Lesson video

Hi everyone.

My name is Ms. Ku and I'm really happy that you're joining me today.

Today, we're going to be looking at bounds, upper bounds, lower bounds, degrees of accuracy, error intervals, and more importantly, the real life application of these.

I hope you enjoy the lesson.

I know I will.

Let's make a start.

Hi everyone and welcome to this lesson on problem-solving with rounding, estimation and bounds and it's under the unit rounding, estimation and bounds.

And by the end of the lesson you'll be able to use your knowledge of rounding, estimation and bounds to solve problems. So, let's have a look at some keywords we'll be using in our lesson.

Firstly, the upper bound.

So remember, the upper bound of a rounded number is the smallest value that would round up to the next rounded value.

For example, 6 has been rounded to the nearest integer, means 6.

5 is the upper bound.

6.

3 has been rounded to one decimal place.

This means 6.

35 is the upper bound.

Now, the lower bound for a rounded number is the smallest value that that number could have taken prior to being rounded.

We'll also be looking at the words error interval.

And an error interval for number X shows the range of possible values of X and it's written as an inequality where A is less than or equal to X, which is less than B.

Today's lesson will be broken into two parts.

Firstly, we'll be looking at estimating in real-life problems and then we'll be looking at other problems involving estimation and rounding.

So, let's make a start estimating in real life.

When estimating in real life, it's so important to ensure we know the appropriate units of measure to use given the context of the question.

For example, what units of measure would you use to measure the length of a football pitch, the weight of a lion, the time before the end of the school day, the height of a spider, the volume of an average house and the area of a bedroom floor? Have a little think.

What do you think would be the appropriate units of measure for each one of these? Well done.

Let's see how you got on.

Well, an appropriate length of measure for a football pitch would be metres.

The weight of a lion would be kilogrammes.

The time before the end of the school day, I guess it depends upon what time it is, but it can either be minutes or hours.

The height of a spider, millimetres.

The volume of an average house, metres cubed and the area of a bedroom floor, metres squared.

So, knowing the appropriate units of measure allows you to then make reasonable estimates and having a general idea of what key measurements look or feel like allows better estimation in real life.

This lesson will give you some examples, but it is always useful to have your own personal references as they help you remember these key measurements.

For example, a width of a pen tip is approximately one millimetre.

The width of a fingernail is approximately one centimetre.

The length of an average guitar is about one metre and the length of a football pitch is approximately a hundred metres.

These are all nice little examples of what one millimetre would look like.

One centimetre would look like.

One metre would look like and 100 metres would look like.

Now, let's have a look at a quick check, a science textbook uses comparisons with humans to show the sizes of different animals.

For part A, can you identify which length would be a good estimate for the width of a hand? Have a little think.

You've got 10 centimetres, 25 centimetres, and five centimetres.

Well, a good estimate would be 10 centimetres.

Now for part B, it does say use your answers to part A to estimate the size of the animals below.

So, if we're saying the estimate width of a hand is about 10 centimetres, what do you think the approximate width of a rodent would be or the approximate length of a spider? Have a little think.

Press pause if you need more time.

Well done.

Well, approximate length of a rodent is about 1.

5 times the width of a hand.

So, if we're saying the width of a hand is 10 centimetres, that means we're saying approximately the rodent is around about 15 centimetres.

What about our spider? Well, by the looks of it, the spider looks like it's about a third of the width of a hand.

So, we know 10 centimetres is a good estimate for the width of a hand.

That means our spider is around about three centimetres.

Well done if you got anything like this or close to it.

Now, let's have a look at another check, using this image, what do you think would be an appropriate height for the average house? Do you think it's eight centimetres, eight metres, 18 metres or 180 metres? Try and think about those references that we looked at before.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, it's certainly not eight centimetres.

Let's think about it.

If a fingernail width is about one centimetre, definitely this house is not eight centimetres.

Eight metres is a really good estimate because if you think about the length of a guitar being about one metre stacking eight guitars, one on top of another might have helped you conclude that eight metres is reasonable.

Now, 18 metres isn't correct given the context of our picture.

Don't get me wrong, there are houses which are 18 metres tall, but they tend to be five stories tall for a height of 18 metres.

And while it definitely isn't 180 metres, let's think about that football pitch.

If we know a football pitch has a length of a hundred metres, then 180 metres would be nearly twice that height.

So, it's definitely not 180 metres.

Now, let's have a look of area.

Well, if you look at a keyboard, each key on a standard keyboard is approximately one centimetre squared.

And this is because each key is approximately one centimetre by one centimetre.

If we're thinking about the size of an average door, the area of an average door is approximately two metres squared and this is because the length is approximately two metres and the width is around about one metre, thus giving us two metres squared.

This is a nice reference so we know what one centimetre squared looks like as well as two metres squared.

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

Here is a scale diagram of a wall.

Some items in the room will help you give an estimate for the area of the green wall only.

I want you to explain your working out as well.

See if you can give an estimate of how much area this green wall occupies.

Press pause for more time.

Great work.

Let's see how you got on.

Well, there's lots of different ways in which you can estimate the area.

Given the fact that the length of the room looks like it's approximately four metres, this is because I visualised how many door widths can fit across the entire room.

Or you can visualise how many guitar lengths fit across that room.

For me, it looks like around about four widths of our door or four lots of our guitar length.

Now, looking at the height, I'm estimating the height to be approximately 2.

5 metres.

This is because we know the height of a standard door is approximately two metres and, for me, it looks like one and a quarter heights of our door.

So, I'm estimating it to be 2.

5 metres.

So, therefore I am estimating the area of the wall to be 4 by 2.

5, which is 10 metres squared.

But we need to take out the area of the windows because I only want the green area.

Now, both windows look like they have the same area as the door.

And given the fact that we know the door is at round about two metres squared, we can say that the windows and the door make approximately four metres squared in total.

So, the green wall is approximately 10 metres squared, which is the area of the entire wall.

Subtract the door and the window, giving me an approximate area of the green wall to be six metres squared.

Any answer in between five metres squared and seven metres squared inclusive is absolutely acceptable as long as you've got the correct explanations in there.

Well done.

Now, let's have a look at mass.

A tablespoon of sugar is approximately 15 grammes.

A bag of flour or a bag of sugar is always labelled as one kilogramme.

So, it's always good to have a good feel of what one kilogramme feels like.

And five bricks approximately weigh 10 kilogrammes.

These are nice little examples of what 15 grammes feels like, one kilogramme feels like, and 10 kilogrammes feels like.

So given this, let's have a look at a quick check.

An adult lion sits proudly in his surrounding area, estimate the mass of the lion.

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, it's 180 kilogrammes.

Given that the average human male weighs between 60 kilogrammes to 90 kilogrammes, the average adult male of a lion can actually weigh between 150 kilogrammes to 250 kilogrammes.

Well done if you got this.

Great work, everybody.

Now, it's time for your task.

Some Oak pupils have given the Oak teachers some measurements.

Which of the following are reasonable? Have a read and press pause for more time.

Well done.

Let's move on to question two.

A builder is quickly estimating the cost to tile this area enclosed in the white lines.

He knows the window has an area of around about 1.

5 metres squared.

So, removing all the units, what is the approximate area of the wall he wants to tile? And for B, if the builder wants to use tiles which cost 27 pounds, 89 pence per metre squared, estimate how much it will cost to tile the wall.

So, you can give it a go.

Press pause for more time.

Well done.

Let's have a look at question three.

I would love you to use your own personal references for one millimetre, one centimetre, one metre, 10 grammes, one kilogramme, and one metre squared.

Well done.

Let's go through these answers.

Well, for question one, hopefully you've realised that A is unreasonable, B is absolutely reasonable, C is unreasonable, D is reasonable and E is not reasonable.

Very good if you got this.

For question two, let's have a look.

Well, given the fact that the window is approximately 1.

5 metres squared, imagining how many windows fit in this space might help.

So, the approximate area is around about five metres squared, but really anything in between four metres squared and six metres squared is acceptable.

Now, if one metre squared cost 27 pound 89, we're asked to estimate how much it will cost to tile the wall.

Well remember, this is an estimate so we have to round, we're going to round 27 pound 89 to one significant figure, which is 30 pounds.

Now, given the area is approximately five metres squared, we multiply each metre squared by 30, giving us 150 pounds.

Using the previous areas, anything between 120 pounds and 180 pounds is acceptable.

Well done.

Well done.

Let's move on to question three.

Remember, having your own personal references of these measurements is an excellent way to remember how to estimate using the correct units.

Well, for me, one millimetre is around about a pen nib width.

One centimetre is the length of a key on a keyboard.

One metre is about the length of a guitar.

10 grammes is about two pieces of A4 paper, one kilogramme, bag of flour, and one metre squared is the area of a standard square window.

Like I said, it's always important to have your own personal references of these measurements.

Great work, everybody.

So now, let's have a look at the second part of our lesson, other problems involving estimation and rounding.

Now, calculating answers is as equally important as reflecting on the reasonableness of the answer.

For example, Laura and Sam are given this calculation, 401.

75 all divided by 4.

8 squared.

Add on 9.

1.

Now, Sam says the answer's 125 and Laura says the answer is 12.

5.

How can you quickly work out who is most likely to be correct without using a calculator? Have a little think.

Well, we can do this using estimation.

So, let's look at our calculation.

We have 401.

75 all over 4.

8 squared at 9.

1.

Let's round, given the fact that we are estimating.

So, here you can see I've rounded 401.

75 to one significant figure giving me 400.

I've rounded 4.

8 all squared to one significant figure giving me five squared and I've rounded 9.

1 to the nearest 10.

From here, I'm using the priority of operations now.

So, calculating five squared first gives me 400 over 25 at 10.

Then working this out, I have 400 over 35, which can then be simplified to 80 over seven, which then can be converted into a mixed number giving me 11 and three sevenths.

So, if I know, through estimation, the answer is approximately 11 and three sevenths, whose answer do you think would be more appropriate? Well, Laura is most likely to be correct.

If you notice our estimated answer is in the tens and so is Laura's answer.

Now, it's time for a quick check, without working out the answer, which of the following is the correct answer to 956 take away 26 squared all divided by 21.

5, take away 9.

63.

See if you can give it a go.

Press pause if you need more time.

Well done.

Well, you should have got C.

Let's have a look why.

For me, I have rounded to one significant figure, 956 I've rounded to 1000, 26 I've rounded to 30, 21.

5 I've rounded to 20.

9.

63 I've rounded to 10.

From here, I've estimated my answer to be around about 10.

So, which answer is in our tens? Well, it's C, 23.

5888795.

Well done.

So, choosing the correct operation for the situation is also equally as important as estimating.

For example, can you match the correct scenario with the correct calculation to estimate the answer? Here we have three scenarios.

The first says, estimate how many 95 millilitre cups a jug holding 821 millilitres can fill.

The second says, estimate the area of a rectangular face of wood, which is 821 millimetres by 95 millimetres.

And the third says, estimate the total mileage when a car has travelled 821 miles and then a further 95 miles.

So, see if you can match the correct calculation with the correct scenario.

Well done.

Let's see how you got on.

Well, we should have the first one is division because we're trying to find out how many times one value fits into another value.

The second one is multiplication because we're using the formula for the area of a rectangle, which is length times width.

And the third is addition.

Because of that keyword total tells us to add.

Well done if you got this.

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

Here's a cardboard box with lengths 0.

54 metres by 0.

29 metres and 0.

41 metres.

And Lucas wants to estimate the number of puzzle cubes of length, 4.

89 centimetres, that will fit in the box.

Lucas thinks it's about 500 puzzle cubes.

Aisha thinks it's about 300 puzzle cubes and Sam thinks it's around about 200 puzzle cubes.

Show with working out who has the best estimate.

See if you can give it a little go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, we're going to change all the lengths to the same units as it'll make the calculation easier.

So, you'll notice how I've converted all my metres into centimetres.

Then I'm going to round because we are estimating.

Now, there's lots of degrees of accuracy you may use, but for me, rounding to one significant figure seems to be the easiest.

Now, from here there were a couple of different ways in which you can estimate.

For me, the most efficient approach is to calculate the number of cubes that will fit across each dimension.

And let's have a look at the length.

We know the length of our box is approximately 50 centimetres and the length of our cube has been rounded to five centimetres.

So, 50 divided by five means 10 of our puzzle cubes will fit along the length.

Let's have a look at the width.

Well, we've rounded it to 30 centimetres and we know the width of our cube is rounded to five centimetres.

30 divided by five is six.

So, that means we expect six of our puzzle cubes to fit along the width.

Let's have a look at the height.

Well, we've rounded the height to 40 centimetres.

The rounded height of our puzzle cube is five centimetres, so that means 40 divided by five is eight.

So, this tells us eight lots of our puzzle cubes can fit along the height.

So then, we simply multiply these lengths together.

10 by six by eight gives us 480.

So therefore, Lucas's estimation was the best estimation as he said around about 500 puzzle cubes.

Another way to do it is using volume.

So, let's work out the estimate volume of our box.

40 multiplied by 50 by 30 gives us 60,000 centimetres cubed.

Let's work out the estimate volume of our puzzle cube, five by five by five gives me 125 centimetres cubed.

So, in dividing the volume of our box by the volume of our puzzle cube gives us exactly the same answer, 480 puzzle cubes.

So, same again.

Lucas is correct.

Well done if you got this one.

Great work, everybody.

So, now it's time for your task.

Aisha and Lucas use a calculator to work out this answer.

Aisha says it's 142.

0827761, but Lucas says it's 14.

20827761.

One of the answers is correct.

Use estimation to find out who's correct.

Press pause for more time.

Well done.

Let's move on to question two.

Sophia has 1.

48 kilogrammes of cat food left for her cat.

Now, if the cat eats three meals a day, where each meal is weighed to be 48 grammes, how many days the 1.

48 kilogrammes of food will last? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's look at question three.

A wall is in the shape of a trapezium and the Oak teacher wants to cover the wall in 20 centimetre square tiles and the tiles come in packs of 12, costing 24 pounds each pack.

Estimate how much it'll cost the teacher to tile the wall.

See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, question one, hopefully you've got this approximation of our answer.

You can see your answer is in the hundreds, so therefore Aisha would be correct.

Well done if you've got this.

For question two, you might notice we have kilogrammes and grammes here.

So, always convert them to the same units, converting them to the same units and working this out, the cat food should last approximately 10 days.

Press pause if you need more time to look at this working out.

Well done.

Let's have a look at question three.

Same again, we need to convert them all to the same units, but we're also estimating, so let's round as well.

So, you might notice I've converted them all to the same units and are rounded to one significant figure.

From here, I can work out the area of each tile as well.

20 centimetres by 20 centimetres gives me each tile to be 400 centimetres squared each.

Now, working out the area of the trapezium gives me an approximate area of 90,000 centimetres squared.

So, let's find out how many tiles we need.

Well, there's 90,000 divided by 400 because it's the area of each tile, means we need approximately 225 tiles.

Well, given the fact that we know tiles are sold in packs of 12, we're dividing our 225 by 12 indicating I need approximately 20 packs.

Given that each pack is 24 pounds, that means I'm estimating the cost to tile is 480 pounds.

Really well done if you got this.

Great work, everybody.

So, knowing the appropriate units of measure allows you to then make reasonable estimates.

And having a general idea of what key measurements look or feel like, allows better estimation in real life.

This lesson gave some examples, but it is always useful to have your own personal references that help you remember these key measurements.

Remember, estimation allows us to reflect on the reasonableness of the answer.

Well done, everybody.

It was great learning with you.