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Hello everyone, my name is Miss 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 other pupils too.

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 will be overestimating versus underestimating.

By the end of the lesson, you'll be able to determine whether calculations using rounding or truncating will give us an underestimate or overestimate.

Now let's have a look at these keywords.

An overestimate.

An overestimate is an estimate for a calculation which is greater than the exact answer, and a underestimate is an estimate for a calculation which is less than the exact answer.

Today's lesson will be broken into three parts.

First, we'll be looking at estimating with multiplication and addition, then we'll be looking at estimating with division and subtraction, and then we'll be looking at overestimating versus underestimating in context.

So let's make a start.

We round to help estimate an answer to a calculation and the answer is never exact, but does give us a rough idea of the answer.

As a result of rounding, the answer can sometimes be an overestimate or an underestimate.

Izzy asks, how do we know if it's an overestimate or underestimate? And Lucas says, well, it depends on the rounding and the operations.

So let's have a look what Lucas means.

What do you think you would use to estimate this calculation? 4.

6, add 2.

What do you think? Well, Izzy says she would use five, add three and Lucas responds, well without calculating any answers, let's look at a number line to see if it's an overestimate or if it's an underestimate.

This number line shows the actual calculation.

6 add our 2.

7 will give us our exact answer.

Now let's have a look at what Izzy chose.

Well, looking at Izzy's estimation, let's see what that looks like.

Well, Izzy chose to round 4.

6 up to five.

So that means we've rounded up and you can see the difference in that little purple arrow.

Lucas then explains, because Izzy's rounded up to five, we know we'll have this much to add onto the exact answer.

Now let's have a look at what she's chose to add.

Given the fact that Izzy has rounded the addend up to three, you might notice we have this much more to add on to the exact value.

So that means the estimate looks something like this.

Now do you think it would be an overestimate or an underestimate? Well done.

It would be an overestimate because it's greater than the exact value.

So let's summarise what we just did.

Well, because we rounded the starting number up, this has made this much of an increase from the exact value and then rounding the addend up has made this much increase from the exact value.

So therefore, given that both numbers are positive increases, we know it will be an overestimate.

Let's have a look at another question.

Lucas gives us 8.

2 add 5.

Izzy has chosen eight add five and she says, I expect it to be an overestimate.

Do you think Izzy is correct? Let's find out.

Let's look at our number line again, this number line shows the exact answer.

2 add our 5.

3 would give us the exact answer here.

Now let's have a look at the numbers that Izzy has chosen.

Well, Izzy has used eight instead of 8.

So it'd be around about here.

That means Izzy's rounded down.

She's rounded down because she's used eight instead of 8.

So what do you think that would do to the exact answer? Well, you can see this estimate will be much less because Izzy has rounded down.

Now let's have a look at that addend.

Originally the calculation was adding 5.

3, but Izzy has rounded it down to five.

So that means Izzy is adding less.

So that means you are adding a smaller number.

So what does this mean with our exact value? Given that the addend is a little bit less and we've rounded down, do you think it will be an underestimate or overestimate? Well, hopefully you can spot it will be an underestimate as it's below the exact value.

Well done if you spotted this.

So in summary, what did we do? Well, first of all, rounding the starting number down has made this much of a decrease from the exact value and then rounding the addend down has also made this decrease from the exact value.

So given the fact that both numbers are decreases, we know it will be an underestimate overall.

Now it's time for your check.

Without calculating the answers, which of the following will be an overestimate or underestimate? We have our exact calculation and our approximate calculation.

D then asks, when using addition, how are you able to identify if an estimate will be an overestimate or an underestimate? See what you think.

Press pause for more time.

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

Well for A, it would be an underestimate.

Look, we've rounded 9.

45 down to nine.

We've rounded 12.

21 down to 12.

So that means it'll be an underestimate.

For B, it'd be an overestimate.

We've rounded 158.

953 up to 160.

We rounded 236.

12 up to 240.

So it'll be an overestimate.

Next it would be a underestimate.

We've rounded 892.

23 down to 890 and we've rounded 112.

998 down to 100.

Great work if you've got this one right.

Now, when using addition, are you able to identify if an estimate will be an over underestimate? Well, if you are rounding up adding another rounded up value, it will give an overestimate.

But if you round down and add it to another rounded down value, it will give an underestimate.

This is a really important point.

Jot it down if it helps.

Great work everybody.

So let's move on.

Here we have Andeep saying, what happens when we round a number up and another number down? And Lucas says it's a really good point.

So let's have a look at an example.

We have 5.

78, add 2.

Andeep is going to use six, add two.

So let's put it in a number line.

While we know this represents the exact value, 5.

78, add 2.

11 will give us the exact value here.

Now given the fact that 5.

78 rounded up to six, it would be around about here.

Now Andeep recognises that it would mean this much of an increase from the exact answer.

Now let's have a look at that addend.

Well, Andeep has rounded 2.

11 down to two.

So this means he's added less.

So this is what he needs to take away from his exact answer.

Let's put this in a number line.

So what do you notice? Well Andeep says he's confused and he can't tell if the estimated answer is more or less than the exact value.

And this is exactly right.

When we round one number up and round another number down with estimation, it's so difficult to know if it's an overestimate or underestimate.

So in summary, given one number gives a positive increase and the other number gives a decrease, without looking at the size of each increase or decrease, it's really difficult to see if it's an overestimate or an underestimate.

Using this, let's see if you can apply it to a check.

See if you can identify which of the calculations gives an overestimate, underestimate, or if it's quite hard to determine.

Give it a go and press pause if you need more time.

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

Well the first one would be an overestimate.

You might see we rounded both numbers up.

The second one would be an underestimate.

We rounded all of those numbers down.

The third one is quite hard to tell.

We rounded 90.

451 up and then we rounded 563 down and then we rounded 23.

2 down.

So it's really quite tricky to tell.

For D, it's an overestimate.

Look how we've rounded each number up.

And the last one, an E is quite a tough one.

It's an overestimate.

This was tricky because yes we did round some numbers down but they're so much smaller to the other number which was in the hundreds which we rounded up.

That was a tricky one.

So well done if you got that one right.

Now let's have a look at multiplication.

Izzy says, is it the same as multiplication? And Lucas says, we'll use area models to help us find out.

What calculation would you use to estimate this answer? 5.

6 multiply by 6.

Izzy says she would use six multiply by seven.

So the exact answer can be represented using an area model.

Here you can see my area model and I'm multiplying 5.

6 by 6.

Now, given the fact that Izzy has rounded 5.

6 up to six, this means she needs to change the area model.

So looking at that 5.

6, it should roughly look like this.

I've still kept my 6.

9 length, but I've changed my 5.

6 into a six.

That means I've got this much added area.

Now, given the fact that she's rounded her 6.

9 to seven, she needs to change her area model again.

So that means it should look roughly like this.

So the 6.

9 has now changed to a seven.

So in comparison, Izzy's area model, six by seven looks like this, and the original area model is the one on the left.

So can you see, do you think it would be an overestimate or underestimate? Well Izzy clearly spots that six multiply by seven is an overestimate as she's increased the area.

Now let's have a look at another calculation.

Alex says, I think this will be an underestimate when we round each number down.

Do you agree with Alex? And do you think you can show your reasoning with an area model? See if you can give it a go and press pause if you need more time.

Let's see how you got on.

Let's look at these area models.

Well a 4.

2 by 6.

42 which would be represented as this.

Now remember what Alex has chosen.

Alex has chosen to round 4.

2 down to four.

So that means our new area model should be this much smaller.

Then he's also chose to round 6.

42 down to six.

So it should be this much smaller.

So from here the original calculation shows 4.

2 multiply by 6.

42.

The estimate calculation shows four multiply by six.

So given the area models, do you think your estimate is an overestimate or an underestimate? Well, hopefully you can spot it's definitely an underestimate.

You can see that as it has less area.

Now let's have a look at a check.

Andeep says when we round a number up and another number down, it's difficult to tell if it's an overestimate or underestimate.

Do you think he's right? Can you explain? Use area models to help.

Well done, well, Andeep is correct.

It is difficult to identify if it's an overestimate or underestimate.

When rounding one number up increases the area, but when rounding one number down decreases the area and it's so hard to tell if the increased area is bigger than the decreased area.

Great work if you spotted this.

Now it's time for your task.

So you can give this a go and identify if it's an overestimate, underestimate, or if it's hard to determine.

Well done.

Let's move on to question two.

Question two gives us a calculation and you need to write a multiplication that gives a product close to but a little bit more than 117,860.

64.

And B wants us to write a calculation which gives a product close to but a little less than 117,860.

64.

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

Great work, let's move on.

Well for question one, here are our answers.

Massive well done if you've got these ones correct.

Press pause if you need a little bit more time to mark these answers.

Well done.

Well for question two A, there are quite a range of different answers here, but basically any values, whether multiplier and or multiplicand have been shifted upwards slightly would give you a number greater than 117,860.

64.

For example, 8,500 times 15, you may have chosen 8,470, multiply by 14.

All of these are absolutely fine, would give an answer close to, but a little bit more than that answer.

Now to identify a product close to, but a little bit less than any values where the multiplier and or multiplicand can have been shifted downwards slightly would have been absolutely fine.

For example, 8,400 times 10.

Great work if you got this one right.

Fantastic work everybody.

That was tough.

So let's move on to estimating with division and subtraction.

Here's a list of calculations and I want you to work out the answers and put them in ascending order, smallest to largest.

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

Well done.

So hopefully you would work out the answers to be two, three, six, 24 and 48.

They are in ascending order.

As we're making that devisor smaller, what do you notice about the answer to the calculation? Do you think you can explain why? Well hopefully you can spot as the divisor gets smaller the answer gets bigger and this is because the divisor fits more times into that dividend.

Let's increase that dividend.

What do you think happens to the answers as we increase the dividend? For example, 24 divide by 2,240 divide by two, 2,400 divide by two and 24,000 divide by two.

Well hopefully you can spot as the dividend gets larger, the answer also gets bigger.

And this is because we have more to share.

Now Jacob spots, so when the dividend gets bigger, the answer gets bigger, but when the divisor gets bigger the answer gets smaller and Lucas agrees and says yes, exactly that.

So let's have a look at this calculation.

489 divided by 0.

61 and we're approximating it to 490 divided by 0.

Lucas says, do you think it will be an overestimate or an underestimate? Now given the fact that the dividend of 489 is now bigger, he's thinking it's going to be an overestimate.

And given that the divisor of 0.

61 has now got smaller, he also thinks it's going to be an overestimate.

So Lucas then asks, what do you think the estimate will be? Well, he thinks it will be an overestimate and he's right.

Well done if you spot this too.

890 has been rounded up to give us a bigger number being divided by a smaller number where we've rounded 0.

61 down to 0.

Let's have a look at a check.

Aisha's done this working out in her maths book.

Do you think it will be an overestimate or underestimate? And I want you to explain.

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, given the fact that dividend has been rounded down, Aisha's expecting an underestimate.

Let's have a look at that divisor.

Well the divisors has been rounded up.

Once again, she expects an underestimate.

So therefore the answer will be an underestimate.

Really well done if you've got this.

So let's have a look at another check.

Which calculation shows an overestimate? And I'd like you to explain.

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

Well done, let's have a look.

Well, the calculation on the left, we do expect an overestimate because 56.

9 has been rounded up to 60, 19.

8 has been rounded up to 20.

Now let's have a look at that divisor.

Well, the divisor has been rounded down, so that means we're dividing by a smaller number, thus we expect an overestimate.

Let's have a look at the calculation on the right.

Well we've rounded down, 56.

9 has been rounded down to 50 and 23.

2 has been rounded down to 20.

Thus we're expecting an underestimate.

Look at that divisor, it's been rounded up.

In other words, we're dividing by a bigger number, thus we're expecting an underestimate.

So that means the calculation on the left shows an overestimate.

Andeep says, let me guess, when we round the dividend up and round the divisor up or vice versa, it's difficult to tell if it's an underestimate or overestimate.

What do you think? And can you explain why? Well done.

Andeep is correct, it's hard to tell.

As a result of rounding or truncating, it is unclear how much of the differences between the overestimate and underestimate of the divisor or dividend will be.

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

Here I want you to put a tick if the statement is true.

Press pause for more time.

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

Well A is correct.

A rounded up value divided by a rounded down value does give you an overestimate.

B is also correct.

A rounded down value over a rounded up value gives you an underestimate.

C and D, it is hard to tell.

So if you want to jot this down as a quick reference, please do.

Well done if you got this one right.

Izzy says, now we need to look at subtraction.

So let's use that number line again to estimate the answer to 23.

6 subtract 5.

Lucas asks, what calculation would you use to estimate the answer? And Izzy says, well I'll round 23.

6 up to 25 and I'd round 5.

8 down to five, thus making 25, subtract five.

So let's see if we can put it on a number line and identify if it's an overestimate or underestimate.

Here you can see it's the subtraction of 5.

So 23.

6, subtract 5.

8 would give us an exact answer round about here.

Now given the fact that Izzy's estimate is 25, subtract five, this means she used 25.

So it'll be around about here.

It shows that there'll be this much difference that she needs to add on to the exact value because she's rounded up.

Now looking at the subtraction, remember she's subtracting a smaller number.

So that means we expect to have this much of a positive difference.

So representing this in a number line, do you expect your estimate to be an overestimate or an underestimate? It would be an overestimate as she's expecting the estimated answer to be greater than the exact value.

Well done if you spotted this.

Now let's summarise all of this in a table.

And using number lines, I'd like you to investigate if you get an overestimate or underestimate or if it's hard to tell when subtracting and rounding up or down.

Please use number lines if it does help you out and fill in this table.

This table will be really useful when you're doing your practise work later on.

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

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

Well, if you have a rounded up value subtracted around a down value, it will give you an overestimate.

A rounded up value subtract a rounded up value, it's quite hard to tell.

And if you have a rounded down value, subtract a rounded up value, it's an underestimate, and a rounded down value subtract a rounded down value is hard to tell.

Great work if you have this.

Make sure you jot it down for reference.

Well done everybody.

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

See if you can identify if it's an overestimate, or underestimate or hard to tell.

Press pause for more time.

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

Well, the first one is an overestimate.

We rounded 87 up to 90 and we've rounded 0.

8 down to 0.

75.

B is also an overestimate.

We rounded up and we rounded that 36 down.

For C, it's an underestimate.

Look how we rounded that 451 down to 450 and then we rounded 12.

5 at 21.

3 up to 15 and 25.

Last one is hard to tell.

You can see how we've rounded some numbers up and some numbers down.

Well done if you got this.

Now it's time for your task.

Question one A says, without calculating you need to identify, is this a little bit more than four, a little bit less than four or exactly four? And B says same again without calculating, is this a little bit more than four, a little bit less than four or exactly four? See if you can give it a go.

Great work everybody.

Let's move on to question two.

Question two wants you to identify if it's an overestimate, underestimate, or if it's hard to tell.

Give it a go and press pause for more time.

Well done.

So the second part is exactly the same.

Identify overestimate, underestimate, or if it's hard to tell.

Great work.

So let's go through these answers.

Well, for one A, it should be a little less than four.

For B, it should be a little bit more than four.

Next, here are our answers.

Well done if you've got any of these right, they were really tough.

Now let's have a look at the second part.

Here are our answers.

Fantastic work, everybody.

If you've got any of these right, these were particularly hard.

Great work, everybody.

Now let's look at our third and final section to our lesson, overestimating versus underestimating in context.

Now the operation and the rounding play an important part when identifying if a number will be an overestimate or an underestimate.

Overestimating and underestimating have a practical role in real life.

So let's have a look at a check.

Three friends are going on holiday and the total cost of the holiday is 1,784 pounds and they're going to share it equally.

Aisha uses a calculator to work out 1,784 divided by three.

Now she thinks there's six different ways to tell her friends how much they will pay.

Either about 600 pounds, about 1,000 pounds, about 590 pounds, about 594 pounds, about 595 pounds, or about 594.

66 pounds.

Which of these do you think is correct? Which do you think is the best way to phrase it? And in this case, is it better to overestimate or underestimate? See what you think.

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

So which of these do you think is correct? Well, all of these could be considered to be correct, depending on the required degree of accuracy.

So for part B, which do you think is the best way to phrase it? Well, 595 pounds might be the best as it doesn't involve dealing with small coins and doesn't end up short of the total.

Now given money is always dealt to two decimal places, 594.

66 pounds should be rounded to 594.

67 pounds and using 594.

66 pounds leaves them two pence short.

So that might not be the amount to tell the friends.

Now in C in this case, it is better to overestimate or underestimate? It's better to overestimate.

So you do not have any money owing or being short of what is owed.

Well done if you got this one right.

Now it's time for your question.

Here we have a question and there are two lifts.

Lift A states the maximum load is eight people or 630 kilogrammes.

Lift B says the capacity of 1,000 kilogrammes is 13 persons.

The two signs on the left are from different lifts.

Each lift can carry a different number of people.

Which lift assumes that people are heavier? Which lift is easier to estimate and why? C says nine identical boxes are under the weight limit for one lift, but over it for another lift.

What do you know about the weight of the boxes? For D, five boxes weighing 23 kilogrammes each are put into each lift.

Approximately how many people can safely get in the lift with them.

And then E, explain why is it better to overestimate all the scenarios above.

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.

Lift A assumes people are heavier and a likely estimation will be 640 divided by eight, which equals 80 kilogrammes for lift A.

And 1,040 divided by 13 gives us 80 kilogrammes on lift B.

Given that as these are the same answer, the actual weight is closer to 80 for the first lift than it is for the second.

Now the actual assumed weights are 78.

75 kilogrammes and 76.

92 kilogrammes to two decimal places respectively.

Well done if you got this one right.

Now for B, it states which lift is easier to estimate? Well, our answers can vary here.

Some may argue that the first lift is easier.

As 630 is closest to 640, which is a multiple of eight.

So 640 divided by eight is quite a straightforward calculation.

However, you may find the second lift easier using a nearby multiplication fact such as 960 divided by 12 or 1,040 divided by 13.

Well done if you've got this one.

Well for C, the boxes must weigh between 70 kilogrammes and approximately 111 kilogrammes.

Next for D, answers will really depend upon how to estimate or calculate with 23.

You might like to take the total weight of the boxes to be 100 rounding 23 to 20, 115 using 23, or 135, rounding to 23 to 25.

Approximately six people can get into the first lift with the boxes and approximately 11 people can get into the second lift with the boxes.

Great work if you got this one right.

Lastly, is it better to overestimate or underestimate in the scenarios above? Well, it's better to overestimate.

If we underestimate, the lift is in danger of exceeding capacity and breaking and we certainly don't want that.

Great work everybody.

So overall, we round to help estimate an answer to a calculation and the answer is never exact and gives us a rough idea of the answer.

As a result of rounding, the answer can sometimes be an overestimate, underestimate, or it's hard to tell.

The operations we use and the type of rounding, for example, rounding up, rounding down, or truncating, will also play an important role in an overestimate or an underestimate.

Great work, everybody.

It was a tough lesson, I really enjoyed learning with you.

Well done.

I've finished the video