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1 NUMERACY ACROSS
THE CURRICULUM
2 NUMERACY SKILLS
and
TEACHING METHODS
A Reference Handbook
for Teachers
3 Number
Contents
Page
Writing and Reading Numbers
4
Language of Operations
4
Order of operation (BODMAS)
5
Writing calculations
5
Addition whole numbers
7
Subtraction whole numbers
7
Addition decimal numbers
8
Subtraction decimal numbers
8
Multiplying and dividing by 10, 100 and 1000
8
Multiplication
9
Division
10
Multiplication and division of decimals
10
Using a calculator
10
Negative numbers
11
Fractions
12
Ratio and proportion
13
Decimal notation
14
Percentages
15
Rounding numbers
16
Standard form numbers
16
Algebra
Using formulae
17
Plotting and drawing graphs
18
Geometry and measures
2D shapes
20
3D shapes
21
Length, mass and capacity unit conversions
21
Time
22
Reading scales
22
Area
23
Volume
23
Handling data
Bar graphs and frequency diagrams
24
Line graphs
25
Pie charts
26
Using data – mean, median, mode and range
27
Correlation and scatter graphs
28
Types of graphs
29
When to use the different type of graphs
33
Numeracy glossary
34
4 Number
Writing and Reading Numbers
Students need to be encouraged to write figures simply and clearly. Zeros with a line
through (∅), and French sevens (7) should be discouraged. Encourage the correct shape
for number 4 and 0 which can sometimes look like six. Also decimal points shown as a “o”
e.g. 1o7 = 1.7
Most students are able to write numbers up to a thousand in words, but there are often
problems with bigger numbers. It common practice to use spaces between each group of
three figures in large numbers rather than commas,
e.g. 34 000 not 34,000.
In reading large numbers students should apply their knowledge of place value working
from the right in groups of three digits. So the first group contains hundreds, tens and
units, next group as thousands and the next group as millions. The number is then read
from left to right.
So 2 084 142 is two million, eighty four thousand, one hundred and forty two.
Language of Operations
All students should be able to understand and use different terms for the four basic
operations but some students with learning difficulties and EAL students may have
difficulty in associating terms with symbols.
+
add
increase
more
plus
sum
total
decrease
difference
less
minus
reduce
subtract
takeaway
x (.)
multiply
of
product
times
lots of
multiplication
÷ (:)
divide
division
quotient
share
divisor
5 Order of Operations
When carrying out a series of arithmetic operations students can be confused about the
order in which the operations should be done, e.g. does 6 + 5 x 7 mean 11 x 7 or 6 + 35?
Some students will be familiar with the mnemonic BODMAS or BIDMAS
Brackets, power Of, Division, Multiplication, Addition, Subtraction
Indices
The important facts are that brackets are done first, then powers, multiplication and
division and finally addition and subtraction, e.g.
(i)
6 + 5 x 7 = 6 + 35 = 41
(ii)
(6 + 5) x 7 = 11 x 7 = 77
(iii)
2 + 43 ÷ 8 - 3 = 2 + 64 ÷ 8 - 3 = 2 + 8 - 3 = 7
Some non-scientific calculators will give incorrect answers because they do operations in
the order in which they are put into the calculator rather than following the BODMAS
order.
Writing calculations
Some pupils are over-dependent on the use of calculators for simple calculations.
Wherever possible pupils should be encouraged to use mental or pencil and paper
methods. It is, however, necessary to give consideration to the ability of the pupil and
the objectives of the task in hand. In order to complete a task successfully it may be
necessary for pupils to use a calculator for what you perceive to be a relatively simple
calculation. This should be allowed if progress within the subject area is to be made.
Before completing the calculation pupils should be encouraged to make an estimate of
the answer. Having completed the calculation on the calculator they should consider
whether the answer is reasonable in the context of the question.
6 Writing Calculations
Students have a tendency to use the ‘=’ sign incorrectly and write mathematical
expressions that do not make sense, e.g.
X
3 x 2 = 6 + 4 = 10 - 7 = 3
It is important that all teachers encourage students to write such calculations correctly,
e.g.
3 x 2 = 6
6 + 4 = 10
10 - 7 = 3
P
The = sign should only be used when both sides of the equation have the same value.
There is no problem with calculations such as
34 + 28 = 30 + 20 + 4 + 8 = 50 + 12 = 62
because each part of the working out has the same value.
The ‘≈’
(approximately equal to) sign should be used when students are estimating
answers, e.g.
1 576 - 312 ≈ 1 600 - 300 = 1 300
Mental Calculations
All students should be able to carry out the following processes mentally, although some
will need time to arrive at an answer:
•
recall addition and subtraction facts up to 20
•
recall multiplication and division facts for tables up to 10 x 10
Students should also be encouraged to carry out other calculations mentally using a
variety of strategies but there will be significant differences in their ability to do so.
34 + 28 = 34 + 30 - 2
163 - 47 = 163 - 50 + 3
23 x 3 = (20 x 3) + (3 x 3)
68 ÷ 4 = (68 ÷ 2) ÷ 2
It is helpful if teachers discuss with students who they might do mental calculations such
as these. Any valid method that produces a correct answer is acceptable.
7 Pencil and Paper Calculations
All students should be able to use pencil and paper methods for calculations involving
simple addition, subtraction, multiplication and division. The following methods are used
by the majority of students, and are generally taught to those students who do not have
a method that they can carry out correctly. It is important to encourage students to set
out their work in columns when adding and subtracting.
Addition and Subtraction
3456 + 975 Estimate 3500 + 1000= 4500 3 4 5 6 + 9 7 5 4 4 3 1 6286 -­‐ 4857 Estimate 6300 -­‐ 4900= 1400 56 127 8 16 -­‐ 4 8 5 7 1 4 2 9 1 1 1 396 -­‐ 178 3 9 16 -­‐ 1 7 8 2 1 8 1 Click on the ? to get extra help with column subtraction.
For weaker students the use of a number line may be useful particularly for subtraction.
512 -­‐ 386 8 Addition and subtraction of decimals are done using the same methods, but students may
need to be reminded that it is necessary to ensure that the decimal points must be
underneath each other.
Decimals
3.96 + 1.78 3. 9 6 + 1. 7 8 5 . 7 4 6. 2 – 4.57 1 1
6. 2 0 -­‐ 4. 5 7 1. 6 3 1 1 1 1 6. 2 – 4.57 5
1 1
6. 2 0 (write the zero as a place filler) -­‐ 4. 5 7 1. 6 3 Multiplying and Dividing by 10, 100, 1000 . . .
The rule for multiplying by 10 is that each of the digits moves one place to the left.
When multiplying by 100 each digit moves two places to the left and so on. In division
the digits move to the right. This rule works for whole numbers and decimals. Decimal
points do not move, e.g.
23 x 100 = 2300
Th
H
2
3
T
2
0
U
3
0
Zeros are needed to fill the empty spaces in the tens and units columns, otherwise when
the number is written down without the column headings it will appear as a different
number i.e. 23 instead of 2 300
9 3.45 x 10 = 34.5
Th
H
T
U
.
3
3
4
.
.
1
10
4
5
1
100
5
Zeros are not needed in empty columns after the decimal point.
260 ÷ 10 = 26
H
T
U
.
2
6
2
0
6
.
.
1
10
1
100
1
10
1
100
3
9
439 ÷ 100 = 4.39
H
T
U
.
4
3
9
4
.
.
Multiplication and Division.
Multiplication
Traditional
357 × 8 Estimate 350 × 2 × 4 = 700 × 4 = 2 800 357 × 8 2856 Grid method
×
8
300
2400
50
400
357 × 8 = 2400 + 400 + 56 = 2856
4 5 Click on the ? to get extra help with grid multiplication.
7
56
10 Division.
810 ÷ 6 Estimate 800÷ 5 = 160 1 3 5 6 8 21 3 0 5832 ÷ 24 Estimate 5800 ÷ 20 = 2900 2 4 3 24 5 8 103 72 When multiplying decimal numbers
(i)
Complete the calculation as if there are no decimal points
(ii)
Insert a decimal point in the answer so that there are the same
number of digits on the right of the decimal point in the answer as
there are in the question.
Two decimal places
in question
(i) 3.2 x 2.4 becomes 32 x 24 = 768
(ii) 3.2 x 2.4 = 7.68
Two decimal
places in answer
When dividing a decimal number by a whole number, the calculation is the same as a
division without decimal points, with the decimal point in the answer being inserted
directly above the decimal point in the question.
Using a Calculator
Many students believe that if they use a calculator, the answer they obtain must be
correct.
It is important that students are encouraged to make an estimate of the
answer they expect to get and, having completed the calculation, complete the answer
with their estimate. They also need to consider whether the answer they have got makes
sense in the context of the question.
Students are taught how to use the various functions on a calculator including:
√ x
2
^
x3
⬚
⬚
xy
11 Students are likely to have various types of scientific calculators that operate in
different ways. In some cases an operation is entered before the number, in others the
operation follows the number, e.g.
Finding √169
In older calculators this is done by
With Casio calculators the order is
1 6 9 √ = √ 1 6 9 = Students should be encouraged to buy the Casio Fx 83 calculator and familiarise
themselves with their own calculator and to learn to use if efficiently.
Negative Numbers
Some students have problems understanding the size of negative numbers and believe
that -10 is bigger than – 5. A number line can be of help when ordering numbers. Moving
to the right, numbers are getting bigger. Moving to the left, numbers are getting
smaller.
Smaller
Bigger
Students in lower ability groups may have difficulty with calculations involving negative
numbers. For addition and subtraction a number line is useful.
-3
Starting point
•
Start at -3
•
Move 8 places to right
•
So -3 + 8 = 5
-
+
Direction
8
Distance to move
+
12 If there are two signs between the numbers, when the signs are the same the move is to
the right and when the signs are different, the move is to the left.
So
+ + or - - is the same as + and means move to the right
+ - or - + is the same as - and means move to the left, e.g.
(i)
-5 + (-6) = -5 -6 = -11
(ii)
6 – (-3) = 6 + 3 = 9
When multiplying and dividing positive and negative numbers, if both numbers are positive
or both numbers are negative, the answer will be positive. If one of the numbers is
positive and the other is negative, the answer will be negative, e.g.
(i)
(-3) x (-2) = 9
(ii) (-8) ÷ 4 = -2
Common Fractions
Most students should be able to calculate simple fractions of quantities. They should
know that to find half of an amount they need to divide by 2.
To find a third of
something they divide by 3 etc. To find a quarter of an amount it is sometimes easier to
divide by two twice rather than to divide by 4.
They should know that the denominator (bottom number of a fraction) indicates the
number of parts the amount is to be divided into and the numerator (top number)
indicates the number of parts required, e.g.
To find
!
!
of 24
•
First find
•
So
!
!
!
!
of 24 = 24 ÷ 4 = 6
= 3 × 6 = 18
Generally, students should be able to do this type of calculation mentally, but some may
need prompting in order to do so. The rule divide by the bottom number and multiply by
the top number does work, but if this rule is used without understanding, it can cause
problems because students tend to forget which numbers they use to divide or multiply.
13 Most students should be able to make equivalent fractions by multiplying or dividing the
numerator or denominator of a fraction by the same number.
×2
e.g.
!
! = ÷5
!
!"
!
!"
×2
= !
!
÷5
Ratio and Proportion
Ratios compare numbers or measurements, e.g.
When making pastry the ration of fat to flour is 1 : 2. This means that for every
gram of fat used, 2 grams of flour are required, or for every ounce of fat there
must be 2 ounces of flour.
Ratios can be simplified in the same way as fractions by dividing all of the parts of the
ratio by the same number, e.g.
The ration 3 : 12 : 6 can be simplified by dividing by 3 giving 1 : 4 : 2
If an amount is to be split into a given ratio, it is first necessary to begin by finding out
how many parts are needed, e.g.
Three people hold shares in a company in the ratio 60 : 100 : 20 and a dividend of
£3 600 is divided between them in the same ratio as their shares. How much does
each person receive?
The three people hold a total of 60 + 100 + 20 = 180 shares. £3 600 ÷ 180 = £20 so the
dividend for each share is £20 and the payments are:
60 x £20 = £1 200
100 x £20 = £2 000
20 x £20 = £400
Ratios are often used to show the scale of a map or plan,
e.g.
If the scale of a map is 1 : 50 000 this means that 1 centimetre on the map
represents a distance of 50 000 centimetres on the ground.
50 000 centimetres = 500 metres = 0.5 kilometre
so a scale of 1 : 50 000 means that 1 centimetre on the map represents 0.5
kilometre on the ground, or 2 cm represents 1 kilometre.
14 Students may experience difficulties in adapting recipes to produce larger or smaller
quantities. They may need practice in modifying recipes and should be encouraged to
adopt a method that is suitable for the particular problem.
If a recipe for four people is to be modified to cater for five people it may be easiest to
find the quantities required for one person and multiply these by five. In other cases it
will be sensible to use other methods to adjust quantities, e.g.
The following recipe will produce 16 flapjacks:
40 grams of margarine
50 grams of sugar
200 grams of rolled oats
100 grams of golden syrup
If 24 flapjacks are required, it is probably easiest to find the recipe for 8 flapjacks (by
halving the given quantities) and add these amounts to the original recipe.
Decimal Notation
All students should be familiar with decimal notation for money although they may use
incorrect notation.
P
£2.35
£3.33
£4.60
£0.25
32p
X
£2.35p
£3.33333..
£4.6
0.25p
0.32p
Students should also be familiar with the use of decimal notation for metric measures,
but sometimes misinterpret the decimal part of the number.
They may need to be
reminded for example that 1.5 metres is 150 centimetres not 105 centimetres.
Students often read decimal numbers incorrectly, e.g. 8.72 is read as eight point seventy
two instead of eight point seven two.
They may also have problems with comparing the size of decimal numbers and may believe
that 2.36 is bigger than 2.8 because 36 is bigger than 8.
If they need to compare
numbers it may help to write all of the numbers to the same decimal places, e.g. 2.36 and
2.80
15 Percentages
Calculating Percentages of a Quantity
Methods for calculating percentages of amounts vary depending on the percentage
required. Students should know that fractions, decimals and percentages are different
ways of representing part of a whole and know simple equivalents.
e.g. find 10% divide by 10
find 1% divide by 100
So to find 12% find 1% then times by 12.
Where percentages have simple fraction equivalents, fractions of amounts can be
calculated, e.g.
(i) To find 50% of an amount, halve the amount
(ii) To find 75% of an amount find a quarter of the amount and multiply it by
three.
10% of an amount can be calculated by dividing by 10. Amounts that are multimples of 10
can then be calculated from this, e.g.
30% is 3 x 10%
5% is half of 10% etc.
When using a calculator it is usually easiest to think of a percentage as a decimal number
and ‘of’ as x, e.g.
17.5% of £36 = 0.175 x £36 = £6.30
The most able students in Years 9, 10 and 11 should be able to increase or decrease
amounts using one operation. This is particularly useful in spreadsheets, e.g.
(i)
Increase 52 by 14%
114% of 52 = 1.14 x 52 = 59.28
(ii)
Reduce 175 by 30%
70% of 175 = 0.7 x 175 = 122.5
16 Calculating an Amount as a Percentage
The most straightforward way to do this is to think of the problem as a common fraction
and then to convert this to a percentage if the fraction is simple. For more complex
questions the fraction can be converted to a decimal number and then a percentage.
e.g. What is 20 as a percentage of 80?
!"
!"
!
= = 25%
!
×2
What is 43 out 50 as a percentage?
!"
!"
= !"
!""
= 86%
×2
What is 123 out 375 as a percentage?
!"#
!"#
= 123 ÷ 375 ×100 = 32.8%
Rounding Numbers
When rounding numbers, digits below 5 round down and digits of 5 or above round up, e.g.
344 ≈ 340
345 ≈ 350
346 ≈ 350
Students are likely to find it easier to understand rounding in terms of rounding to the
nearest hundred, whole number, tenth, etc than rounding to a specified number of
significant figures or decimal places.
Standard Form
Only the students in Set 1 in Years 8 and 9 and Sets 1 and 2 in Years 10 and 11 are likely
to be familiar with standard form notation. They should be able to use calculators for
calculation in standard form using the EXP or the xy button.
Students often have
difficulty in interpreting the result of a standard form calculation on some calculators
and need to be encouraged to write for example 3.42 x 108 and not 3.428.
17 ALGEBRA
Using Formulae
Most students should be able to cope with using simple formulae. Those who find it
difficult might find it helpful if the formula is written with boxes to insert the relevant
numbers.
S
e.g.
D=ST
If S = 40 and T =2 find D.
D=
40 T
×
2 = 80
Rules such as the triangle rule, work for specific types of questions and may be useful
for weaker students in some subjects as Science, PE
So with Distance, Speed and Time:
Speed = Distance
Time
,
Distance = Speed x Time , Time = Distance
Speed
Because speed is calculated by dividing a distance by a time the units of speed involve a
distance and a time e.g. metres per second or m/s or m s-1.
18 Also with Mass, Density and Volume:
Density = Mass ,
Volume
Mass = Density x Volume , Volume = Mass
Density
Because density is calculated by dividing a mass by a volume the units of density involve a
mass and a volume e.g. grams per cubic centimetre or g/cm3 or g cm-3
Plotting Points and Drawing Graphs
When drawing a diagram on which points are to be plotted, some students will need to be
reminded that numbers on the axes are written on the lines and not in the spaces, e.g.
P
0
1
2
3
4
5
X
Not
0
1
2
3
4
And equally spaced
X
0
1
2
3
4
5
6
Not
X
0
1
2
3
4
19 When drawing graphs for experimental data it is customary to use the horizontal axes
for the variable which has a regular interval, e.g.
(i)
In an experiment in which the temperature is taken every 5 minutes the
horizontal axis would be used for time and the vertical axis for
temperature/
(ii)
If the depth of a river is measured every metre, the horizontal axis would
be used for the distance from the bank and the vertical axis for the depth.
Having plotted points correctly students can be confused about whether they should join
the points or draw a line of best fit. Generally when the students have calculated the
points from an equation the points should be joined. If the points have been obtained
through experiment, a line of best fit is probably needed. Further details appear in the
following section on Data Handling.
20 GEOMETRY & MEASURES
Shapes
It is important to use correct names of shapes. 2D and 3D shapes and their properties
are below.
2D Shapes
A polygon is a 2D shape consisting of 3 or more straight sides. A regular polygon has all
sides and angles the same size. Specific names of polygons are shown below the table.
Number of sides
3
4
5
6
7
8
9
10
Name of polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Some triangles have special names:
Equilateral triangle
Isosceles triangle
Scalene triangle
All sides and angles
Two sides and two
All sides and angles are
are equal.
angles are equal.
different.
Right-angled triangle
One angle is a right angle (90º)
21 3D Shapes
Students should know all the names of the shapes below
Length, Mass and Capacity
All students should be familiar with metric units of length, mass and capacity but some
students will have difficulty with notation and converting from one unit to another.
They should know the following facts:
Length
Mass
10mm = 1 cm
1 000 g = 1 kg
100 cm = 1 m
1 000 kg = 1 tonne
Capacity
1 000 ml = 1 litre
1 000 m = 1 km
Many students will be unfamiliar with imperial units and the metric equivalents.
Length
Mass
Capacity
1 inch = 2.5 cm
1 ounce = 30 g
1 pint = 570 ml
1 foot = 30 cm
1 pound = 450 g
1 gallon = 4.5 litres
1 yard = 90 cm
1 stone = 6.4 kg
1 mile = 1.6 km
1 m = 40 inches
1 km =
5
8
mile 1 kg = 2.2 pounds
22 Students are encouraged to develop an awareness of the sizes of units and an ability to
make estimates in everyday contexts. They are also expected to consider the
appropriateness of their solution to a problem in the particular context of the question.
Time
All Students should be able to tell the time but a small number of students with learning
difficulties may experience problems in using analogue clocks and equating analogue and
digital times. Some students will need help with problems involving time, e.g.
A recipe says that a cake should be in the oven for 35 minutes.
If it is put into the oven at 10.50 a.m. when will it be ready?
With students who have difficulty this is usually best approached by splitting up the
time, e.g.
10.50 + 10 mins à 11.00
11.00 + 25 mins à 11.25
Most students should be familiar with 12 and 24 hour clock times but support may be
necessary when interpreting timetables.
When writing 24 hour clock times, four figures are used without any punctuation.
e.g.
0650 or 1427
24 hour clock times often appear on digital clocks with a colon between the hours and
minutes.
e.g.
06:50
or
14:57
12 hour clock times are usually written with a dot between the hours and the minutes, e.g.
6.50 am or 2.27 p.m.
Reading Scales
Students in lower ability groups often have difficulty in reading scales on graphs. They
are inclined to assume that one division on the scale represents one unit. It may help if
they begin by counting the number of divisions between each number on the scale and
then determine what each division of the scale
represents.
There is 5 units between each end of the scale
that has 10 divisions. So each division is worth
0.5
!
!"
=
23 Area and Volume
Problems can arise when calculating areas and volumes if the lengths give are not in the
same units. It is usually easiest to begin by converting all of the units of length to the
units that are required for the answer, before doing any calculation, e.g.
16 mm
Either
Area = 16 x 20 = 320 mm2
2 cm
or
Area = 1.6 x 2 = 3.2 cm2
A common error when converting square and cubic units is to use the wrong conversion
factor.
Area
2
Volume
2
3
1 cm = 100mm
1 cm = 1 000 mm3
1 m2 = 10 00 0 cm2
1 m3 = 1 000 000 cm3
1 km2 = 1 000 000 m2
Some higher ability students will be aware that 1 litre = 1 000 cm3
Area
1cm
10 mm
1cm
=
10mm
1cm × 1cm = 1 cm²
10mm × 10mm = 100 mm²
Volume
1m
100cm
1m
1m
1m × 1m × 1m = 1m³
=
100cm
100cm
100 cm × 100cm × 100cm = 1 000 000cm³
24 HANDLING DATA
Graphs and Diagrams
Bar graphs and line graphs should always be drawn on squared paper or graph paper.
Bar Graphs
All students should be able to draw bar graphs, but they will frequently draw them
incorrectly.
The way in which a graph is drawn depends on the type of data being
represented. Graphs should be drawn with gaps between the bars if the data categories
are not numeric (colours, types of vehicle, names of football teams etc). There should
also be gaps between the bars if the categories are numeric but can only take particular
values (shoe size, Key Stage 3 level etc).
If the data categories are numeric and
continuous they can take any value within the range (heights of children, time taken to
complete a marathon etc). In this case the data will be grouped, the horizontal axis
should be labelled as when plotting co-ordinates rather than under the bars and there
should be no gaps between the bars.
It is also important to emphasise the need to label axes correctly. In a bar graph with
separate columns, the labels should be underneath the columns on the horizontal axis. On
the vertical axis the labels should be on the lines, e.g.
Favourite mobile phone
Frequency
10 8 6 4 2 0 iphone HTC Samsung Sony Phone
25 Where data are continuous e.g. lengths, the horizontal scale should be like the scale used
for a graph on which points are plotted, e.g.
P
0
10
20
30
40
50
x
not
0-10
11-20
21-30 31-40 41-50
Line Graphs
Line graphs are used to show trends. They should only be used with data in which the
order in which the categories are written is significant.
Number of pupils 20 Graph to show number of text messages sent 15 10 5 0 1 2 3 4 5 Text messages sent in 1 hour 6 7 P
26 Graph to show favourite chocolate bar bought 10 Number of pupils 8 6 4 2 0 Mars Flake Aero Twix Favourite chocolate bar Twirl Computer Drawn Graphs
Students throughout the school should be able to use Excel to draw graphs to represent
data, although students in lower ability groups will need assistance. Because it is easy to
produce a wide variety of graphs, there is a tendency to produce diagrams that have
little relevance.
In most cases diagrams other than bar charts, column graphs, line
graphs, scatter graphs and pie charts should be avoided.
Pie Charts
The way in which students should be expected to calculate the angles for pie charts will
depend on the complexity of the question. If the numbers involved are simple it will be
possible to calculate simple fractions of 360o e.g.
Type of computer
Desk top
Frequency
20
Lap top
10
Tablet
6
Tower
4
Fraction
20 1
=
40 2
10 1
= 40 4
6
3
=
40 20
4
1
=
40 10
Angle
180º
90º
54º
36º
Total
40
360º
With more difficult numbers, students should first work out the share of 360o to be
allotted to one item and then multiply this by the number in each category, e.g.
students were asked what they intended to do after leaving school.
360o ÷ 180 = 2o so each student has 2o of the pie chart.
180
27 If data is in percentage form is to presented as a pie chart it is probable easiest to
multiply 360o by the decimal equivalent, e.g.
32% = 0.32
0.32 x 360o
= 115.20
≈ 1150
Alternatively a circular protractor, scaled in percentage divisions(pie chart scale) rather
than degrees, can be used.
Using Data
Range
The range of a set of data is the difference between the highest and lowest data values,
e.g.
If in an examination the highest mark is 80% and the lowest mark is 45%, the
range is 35% because 80% - 45% = 35%
The range is always a positive number, so it is not 45% - 80%.
Averages
Three different averages are commonly used:
•
The mean is calculated by adding up all of the values and dividing by the
number of values
•
The median is the middle value when the values have been arranged in order
•
The mode is the most common value. It is sometimes called the modal value,
e.g.
for the values
The mean is
3, 2, 5, 8, 4, 3, 6, 3, 2
!!!!!!!!!!!!!!!!!
!
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The median is 3 because 3 is the middle number when the values are put in order.
2 , 2 , 3 , 3, 3 , 4 , 5 , 6 , 8
The mode is 3 because 3 occurs most often.
28 Correlation
Correlation is a measure of how strongly two sets of data are connected. (It is important
to remember that a high correlation might occur coincidentally and does not necessarily
indicate a relationship between the two sets of data).
A high positive correlation (0.8 to 1.0) indicates that generally, as one data value
increases the other increases. A high negative correlation (-0.8 to -1.0) indicates that
generally as one value gets bigger the other value gets smaller. A correlation that is near
to zero suggests that there is no connection between the two sets of data.
Scatter graphs are used to indicate the degree of correlation. If the points are near to
the line of best fit there is a good correlation between the two sets of data.
2 2 2 1 1 1 0 0 0.5 1 1.5 2 0 0 Good posi<ve correla<on 0 0.5 1 1.5 2 0 Good nega<ve correla<on 0.5 1 1.5 2 No correla<on If there is a good correlation the line of best fit can be used to estimate an unknown
data value if one value is known.
Ice cream sales
Line of best fit
Temperature
29 Types of graphs
Favourite mobile phone
Bar Graph
Useful for comparing data in
different categories
•
The length of bars indicate the
size of the data categories
•
For qualitative data eg. colour, names and
discrete quantitative data eg. number of
children in family gaps are left between the
bars.
Frequency polygon
•
Useful for comparing data in
different categories and showing
trends.
•
The mid-points of the frequencies of
each group are joined.
Vertical line graph
•
Use for comparing data in different
categories.
•
This type of graph is generally just seen in
Maths.
•
The lengths of the lines indicates the frequency
of the data category.
•
For quantitative and discrete quantitative data.
•
Really a bar chart with narrow columns.
Frequency
•
10 8 6 4 2 0 iphone HTC Samsung Sony Phone
30 Line graph
•
Useful for showing upward
And downward trends in data.
•
Intermediate points might be
unreliable but might be used for
estimates.
Temperature(degrees Celsius)
38 37 36 35 34 33 32 31 30 29 Mon am Mon pm Time
Histograms
•
Useful for showing data in grouped
frequencies when the class intervals
are not all the same.
•
The area of the columns represent
the frequencies.
Pie chart
•
Used to show how something is divided up.
•
Useful when proportions are more important
than numbers.
Tues am Tues pm 31 Cumulative frequency diagram(polygon)
•
Useful for finding the number of values
above or below a specified point and to find the
median, quartiles and percentiles.
•
A cumulative frequency table is created from a
frequency table by adding each frequency to
the sum of its previous value.
•
In a cumulative frequency graph points
are plotted at the top of the range for
the class.
Scatter graph
•
Used to examine the relationship between
two variables.
•
If the points generally lie near to a line of
best
fit(trend
line)
there
is
a
good
correlation between the variables.
•
The line of best fit can be used to estimate
one variable given the other variable.
•
Estimates of variables found by extending
the line of best fit beyond the range of the
data are unreliable.
Pictogram
•
Pictograms
represent
are
charts
numbers
to
in
which
make
it
icons
more
interesting and easier to understand.
•
A key is often included to indicate what
each icon represents. All icons must be of
the same size, but a fraction of an icon can
be used to show the respective fraction of
that amount.
32 Function graph
•
Used to show the relationship between two
variables.
•
The function is represented by an equation
which can be used to produce a table of
values for the graph
•
Rienforce
coordinates,
the
correct
particularly
plotting
those
Travel graph
Used to show changes in distance or
velocity over a period of time,
•
The gradient of a distance time graph
gives the velocity.
•
The gradient of a velocity time graph
gives the accelration.
•
The area under a velocity time graph
gives the distance travelled.
2
involving
negatives.
•
of
-9 33 34 Numeracy Glossary
Acute angle – An angle measuring less than 90°
Add/addition – To join two or more quantities to get the sum or total
Adjacent – Next to
Algebra – An area of maths where unknown quantities are represented by letters
Alternate angles – Equal angles within parallel lines that are identified by a Z shape
Angle – The amount of turning between two lines meeting at the same point
Anti-clockwise – The opposite direction to which hands move round a clock
Approximate – To estimate a number, usually through rounding
Arc – A section of the circumference of a circle
Area – The size of the space a surface takes up, measured in units²
Ascending – Going up
Average – A summary of a set of data, either mode, median and mean
Axis – Reference lines on a graph
Bar graph – A graph using bars to show quantities for easy comparison
Bisect – To divide into two equal sections
Box plot – A diagram that uses a number line to show the distribution of data through the
minimum, lower quartile, median, upper quartile and maximum
Brackets – Symbols used to enclose an expression, ( )
Calculate – Work out, find the value of
Calculator – A device that performs mathematical operations
Capacity – The amount a container can hold
Centimetre – A metric unit for measuring length (10 millimetres)
Centre – The middle
Certain – Inevitable, will definitely happen
Chance – The likelihood that a particular outcome will occur
Circle – A 2D shape whose edge is always the same distance from the centre
Circumference – The perimeter of the circle
Chord – A straight line joining two points at the edge of the circle, not through the centre
Clockwise - The direction which hands move round a clock
Common denominator – A denominator which is a multiple of the other denominators
Compasses (pair of) – A mathematical instrument used to draw circles
Cone – A 3D shape with a circular base which tapers to a single vertex at the top
Congruent – Having the same shape and the same size
Continuous data – Data which could have an infinite number of values with a particular range
Coordinates – Pairs of numbers used to show a position of a graph with axes, eg (2,-4)
Corresponding angles– Equal angles within parallel lines that are identified by a F shape
Cross section – The face that results from slicing through a prism
Cube – A 3D shape with 6 square faces
Cuboid A 3D with 3 pairs of rectangular faces
Cube number – A number found by multiply a number by itself 3 times, eg 43 = 4 x 4 x 4 = 64
Cylinder – A prism whose cross section is a circle
Data – A collection of information
Decagon – A 2D shape with 10 sides
Decimal – A part of a number or a whole, 0.4 or 3.279
Decrease – To make smaller
Degree – The unit with which angles are measured, eg 67°
35 Denominator – The bottom number of a fraction
Density - The degree of compactness of a substance, found by mass ÷ volume
Descending – Going down
Diagonal – A straight line joining two non-adjacent vertices
Diameter – A line going through a circle edge to edge that passes through the centre
Dice – A cube marked with dots or numbers
Digit – A symbol used to show a number, 1 2 3...
Discrete data - Data which has only a finite number of values
Divide/division – To share equally, ÷
Double – To multiply by 2
Edge – The part of a 3D shape where 2 faces meet
Equal to/equals – To have the same value, =
Equation - Two expressions that are equal to each other
Equilateral triangle – A triangle with 3 equal sides and 3 equal angles
Equivalent fractions – Two fractions representing the same proportion
Estimate – To find a close answer by rounding
Even number – A number in the 2x table
Even chance – An outcome shares the same probability of occurring with another
Expression (algebraic) – Made up of terms and operations (algebra)
Exterior angle – The angle formed outside a polygon when a side is extended
Face – The flat part of a 3D shape
Factor – A number that divides exactly into another
Formula – A mathematical rule to describe a relationship between quantities
Fraction – A part of a number or a whole,
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Frequency – The number of times a particular value appears in a set of data
Gradient – The slope of a line
Gram – A metric unit for measuring mass
Graph – A drawing or diagram used to record information
Half – To divide by 2
Hexagon – A 2D shape with 6 sides
Heptagon – A 2D shape with 7 sides
Highest common factor – The greatest of all the factors shared by a pair of numbers
Horizontal – A straight line parallel to the horizon
Hypotenuse – The longest side of a right-angled triangle
Impossible – Will not happen
Improper fraction – A fraction with a larger numerator than denominator
Increase – To make bigger
Index/indices – Numbers or letters raised to a power, 4² or a6
Inequality – Two amounts not equal to each other, < ≤ ≥ >
Infinite/infinity – Unlimited, goes on forever
Integer – A whole number
Interior angle – An angle inside a polygon
Intersect – The point where two lines cross
Inverse operations – Opposite operations, + inverse to -, x inverse to ÷
Irregular (polygon) – A polygon with different sized sides and angles
Isometric (paper) – equal dimensions between dots
Isosceles triangle – A triangle with 2 equal sides and 2 equal angles
Kilogram – A metric unit for measuring mass (1000 grams)
36 Kilometre – A metric unit for measuring length (1000 metres)
Kite – A 2D shape with two pairs of equal sides and one pair of opposite angles that are equal
Line of symmetry – Divides a shape into two congruent sides
Linear – Has one dimension
Litre – A metric unit for measuring capacity (1000 millilitres)
Lowest common multiple - The smallest of all the multiples shared by a pair of numbers
Maximum – The greatest possible value
Mean – An average found by finding the sum of the data and dividing by the number of values
Median – An average found by locating the middle value of an ordered set of data
Metre – A metric unit for measuring length (100 centimetres, 1000 millimetres)
Midpoint – The middles point between 2 values or 2 coordinates
Millilitre – A metric unit for measuring capacity
Millimetre – A metric unit for measuring length
Minimum – The smallest possible value
Minus - Negative
Mixed number – A number comprised of an integer and a fraction
Mode – An average found by identifying the value with the highest frequency
Multiply/multiplication – A number is added to itself a number of times, x
Multiple – A number in another number’s times table
Negative – Below/less than zero/0, -4
Net – A 2D shape that can be folded into a 3D shape
Nonagon – A 2D shape with 9 sides
Number line – A line marked with numbers
Numerator – The top number of a fraction
Obtuse angle - An angle measuring more than 90° but less than 180°
Octagon – A 2D shape with 8 sides
Odd number – A number not in the 2 x table
Operations – Add, subtract, multiply, divide
Opposite angles – A pair of equal angles directly opposite each other formed by the intersection
of 2 straight lines
Origin – Coordinate (0,0)
Outcome – One of the possible results of a probability experiment
Outlier – A value far away from the others in a set of data (also called anomaly)
Parallel – Lines that are the same distance apart
Parallelogram – A 2D shape with 2 pairs of parallel lines
Pentagon – A 2D shape with 5 sides
Percent/percentage – A part of a number or a whole. Per cent means out of 100, 46%
Perimeter – The distance around the edge of a 2D shape
Perpendicular – Two lines meeting at a right-angle
Pi – Ratio of the circumference to a circle’s diameter, , 3.141592...
Pictogram – A graph using pictures to represent frequency
Pie chart – A graph using a divided circle where each section represents a part of the total
Place value – The value of a digit depending on its place in the number
Plan – A diagram showing the view from directly above
Plane – A flat surface
Polygon – A 2D shape with straight sides
Population – Whole set from which a sample is taken
Positive – Above/greater than zero/0
Prime – a number with only two factors, 1 and itself
37 Prime factor – A number which is both a factor of something and a prime
Prism – A 3D shape with a constant cross section throughout
Probability – The chance that a particular outcome will occur
Product – The result of multiplying
Proportion – A part to whole comparison
Protractor – An instrument used to measure the size of angles
Pyramid - A 3D shape with a polygon base which tapers to a single vertex at the top
Pythagoras – In any right-angled triangle where c is the hypotenuse, a² + b² = c²
Quadrant – Any quarter of a plane divided by an x- and y-axis
Quadrilateral – A 2D shape with 4 sides
Qualitative data – Non-numerical data
Quantitative data – Numerical data
Quantity – A number of something
Radius – The distance from the centre of a circle to its edge
Random – A chance pick from a number of items
Range – The smallest value subtracted from the greatest value
Ratio – Comparative value of 2 or more amounts
Reciprocal – One of two numbers whose product is 1, ½ and 2
Rectangle – A quadrilateral with two pairs of parallel sides with different lengths and all vertices
are right-angles
Recurring decimal – A decimal which has repeating digits or a repeating pattern of digits
Reflection – A mirror view
Reflex angle – An angle measuring more than 180° and less than 360°
Regular polygon – A polygon with all sides and angles equal
Remainder – The remaining amount after dividing a quantity by a number that is not a factor
Rhombus – A parallelogram with all sides equal
Right-angle – An angle measuring exactly 90°
Right-angled triangle – A triangle with one right-angle
Rotation – To turn an object
Rotational symmetry – When a turning shape has the same outline as the original shape
Round/rounding – Change the number to a more convenient value
Sample – A part of the population to be used
Scale factor – The ratio of two corresponding edges on a scaled drawing
Scalene triangle – A triangle with all different sides and all different angles
Scatter diagram – A diagram with coordinates plotted to show the relationship between two
variables
Sector – A section of a circle bounded by two radii and an arc
Segment – A section of a circle bounded by a chord and an arc
Semi-circle – Half a circle
Sequence – An ordered set of numbers or objects arranged according to a rule
Set (of data) – A collection of items
Similar - Having the same shape but a different size
Simplify (algebra) – To remove brackets, unnecessary terms and numbers
Simplify (fractions) – To reduce the numerator and denominator in a fraction to the smallest
numbers possible
Solve/solution – To work out the answer
Sphere – A 3D shape that is perfectly round, a ball
Square – A 2D shape with all equal sides and all angles 90º
Square number – A number that results by multiplying another number by itself
38 Square root – The opposite of squaring a number
Subtract/subtraction – To take one quantity away from another, Sum – The result of adding
Surface area – The area of the surface of a 3D shape
Symmetry – An object is symmetrical when one half is a mirror image of the other
Tally – Use of sets of 5 marks to record a total,
Term (nth) – One of the numbers in a sequence
Tessellation – Patterns of shapes that fit together without any gaps
Tetrahedron – A 3D shape with four triangular faces, a triangular-based pyramid
Three-dimensional (3D) – Having three dimensions, length, width and height
Transformation – A change in position or size
Translation – To move an item in any direction without rotating it
Trapezium – A 2D shape with four sides, two of them being parallel
Tree diagram – A diagram used to display the probability of different outcomes with each branch
representing one possible outcome
Triangle – A 2D shape with three sides
Triple/treble – To multiply by three
Two-dimensional (2D) - Having two dimensions, length and width
Unit - One
Unit of measure – Standard amount or quantity
Variable – Something that varies, represented by a letter in algebra
Venn diagram – A diagram using circles to show relationships between sets
Vertex/vertices – The point where two sides meet, or three or more faces
Vertical – Perpendicular to the horizon
Volume – The amount of space occupied by a 3D object
X-axis – The horizontal axis on a graph
Y-axis – The vertical axis on a graph
Y-intercept – Where a line intersects the y-axis
39 Index
2D shapes
Area
20 21 8 7 23 Bar graphs and frequency diagrams
24
Correlation and scatter graphs
28
Decimal notation
14 10 12 3D shapes
Addition decimal numbers
Addition whole numbers
Division
Fractions
Language of Operations
Length, mass and capacity
Conversions
Line graphs
Multiplication
Multiplication and division of decimals
Multiplying and dividing by 10, 100 and 1000
Negative numbers
Numeracy glossary
Order of operation (BODMAS)
Percentages
Pie charts
Plotting and drawing graphs
Ratio and proportion
Reading scales
Rounding numbers
Standard form numbers
Subtraction decimal numbers
Subtraction whole numbers
Time
Types of graphs
Using a calculator
Using data – mean, median, mode and range
Using formulae
Volume
When to use the different type of graphs
4 21 21 25 9 10 8 11 34 5
15 26 18 13 22 16 16 8 7 22 29 10 27 17 23 33 Writing and Reading Numbers
4
Writing calculations
5