Lesson video

Hello, my name's Mr. Southall, and in this maths lesson, I'll be teaching you about estimating and rounding in context.

Before we start, please make sure that you're in a quiet space away from any distractions and any devices are placed on silent mode.

You'll need a pencil and paper for this lesson for some activities, so if you don't have those available, then please press pause now and gather those resources.

Otherwise, let's begin.

So in this lesson, we're going to look at estimating and rounding contexts, and we'll structure this in the following ways.

We'll start by looking at real life contexts, then we'll move on to appropriate degrees of accuracy, then we'll move on to making estimates, and finally, we'll have an independent task.

As I mentioned before, you'll need a pencil and a ruler and some paper or a book to work from.

Please press pause if you need to collect those resources.

Real life contexts.

With estimating, we need to think about the context of what we're estimating and think about how to make the right choices about units and size.

So here we have four different examples of contextual estimation.

We have a jug of water, we have a jar of sweets, we have a box of cherries, and we have a scale or a number line.

We need to think about the magnitude of what we're going to be estimating.

For example, do we need to think about how many hundreds of cherries there are in this context, or tens of cherries? Do we need to think about how many litres of water we're talking about or millilitres? These are all questions we need to decide upon before we make our estimate.

Appropriate degrees of accuracy.

In this example, we have Dr.

Livingstone's expedition to Victoria Falls, and it's listed here as four different points and four different distances in hundreds of miles.

So an appropriate degree of accuracy for this might be to the nearest hundred miles.

An inappropriate degree of accuracy might be to the nearest metre, because that's much too small for the units that are being used.

Similarly for the box of cherries, I don't think it would be appropriate to be thinking about this in thousands of cherries or hundreds of cherries, because the box contains less than those quantities, or it seems to.

So we need to make a decision about how accurate our estimation is going to be and what sort of units we should use each time.

The population of Paris is 2,206,488.

And this girl says, "The population of Paris is 2,206,488.

We need to round it so that we can compare it easily to the population of other capital cities." So that final sentence is giving us context about why we want to estimate.

She says we need to round it so we can compare it to the population of other capital cities.

So we need to be thinking about other capital cities and the likely population sizes of those and think, well, what would be useful as a number to compare them? The girl here is saying that this number is too accurate.

There's too many digits to think about.

So we need to round it and make it easier.

Then the boy says, "This is a seven-digit number, and it's only going to be used for a rough comparison.

I think that we could round it to the nearest multiple of a million.

So that would be two million." In other words, he's suggesting that to make this an easy comparison, we should just talk about the millions, so in this case two million, and we can ignore the hundreds of thousands, the tens of thousands, the thousands, the hundreds, the tens, and the ones.

Her response is that, "There are probably quite a lot of cities with populations around 2 million, so our estimate may not be accurate enough.

200,000 people is a lot of people to forget about.

What about rounding it to the nearest multiple of 100,000?" So what she's saying here is if we stuck with the boy's idea and just said it's about two million, we're likely to have problems when we're comparing with other countries with similar population sizes, because many of them might be rounded to 2 million and therefore we can't see which ones are bigger or smaller than each other.

So her suggestion is we just add a bit more accuracy and we include the hundreds of thousands in our estimate.

So 2,200,000.

"So a good estimate for the population of Paris is 2,200,000" It seems like they're agreeing now.

What would be a reasonable estimate and why? Here we have two different examples to consider.

The first one says children in need have a target of 50 million pounds.

Raised so far is 7,800,000, sorry, 7,598,912 pounds.

Report how much is raised and how much is still needed.

Now, contextually here, it's important to think about what is the most useful part of that number, okay? Do people need to know about the 912 pounds, for example? Or the two pounds at the end? We need to make a decision about the minimum amount of information we can present that makes it still useful.

So in this example, I would suggest that we change this number and round it to either 8 Million and leave it at that, or potentially 7.

6 million.

Okay? And in that second example, this one, I've just rounded it to the nearest hundred thousand pounds.

Now, both of those are useful.

I think if we go much beyond that, then we're just giving maybe too much information compared to what's really needed to communicate the idea that we still have some money to raise.

Now, our target is 50 million.

Now, to get that 50 million from 8 million is very easy or much easier than this number.

So that would be a target of an additional 42 million pounds.

So if I'm the newsreader, I can say we've raised nearly 8 million pounds and we need to raise another 42 million pounds to hit our target.

Similarly, if I use the slightly more accurate one of 7.

6 million pounds, then I would need to report that I need to raise a further 42.

4 million pounds.

Okay, so that calculation is a little bit more difficult, but again, the figure works quite nicely.

If you want an absolute answer, preferably I would say the 8 million is going to be easier for calculations.

And I don't think you lose too much accuracy.

In the second example, we need to buy wood to make a shelf for 160.

7 centimetres of length.

The key thing here is that we don't buy too little and we're aware that you can't buy wood to the decimal place.

So in all likelihood you're going to order something like 165 centimetres of wood, or potentially 170 centimetres of word.

It depends what the shop is selling.

Either of those would be a good answer.

Two more examples to look at.

We have an attendance at an England game listed as 88,795 and the newspaper reports this as 89,000.

And we have a length of walk that's 5.

2 kilometres and Dev says, "I think it's about 10 kilometres." So have a think, pause the video, and decide whether you think these estimates are appropriate in this context.

And press play when you're ready to resume.

Welcome back.

If we take that first example, this is the original figure, this is the estimate.

We've rounded to the nearest thousand.

I think that's appropriate, okay? It clearly indicates a close number to the quantity of people that was actually there without having to get too precise and bogged down in the sort of minutia detail.

In the second example, the length of the walk is 5.

2 kilometres, but this is almost double the length, okay? So if I was expecting to go on a 10 kilometre walk and it was actually 5.

2, I've only walked just about half the distance.

So proportionally, I think this is too high.

I would suggest a better and more accurate estimate would be five kilometres.

It's a little bit less, but it's much closer to the actual number.

Making estimates.

Here, we're revisiting Dr.

Livingstone's expedition to Victoria Falls.

And you can see it's in four segments, one, two, three, four.

And each segment is listed as hundreds, tens, and ones as the miles.

So we have 527 miles, we have 586 miles, we have 427 miles, and we have 636 miles.

And we want to know how far Dr.

Livingstone has travelled altogether.

And we need to make an estimate.

And we have three choices here.

So what I'd like you to do is pause the video, decide which of those choices is the most accurate and why.

And press play when you're ready to resume.

All right, let's look at these then.

If we take this first one, that's close to 500 miles.

And this is close to 600 miles and this is close to 400 miles and this is close to 600 miles, okay? And if we add those together, we get 2,100 miles.

Okay? So a good estimate is about this number.

And if we look at our choices here, from those provided, this one is much too low, this one's very high, but this one is very close.

Okay? So a good estimate of the total length of this journey is about 2,150 miles.

What about this number line? Imagine this is a scale on a map or on a measuring cylinder or something like that.

What number do you think is represented here? Press pause, have a think, justify your decision, and press play to resume.

Okay, so if we think about this logically, the halfway point is about there, so that would be about 50.

And then the halfway point between here and here is probably about there-ish.

So this number is going to be close to 75.

Okay? And in fact, it's 80.

So that's the exact value.

But remember, we were just estimating.

So if you got close to 75, then you've done a really good educated guess.

Well done.

Estimate the total of these three shopping receipts.

So we have three slightly awkward numbers, 74 pound 68, 65 pound 90, and 59 pounds and five pence.

So if I was going to estimate the total, that means I want to round these numbers to something more manageable, but without losing too much accuracy.

So I would change this to 75, this to 65, and this to 60.

Okay? That way I'm not too far off any of these numbers, but now I've got easier numbers to calculate with, okay? And I can do 75 added to 65 added to 60, which gives me a total of 200 pounds.

So an estimated total is 200 pounds.

Remember, it's not the exact total.

That's not what I'm trying to do.

I'm just trying to get a good educated guess.

Graeme wants to buy a shelving unit and needs to know if it would fit his lounge.

He only had a ruler instead of a tape measure so he measured the width of one square section.

His measurement was 29.

3 centimetres.

Sensibly estimate the height and the width of the entire unit.

Okay.

So we have two key pieces of information here.

Each section is square and his measures were 29.

3.

So if I want to estimate this, have a think, what do you think is going to be a suitable estimate for this number? And then how are you going to calculate with that an estimated guess for that length and that length? Pause the video, have a think, do some calculations, and press play when you're ready to resume.

Okay, I think a suitable estimate here would be 30.

And I think if you remember that this is sectioned into squares, then we have one, two, three, four sections there, so that's going to be four times 30.

And we have one, two, three, four sections there, so that's also going to be four times 30.

How do I know that they're the same size? Because it's square.

So this is going to give me 120 and this is going to give me 120.

My units are centimetres, so the length of the unit, sorry, the height of the unit is 120 and the width of the unit is a 120 as an estimate.

Romesh, Joe, and Mel all raised money for Comic Relief.

Romesh raised 450,280 pounds, Joe raised 733,012 pounds, and Mel raised 191,102 pounds.

What would be a sensible estimate of the total? Have a read, pause the video, do some calculations.

Remember, this is an estimate so you need to round your numbers first.

And then press pause and play when you are ready to resume.

Welcome back.

So if we look at the leading numbers for these quantities, we have four, seven, and one.

So if we add those together, remember this is 400,000, 700,000, and 100,000, then we're going to get 1.

2 million pounds or 1,200,000 pounds.

Okay? Now that's before we even look at the rest of the numbers.

And something stands out to me: this nine, okay? So that's really close to another 100,000 pounds.

So that would take us to 1,300,000 pounds.

Okay? And all I'm looking at at the moment is the four, the seven, the one here, and this nine here.

Okay? So already, it looks to me as though this is the closest estimate, okay? This is looking too low and this is looking a bit high.

Now, you may have done this in a number of different ways.

You may have rounded each one of these numbers to the nearest 10,000 pounds.

But I think whatever method you use, you're going to get close to this answer here, okay? And that is the most accurate estimate in this example.

Independent learning.

Here is a challenge question.

Bethany and Ahmed each start with the same positive whole number, or integer.

Bethany then rounds her number to the nearest ten and adds it to Ahmed's number to get 58 in total.

What is Ahmed's number? Pause, have a think, do some calculations, and press play to resume once you are finished.

Welcome back.

Okay, the way I approached this problem was to think about these three statements, okay? This represents the final calculation where Bethany's rounded number is added to Ahmed's unrounded number, okay? And you should notice that the total ended in an eight.

Now, if Bethany rounded her number to the nearest ten, then it must end in a zero.

So this is Bethany's number, something zero, and this is Ahmed's number, something eight.

Okay? Now, we know that they started with the same number.

So they started with something eight and something eight.

But when Bethany rounded hers, she added two.

In effect, she added two on.

And therefore the total must've been two greater than it is in the second line.

So that means we have something eight plus something eight equals 56.

And we know that those two numbers are the same, so we could just do 56 divided by two and we get 28 plus 28 is 56.

So Ahmed's number is 28.

Bethany started with 28, but she rounded it to 30 before she did her addition.

So that brings us to the end of today's lesson.

Really well done on all the learning that you've done today.

It's fantastic.

And just before we finish, have a think about what the most important learning was for you in that lesson, and maybe take some notes on that too.

Anyway, thank you so much for participating today.

Enjoy the rest of your day of learning and I'll see you again soon.