VCE 11 General Chapter 1

VCE

General Mathematics Units 1 & 2 Accreditation period from 2023

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Title: Mathspace VCE General Mathematics Units 1 & 2 Print run 1 - January 2026 ISBN: 978-1-963022-96-4 Editors and lead authors: Erin Gallagher, Jeremy Hunter, Elisha Templin, Nazeel Ali, Katrina Du, Neeti Meghani, Jessica Purton, Isha Tiwari Writing and development team: Adam Humphreys, Ma Lourdes Ona, Sheramie Castillo, Regine Jara, Lesley Ann Lopez, Adriane Abunda, Jady Amarillas, Joy Ann Antonio, Macy Castillo, Christian Dinglasan, Adrian Doctolero, Dharent Fernandez, Dhave Fernandez, Rona Josue, Jessa Lambinicio, Eric Malabag, Julie Ann Manzano, Kyra Joy Manzano, Mary Joy Tacud, Albert Jara Images and design team: Scott Nolan, Chastine Marquez, Keith Gimeno, Jasper Jumawan, Mary Matheu, Jemark Orevillo, Jessa Ortega

Contents 1

Data distributions

2

1.01

Types of data

4

1.02

Categorical data displays

13

1.03

Numerical data displays

26

1.04

Measures of centre

42

Investigation: The problem with average 1.05

Data distribution

56

1.06

Quartiles and spread

72

1.07

Box plots

85

1.08

Outliers

102

1.09

Standard deviation

118

1.10

Compare sets of data

138

Chapter 1 review

2

Linear relationships

160

2.01

Solve linear equations

162

2.02

Linear equation applications

168

2.03

Graphs from tables

175

2.04

Slopes and intercepts

187

2.05

Sketch linear functions

200

2.06

Equations of straight lines

208

Investigation: Filling a pool Chapter 2 review

3

Linear applications

224

Linear models

226

3.01

Investigation: Intersection of lines 3.02

Graphical simultaneous equations

237

3.03

Algebraic simultaneous equations

247

3.04

Applications of simultaneous equations

256

3.05

Piecewise functions

262

3.06

Step graphs

276

Chapter 3 review

Contents mathspace.co

iii

4

Financial arithmetic

288

4.01

Percentages

290

4.02

Rates and ratios

301

4.03

Unit cost comparisons

309

4.04

Percentage change and GST

315

4.05

Simple interest

326

4.06

Compound interest

337

Investigation: Spreadsheets and interest 4.07

Financial applications

346

4.08

Purchase options

359

Chapter 4 review

5

Number patterns and recurrence relations

374

5.01

Number patterns and sequences

376

5.02

First-order linear recurrence relations

387

5.03

Arithmetic sequences

400

5.04

Applications of arithmetic sequences

410

5.05

Geometric sequences

421

5.06

Applications of geometric sequences

431

Investigation: Saving money Chapter 5 review

6

Matrices

440

6.01

Introduction to matrices

442

6.02

Represent information in matrices

449

6.03

Addition and subtraction of matrices

460

6.04

Scalar multiplication

467

6.05

Matrix multiplication

473

6.06

Powers of matrices

486

6.07

Applications of matrices

498

6.08

Inverse matrices

510

6.09

Simultaneous equations with matrices

518

Investigation: Is there always a solution?

6.10

Transition matrices Chapter 6 review

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528

7

Graphs and networks

542

7.01

Introduction to graphs

544

7.02

Degree and adjacency matrices

556

7.03

Planar graphs and Euler’s formula

569

7.04

Walks and paths

583

7.05

Traverse connected graphs

600

7.06

Weighted graphs and networks

612

7.07

Trees

627

Investigation: Networks and coding Chapter 7 review

8

Compare numerical variables

642

8.01

Explanatory and response variables

644

8.02

Scatterplots and correlation

652

8.03

Association and causation

666

8.04

Line of good fit

675

Investigation: Answer questions with data Chapter 8 review

9

Variation 694 9.01

Direct and inverse variation

696

9.02

Orders of magnitude and logarithms

707

Investigation: Represent Australian populations 9.03

Log scales

717

9.04

Data transformations

725

9.05

Data models

746

Chapter 9 review

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v

10

Shape and measurement

760

10.01

Significant figures

762

10.02

Measurement units

770

10.03

Pythagoras’ theorem

781

10.04

Perimeter

793

Investigation: Circumference 10.05

Arc length and perimeter of sectors

803

10.06

Area

814

10.07

Area of composite shapes

823

10.08

Total surface area

835

10.09

Volume

851

10.10

Composite solids

863

10.11

Similarity and scale factors

877

Investigation: Scale diagrams 10.12

Area and volume in similar figures

890

Chapter 10 review

Chapter 10 review

11

Applications of trigonometry

902

11.01

Trigonometric ratios

904

11.02

2D Applications of trigonometry

920

11.03

Trigonometry in 3D

933

11.04

Trigonometry of obtuse angles

944

11.05

Sine rule

950

11.06

Cosine rule

966

11.07

Area of non-right angled triangles

977

11.08

Mixed applications

990

Investigation: Triangle inequality Chapter 11 review

Answers

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1002

“Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”

– Paul Lockhart

Big ideas • Effective data analysis begins with classifying data by type (categorical or numerical) and organising it through appropriate visual displays to reveal initial patterns and frequencies.

• Key statistical measures are used to summarise and interpret the essential features of a dataset, including its central tendency, spread, and overall shape.

• Visual summaries, such as box plots, and comparative statistics enable the analysis of multiple datasets, allowing for clear interpretation of their similarities and differences in centre, spread, and distribution.

1 Data distributions Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10

Types of data Categorical data displays Numerical data displays Measures of centre Investigation: The problem with average Data distribution Quartiles and spread Box plots Outliers Standard deviation Compare sets of data Chapter 1 review

4 13 26 42 56 72 85 102 118 138

Box plots help detect outliers! An outlier can be a surprising piece of data, like finding penguins in a grassy, rocky area– totally unexpected but real!

1.01 Types of data After this lesson, you will be able to…

• distinguish between nominal, ordinal, discrete, and continuous data.

• classify real-world data sets into appropriate data types.

• compare and contrast the characteristics of nominal, ordinal, ratio, and interval scales.

Types of data A statistical variable is a measurable or observable characteristic or attribute of an item or person.

Statistical data are the actual values or observations collected for these variables. Data is collected to observe and analyse changes in these variables. For example: Variable

Data

Height of students

162 cm, 168 cm, 173 cm, 183 cm

Type of animal

Dog, Cat, Bird, Fish

Univariate data Data relating to a single variable.

Statistical variables are classified as either numerical or categorical: Numerical data Data where each data point is represented by a number. Also known as quantitative data.

• Examples of numerical data include: the number of items sold each month, daily temperatures, heights of people, and ages of a population.

• The data can be further defined as either discrete (associated with counting) or continuous (associated with measuring).

Categorical data Data where each data point is represented by a word or label. Also known as qualitative data.

• Examples of categorical data include: brand names, types of animals, favourite colours, and names of countries.

• The data can be further defined as either ordinal (it can be ordered) or nominal (unordered).

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Data

Numerical

Discrete

Continuous

Categorical

Ordinal

Nominal

Discrete data Numerical data with distinct, separate values. There is a definite ‘gap’ separating one data point from the next.

Discrete data usually, but not always, consists of whole numbers, and is often collected by some form of counting.

Examples of discrete data Number of goals scored per match

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, . . .

Number of products sold each day

386, 397, 401, 410, 411, 422, 437, . . .

Shoe size

5, 5.5, 6, 7, 8.5, 9, 10, 10, 12.5

In each of these cases, there are no in-between values. For example, there is no way to score 2.5 goals. Discrete data can involve decimals as long as there is a definite gap between possible values. For example, a shoe will not ever come in size 8.145, so it is discrete.

Continuous data Numerical data that can take any value within a range.

Continuous data often involves the use of decimal numbers, and is often collected using some form of measurement.

Examples of continuous data Height of trees in a forest (in metres)

12.359, 14.022, 14.951, 18.276, 11.032, . . .

Times taken to run a 10 km race (minutes)

55.34, 58.03, 57.25, 61.49, 66.11, 59.87, . . .

Daily temperature (°C)

24.4, 23.0, 22.5, 21.6, 20.7, 20.2, 19.7, . . .

Since continuous data is typically measured, it is limited by the accuracy of the measuring device being used. So continuous data may appear discrete when it is rounded. For instance, it makes sense that time is continuous, even though it can only be measured to the nearest second (or millisecond, and so on).

Ordinal data Categorical data with a natural order or rank.

1.01 Types of data mathspace.co

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The word ‘ordinal’ means ‘ordered’.

Examples of ordinal data Product rating on a survey

satisfactory, good, good, good, good, excellent, excellent, . . .

Exam grades

C, C, C, C, B, B, B, B, B, B, B, B, A, A, A, A, A, A, A, . . .

Size of fish in a lake

small, small, small, medium, medium, medium, large, large, . . .

Nominal data Categorical data with no inherent order.

The word ‘nominal’ basically means ‘name’.

Examples of nominal data Nationalities in a team

German, Austrian, Italian, Spanish, Dutch, Italian, . . .

Make of car driving through an intersection

Toyota, Holden, Mazda, Toyota, Ford, Toyota, Mazda, . . .

Hair colour of students in a class blonde, red, brown, blonde, black, brown, black, red, . . .

Example 1 Which one of these variables is discrete?

A Height

B

The time it takes to swim 200 metres

C Daily temperature

D

The number of family pets

Create a strategy

Apply the idea

Choose the option that can be counted but is distinct and separate from one another.

The correct answer is option D: The number of family pets.

Example 2 Classify this data into its correct category: Weights of kittens A Numerical discrete

B

Categorical nominal

C Numerical continuous

D

Categorical ordinal

Create a strategy

Apply the idea

Determine if the data is represented by a number, or is separated into categories.

The weight of a kitten can be measured. So it is numerical or quantitative.

Weight is a measurement that can have any number of decimal places, so it is continuous.

The correct answer is option C.

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Idea summary A statistical variable is a measurable characteristic being studied. Data collected for one variable is called univariate data. Variables can be classified as either numerical or categorical.

• Numerical data (quantitative) consists of numbers. It is further divided into: • Discrete data: Values are distinct and separate, often found by counting (e.g., shoe size).

• Continuous data: Can take any value within a range, often found by measuring (e.g., height).

• Categorical data (qualitative) consists of words or labels. It is further divided into: • Ordinal data: Categories have a natural, meaningful order (e.g., product ratings).

• Nominal data: Categories have no inherent order (e.g., hair colour).

Level of measurement To perform statistical analysis of data, it is important to understand what summary statistics can be meaningfully calculated, interpreted, and compared from a given set of data. Different levels of measurement can be applied depending on the properties of the given data.

The four widely applied levels of measurement are: Nominal scale

True zero

A level of measurement with categories that do not have a natural order or ranking.

A point on a scale representing the complete absence of the measured quantity.

Example: Employment status, blood type, or eye colour Ordinal scale A level of measurement with categories together with an explicit ranking.

Example: Customer satisfaction ratings (very low, low, average, high, very high)

Interval scale A numerical scale for numerical data with equal intervals but no true zero.

Example: Temperature in Celsius, as 30°C is not ‘double as warm’ as 10°C, but is 20°C warmer

Other examples of interval scale include: • pH scale - 0 does not mean an absence of pH, it means as acidic as possible • Dates - for the year 2025, 0 is not the ‘beginning of time’ • Time during the day using a 12-hour clock

1.01 Types of data mathspace.co

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Ratio scale A numerical scale for numerical data with equal intervals and a true zero.

Example: Mass, as 5 g is half as heavy as 10 g and 0 g means there is no mass Other examples of ratio scale include: • Temperature measured in Kelvin as 0 K has no molecular movement, so is a true zero • Speed, because 40 km/h is indeed twice as fast as 20 km/h • Time using a stopwatch The significant difference between an interval scale and a ratio scale is the inclusion of a ‘true zero’ or ‘absolute zero’.

RATIO Named + Ordered + Equal scale + Can accommodate absolute zero

INTERVAL Named + Ordered + Equal scale

ORDINAL Named + Ordered variables

NOMINAL Named variables A C D B

0 0.5 1 1.5 2 2.5 3

2 2.5 3 3.5 4 4.5 5

A<B<C<D

The properties of each scale can be compared in terms of how they lend themselves to being used to calculate and compare statistics:

Categorises the values

Nominal

Ordinal

Interval

Ratio

Y

Y

Y

Y

Ranks values in order

Y

Y

Y

Frequency distribution

Y

Y

Y

Y

Mode

Y

Y

Y

Y

Y

Y

Y

Mean

Y

Y

Can say one value is _ units greater or smaller than the other

Y

Y

Median

Can say one value is _ times greater or smaller than the other

Y

Note: The mean of ordinal data can sometimes be calculated by assigning numbers to the outcomes. For example, for a survey asking for a customer satisfaction rating of very low, low, average, high, and very high, the numbers 1 to 5 can be assigned to the outcomes, and the mean level of satisfaction reported can be calculated. However, it is debatable whether this can be meaningfully interpreted, as the original categories are not separated into equal intervals.

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Example 3 Classify this data into its correct category: Population of a town A Nominal

B

Ordinal

C

Interval

Ratio

D

Create a strategy Use the table:

Categorises the values

Nominal

Ordinal

Interval

Ratio

Y

Y

Y

Y

Y

Y

Y

Ranks values in order Frequency distribution

Y

Y

Y

Y

Mode

Y

Y

Y

Y

Y

Y

Y

Mean

Y

Y

Can say one value is _ units greater or smaller than the other

Y

Y

Median

Can say one value is _ times greater or smaller than the other

Y

Apply the idea The population can be compared through a ratio of values. For example, a population of 2000 is considered twice as large as a population of 1000. The correct answer is option D.

Idea summary The four levels of measurement classify data based on their properties, which determines the types of statistical analysis that can be performed.

• Nominal scale: Used for categorical data where categories have no natural order or rank (e.g., blood type).

• Ordinal scale: Used for categorical data where categories have a meaningful, explicit ranking (e.g., satisfaction ratings).

• Interval scale: A numerical scale with equal intervals between values, allowing for meaningful differences. It lacks a true zero (e.g., temperature in Celsius).

• Ratio scale: A numerical scale that has all the properties of an interval scale but includes a true zero, allowing for meaningful ratios (e.g., height, mass).

The critical distinction between interval and ratio scales is the concept of a true zero (also known as an absolute zero), which represents the complete absence of the quantity being measured.

1.01 Types of data mathspace.co

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1.01 Practice questions What do you remember?

1

Fill in the blanks with the terms from this list: • a b c d

2

3

Interval

•

Numerical

•

Categorical

•

Variable

Univariate data can be numerical or ⬚ data.

We can find the mean of ⬚ data.

Univariate means one ⬚.

There is no true zero on an ⬚ scale.

Determine whether each statement is true or false: a

A ratio scale has a true zero.

b

An interval scale is a special type of ratio scale.

c

Discrete data is a special type of numerical data.

d

Categorical data can be broken down into ordinal and interval data.

e

The mode of categorical data can be determined.

Identify whether these examples are numerical or categorical data: a

Favourite flavours

b

Maximum temperature

c

Daily temperature

d

Types of horses

e

Amount owed on lay-by

f

Types of vegetables

g

Daily UV index

h

Maximum rainfall

i

Favourite colours

j

Brands of phones

k

Types of dogs at the park

l

Maximum snowfall

Practice Ex 1

4

5

10

Identify whether these data variables are discrete or continuous: a

The number of classrooms in a school

b

Daily humidity

c

The time taken to run 200 metres

d

Lengths of cats’ whiskers

e

The distance from the Earth to nearby galaxies

f

The number of people taller than you in each class

g

Marks on the most recent Physics test

h

Time it takes to fall asleep at night

i

The amount of water consumed in a day

Identify whether these are nominal or ordinal categorical data: a

Eye colour

b

Birthdate

c

Temperature

d

Favourite song

e

State/territory of birth

f

Year level

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Ex 2

6

Classify these data examples as one of these types: • Numerical discrete • Numerical continuous • Categorical ordinal • Categorical nominal

7

Ex 3

8

9

a

Types of trailers in a shopping centre car park

b

Heights of people at an athletics carnival

c

Weights of dogs

d

The languages spoken in a class

e

The number of languages students in a town speak

f

The number of people at an athletics carnival

g

The time spent playing games each day

h

The number of sports people play

Classify these data examples as one of these types:

a b c d

• Numerical continuous • Numerical discrete • Categorical The workplaces of people living in Sydney The body temperature of a hospital patient taken over a 21-hour period The brands of breakfast foods in a supermarket The weights of club members

e

The number of people attending a netball game

Identify the level of measurement that can be applied to these data as nominal, ordinal, interval or ratio: a

Number of pets in a household

b

pH of liquid

c

Make of car

d

Weights of eggs

e

Daily rainfall in a city

f

Military rank

g

Temperature measured in degrees Celsius

h

Letter grades on an assignment (A to F)

i

Year of birth

Identify the level of measurement that can be applied to the data generated in each situation as nominal, ordinal, interval or ratio: a

The time taken for a team of swimmers to complete one lap of a swimming pool is recorded by their coach.

b

The shoe sizes of shoe prints left at the scene of a crime are recorded by a detective.

c

The religion practised by a person is recorded during a census.

d

A driver is being rated by customers on a scale of 1 star to 5 stars.

1.01 Types of data mathspace.co

11

10

Determine whether these statistics can be meaningfully computed, interpreted, and compared, from the data in each situation: i

Mode

iii a

c

Differences between outcomes iv Ratios between outcomes The temperature, in degrees Celsius, of the ocean at a particular location is monitored and fed back to a weather station.

A group of people is surveyed and asked to indicate their attitude on a recent government policy by selecting a number from 1 to 5, where 1 = strongly disagree, 2 = disagree, 3 = neutral, 4 = agree and 5 = strongly agree.

The flavour of ice cream selected by customers at a market stall is recorded over a day.

d

The weight of puppies at a puppy school is recorded each week.

b

Median

The graph shows the height of each student in Kate’s class:

Number of students

11

ii

10 9 8 7 6 5 4 3 2 1 0

155 160 165 170 175 180 185 190 195 200 Height (cm)

a

Is the variable ‘height of student’ continuous or discrete? Explain your answer.

b

Is the variable ‘number of students in a given height range’ continuous or discrete?

Explain your answer.

Extend your thinking 12

Patients at a hospital are asked to give their pain a rating from 1 to 10.

Vera says: “The pain ratings are interval data because they are numerical and have a scale going up in equal amounts.”

Critique Vera’s reasoning and give a correct explanation for the level of measurement that can be applied to the pain rating data.

13

Recording the distance, in kilometres, between towns is supposed to produce continuous data. However, distances listed on road signs are always represented as whole numbers.

Explain how this can be.

14

Explain in your own words why the amount of money spent on groceries is a discrete variable even though it involves decimal values.

15

Explain in your own words why postcode is a categorical variable while the number of houses in a postcode is a numerical variable.

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