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

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<strong>Geometry</strong> <strong>Measurement</strong> Chance <strong>and</strong> data<br />

<strong>Maths</strong><br />

Units of<br />

measurement<br />

Length<br />

mm<br />

cm<br />

m<br />

km<br />

Area<br />

mm 2<br />

cm 2<br />

m 2<br />

km 2<br />

ha<br />

Capacity<br />

mL<br />

L<br />

kL<br />

Mass<br />

mg<br />

g<br />

kg<br />

t<br />

millimetre<br />

centimetre<br />

metre<br />

kilometre<br />

square millimetre<br />

square centimetre<br />

square metre<br />

square kilometre<br />

hectare<br />

millilitre<br />

litre<br />

kilolitre<br />

milligram<br />

gram<br />

kilogram<br />

tonne<br />

Temperature<br />

°C degree Celsius<br />

°F degree Fahrenheit<br />

K<br />

Time<br />

s<br />

min<br />

h<br />

d<br />

kelvin<br />

second<br />

minute<br />

hour<br />

day<br />

m. month<br />

yr or y.<br />

cent. or C<br />

year<br />

century<br />

<strong>Measurement</strong><br />

Mathematical terms<br />

arbitrary units<br />

Non-st<strong>and</strong>ard units of measure; e.g. h<strong>and</strong><br />

spans, counters, tiles, claps.<br />

area (a)<br />

The measurement of surfaces <strong>and</strong> regions,<br />

usually expressed in square units of<br />

measurement.<br />

base (b)<br />

A particular side of a shape from which the<br />

height of an object can be measured.<br />

capacity<br />

The amount of space inside a container.<br />

What it holds, measured in mL, L etc.<br />

circumference (c)<br />

The perimeter of a circle; the distance<br />

measured around a circle.<br />

diameter (d)<br />

Straight line passing through the centre of<br />

a circle <strong>and</strong> terminating at each end by the<br />

circumference.<br />

height (h)<br />

The highest part of a shape from the base.<br />

length (l)<br />

The distance measured from end to end.<br />

mass (weight)<br />

The amount of matter in an object.<br />

Rules <strong>and</strong> formulas<br />

Circle<br />

Circumference:<br />

= 2 r or x d<br />

diameter<br />

radius<br />

Rectangle<br />

Prisms<br />

Area:<br />

= r 2<br />

Perimeter:<br />

= 2 x (l + w)<br />

Area:<br />

= l x w<br />

length<br />

V = A x H<br />

V = (l x w) x H<br />

side<br />

width<br />

Triangle<br />

Perimeter:<br />

= s + s + s<br />

Area:<br />

= 1 2<br />

x (b x h)<br />

base<br />

V = A x H<br />

V = b x h x H<br />

2<br />

perimeter (p)<br />

The length of the boundary of a plane region. In a<br />

polygon, it is the sum of the lengths of all sides.<br />

pi ( )<br />

The ratio of the circumference of a circle to its<br />

diameter. It is approximately 3.14<br />

Pythagoras’s theorem<br />

The square of the hypotenuse is equal to the sum<br />

of the squares of the other two sides.<br />

c 2 = a 2 + b 2<br />

c 2 = a 2 + b 2<br />

b c = hypotenuse<br />

a 2 = c 2 – b 2<br />

b 2 = c 2 – a 2<br />

a<br />

radius (r)<br />

The distance from the centre of a circle to a<br />

point on its circumference.<br />

side (s)<br />

The edge of a 2-D shape.<br />

surface area (sa)<br />

The total area of the outside surface of a 3-D<br />

shape.<br />

volume (v)<br />

The amount of space occupied by an object.<br />

width (w)<br />

The distance from one side to another; the<br />

breadth (b).<br />

height<br />

Parallelogram<br />

A = bh<br />

Area = base x height<br />

The surface area (TSA)<br />

The total area of the outside of a 3D shape;<br />

e.g. for a rectangular prism<br />

TSA = (2 x Area 1) + (2 x Area 2) + (2 x Area 3)<br />

rectangular<br />

prism<br />

triangular<br />

prism<br />

b<br />

cylinder<br />

h<br />

V = A x H<br />

V = ( r 2 ) x H<br />

Trapezium<br />

a<br />

h(a + b)<br />

A =<br />

2<br />

Area = sum of two<br />

parallel sides<br />

x height, halved<br />

b<br />

Square<br />

Perimeter:<br />

= 4 x l<br />

Area:<br />

= l x l<br />

or<br />

= l 2<br />

1<br />

3<br />

cube<br />

h<br />

2<br />

V = l x w x h<br />

Surface area = 6 x (l x h)<br />

length<br />

Presenting data<br />

Pie<br />

graph<br />

Portions of<br />

a circle are<br />

used to show<br />

a whole<br />

divided into<br />

parts.<br />

Line graph<br />

A graph which has a vertical <strong>and</strong><br />

a horizontal axis <strong>and</strong> is formed by<br />

joining points with straight lines<br />

or a curve to represent data.<br />

Box-<strong>and</strong>-whisker plot<br />

Chance <strong>and</strong> data<br />

Histogram<br />

Is used to represent small or large<br />

amounts of data.<br />

For example:<br />

The box shows the median, upper <strong>and</strong> lower quartiles (<strong>and</strong> interquartile range).<br />

The ends of the whiskers show the lowest <strong>and</strong> highest values in the data (the<br />

range).<br />

Multiple bar graph<br />

Can be used to graphically<br />

compare two sets of data.<br />

Fixed data is placed on the<br />

horizontal axis.<br />

Statistics<br />

Sport played<br />

No. of<br />

students<br />

Soccer 34<br />

T-ball 8<br />

Swimming 50<br />

Netball 34<br />

football 24<br />

basketball 50<br />

Collecting <strong>and</strong> classifying<br />

information <strong>and</strong> data from a<br />

sample for a specific purpose<br />

Data can be collected from a set<br />

group of people or from a ‘r<strong>and</strong>om<br />

sample’.<br />

A r<strong>and</strong>om sample is where every<br />

member of the sample has an equal<br />

chance of being chosen.<br />

For example:<br />

Test scores from a r<strong>and</strong>om group<br />

of students were collected. They are<br />

as follows:<br />

<strong>11</strong> 18 13 16 16 16 19<br />

12 <strong>15</strong> 14<br />

Percentage<br />

34<br />

x 100 200<br />

8<br />

200 x 100<br />

50<br />

x 100 200<br />

34<br />

x 100 200<br />

24<br />

x 100 200<br />

50<br />

x 100 200<br />

Information can be presented in many different ways.<br />

Bar graph<br />

A graph which represents information regarding frequency of outcomes<br />

using bar lengths. The graph has a vertical <strong>and</strong> horizontal axis. The bars<br />

may be vertical or horizontal.<br />

Stem-<strong>and</strong>-leaf plots<br />

Arranges data to show its shape <strong>and</strong><br />

distribution. It may be used to calculate the<br />

mean, median <strong>and</strong> mode of a set of data.<br />

Each data value is split into a ‘stem’ <strong>and</strong> a<br />

‘leaf’. In this case the leaf represents the unit<br />

digit <strong>and</strong> the stem represents all others.<br />

23, 25, 21<br />

32, 35<br />

47, 49<br />

represented as:<br />

stem leaf<br />

2<br />

3<br />

4<br />

3 5 1<br />

2 5<br />

7 9<br />

Scatter graph<br />

Used to compare two sets of data to determine if there is a<br />

correlation between them. The dots on the scatter graph represent<br />

the data points for each person. A straight line of best fi t may be<br />

drawn.<br />

NOTE<br />

Scatter graphs may<br />

show:<br />

weak strong<br />

positive relationships<br />

weak strong<br />

negative relationships<br />

no relationship<br />

©R.I.C. Publications<br />

Low Resolution Images<br />

Display Copy<br />

This graph shows a weak<br />

positive relationship.<br />

Once the data is collected it can be used to calculate:<br />

range (of a distribution)<br />

The difference between the greatest <strong>and</strong> least<br />

values in a set of data.<br />

For example:<br />

The greatest value is 19, the least value is <strong>11</strong>.<br />

Therefore, the range is 19 – <strong>11</strong> = 8.<br />

mode<br />

The measurement or observation that<br />

occurs most often; the item with the highest<br />

frequency.<br />

For example:<br />

The test score which occurs most frequently is<br />

16. The mode is 16.<br />

bimodal: can have more than one mode.<br />

mean (average)<br />

The mean of a set of numbers is a single number used to<br />

represent the set. It is calculated by dividing the sum of the<br />

numbers by the number in the sample.<br />

For example:<br />

<strong>11</strong> + 18 + 13 + 16 + 16 + 16 + 19 + 12 + <strong>15</strong> + 14 = <strong>15</strong>0<br />

<strong>15</strong>0 ÷ 10 (students) = <strong>15</strong> The mean is <strong>15</strong>.<br />

median<br />

The middle measurement, or score, when items are<br />

arranged in order of size. Where there is no middle score,<br />

the mean of the two central scores is taken.<br />

For example:<br />

<strong>11</strong>, 12, 13, 14, <strong>15</strong>, 16, 16, 16, 18, 19<br />

The median is<br />

<strong>15</strong> + 16<br />

= <strong>15</strong>.5<br />

2<br />

Chance<br />

The likelihood of an event occurring.<br />

For example:<br />

We have a 1 in 6 chance of throwing a 6 on a die. This would be written as 1 6 .<br />

Some common terms used in chance are:<br />

probability (P)<br />

The likelihood of a particular outcome in a chance event.<br />

For example:<br />

The probability of throwing a ‘head’ on the toss of a coin is 1 2 .<br />

Probabilities are always written as fractions or decimals between 0 <strong>and</strong> 1.<br />

Anything which has no chance of happening has a probability of 0.<br />

Things that are certain to happen have a probability of 1.<br />

To fi nd the probability of an event, use this calculation:<br />

P (event) =<br />

This is broken into:<br />

experimental probability<br />

When experiments need to be undertaken to gather<br />

data in order to calculate the probability of an event.<br />

Diagrams <strong>and</strong> tables<br />

Tree diagram<br />

These are used for classifi cation activities<br />

or to show possible outcomes of chance<br />

events.<br />

Tally<br />

Surveys<br />

Develop an<br />

objective question<br />

to ask the sample.<br />

Provide categories<br />

for answers.<br />

Use tallies <strong>and</strong><br />

tables to record<br />

data.<br />

Present data in a<br />

format which is<br />

easy to read.<br />

This is a stroke which is used to record<br />

items. Tallies generally use a diagonal stroke<br />

on the fi fth item.<br />

For example:<br />

Some data is best displayed in a diagram. There are many different types to choose from.<br />

Table<br />

These are used to organise data for a<br />

particular purpose.<br />

Number of children in each team<br />

Blue Red Green Gold<br />

Boys 46 43 49 32<br />

Girls 40 50 35 49<br />

Total 86 93 84 81<br />

Chance <strong>and</strong> data<br />

number of favourable outcomes<br />

total number of outcomes<br />

complement<br />

Every outcome has a complement.<br />

For example:<br />

‘winning’ <strong>and</strong> its complement, ‘not winning’ or ‘rolling a six’ <strong>and</strong> its complement,<br />

‘not rolling a six’.<br />

P(event occurring) + P(event not occurring) = 1 or<br />

outcome<br />

P(event not occurring) = 1 – P(event occurring)<br />

A result.<br />

Array<br />

An arrangement of numbers<br />

in rows <strong>and</strong> columns; eg a<br />

matrix.<br />

Pascal’s triangle is<br />

another example.<br />

Each entry is the sum<br />

of the two numbers<br />

directly above it.<br />

x x x x<br />

x x x x<br />

x x x x<br />

sample space (S)<br />

A list or diagram of all possible outcomes.<br />

For example:<br />

Tossing a coin:<br />

the sample space is the set {head, tail}.<br />

Tossing a single die:<br />

the sample space is {1, 2, 3, 4, 5, 6}<br />

theoretical probability<br />

To calculate the probability of an<br />

event.<br />

Two-way table<br />

These are used to display data that is<br />

related to each other.<br />

Even<br />

= 12<br />

Venn diagram<br />

Named after John Venn, an English logician.<br />

This diagram is used to represent the<br />

classifi cation of sets of items. It is possible<br />

to make a Carroll diagram from any Venn<br />

diagram.<br />

Carroll diagram<br />

Named after Lewis Carroll; author,<br />

mathematician <strong>and</strong> logician. This diagram is<br />

useful when recording classifi cation data.<br />

Not even<br />

Square 4, 16 1, 9<br />

Not<br />

square<br />

2, 6, 8, 10, 12,<br />

14, 18, 20<br />

3, 5, 7, <strong>11</strong>,<br />

13,<strong>15</strong>, 17, 19<br />

<strong>6835</strong>RE maths 2 y9.indd 1<br />

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Equivalent measures<br />

Capacity<br />

Length<br />

1 000 000 cm 3 = 1 m 3<br />

10 mm = 1 cm<br />

100 cm = 1 m<br />

1000 m = 1 km<br />

Time<br />

Area<br />

100 mm 2 = 1 cm 2<br />

10 000 cm 2 = 1 m 2<br />

10 000 m 2 = 1 ha<br />

100 ha = 1 km 2<br />

Mass<br />

1000 mg = 1 g<br />

1000 g = 1 kg<br />

1000 kg = 1 t<br />

Volume<br />

1000 mm 3 = 1 cm 3<br />

Angles of elevation<br />

The angle of elevation is the angle ray AB<br />

makes with the horizontal.<br />

Angles of depression<br />

The angle of depression is the angle ray CD<br />

makes with the horizontal.<br />

Trigonometric ratios<br />

How to choose a trigonometric ratio:<br />

For right angled triangles:<br />

θ<br />

Hypotenuse<br />

Adjacent<br />

Opposite<br />

1. Label each side of the triangle with O, A <strong>and</strong> H.<br />

2. Decide which two sides are part of the problem.<br />

3. Use: SOH sin θ = O<br />

H or<br />

CAH cos θ = A<br />

H or<br />

O<br />

TOA tan θ =<br />

A<br />

to solve the problem.<br />

<strong>Measurement</strong><br />

<strong>Measurement</strong><br />

1000 mL = 1 L<br />

1000 L = 1 kL<br />

60 seconds = 1 minute<br />

60 minutes = 1 hour<br />

24 hours = 1 day<br />

7 days = 1 week<br />

14 days = 1 fortnight<br />

28–31 days = 1 month<br />

~ 52 weeks = 1 year<br />

12 months = 1 year<br />

365 days = 1 year<br />

366 days = 1 leap year<br />

10 years = 1 decade<br />

100 years = 1 century<br />

1000 years = 1 millennium<br />

Unknown side<br />

Label it with a pronumeral (x) <strong>and</strong><br />

calculate.<br />

<strong>15</strong>º<br />

x mm (H)<br />

For example:<br />

20<br />

sin <strong>15</strong>° = x<br />

20<br />

x = sin <strong>15</strong>°<br />

x = 77.3 mm<br />

20 mm<br />

(O)<br />

Decimal measures<br />

Decimal measures<br />

<strong>Measurement</strong>s should be written in decimal<br />

form where possible.<br />

For example:<br />

10 cm, 3 mm 10.3 cm<br />

2 m, 7 cm, 5 mm 2.075 m<br />

3 km, 408 m, 52 cm 3.40852 km<br />

6 L, 430 mL 6.430 L<br />

<strong>15</strong> kL, 675 L <strong>15</strong>.675 kL<br />

12 g, 325 mg 12.325 g<br />

4 kg, 75 g 4.075 kg<br />

5 t, 620 kg 5.620 t<br />

Temperature<br />

A thermometer is used to measure<br />

temperature.<br />

Freezing<br />

point of<br />

water:<br />

0 °C<br />

Measuring tools<br />

Various tools are used to measure in<br />

different situations. For example:<br />

length<br />

mass<br />

capacity<br />

Unknown angle<br />

Choose the trigonometric ratio<br />

using the steps shown <strong>and</strong> calculate.<br />

For example:<br />

θº<br />

2<strong>15</strong> cm (A)<br />

Boiling<br />

point of<br />

water:<br />

100 °C<br />

ruler<br />

tape measure<br />

trundle wheel<br />

balance<br />

kitchen scales<br />

bathroom scales<br />

measuring spoons<br />

measuring cups<br />

measuring jugs<br />

graduated cylinders<br />

tan θ = O A<br />

84<br />

= 2<strong>15</strong><br />

θ = tan ( 2<strong>15</strong>)<br />

–1 84<br />

θ = 21.3°<br />

84 cm<br />

(O)<br />

2-D shapes<br />

2-D shapes have two dimensions—width <strong>and</strong> length.<br />

They may have curved or straight sides.<br />

Shapes with straight sides are called<br />

‘polygons’. Polygons have three or more<br />

sides <strong>and</strong> angles.<br />

Polygons with sides of an equal length <strong>and</strong><br />

angles of an equal size are called regular<br />

polygons.<br />

Polygons with three sides are called ‘triangles’.<br />

3-D shapes<br />

Parts of a 3-D shape include:<br />

faces<br />

The surfaces of a three-dimensional fi gure.<br />

edges<br />

The intersections of two faces of a<br />

three-dimensional fi gure.<br />

vertices<br />

The intersections of three edges of a threedimensional<br />

fi gure.<br />

<strong>Geometry</strong><br />

<strong>Geometry</strong><br />

Polygons with four sides are called<br />

‘quadrilaterals’.<br />

Quadrilaterals with opposite sides parallel<br />

are called ‘parallelograms’.<br />

Other shapes include circles, semicircles <strong>and</strong><br />

ellipses.<br />

prisms<br />

If the two ends of a prism are<br />

the same size <strong>and</strong> shape; they<br />

are congruent.<br />

Nets<br />

A net is a 2-D plan which can be used to<br />

make a 3-D shape.<br />

Three-dimensional shapes are also called ‘solid figures’. A 3-D shape has length, width<br />

<strong>and</strong> height.<br />

©R.I.C. Publications<br />

Low Resolution Images<br />

Display Copy<br />

regular polyhedra<br />

The faces of these 3-D shapes are all the same shape<br />

<strong>and</strong> size. There are fi ve: tetrahedron, octahedron,<br />

hexahedron, icosahedron <strong>and</strong> dodecahedron.<br />

pyramids<br />

A pyramid is made up of a base shape such as a<br />

triangle, square, hexagon or decagon.<br />

The opposite end of the shape forms a point<br />

called an ‘apex’.<br />

Angles<br />

An angle (<br />

acute angle<br />

An angle less than 90°.<br />

right angle<br />

An angle of exactly 90°.<br />

obtuse angle<br />

An angle between<br />

90° <strong>and</strong> 180°.<br />

reflex angle<br />

An angle between<br />

180° <strong>and</strong> 360°.<br />

The four angles in a<br />

quadrilateral add up to<br />

360º.<br />

rectangle<br />

90º + 90º + 90º + 90º = 360º<br />

rhombus<br />

a + b + a + b = 360°<br />

parallelogram<br />

a + b + a + b = 360º<br />

acute<br />

angle<br />

right<br />

angle<br />

obtuse angle<br />

reflex angle<br />

straight angle<br />

An angle which is exactly<br />

180°.<br />

straight angle<br />

complementary<br />

angles<br />

Two angles which add to<br />

90° (a right angle).<br />

) is formed when two lines meet. Angles are measured in degrees.<br />

A degree ( ) is a unit of angular measure. There are 360° in one complete<br />

rotation.<br />

A protractor is a tool used to measure the size of an angle.<br />

Quadrilaterals<br />

50°<br />

40°<br />

supplementary angles<br />

Two angles which<br />

add to 180° (a<br />

straight angle).<br />

corresponding<br />

angles<br />

alternate angles<br />

co-interior<br />

angles<br />

<strong>Geometry</strong><br />

<strong>Geometry</strong><br />

50°<br />

130°<br />

vertically opposite angles<br />

Congruent angles formed when two lines<br />

intersect.<br />

Mathematical terms<br />

attribute<br />

A characteristic of an object. A<br />

way to classify objects; e.g. round,<br />

red, thick.<br />

classification<br />

Arrangement into classes<br />

(sets or groups) according to<br />

attributes.<br />

congruent<br />

Two fi gures are congruent if they<br />

are the same size <strong>and</strong> shape.<br />

coordinates<br />

A referenced point on a grid can<br />

be found using an ‘ordered pair’<br />

of numbers. These are called the<br />

coordinates of the point. The<br />

horizontal axis is always read or<br />

written before the vertical axis.<br />

line<br />

Made up of an infi nite set<br />

of points extending in both<br />

directions.<br />

line segment<br />

A line with two end points.<br />

model<br />

A representation of an object<br />

preserving the signifi cant<br />

features.<br />

net<br />

A fl at pattern that can be folded<br />

to make a three-dimensional<br />

model.<br />

network<br />

A system of lines (paths) <strong>and</strong><br />

nodes (points representing<br />

intersections).<br />

parallel lines<br />

Two or more straight lines in<br />

the same plane which will never<br />

meet. They are always the same<br />

distance apart.<br />

path<br />

A line connecting points (nodes)<br />

in a network.<br />

perpendicular lines<br />

Two lines which form right angles.<br />

plane<br />

A surface which is fl at <strong>and</strong> has no<br />

boundaries.<br />

Triangles<br />

Triangles<br />

angles in a triangle<br />

Triangles can be named according to the size<br />

of their angles.<br />

exterior angles<br />

The exterior angle<br />

of a triangle is<br />

equal to the sum of<br />

the two opposite<br />

interior angles.<br />

congruent triangles<br />

Triangles are congruent if they are identical<br />

in shape <strong>and</strong> size.<br />

Two triangles can be called congruent if:<br />

SSS: all sides the same length or<br />

SAS: two sides the same <strong>and</strong> the angle<br />

between those sides the same or<br />

ASA: two angles the same <strong>and</strong> the side<br />

between those angles the same length<br />

or<br />

RHS: If the hypotenuse <strong>and</strong> a side of one<br />

triangle are respectively equal to the<br />

hypotenuse <strong>and</strong> a side of the other<br />

triangle, then the two right triangles<br />

are congruent.<br />

ray<br />

Made up of an infi nite set of<br />

points emanating from a point<br />

<strong>and</strong> going in one direction.<br />

scale<br />

The ratio of measurements<br />

of a model or diagram to<br />

corresponding measurements of<br />

an enlarged or reduced version.<br />

section<br />

A fl at surface made by cutting<br />

through a solid in any direction.<br />

similar<br />

Two shapes are similar when<br />

they have the same shape but are<br />

different in size.<br />

tessellation<br />

A repeating pattern of congruent<br />

shapes that completely cover an<br />

area leaving no gaps or overlaps.<br />

traversable<br />

A network is traversable if all<br />

paths can be traced over without<br />

going over the same path twice.<br />

<strong>Maths</strong><br />

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<strong>RIC</strong>–<strong>6835</strong><br />

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