3. Why do we sometimes round figures rather than giving an exact figure?
Rounding
We do not always need to know the exact value of a number.
There are 1432 pupils at Eastpark Secondary School.
There are about fourteen hundred pupils at Eastpark Secondary School.
There are about one and a half thousand students at Eastpark Secondary School.
Teacher notes
Talk about rounding in real life contexts, for example the number of people at a football match. Ask pupils to give examples.
State that we can also use rounding in maths to give approximate answers to calculations.
We can round numbers to the nearest 1000, 100, 10, whole number, 0.1, , and so on, depending on the level of accuracy required.
Why do we sometimes round figures rather than giving an exact figure?

4. Rounding There are four main ways of rounding a number:
to the nearest 10, 100, 1000, or other power of ten
to the nearest whole number
to a given number of decimal places
to a given number of significant figures.
The method of rounding used usually depends on what kind of numbers we are dealing with and how the numbers are being used.
Teacher notes
Point out that in some cases we may by required to give a whole number to a given number of decimal places. When this happens the decimal places are filled with 0’s. For example, 8 to three decimal places is
In some cases we may round to other numbers, for example to the nearest 0.5, 5, 50, 500 etc. but this is less usual.
For example, whole numbers might be rounded to the nearest power of ten or to a given number of significant figures.

5. Rounding to powers of ten
Ten thousands
Thousands
Hundreds
Tens
Units
100
101
102
103
104 1 7 8 4 3
Round to the nearest 100.
Look at the digit in the hundreds position.
The number will be rounded to either or as it is between these two.
Look at the digit in the tens position.
If this digit is 5 or more then we round up to the larger number (34900). If it was less than 5 we would round down to
Teacher notes
Talk through the example.
Note that is not equal to is an approximation. We must include ‘to the nearest 100’ to ensure that the equals sign is not used incorrectly.
34871 = (to the nearest 100)

6. Rounding to powers of ten

7. Rounding to decimal places
Round to one decimal place.
Hundredths
Thousandths
Ten thousandths
Units
Tenths
100
10-1
10-2
10-3
10-4 2 7 5 2 4
Look at the digit in the first decimal place.
The number will be rounded to either 2.7 or 2.8 as it is between these two.
Look at the digit in the second decimal place.
If this digit is 5 or more then we round up to the larger number (2.8). If it was less than 5 we would round to 2.7.
Teacher notes
Talk through the example.
Emphasize that is not equal to is an approximation.
= 2.8 (to 1 decimal place)

8. Rounding to decimal places
Teacher notes
Talk through each answer in the table.
Stress that the number of decimal places tells you the number of digits that must be written after the decimal point, even if the last digit (or digits) are zero.

9. Rounding to significant figures
Numbers can also be rounded to a given number of significant figures.
The first significant figure of a number is the first digit which is not a zero.
For example,
This is the first significant figure
and
This is the first significant figure

10. Rounding to significant figures
The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros.
For example,
This is the first significant figure.
This is the second significant figure.
This is the fourth significant figure.
This is the third significant figure.
and
This is the first significant figure.
This is the second significant figure.
This is the third significant figure.
This is the fourth significant figure.

11. Rounding to significant figures
Teacher notes
Discuss each example, including the use of zero place holders and zeros that are significant figures.

13. Discrete and continuous quantities
Numerical data can be discrete or continuous.
For example:
boot sizes
the number of children in a class
amounts of money.
Discrete data can only take certain values.
Continuous data comes from measuring and can take any value within a given range. It is only as accurate as your method of measuring.
For example:
the weight of an apple
the time it takes to get to school
heights of 15 year-olds.
Photo credit (upper): © MonkeyBusinessImages, Shutterstock.com
Photo credit (lower): © MichalMrozek, Shutterstock.com

14. Upper and lower bounds for discrete data
The population of the United Kingdom is 59 million to the nearest million. In what range of values could the population be in?
The most this could be before being rounded down is:
The least this could be before being rounded up is:
We can give the possible range for the population as:
≤ population ≤
or ≤ population <
Teacher notes
Discuss the fact that if the population has been given to the nearest million the actual amount could be half a million more or less either way. Because of the convention of rounding up when a number is half way between two others, the most the population could be is: – 1 =
This value is called the lower bound…
… and this value is called the upper bound.

15. Upper and lower bounds for discrete data
Last year a shopkeeper made a profit of £43 250, to the nearest £50. What range of values could this amount be in?
The lower bound is half-way between £ and £43 250:
£43 225
The upper bound is half-way between £ and £43 300, minus 1p:
Teacher notes
Explain that because the amount has been rounded to the nearest £50, the lower bound must be half-way between the given amount and the given amount – £50.
The upper bound must be half-way between the given amount and the given amount + £50. We then subtract 1p because if the amount was equal to £ it would be rounded up to £ p is the smallest amount we can subtract because we consider amounts of money to be discrete since they are given in a whole number pence.
Photo credit: © MonkeyBusinessImages, Shutterstock.com £
The range for this profit is:
£ ≤ profit ≤ £

16. Upper and lower bounds for discrete data

17. Bounds for continuous data
The height of the Eiffel Tower is 324 metres to the nearest metre. What range of values could its height be in?
The least this measurement could be before being rounded up is:
Lower bound = m
The most this measurement could be before being rounded down is up to but not including:
Upper bound = m

18. Bounds for continuous data
The height of the Eiffel Tower is 324 metres to the nearest metre. What range of values could its height be in?
323.5 m ≤ height < m
The height could be equal to m so we use a less than or equal to symbol.
If the length was equal to m however, it would have been rounded up to 325 m. The length is therefore “strictly less than” m and so we use the < symbol.
Teacher notes
We can read the inequality as “the height is between m and m, but doesn’t include m”.
Explain that if the length is strictly less than cm, it can be any value up to …
It is interesting to note that if we convert … using the method shown in the “Terminating and Recurring Decimals” chapter of Unit 2 Number and Algebra: Decimals, we can show that it is actually equal to
See Unit 3 Geometry and Measures: More Measures to investigate the combined effect of bounds in calculations.

19. Upper and lower bounds Teacher notes
Use this activity to demonstrate how to locate the upper and lower bounds of measurements given to the nearest 0.01, 0.1, 1, 5, or 10 centimeters. The measurement can be changed by dragging on the ruler.
Stress that when we illustrate an inequality on a number line the value indicated by the empty circle is not included.
Upper and lower bounds for measurements are also covered in Unit 3: Geometry and Measures, More Measures.
You may wish to go over the conventions that are used when illustrating inequalities. For more details on this, see the presentation Simple inequalities in Unit 2: Number and Algebra.

21. Village signs

22. Child’s toy
A child’s toy has different shaped pieces that go into the same shaped holes in a box.
One shape is a cylinder.
One company makes the pieces 4 cm diameter to the nearest 0.5 cm.
Another company makes the box with circular holes 4.1 cm diameter to the nearest 0.1 cm.
Teacher notes
The answer should show that the range of values for the piece is 3.75 cm to 4.25 cm and for the hole is 4.05 to 4.15 cm. The piece can therefore be larger than the hole so it will not fit.
They test the pieces and they do not fit. Explain how this happened using upper and lower bounds.

23. Carpet fitting A bedroom is measured for a new carpet.
It is 4.2 m by 5.4 m measured to one decimal place.
The carpet is cut in the shop and can be cut accurately to the same accuracy.
What is the smallest size of carpet that can be cut so that it fits the room with no gaps round the edge?
Teacher notes
The solution is that the room is 4.15 – 4.25 m by 5.35 – 5.45 m.
The carpet must cater for the worst case (i.e. the biggest size possible), so it will need to be: 4.25 m by 5.45 m.
It must be cut so that it is at least that size. As a result, the lower bounds of the carpet size must be 4.25 m by 5.45 m.
Therefore, the carpet must be 4.3 m by 5.5 m
If the shop is cutting a piece 4.3 m by 5.5 m then the largest it could be is:
4.35 m by 5.55 m so this is the size that will need to be charged for.
4.35 m × 5.55 m × £12 = £
The carpet costs £12 per square metre. What should the shop charge?

24. In the sweet shop
A sweet shop has some jars of sweets with the number of sweets on the front, but the numbers have been rounded to one significant figure.
What is the lowest and highest number of sweets possible in each jar? Explain your answers using some or all of the terms:
upper bound
significant figures
Teacher notes
80 sweets is correct to 1 significant figure so the number of sweets in the jar could be between 75 (the lower bound) and 84 (the upper bound – note it is not 85 as this would round up).
200 sweets is correct to 1 sig fig so the possible number of sweets in this jar ranges from 150 to 249 sweets.
1000 sweets is more complicated. The lower bound is 950 to still round up to 1000.
The upper bound is 1499 to round down to Students may think that the upper bound is 1100 sweets. They should be reminded that 1100 is an example of rounding to 2 significant figures. To round to 1 significant figure, any number of sweets between 1001 and 1499 will be rounded down to 1000.
It is important to stress that 500 is already to 1 significant figure so it is not correct as the lower bound. As a result, the limits on the number of sweets in this jar is 950 ≤ sweets < 1499.
lower bound
rounded