RIC-6836 Maths Essentials - Geometry and Measurement 2 (Ages 11-15)

Geometry Measurement Chance and data
Maths
Units of
measurement
Length
Area
mm
cm
m
km
mm 2
cm 2
m 2
km 2
ha
Capacity
mL
L
kL
Mass
mg
g
kg
t
Temperature
millimetre
centimetre
metre
kilometre
square millimetre
square centimetre
square metre
square kilometre
hectare
millilitre
litre
kilolitre
milligram
gram
kilogram
tonne
°C degree Celsius
°F degree Fahrenheit
K
kelvin
Capacity
To calculate the capacity
of a container, calculate its
volume and then convert to
units of capacity.
1 cm 3 = 1 mL
1000 cm 3 = 1 L
1 m 3 = 1000 L = 1 kL
Density
Density is the mass per unit
volume.
density =
mass
volume
d = m V
Units: g/cm 3 or kg/m 3
Measurement
Rules and formulas
Composite shapes
A composite shape is a combination
of other shapes. The area can be
calculated by adding the area of the
individual shapes.
Area formulas
Circle
Circumference:
= 2 r or x d
diameter
radius
Area:
= r 2
Parallelogram
A = bh
Area = base x height
Rectangular
prism
b
V = A x H
V = (l x w) x H
side
Triangle
h
Perimeter:
= s + s + s
Area:
= 1 2
x (b x h)
base
Triangular
prism
For example:
height
The surface area (TSA)
is the total area of the outside surfaces of three-dimensional shapes.
Cube
TSA = 6l 2
Rectangular prism
TSA = 2(lw + hw + hl)
Volume of prisms
Volume of non-prisms
Cone
V = 1 3 AH
= 1 3 r2 H
Cylinder
TSA = 2 r(r + h)
Cone
TSA = r 2 + rs
Some solids are irregular and require specifi c formulas.
Trapezium
V = A x H
V = b x h x H
2
Square pyramid
1 2
Area = Area 1 + Area 2 + Area 3
Rectangle
Perimeter:
= 2 x (l + w)
Area:
= l x w
length
Cylinder
3
Sphere
TSA = 4 r 2
Square-based pyramid
TSA = b 2 + 2bh
V = 1 3 AH
= 1 3 lwH Sphere
a
A = 1 2
(a + b) x h
Area = sum of two
parallel sides
x height, halved
b
h
width
Presenting data
Pie
graph
Portions of
a circle are
used to show
a whole
divided into
parts.
Line graph
A graph which has a vertical and
a horizontal axis and is formed by
joining points with straight lines
to represent data.
Box-and-whisker plot
Chance and data
Histogram
Is used to represent small or large
amounts of data.
For example:
The box shows the median, upper and lower quartiles (and interquartile range).
The ends of the whiskers show the lowest and highest values in the data (the
range).
Multiple bar graph
Can be used to graphically
compare two sets of data.
Fixed data is placed on the
horizontal axis.
Statistics
Sport played
No. of
students
Soccer 34
T-ball 8
Swimming 50
Netball 34
football 24
basketball 50
Bar graph
A graph which represents information regarding frequency of outcomes
using bar lengths. The graph has a vertical and horizontal axis. The bars
may be vertical or horizontal.
Stem-and-leaf plots
Arranges data to show its shape and
distribution. It may be used to calculate the
mean, median and mode of a set of data.
Each data value is split into a ‘stem’ and a
‘leaf’. The leaf represents the unit digit and
the stem represents all others.
23, 25, 21
32, 35
47, 49
represented as:
stem leaf
2
3
4
3 5 1
2 5
7 9
Scatter graph
Used to compare two sets of data to determine if there is a
correlation between them. The dots on the scatter graph represent
the data points for each person. A straight line of best fi t may be
drawn.
NOTE
Scatter graphs may
show:
weak strong
positive relationships
weak strong
negative relationships
no relationship
V = A x H
Collecting and classifying information and data from a sample for a specific purpose.
V = ( r 2 ) x H
convenience sampling
median
Choosing a sample the most costeffi
cient/easiest way.
are arranged in order of size. Where there is no
The middle measurement, or score, when items
middle score, the mean of the two central scores
cumulative frequency
is taken.
Number of results below a given value.
For example:
discrete variables
11, 12, 13, 14, 15, 16, 16, 16, 18, 19
V = 4 Variables that can be measured exactly.
The median is
3 r3 interquartile range (IQR)
15 + 16
= 15.5
Range of 50% of a distribution. Elimates mode 2
extreme values.
IQR = upper quartile – lower quartile
mean (average) ( x )
Add up all the results and divide by the
number of results.
values
mean =
number of values
Percentage
34
x 100 200
8
200 x 100
50
x 100 200
34
x 100 200
24
x 100 200
50
x 100 200
Information can be presented in many different ways.
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This graph shows a weak
positive relationship.
The result that occurs most frequently. The
modal class is the class interval with the highest
frequency.
bimodal: can have more than one mode.
outliers
Surveyed results that are outside a certain range.
random sampling
Each member in the population has the same
chance as every other member of being
included in the sample.
range (of a distribution)
The difference between the greatest and least
value in a set of data.
sample
A portion of the population chosen to take
part in a questionnaire/survey.
systematic sampling
Choosing a sample in an organised way (e.g.
every fi fth person).
Chance
addition law of probability
When A and B are mutually exclusive
events:
P(A or B) = P(A) + P(B)
complement
A situation with two possible outcomes
has one outcome and its complement; e.g.
‘winning’ and its complement ‘not winning’
or ‘rolling a six’ and ‘not rolling a six’.
P (event not occurring) =
1 – P (event occurring)
or P(A´) = 1 – P(A)
A´ is the complement of A
experimental probability
When experiments need to be undertaken
to gather data in order to calculate the
probability of an event.
independent events
Knowing the outcome of one event does
not affect the probability of another event.
An example is two separate coin tosses.
The fi rst toss being a ‘heads’ does not
affect the probability of the second toss.
Diagrams and tables
Array
An arrangement of numbers
in rows and columns; eg a
matrix.
Pascal’s triangle is another
example.
Each entry is the
sum of the two
numbers directly
above it.
Tables
These are used to organise data for a
particular purpose.
Number of children in each team
Blue
Red
Green
Some data is best displayed in a diagram.
There are many different types to choose from.
x x x x
x x x x
x x x x
sample space (S)
A list or diagram of all possible outcomes.
For example:
Tossing a coin:
the sample space is the set {head, tail}.
Tossing a single die:
the sample space is {1, 2, 3, 4, 5, 6}
Gold
Boys 46 43 49 32
Girls 40 50 35 49
Total 86 93 84 81
Chance and data
odds
Used in gambling or sport.
a = number of ways to lose
b = number of ways to win
Odds against an event occurring = a:b
a + b = total number of outcomes
a
Probability of winning = a + b
probability (P)
The likelihood of a particular outcome in a
chance event.
of rolling a six etc.
1
6
The likelihood of an event occurring.
Probabilities are always written as fractions
or decimals between 0 and 1.
0 P (event) 1
To fi nd the probability of an event, we use
this calculation:
P (successful event)
number of favourable outcomes
=
total number of outcomes
Tree diagram
These are used for classifi cation activities
or to show possible outcomes of chance
events.
Two-way table
These are used to display data that are
related to each other.
outcome
A result.
multiplication law of probability
When A and B are independent:
P(A and B) = P(A) x P(B)
mutually exclusive
If one event occurs, the other does not. An
example is tossing a coin. Either it is ‘heads’
or ‘tails’. It can not be both.
theoretical probability
To have an idea of what the probability of
an event is without collecting the data. In
the long run experimental and theoretical
probabilities become much closer.
probability tree diagram
A probability tree diagram has probability
values assigned to each branch.
Venn diagram
Named after John Venn, an English logician.
This diagram is used to represent the
classifi cation of sets of items. It is possible
to make a Carroll diagram from any Venn
diagram.
Carroll diagram
Named after Lewis Carroll; author,
mathematician and logician. This diagram is
useful when recording classifi cation data.
Even
Not even
Square 4, 16 1, 9
Not
square
2, 6, 8, 10, 12,
14, 18, 20
3, 5, 7, 11,
13,15, 17, 19
6836RE maths 2 y10.indd 1
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