RIC-6835 Maths Essentials - Geometry and Measurement 1 (Ages 11-15)

Geometry Measurement Chance and data
Maths
Units of
measurement
Length
mm
cm
m
km
Area
mm 2
cm 2
m 2
km 2
ha
Capacity
mL
L
kL
Mass
mg
g
kg
t
millimetre
centimetre
metre
kilometre
square millimetre
square centimetre
square metre
square kilometre
hectare
millilitre
litre
kilolitre
milligram
gram
kilogram
tonne
Temperature
°C degree Celsius
°F degree Fahrenheit
K
Time
s
min
h
d
kelvin
second
minute
hour
day
m. month
yr or y.
cent. or C
year
century
Measurement
Mathematical terms
arbitrary units
Non-standard units of measure; e.g. hand
spans, counters, tiles, claps.
area (a)
The measurement of surfaces and regions,
usually expressed in square units of
measurement.
base (b)
A particular side of a shape from which the
height of an object can be measured.
capacity
The amount of space inside a container.
What it holds, measured in mL, L etc.
circumference (c)
The perimeter of a circle; the distance
measured around a circle.
diameter (d)
Straight line passing through the centre of
a circle and terminating at each end by the
circumference.
height (h)
The highest part of a shape from the base.
length (l)
The distance measured from end to end.
mass (weight)
The amount of matter in an object.
Rules and formulas
Circle
Circumference:
= 2 r or x d
diameter
radius
Rectangle
Prisms
Area:
= r 2
Perimeter:
= 2 x (l + w)
Area:
= l x w
length
V = A x H
V = (l x w) x H
side
width
Triangle
Perimeter:
= s + s + s
Area:
= 1 2
x (b x h)
base
V = A x H
V = b x h x H
2
perimeter (p)
The length of the boundary of a plane region. In a
polygon, it is the sum of the lengths of all sides.
pi ( )
The ratio of the circumference of a circle to its
diameter. It is approximately 3.14
Pythagoras’s theorem
The square of the hypotenuse is equal to the sum
of the squares of the other two sides.
c 2 = a 2 + b 2
c 2 = a 2 + b 2
b c = hypotenuse
a 2 = c 2 – b 2
b 2 = c 2 – a 2
a
radius (r)
The distance from the centre of a circle to a
point on its circumference.
side (s)
The edge of a 2-D shape.
surface area (sa)
The total area of the outside surface of a 3-D
shape.
volume (v)
The amount of space occupied by an object.
width (w)
The distance from one side to another; the
breadth (b).
height
Parallelogram
A = bh
Area = base x height
The surface area (TSA)
The total area of the outside of a 3D shape;
e.g. for a rectangular prism
TSA = (2 x Area 1) + (2 x Area 2) + (2 x Area 3)
rectangular
prism
triangular
prism
b
cylinder
h
V = A x H
V = ( r 2 ) x H
Trapezium
a
h(a + b)
A =
2
Area = sum of two
parallel sides
x height, halved
b
Square
Perimeter:
= 4 x l
Area:
= l x l
or
= l 2
1
3
cube
h
2
V = l x w x h
Surface area = 6 x (l x h)
length
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
or a curve 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
Collecting and classifying
information and data from a
sample for a specific purpose
Data can be collected from a set
group of people or from a ‘random
sample’.
A random sample is where every
member of the sample has an equal
chance of being chosen.
For example:
Test scores from a random group
of students were collected. They are
as follows:
11 18 13 16 16 16 19
12 15 14
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.
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’. In this case 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
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This graph shows a weak
positive relationship.
Once the data is collected it can be used to calculate:
range (of a distribution)
The difference between the greatest and least
values in a set of data.
For example:
The greatest value is 19, the least value is 11.
Therefore, the range is 19 – 11 = 8.
mode
The measurement or observation that
occurs most often; the item with the highest
frequency.
For example:
The test score which occurs most frequently is
16. The mode is 16.
bimodal: can have more than one mode.
mean (average)
The mean of a set of numbers is a single number used to
represent the set. It is calculated by dividing the sum of the
numbers by the number in the sample.
For example:
11 + 18 + 13 + 16 + 16 + 16 + 19 + 12 + 15 + 14 = 150
150 ÷ 10 (students) = 15 The mean is 15.
median
The middle measurement, or score, when items are
arranged in order of size. Where there is no middle score,
the mean of the two central scores is taken.
For example:
11, 12, 13, 14, 15, 16, 16, 16, 18, 19
The median is
15 + 16
= 15.5
2
Chance
The likelihood of an event occurring.
For example:
We have a 1 in 6 chance of throwing a 6 on a die. This would be written as 1 6 .
Some common terms used in chance are:
probability (P)
The likelihood of a particular outcome in a chance event.
For example:
The probability of throwing a ‘head’ on the toss of a coin is 1 2 .
Probabilities are always written as fractions or decimals between 0 and 1.
Anything which has no chance of happening has a probability of 0.
Things that are certain to happen have a probability of 1.
To fi nd the probability of an event, use this calculation:
P (event) =
This is broken into:
experimental probability
When experiments need to be undertaken to gather
data in order to calculate the probability of an event.
Diagrams and tables
Tree diagram
These are used for classifi cation activities
or to show possible outcomes of chance
events.
Tally
Surveys
Develop an
objective question
to ask the sample.
Provide categories
for answers.
Use tallies and
tables to record
data.
Present data in a
format which is
easy to read.
This is a stroke which is used to record
items. Tallies generally use a diagonal stroke
on the fi fth item.
For example:
Some data is best displayed in a diagram. There are many different types to choose from.
Table
These are used to organise data for a
particular purpose.
Number of children in each team
Blue Red Green Gold
Boys 46 43 49 32
Girls 40 50 35 49
Total 86 93 84 81
Chance and data
number of favourable outcomes
total number of outcomes
complement
Every outcome has a complement.
For example:
‘winning’ and its complement, ‘not winning’ or ‘rolling a six’ and its complement,
‘not rolling a six’.
P(event occurring) + P(event not occurring) = 1 or
outcome
P(event not occurring) = 1 – P(event occurring)
A result.
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.
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}
theoretical probability
To calculate the probability of an
event.
Two-way table
These are used to display data that is
related to each other.
Even
= 12
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.
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
6835RE maths 2 y9.indd 1
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Equivalent measures
Capacity
Length
1 000 000 cm 3 = 1 m 3
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
Time
Area
100 mm 2 = 1 cm 2
10 000 cm 2 = 1 m 2
10 000 m 2 = 1 ha
100 ha = 1 km 2
Mass
1000 mg = 1 g
1000 g = 1 kg
1000 kg = 1 t
Volume
1000 mm 3 = 1 cm 3
Angles of elevation
The angle of elevation is the angle ray AB
makes with the horizontal.
Angles of depression
The angle of depression is the angle ray CD
makes with the horizontal.
Trigonometric ratios
How to choose a trigonometric ratio:
For right angled triangles:
θ
Hypotenuse
Adjacent
Opposite
1. Label each side of the triangle with O, A and H.
2. Decide which two sides are part of the problem.
3. Use: SOH sin θ = O
H or
CAH cos θ = A
H or
O
TOA tan θ =
A
to solve the problem.
Measurement
Measurement
1000 mL = 1 L
1000 L = 1 kL
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
14 days = 1 fortnight
28–31 days = 1 month
~ 52 weeks = 1 year
12 months = 1 year
365 days = 1 year
366 days = 1 leap year
10 years = 1 decade
100 years = 1 century
1000 years = 1 millennium
Unknown side
Label it with a pronumeral (x) and
calculate.
15º
x mm (H)
For example:
20
sin 15° = x
20
x = sin 15°
x = 77.3 mm
20 mm
(O)
Decimal measures
Decimal measures
Measurements should be written in decimal
form where possible.
For example:
10 cm, 3 mm 10.3 cm
2 m, 7 cm, 5 mm 2.075 m
3 km, 408 m, 52 cm 3.40852 km
6 L, 430 mL 6.430 L
15 kL, 675 L 15.675 kL
12 g, 325 mg 12.325 g
4 kg, 75 g 4.075 kg
5 t, 620 kg 5.620 t
Temperature
A thermometer is used to measure
temperature.
Freezing
point of
water:
0 °C
Measuring tools
Various tools are used to measure in
different situations. For example:
length
mass
capacity
Unknown angle
Choose the trigonometric ratio
using the steps shown and calculate.
For example:
θº
215 cm (A)
Boiling
point of
water:
100 °C
ruler
tape measure
trundle wheel
balance
kitchen scales
bathroom scales
measuring spoons
measuring cups
measuring jugs
graduated cylinders
tan θ = O A
84
= 215
θ = tan ( 215)
–1 84
θ = 21.3°
84 cm
(O)
2-D shapes
2-D shapes have two dimensions—width and length.
They may have curved or straight sides.
Shapes with straight sides are called
‘polygons’. Polygons have three or more
sides and angles.
Polygons with sides of an equal length and
angles of an equal size are called regular
polygons.
Polygons with three sides are called ‘triangles’.
3-D shapes
Parts of a 3-D shape include:
faces
The surfaces of a three-dimensional fi gure.
edges
The intersections of two faces of a
three-dimensional fi gure.
vertices
The intersections of three edges of a threedimensional
fi gure.
Geometry
Geometry
Polygons with four sides are called
‘quadrilaterals’.
Quadrilaterals with opposite sides parallel
are called ‘parallelograms’.
Other shapes include circles, semicircles and
ellipses.
prisms
If the two ends of a prism are
the same size and shape; they
are congruent.
Nets
A net is a 2-D plan which can be used to
make a 3-D shape.
Three-dimensional shapes are also called ‘solid figures’. A 3-D shape has length, width
and height.
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regular polyhedra
The faces of these 3-D shapes are all the same shape
and size. There are fi ve: tetrahedron, octahedron,
hexahedron, icosahedron and dodecahedron.
pyramids
A pyramid is made up of a base shape such as a
triangle, square, hexagon or decagon.
The opposite end of the shape forms a point
called an ‘apex’.
Angles
An angle (
acute angle
An angle less than 90°.
right angle
An angle of exactly 90°.
obtuse angle
An angle between
90° and 180°.
reflex angle
An angle between
180° and 360°.
The four angles in a
quadrilateral add up to
360º.
rectangle
90º + 90º + 90º + 90º = 360º
rhombus
a + b + a + b = 360°
parallelogram
a + b + a + b = 360º
acute
angle
right
angle
obtuse angle
reflex angle
straight angle
An angle which is exactly
180°.
straight angle
complementary
angles
Two angles which add to
90° (a right angle).
) is formed when two lines meet. Angles are measured in degrees.
A degree ( ) is a unit of angular measure. There are 360° in one complete
rotation.
A protractor is a tool used to measure the size of an angle.
Quadrilaterals
50°
40°
supplementary angles
Two angles which
add to 180° (a
straight angle).
corresponding
angles
alternate angles
co-interior
angles
Geometry
Geometry
50°
130°
vertically opposite angles
Congruent angles formed when two lines
intersect.
Mathematical terms
attribute
A characteristic of an object. A
way to classify objects; e.g. round,
red, thick.
classification
Arrangement into classes
(sets or groups) according to
attributes.
congruent
Two fi gures are congruent if they
are the same size and shape.
coordinates
A referenced point on a grid can
be found using an ‘ordered pair’
of numbers. These are called the
coordinates of the point. The
horizontal axis is always read or
written before the vertical axis.
line
Made up of an infi nite set
of points extending in both
directions.
line segment
A line with two end points.
model
A representation of an object
preserving the signifi cant
features.
net
A fl at pattern that can be folded
to make a three-dimensional
model.
network
A system of lines (paths) and
nodes (points representing
intersections).
parallel lines
Two or more straight lines in
the same plane which will never
meet. They are always the same
distance apart.
path
A line connecting points (nodes)
in a network.
perpendicular lines
Two lines which form right angles.
plane
A surface which is fl at and has no
boundaries.
Triangles
Triangles
angles in a triangle
Triangles can be named according to the size
of their angles.
exterior angles
The exterior angle
of a triangle is
equal to the sum of
the two opposite
interior angles.
congruent triangles
Triangles are congruent if they are identical
in shape and size.
Two triangles can be called congruent if:
SSS: all sides the same length or
SAS: two sides the same and the angle
between those sides the same or
ASA: two angles the same and the side
between those angles the same length
or
RHS: If the hypotenuse and a side of one
triangle are respectively equal to the
hypotenuse and a side of the other
triangle, then the two right triangles
are congruent.
ray
Made up of an infi nite set of
points emanating from a point
and going in one direction.
scale
The ratio of measurements
of a model or diagram to
corresponding measurements of
an enlarged or reduced version.
section
A fl at surface made by cutting
through a solid in any direction.
similar
Two shapes are similar when
they have the same shape but are
different in size.
tessellation
A repeating pattern of congruent
shapes that completely cover an
area leaving no gaps or overlaps.
traversable
A network is traversable if all
paths can be traced over without
going over the same path twice.
Maths
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RIC–6835
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