Lesson video

Hi there.

My name's Ms. Lambell.

You've made such a super fantastic choice deciding to join me today and do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Checking and Securing Understanding of Estimating Using Rounding".

And this is within the unit rounding, estimation and bounce.

By the end of today's lesson, you'll be able to estimate the value of an expression through rounding.

Keywords that we'll be referring to in today's lesson are estimate, significant figures, overestimate and underestimate.

All of these should be familiar to you, but it's always worth a quick recap, isn't it? A quick estimate for a calculation is obtained from using approximate values, often rounded to one significant figure.

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

The first significant figure is the first non-zero digit.

An overestimate is an estimate for a calculation which is greater than the exact answer.

And on the flip side obviously, an underestimate is an estimate for a calculation which is less than the exact answer.

Today's lesson is split into two separate learning cycles.

In the first one, we will just concentrate on estimating the value of different calculations.

And in the second one we'll look at whether our answers were overestimates or underestimates.

Let's get going with that first one then.

And that's estimating a calculation.

OK Academy year 11, pupils are planning their prom.

It starts at 7:30 PM and ends at 11:00 PM, the head teacher wants to know an approximate cost to hold the event.

Here is a list of the costs and the number of pupils and teachers that are attending the event.

We've got the venue is 380 pounds.

The DJ costs them 72 pounds per hour.

A photo booth costs 158 pounds.

The food is 18 pounds per person and there are 182 pupils going and 17 teachers.

How many significant figures should we use when estimating calculations? And if you were listening really hard when I went through those key words, you will have said one significant figure.

I'm gonna round numbers to one significant figure.

Okay, let's do that.

First we're gonna round, like I said, each number to one significant figure.

What is 380 pounds rounded to one significant figure? Yeah, that's right, it's 400 pounds.

So I'm going to replace my 380 with 400.

That's the estimated figure we are going to use for the cost of the venue.

What is 72 pounds rounded to one significant figure? Yeah, that's right, it's 70 pounds.

So let's replace that in our costs.

What's 158 pounds rounded to one significant figure? That's 200 pounds.

So let's change that in our list of costs.

Now, 18 pounds, the cost of the food per person to one significant figure and that's 20 pounds.

Now let's round the number of pupils.

So what's 182 rounded to one significant figure? That's 200.

And then finally the number of teachers which is 17.

Let's round that to one significant figure and that's 20.

Next, we need to work out the length of time as the DJ's cost is per hour.

How many hours does the prom last? I've given you the information that it starts at 7:30 PM and ends at 11:00 PM.

How many hours does the prom last? Lasts for 3.

5 hours or you may have said three and a half.

What is 3.

5 hours rounded to one significant figure? That's right, the 0.

5 means we're going to round up to four hours.

Here we've got the length of the prom now we know is four hours or that's our estimated value we are going to use for the calculation for the estimate.

Now we can calculate an estimate for the total cost of the DJ.

The total cost of the DJ is the number of hours multiplied by the cost per hour.

What is the approximate cost of hiring the DJ for the evening? It's 280 pounds.

We're gonna hire the DJ for our estimated number of hours, which is four, and our estimated cost is 70 pounds per hour giving us 280 pounds.

So I've now changed my value for DJ to 280.

Notice here that we do not round this value.

Now we'll calculate the cost of the food.

Total cost of the food will be found by multiplying the number of people by the cost per person.

What is the approximate cost of the food for the event? And that's 220 multiplied by 20 pounds, which is 4,400 pounds.

Where did that 220 come from.

Yeah that's right, it was the number of pupils plus the number of teachers, the total number of people that are going to the prom.

Okay, but remember that's our estimate.

We now know the food costs 4,400 pounds.

Now we can calculate the total cost of the event.

So that's the venue plus the DJ, plus the photo booth plus the food.

What is the approximate cost of the prom? And hopefully you got 5,280.

We take the cost of the venue, which we've approximated is 400 pounds.

The approximate cost of the DJ is 280 pounds.

The approximate cost of the photo booth is 200 pounds and the approximate cost of the food is 4,400 pounds, meaning the approximate cost of the prom is 5,280 pounds.

This is the figure they would need to share with the head teacher.

He didn't wanna know an exact amount, he just wanted to know an estimated value so he could get an idea of how much the prom was going to cost.

Now what I want you to think about is would it have made a difference if we had found the total number of people, rounded that to one significant figure and then found the total cost of the food? What do you think? What is the total number of people? Total number of people is 199, 182 pupils plus the 17 teachers, that's 199.

What is 199 rounded to one significant figure And that's 200.

So yes it would, and our estimate would've been closer to the exact amount as we have used a reduced number of roundings.

The number of we've done is smaller, giving us an answer which will be closer to the exact value.

I'd like you now to choose the correct calculation to estimate the cost of hiring a plumber for 3.

25 hours.

Whether total cost is the cost per hour, which is £54.

75 multiplied by the number of hours plus an additional 38 pounds.

Remember here we are calculating just an estimate.

So sometimes people use that term ballpark.

We wanna know a ballpark figure of how much it's going to cost us.

Pause the video, make your decision, which of those is the correct estimate and then when you've got your answer, come back.

What did you decide? Hopefully you decided on C.

Cost per hour £54.

75 is 50 pounds to one significant figure.

The number of hours 3.

25, we would round to one significant figure is three hours and 38 pounds rounded to one significant figure is 40 pounds giving us C.

Lucas and Sofia are working on this question, estimate the value of 1.

68 multiplied by 23.

4 and we can see here that they've been given how many marks it's worth.

So they're thinking about this in terms of an exam question.

It's worth three marks, here's Lucas's workings.

Lucas has done 168 multiplied by 234.

He must have done his work on another bit of paper, but he is written down that answer which is 39,312.

So 1.

68 multiplied by 23.

4 is equal to 39.

312.

So my estimated answer is 40.

That's Lucas's workings.

If you need to pause the video and just have a look through them at a bit of a slower pace, obviously you can.

Let's take a look at what Sofia has done.

Sofia has written 1.

68 as two to one significant figure, 23.

4 as 20 to one significant figure and then she's multiplied her two estimated values, her two approximate values and given an answer of 40, they both get an answer of 40, but one of them unfortunately gets zero marks while the other gets three marks.

I'd like you to pause the video and decide which pupil gets which marks.

But remember the most important thing is I'd like you to be able to tell me why.

Pause the video and then we'll check whether you've got the correct understanding when you get back.

Superb work.

Well done, let's see whether you agree with me.

Unfortunately it's Lucas who is going to get zero marks.

Oh dear, poor Lucas.

Let's take a look at why Lucas gets zero marks.

And the reason Lucas gets zero marks is because he has worked out an exact answer and then rounded that.

If a question is asking us to estimate, you must estimate right from the beginning.

We cannot work out the exact answer once we've worked out the exact answer.

Surely we would just use this.

So Sofia, she's the one who gets three marks.

Sofia gets all three marks as she correctly rounded the initial values to one significant figure and then multiplied those together.

Really important when you are answering an estimate question that you round all of the values in the question to one significant figure before you carry out any calculations.

Now here's a quick check for understanding.

The entry cost to a museum is £8.

25.

Estimate the cost of entry when a school trip takes four coaches each with 52 seats.

And this again is a three mark question.

I'd like you to decide which the following gets three marks and also I'd like you to have a think about how many marks do you think the other response gets.

Pause the video, take a really good look at each of the methods, see where they're the same, where they're different, and can you decide why one gets three marks, which one, and also how many marks do you think the other is worth? Pause the video and then when you are ready we'll come back and we'll check.

Superb, well done.

My estimated answer is 1,600.

That is correct and that gets three marks.

That is a perfect answer.

Notice how it is set out.

We've clearly written all of the values that are given in the question, £8.

25, four coaches and 52 seats.

Then we've clearly rounded each of those to one significant figure and then we've shown the calculation for the total number of people and we've shown the calculation for the entry cost.

My estimated answer for the second one is 2000, which unfortunately is wrong.

Here, this gets two marks.

Their only mistake is that they've rounded the final answer again, once you get your final answer in an estimation question, it is really important then not to round that answer again.

So here, because they have decided round 1,600 to 2000, which isn't wrong, that's a correct rounding to one significant figure, but we don't round our final answer.

We'll now take a look at this calculation, first of all, we're going to round all of the numbers to one significant figure.

We're gonna write each number to one significant figure.

I say we, you are, and then you're gonna check, here are three numbers.

I'd like you to pause the video, write each of them to one significant figure and then come back when you've got those answers.

Super, you should have 300, 20 and 0.

5.

You did, brilliant.

Now we substitute these rounded numbers into our calculation.

So we end up with 300 multiplied by 20 over 0.

5.

We're now gonna calculate the numerator.

300 multiplied by 20 is 6,000.

I don't like dividing by decimals.

So I am going to multiply my numerator and my denominator by 10 over 10, which remember is the same as multiplying by one.

So I've not changed its value, to give me 60,000 over five.

Now I'm going to do 60 divided by five, which is 12.

So my final answer is 12,000.

Remember here, you may have answered this question in a slightly different way.

What's most important is this would be a non calculator question.

So you must make sure you've not used your calculator, but there are many alternative ways to get to the answer of 12,000.

Now you're ready for your task A, you're gonna estimate the value of each of those following calculations.

Remember when you are estimating, you should not need a calculator, 'cause all of the calculations are going to be made much simpler by writing each number to one significant figure first.

Okay, pause the video, good luck with these and then when you are ready, I'll be here waiting to reveal question number two, super work.

Question number two and question number three.

The entry cost to a theme park is 37 pounds 50, estimate the cost of entry when a school trip takes five coaches each with 57 seats.

And question number three, you're gonna estimate the cost per person to go on the school trip to a museum.

And I've given you all of the information that you need to answer that question.

Again, remember this is non calculator and make sure you show all of your steps.

Set your workout really, really carefully like Lucas and Sofia did earlier.

Pause the video and I'll be here when you get back.

And question number four, we've got another prom.

So a slightly different prom, okay, this time it's starting at 7:15 and finishing at 11:30.

I've given you all of the costs.

Lucas is going to find the total number of people and then calculate the cost of food.

Whereas Sofia is going to find the total cost of food for pupils and the total cost of food for teachers separately, pause the video, no calculators, and then when you come back we'll check your answers.

Great work, let's check those answers then.

1A 24,000, B is 50, C, 1201, D, 14,000 E, 3000 and F 200,000.

I have shown you each of the steps of working.

So if you've made any errors, just pause the video, have a look at my workings and then I'm sure you'll be able to see what error you've made.

Question number two, the answer is 12,000 pounds.

Wow, that's a lot of money, isn't it? And then question number three, the correct answer was 18 pounds.

And again, all of the calculations are there for you if you need to pause the video because you didn't get 18 pounds, but I'm sure you did.

And then finally, question number four.

A is 2,820 pounds, B is 3020 pounds.

Question C, Lucas's method was best or would be closer to the exact value because it uses one less rounded value.

Well done on those, now we'll move on to our second learning cycle.

So we're going to be looking at whether our answers are overestimates or underestimates.

Here is a question we looked at earlier in the first learning cycle, so it's the one about the trip to a museum where tickets cost £8.

25.

We've taken four coaches and each coach has 52 seats.

We are now going to consider whether our answer was an overestimate or an underestimate.

Lucas says we'll have to work out the exact answer and compare it to our estimate of 1,600 pounds.

Do you agree with Lucas? Sofia's response to Lucas is, I'm pretty certain that when I did that in my assessment, I didn't get the marks.

Sofia is right.

You do not get the marks if you compare your estimated value to the exact value.

Lucas says, how are we gonna answer this question then? Can you see a way that we would be able to tell if our estimate is an overestimate or an underestimate? Sofia says it will be an underestimate because we round both the price and the number of seats on the coach down.

We rounded eight pounds 25 down to eight and the number of seats down to 50.

so we have underestimated as we have round both of those numbers down.

Sofia's uncle has a market stall which sells these items. We've got T-shirts costing 18 pounds 65 each and got sunglasses that cost five pounds 95 each.

He records how many he sells on a Saturday and estimates his takings.

He sells 38 T-shirts and 15 pairs of sunglasses.

What is his estimated takings for Saturday? Step one, we're going to round each of the numbers in the question to one significant figure.

numbers in the question then are the 38 T-shirts, the 15 pairs of sunglasses, the cost of the T-shirt and the cost of the sunglasses? I'd like you now please to pause the video and round each of those to one significant figure.

Great, Should have 40, 20, 20 pounds and six pounds.

you have, Brilliant, now we're going to work out the cost of the T-shirts.

Well we've got 40 T-shirts sold at 20 pounds each giving us 800 pounds and the cost of the glasses, 20 pairs sold at six pounds each is 120, meaning the estimated take-in for Sofia's uncle on the market stall on Saturday was 920 pounds.

Now we need to consider whether that is an overestimate or an underestimate.

What do you think? His estimate is an overestimate as he rounded both prices up and the number of each sold up, every single number in the question he rounded up.

So clearly this is going to be an overestimate.

It's really important that we refer to every number in the question when we're given our answer as to whether it is an overestimate or an underestimate.

Now I'd like you to do this check, when estimating the answers to the following questions, which will give an overestimate, pause the video.

I don't want you to work out any answers.

I just want you to decide whether our answer would be an overestimate or an underestimate.

Pause the video and then when you're ready, come back and we'll check.

Great A, no, that would not be an overestimate, B would because we were around 3.

8 to four kilometres, 6.

5 to seven kilometres.

We've overestimated how far we've run and C would not be an overestimate.

Let's take a look at this question now.

Oak Academy organised a charity evening.

They sold tickets for 18 pounds each.

They sold 195 tickets.

The cost of organising the event was 900 pounds.

Any profits will go to charity.

We are going to work an estimate for the amount of money donated to charity.

We also need to decide whether this is going to be an over or underestimate and I should really have put there also, we need to make sure that we give a reason for that answer.

Step one hasn't changed.

We're gonna round all numbers to one significant figure.

I'd like you please to round each of those to one significant figure and when you've got those answers come back and you should have 20 and notice here, because I'm gonna be considering whether I'm overestimating or underestimating.

I've just put a little note in the brackets to say what I've done.

I've rounded 18 pounds up to 20 pounds.

I've rounded 195 up to 200 and I'm going to use the exact figure here because it was already to one significant figure.

The profit is going to be the total amount taken in ticket sales, which is 20 multiplied by 200.

Each ticket was 20 pounds and we sold 200.

That's when we're estimating.

Subtract how much it costs to organise it because obviously that needs to come off, giving us 4,000, subtract 900, which is 3,100 pounds.

The charity will receive approximately 3,100 pounds and this is going to be an overestimate because we rounded both, notice both.

I've highlighted there, it's really important we refer to both the price and the number of tickets up.

Estimate the area and circumference of this circle.

Sofia says, Lucas, please could you help me out with this question on my homework? And Lucas, because he's such a brilliant guy, says, I'll certainly try my best.

Brilliant, thanks Lucas.

He says, firstly we need to recall the formula for the area and circumference of a circle.

Can you recall the two formula? Circumference is PI multiplied by the diameter.

An area is PI multiplied by the radius squared.

Sofia now says, I need a calculator to do that because I need the PI button, but I forgot to say that this is a non calculator question.

All right, we're not allowed to use a calculator.

Lucas says, that's okay Sofia, the question says estimate.

So we do not need to use an exact value of PI, well spotted Lucas.

It says estimate.

We don't want to know the exact value.

I'm asking you now then, what value will we use as an approximation for PI? What do you think? Well, PI is 3.

14159265359 and we know that that continues.

To one significant figure, that's three.

So any question where it asks you to use a value of PI in your estimating, you just use a value of three.

Let's start with the circumference.

Now we know that the formula is PI multiplied by D.

What is the diameter of our circle rounded to one significant figure? The diameter is 8.

2 centimetres.

Remember, diameter goes whole way across the circle, making sure it goes through the centre.

So from circumference to circumference, the diameter is 8.

2.

We round that to one significant figure and that's eight centimetres.

We know the circumference is PI multiplied by the diameter.

We're using a value of three for PI and we're using a value of eight for the diameter, meaning the circumference of this circle is approximately 24 centimetres.

Is that an overestimate or an underestimate? It's an underestimate as both the diameter and PI have been rounded down.

Diameter went from 8.

2 to eight and PI went from 3.

14 et cetera, to three.

It was an underestimate.

Now let's take a look at the area.

What is the radius of our circle rounded to one significant figure? R, the radius is half of the diameter, isn't it? 8.

2 divided by two is 4.

1, to one significant figure that's four centimetres.

The radius for our calculation, we are going to use a value of four centimetres.

A is PI multiplied by R squared.

PI, remember we're going to use an approximate value of three.

So it's three multiplied by four squared, which is 48 centimetre squared.

The estimated area of this circle is 48 centimetre squared.

Again, is this an overestimate or an underestimate? That's right, it's an underestimate.

Remember, we need to refer to both values we used and we used the radius and we used PI as we'd find it.

Both those down is going to be an underestimate.

Now I'd like you to have a go at this quick check for understanding.

I'd like to estimate the area of the following and then write them in order from smallest to largest.

You'll need to estimate each of the areas first.

No guessing remember, I don't want you just to guess what order they go in.

Also, it might be worth me saying here that you can clearly see that these diagrams are not drawn to scale.

Now you can pause the video, work out your estimated areas, order them, and then come back.

Remember no calculators and you shouldn't need a calculator if you've done your rounding correctly.

How did you get on, A? The approximate area is 27 centimetres squared B was 28 centimetres squared and A was 24 centimetres squared.

Meaning in order from the smallest to the largest, we should have C, A, and then B.

Now for task B, you're gonna pause the video, have a go at this question and then when you are ready, come back.

So no calculators, everything we've done today's lesson has set you up perfectly for being able to answer these questions successfully.

So you don't need good luck, but I always like to say good luck anyway, pause the video and then when you come back I'll reveal the next question.

And question number two, again, pause the video when you've got your answers, come back.

Great work, question number three, finally, question number four, superb work on those.

Let's check our answers.

Question one A, 920, B, the answer will be an underestimate as both the price and the number of tickets were rounded down.

Make sure you've used that word price and you've referred to both the price of the tickets and the number of tickets.

Question 2A, 160 pounds.

This time it would be an overestimate as both the price and the number of protractors needed were rounded up.

Question three, the area of A is 20 metres squared.

The area of B is six metres squared.

The total area therefore is 26 metres squared.

This is going to be an underestimate as all of the measurements were rounded down.

Notice the reference there to all.

And then question 4A, you may have two different answers here.

It will depend on whether you decided to find the radius before you round it or the radius after you'd round it.

So you may have 1,200 metres squared or 2700 metres squared.

The answer if you use the first method would be an underestimate as both PI and the radius were rounded down.

Using the second method, it is an overestimate as although PI is rounded down, the radius is rounded up and is squared, which will have a greater effect on the area.

Great work today, let's now summarise our learning.

When estimating calculations, all numbers should be rounded to one significant figure before the calculation is carried out.

So remember back to Lucas.

Lucas worked out the exact answer and then rounded that, that's not okay.

We have to make sure we round all of the initial values first before carrying out our calculation.

And there's our example there.

An assumption can be made on whether an estimation is an underestimate or overestimate.

And we can consider this by looking at whether we have rounded the values up or down.

If we rounded them up, it's an overestimate.

If we've rounded them down, it's an underestimate.

Thank you so much for joining me today.

You've done fantastically well and I'm looking forward to seeing you again really soon 'cause I know you will, take care of yourself.

Goodbye.