you've done.

Well, you would be correct if you looked at 4.

399 metres and thought, I'm going to round it to one significant figure, thus making each car approximately 4 metres.

And then, given the fact that we have 5 cars here, 4 multiplied by the 5 is 20 metres.

So you've approximated the length of the traffic jam of 4.

399 multiplied by 5 to be approximately 4 times 5.

You may also be correct if you looked at the 4.

399 metres and rounded to two significant figures, making 4.

4, and then multiplying it by the number of cars, thus giving you 4.

4 multiplied by 5, which is 22 metres.

That would be fine as well as both of those calculations are approximates of 4.

399 multiplied by 5.

So rounding to one significant figure gave an estimated length of 20 metres, and rounding to two significant figures gave an estimated length of 22 metres.

Both estimations are absolutely fine, but what do you think the benefit of rounding to one significant figure is or two significant figures? Well, rounding to one significant figure was quicker, but rounding to two significant figures was more accurate but perhaps took a little longer.

Well done if you got this.

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

Both Aisha and Alex are given this calculation to estimate the answer.

253 divided by 48.

Aisha says, "I would use 300 divided by 50," and Alex says, "I would use 250 divided by 50." Whose calculation would you use and why? 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, Aisha and Alex are both correct, so you're really free to use either calculation.

But I'm going to ask another question now.

Whose calculation would be more accurate? And I want you to think of the reason why as well.

Well done.

Well, hopefully you spotted Alex is more accurate as he's rounded to more significant figures.

So let's have a look at what Aisha did.

Well, Aisha had 253 and she rounded it to one significant figure, and 48 is also rounded to one significant figure, thus making the very simple calculation of 300 divided by 50, which is 6.

Alex, on the other hand, rounded 253 to two significant figures, and 48 was rounded to one significant figure, thus he made the simple calculation of 250 divided by 50, which is 5.

So Alex will be more accurate as he rounded to more significant figures.

Let's have a look at another check question.

Izzy, Laura, and Alex are given this calculation.

Who has correctly estimated the answer to the calculation? Have a little look, pay attention to that working out and the answer, and have a think who is correct.

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

Well done.

So let's see how you got on.

Well, the rounded numbers by all three students are correct.

So 32.

9 multiplied by 8.

56 divided by 0.

498.

The numbers that each student rounded to was absolutely fine, there were no problems there.

But where on earth was the mistake? Well, the mistake came from the division of the 0.

5.

Izzy and Laura made the same error when calculating the answer.

Just remember, when dividing by a number, it's the same as multiplying by the reciprocal.

So when we're dividing by 1/2 it's the same as multiplying by 2.

So that means Alex is correct.

A huge well done if you got this one right.

It's really important to remember that the priority of operation still applies, even when we estimate.

This lovely image helps us remember those priority of operations.

The very top of this pyramid are the brackets, then we move on to exponents and roots, then we move on to multiplication and division, and then, finally, addition and/or subtraction.

So let's apply this with estimation.

So we're going to have a look at the question 39.

7 multiplied by 9.

7 and divided by 0.

779 subtract 0.

601.

So let's see if we can round so we can come up with an approximate answer.

Well, first of all, let's suitably round.

I'm going to change my 39.

7 to 40, 9.

7 into 10, 0.

779 into 0.

8, and 0.

601 into 0.

6.

These numbers are much easier to use in our calculation.

Then, remember those implicit brackets.

Because of that long dividing line, we have implicit brackets.

So that means we need to work out 40 multiplied by 10, and we need to work out 0.

8 subtracted 0.

6, thus giving us 400 over 0.

2.

Just remember those implicit brackets because of that long dividing line.

Now we're gonna simply divide.

400 divided by 0.

2.

Or remember, division is the multiplication by the reciprocal.

So if you think of 0.

2 as a fraction, that's 1/5.

So that means dividing by 1/5 is the same as multiplying by 5, therefore giving us a final answer of 2,000.

A huge well done if you spotted that.

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

Here, I want you to estimate the answer to this.

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, we have some awkward numbers here.

887 subtract 67.

8, multiply by the square root of 4.

501.

So let's round, making nicer numbers.

I'm going to choose 900, 70, and 4.

So let's have a look at why I've chosen those numbers.

Well, I rounded 887 to one significant figure, I rounded 67.

8 to one significant figure, and I looked at that square root of 4.

501 and thought, well, I need to round to the nearest square number because square root of, it's gonna be so much easier.

So that's why I chose 900, 70, and 4.

Now let's apply those priority of operations.

We've got to apply our roots first, so that means it'll be 900 subtract our 70 multiply by our 2.

Then, we're going to be using multiplication next.

So working out that 70 multiply by 2 gives us 140.

So our next approximate calculation would be 900 subtract 140.

Then, I can work out my answer to be simply 760.

So the approximate answer to 887 subtract 67.

8 multiplied by the square root of 4.

501 is approximately 760.

Really well done if you got this one right.

Great work, everybody.

So let's have a look at your task.

Here, I want you to fill in the gaps to make the approximations correct.

You can only use each number once.

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

Great work.

Let's move on to question two.

Question two gives us this calculation, 19.

173 multiplied by 338.

And A says, which of these calculations would you use and explain your choice.

Would you use 20 multiplied by 350, 20 multiplied by 300, 19 multiplied by 350, or 19 multiplied by 300? B wants you to identify which of those calculations would give the closest estimate.

And C asks, which gives the estimate that is furthest away? See if you can give it a go, and press pause for more time.

Well done.

Let's move on to question three.

Now, question three consists of four calculations where you have to estimate your answer, ensuring you show your working out.

Remember those priority of operations? See if you can give it a go, and press pause for more time.

Great work.

So let's move on to the next part of this question.

It wants you to continue your work with estimation and the priority of operations.

See if you can give it a go.

Press pause for more time.

Well done.

Let's go through these answers.

Well, for question one, remember you could only use each number once.

So that means 5 should go here, 4 should go here, 900 should be here, 200 is here, 4 is here, and 50 is there.

Great work if you got this one right.

For question two, which of these calculations would you use to explain your choice? Well, there's no right answer to this because as long as there's a justification to the calculation chosen, you are absolutely fine.

You could have said, "I'm going to use 20 multiplied by 350," and this is acceptable as 19.

173 has been rounded to one significant figure and 338 has been rounded to the nearest 50.

So that would be fine.

20 multiplied by 300 is absolutely fine as a estimate calculation too because we rounded to one significant figure and we rounded 338 to one significant figure.

So that would be fine as well.

You may have also used 19 multiplied by 350.

Same again, it's absolutely fine because you may have rounded 19.

173 to two significant figures, and 338, it's been rounded to the nearest 50.

Or you may have chosen 19 multiplied by 300 is fine.

This is because you rounded 19.

173 to two significant figures and 338 has been rounded to one significant figure.

So any of those calculations would've been absolutely fine as long as you explained your choice.

Well done if you got this one.

Now, B says, which one gives you the closest estimate? Now, it'll be 19 multiplied by 350.

This is because 19.

173 is closer to 19 than 20 and 338 is closer to 350 than the 300.

Very well done if you got that one.

And part C, which estimate gives the one which is furthest away? Well, it would be the 20 multiplied by 300.

This is because 20 is the furthest away from 19.

173 and the 300 is the furthest away from 338 rather than using 350.

Really well done if you got this.

Now let's have a look at question three.

All I'm going to do now is show you some example estimates that I've used and the working out.

Yours might be a little bit different, as long as you justified your rounding to one or two significant figures or another degree of accuracy.

For A, I've used 6,000 multiplied by 30 divided by 900.

Then I've got this calculation to give me a final approximate answer of 200.

B, I've used 60 and 80.

Then I've got -20 all squared, thus giving me a approximate answer to be 400.

Well done.

For C, I've rounded, giving 20 subtract 10 over 0.

2.

And remember those implicit brackets, so that means I have to do the 20 subtract 10, which is 10, divided by the 0.

2.

Remember, the division of a number is the same as the multiplication of the reciprocal.

So therefore dividing by 1/5 is the same as multiplying by 5, giving me an answer of 50.

For D, I've chosen these rounded numbers.

Remember those implicit brackets, thus giving me this.

And then from here, I'm going to simplify, just to give me an approximate answer to the calculation to be 1/4.

For E, I've chosen these rounded numbers.

Recognise those implicit brackets.

Now remember, we're dividing by 1/2, which is the same as multiplying by 2, thus giving me an approximate answer of 12.

For F, big numbers here, but let's round.

I've chosen 30 squared subtract the root of 144, giving me 900 subtract 12, giving me an approximate answer to the calculation to be 888.

Amazing work, everybody.

Great work, everybody.

So let's have a look at estimation in context.

Now, having a general estimate of everyday things can really allow us to estimate other calculations.

For example, did you know a standard full size guitar is always considered to be generally about one metre in length? And using this, do you think you can estimate the height of our little cat? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

And more importantly, how did you decide on the estimate? Now, there's lots of different ways.

So for example, you could have looked at the guitar and said, "Well, perhaps because this is about one metre in height, if I find the middle of this guitar, that means that would be 50 centimetres.

The cat is below the 50 centimetres, so I can come up with an estimate." Or maybe you thought of the estimate in a different way.

Maybe you imagined how many cats it would take to make one metre.

Either way, you probably would've got something around about 30 to 35 centimetres.

Well done.

But it's more the strategy that you use to estimate which is more important.

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

A science textbook uses comparisons with humans to show the sizes of different animals.

Here we have Alex, and Alex is of average height for a pupil in secondary school.

Which of these do you think is a good estimate for Alex's height? And then I want you to use the answer to part A to estimate the height of the animals below.

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, the answer to part A is 1.

5 metres.

That's the average height of a secondary school pupil.

Now using that 1.

5, do you think you've got an idea of how tall the giraffe is? Well, the height of the giraffe is approximately three metres high, and this is because if you look at the image, the giraffe is approximately twice the height of Alex, and we know Alex is 1.

5 metres.

Now let's have a look at our little duck.

What do you think the height is there? Well, if Alex is 1.

5 metres, that means the duck is around about 1/5 of Alex's height.

So I'd expect the duck to be around about 30 centimetres.

Really well done if you've got something similar to these answers.

Remember, they don't have to be exactly the same but not too far out either.

Really well done.

So alongside the general knowledge of approximate sizes of everyday things, it's important to know the most appropriate units of measure.

Can you name as many metric units of measure for length, mass, and capacity as you can? See if you can give it a go.

Press pause if you need more time.

So let's recap.

Length can be measured in millimetres, centimetres, metres, kilometres, for example.

We've got mass can be measured in milligrammes, grammes, and kilogrammes, and capacity can be measured in millilitres, centilitres, and litres, so on and so forth.

So now we know the units of measure for length, mass, and capacity.

Let's see how we can use it in an estimation question.

I want you to look at this question and identify what would be a suitable unit of capacity to measure each of the items in the table.

And question B says, each of the values below matches one of the six items above.

Which is which? 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, a good measure for a drink bottle would be millilitres.

A good measure of a pond would be litre.

A good measure of a swimming pool would also be litres.

A petrol tank, a good measure would be litres.

Cup of tea, millilitres, and a dose of medicine, millilitres.

Well done if you got this.

Now let's associate those quantities with those correct units of measure.

Well, we have a huge number, 375,000, so that would be good with our swimming pool.

We have 5, which would be good for a dose of medicine.

55 litres would be a good approximate estimate for a petrol tank.

We have 500 millilitres for a water bottle, and we have 1,500 litres for a pond.

And lastly, 300 millilitres for a cup of tea.

Well done if you got this.

Now it's time for your task.

See if you can do these questions.

Part A will help you with part B and C.

Press pause if you need more time.

Well done.

Let's move on to question two.

Question two says, what would be a sensible unit of mass to measure each of the items in the table? And B says, each of the values below matches one of the six items. Which is which? And then part C says, an orange weighs approximately 120 grammes.

How many oranges do you think would be equivalent to one of each of the things in part A? This is a great question, see if you can give it a go, and press pause if you need more time.

Well done.

So let's move on to question three.

Did you know horses are measured using hands as a unit, as shown in the diagram? And what we're asked to do is estimate the height in centimetres of each of the horses.

Now, B states that the standard metric equivalent for a hand is actually 10.

2 centimetres.

So does this mean your estimate will be too big or too small or just right from part A? And C states, how many hands high are you to the top of your head? And D states, estimate some other objects using your hands as a unit.

See if you can give this a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for question 1A, a good estimate for the width of a hand is 10 centimetres, and remember that for question three as well.

Then question B says, using that answer, it has to estimate the length of, in this case, a rodent.

So given the fact that the width of a hand is around about 10 centimetres, a rodent is approximately 1.

5 times the width of the hand.

So that means I'm expecting it to be approximately 15 centimetres.

Let's have a look at that spider and see.

Well, remember the width of a hand is 10 centimetres and the spider is approximately about 1/3 of the width of a hand.

So that means we're expecting around about three centimetres.

Don't worry if you didn't get the exact values as me.

As long as they're approximately the same, that's fine.

For question two, what would be a sensible unit of mass to measure each of the items in the table? Well, hopefully you spotted we have these units of measure.

Now we're going to associate those values to those units, and this is what you should have got.

Massive well done if you got this one right.

Interestingly, part C states that an orange weighs 120 grammes, and we're asked to identify how many oranges do you think would be equivalent to one of each of the things in part A.

Well, so if we were looking at a baby, it'd be around about 25 oranges.

For an elephant, it'll be huge, 32,000 oranges would make an elephant.

If you were talking about a full suitcase, it'll be 160 oranges.

A cargo ship container, that is 24,000 oranges.

A newborn puppy is approximately one orange.

And a packet of crisps, that's about 1/4 of an orange.

So these are approximately the number of oranges that we'd expect in each item in part A.

So if you've got something slightly different but not too far, don't worry, you're absolutely fine.

Now let's have a look at question three.

Now for question 3A, we had to estimate the height in centimetres of each of the horses.

Remember the width of a hand that we looked at in question one.

Well, question one says the width of a hand was 10 centimetres.

Now, for part B, it states that the standard metric equivalent for a hand is actually 10.

2 centimetres.

So does this mean our estimate in part A was too big, too small, or just right? Now, given the fact that we used an estimate of a hand to be less than 10.

2, then that means it will be too small because we used 10 centimetres.

But perhaps you used a width of a hand to be greater than 10.

2, that means your estimate would be too big.

And for part C and D, they're really nice little activities.

Why don't you check with a ruler to find out how far your estimation is from the accurate answer? Great work.

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

When showing estimations, we use the approximation sign, these two wiggly lines, and they identify that the calculations are approximately the same but not equal.

It's also important to remember that in today's lesson we looked at rounding to one significant figure, and that is general practise and quicker, but please do remember that rounding to two significant figures or more is more accurate but perhaps takes a little longer.

Also, remember having a general estimate of everyday things, whether it be the height of an average secondary school pupil or the width of your hand or the height of a guitar, can allow us to estimate other calculations.

A massive well done.

It was great learning with you.