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  Explanations and practice for each high-school maths topic

A free online textbook

The full explanations in simple language make it suitable for independent study.

Studying online

Help is offered (no cost) to schools thinking of using these materials. Click Home>Contact.

Reviews

This website is absolutely AMAZING and we use it regularly! Thank you for such a wonderful resource!

Michelle Pacey, Middle School Math Facebook Group

Thank you so much! So much effort but blessing so many especially me! The topics and levels are well-laid out, the font, size and spacing are reader-friendly. Love the interesting bits of background info that makes self-learning easy. Suitable for WA OLNA test revision, Primary Maths Extension (which I'll be using) and also good recap for Year 11/12 Essential/General Maths. I'd also be using the Maths Games! If you print/publish it into loose leaf folders, I'd buy it! For the love of Maths and learners, thank you!

Anna Tan, Middle School Math Facebook Group

More Reviews

Overview of M1Maths

The M1Maths materials cover school maths from whole numbers to calculus. They should be useful for students from Year 5 or 6 to the end of high school.

The core materials consist of about 130 Knowledge modules, each dealing with one mathematical topic and 4 Skills modules (all accessible through the Modules menu). There are also various other materials that could be useful for students and teachers (accessible through the Extras buttons).

Knowledge Modules

The Knowledge modules are divided into 7 strands (Number, Algebra, Measurement, Geometry, Statistics, Probability and Calculus) and 6 levels (Level 1 corresponding to maths from about Year 5 to Year 7 and Level 6 corresponding to about Years 11 to 12 for students who do calculus).

The assignment of topics to levels won’t necessarily correspond exactly to any particular curriculum, but modules to cover all aspects of most curricula should be easy to find at about the expected level with a short search of the module list for the relevant strand.

Each Knowledge module has 5 sections:

Summary: This is a statement of the main ideas. It is suitable for reference or revision after the module has been studied.

Learn: This section is a fuller explanation of the ideas along with practice exercises. The exercises could be used after classroom teaching or self-guided students may be able to use the explanations and exercises to learn independently.

Solve: This section consists of a number of problems which require students to apply the knowledge gained from the module as well as to use some original thinking.

Revise: This section consists of a number of Revision Sets, each with a few questions designed to maintain knowledge of that module after the work on it has been completed. There is at least one set for each module; the plan is to eventually have three or four sets for each. Cycling through the revision sets for the modules that have been covered so that each module is revisited every few weeks might be helpful. This could be a regular homework task.

Answers: This section contains the answers to the questions in the Learn, Solve and Revise sections.

Some modules also have a Lead-In which contains an activity to prepare for the learning.

Learning by Thinking Modules

These are alternative versions of some of the knowledge modules. They are distinguished by being green on the Modules menu rather than the usual straw colour and are labelled LbT.

They develop the same knowledge, but do it by leading the student through a sequence of problems in which they should develop the required knowledge themselves rather than being given the procedures.

The idea is that this will improve the student’s problem-solving skills, but, more importantly, it will ensure that the knowledge built forms a logical structure and avoids the common problem of students memorising procedures rather than understanding the ideas behind them. Memorised procedures are easily forgotten, whereas knowledge worked out logically is generally going to be reproduceable at any time later.

Learning Approach taken in the Knowledge Strands

The learning sequence in the strands varies from fairly conventional in the Number, Measurement, Geometry and Calculus strands to somewhat unconventional in the Algebra, Statistics and Probability strands. The deviation from convention is designed to ensure that the concepts build upon each other in ways that are always meaningful and applicable to life and to ensure that concepts make sense and, as far as possible, are obvious to the students in light of what they already know.

Clicking on the strand names in the drop-down navigation menu will provide more information on the approach taken in each strand.

Skills Modules

The Skills modules cover Mental Arithmetic, Problem Solving, Investigating and Communication. Each has sections corresponding to the same levels as the knowledge modules. They contain explanations of the skills and how to acquire them, and sets of practice questions with answers. Use of the skills modules would allow students to pay explicit attention to developing these aspects of their mathematical prowess.

Other Materials

The Extras buttons lead to other materials which could be useful for students and teachers.

Uses for the Modules

The M1Maths modules could be used in a number of ways.

  • They could be used as an online textbook. The site contains everything that a textbook contains as well as quite a few things that textbooks don’t. The explanations are fuller than those given in most textbooks. The aim of this is to allow students to learn independently if they wish or need to. Using M1Maths in place of a commercial textbook could save a lot of money which could then be spent on other resources and experiences.
  • As with all textbooks, M1Maths won’t follow the school program exactly. M1Maths modules can be assigned to each work unit in the school program.
  • Alternatively, M1Maths could be used as the work program. Students would work through each level in turn. The sequence of modules within a level would need to be decided, but some model sequences are offered in the Programs section under Extras for Teachers. Programs could be curriculum-paced or student-paced as explained in the Programs section
  • Students could use the practice exercises, problems and revision sets as an alternative to some or all of the exercises in a commercial textbook. This could act as a trial for schools thinking of replacing their commercial textbook with the M1Maths materials.
  • The explanations could be used when teachers introduce and explain new concepts. Teachers could refer to or work through the explanations with the class. Those students who are up to it might learn to work through the explanations by themselves and thus increase their independence in learning.
  • Students who can use the explanations independently could go through them at home prior to teacher explanation of the concepts in a ‘Flipped Classroom’ approach.
  • Students who can and wish to could also use the explanations and exercises to go ahead of what their class is doing or to familiarise themselves with a topic before learning it formally.
  • Students could use the explanation after the topic has been taught to revise or to remind themselves of things they have forgotten. This could be particularly valuable for students who have forgotten ideas from earlier years which may not be in the current textbook.
  • Students who did not fully grasp concepts when taught could use the explanations at their own pace to get a different perspective of a concept, which might possibly make the penny drop.
  • The materials could be useful for students who miss lessons.
  • The explanations, practice questions and problems provide an excellent source of material for maths tutors (or parents) working with students from about Year 5 to Year 12.

Contact

M1Maths is produced, maintained and progressively improved by David Ilsley in Brisbane, Australia.

Please feel free to make contact if you are a teacher thinking of trialling or using the materials in your school and would like some help. Support can be provided at no cost. Support will need to be mostly online, though it could include school visits for schools in Southeast Queensland.

Also feel free to make contact if you have any questions, comments, feedback or suggestions. These will help to improve the site. Comments about things like incorrect answers, unclear language, poor explanations and even typos are very welcome, will be acted upon and will be acknowledged unless requested otherwise.

The following have helped to improve the site: Rex Boggs, Dario Lucisano.

Contact can be made using the boxes below. Your comment will not be visible to other users.

       

Alternatively, you can send an email to d.ilsley@gmail.com.

Or you can use a Microsoft Form.

There is also a Facebook Group for people who are happy for their comments to be seen by others and to engage in discussions. This can be accessed via this link: M1Maths Facebook Group

Brief Biography

David Ilsley grew up and went to school in England. He studied Natural Science at Cambridge University, majoring in Geology and obtaining 1st Class Honours and a Masters Degree.

He moved to Australia to pursue research work in Geology at the Australian National University in Canberra and then worked for mining companies for a few years.

After marrying, he decided that the life of an exploration geologist (with many long periods away from home) wasn’t totally compatible with married life or bringing up children and enrolled in a Diploma of Education at the University of Queensland.

After qualifying, he taught at a number of high schools in SE Queensland. This work was interspersed with periods as a regional mathematics adviser, a writer of Senior Mathematics materials for Open Learning, a curriculum developer for the 2002 Queensland P-10 Maths Syllabus and a classroom coach training teachers in middle schools in the Bronx, New York.

David has published articles in professional journals and presented at conferences and was vice president of the Queensland Association of Mathematics Teachers and editor of their journal Teaching Mathematics from 1998 to 2000.

The M1Maths materials were developed progressively over much of this teaching career and were initially published on the Internet in 2019.

Now retired and just doing occasional relief teaching, he works on further development of the materials along with pursuing his interests in travel, astronomy and gardening his hectare of land in Logan City.

Usage Rights

Except where otherwise stated, the M1Maths website (m1maths.com) is copyrighted under

Creative Commons Attribution 4.0 License CC BY-NC 4.0.

This allows the materials to be printed, copied, adapted and stored locally. It also allows them to be re-published for non-commercial purposes as long as clear attribution is made to M1Maths (m1maths.com).

Please contact David Ilsley on d.ilsley@gmail.com with inquiries re any other uses.

Some of the resources on this site are based on material the author produced for classroom use while working for the Queensland Department of Education. Queensland Department of Education material is used under the Creative Commons Attribution 4.0 Licence CC BY 4.0.

Also, some of the resources are based on material the author produced for classroom use while working for Canterbury College. These are used with permission.

The clip art is from Art Explosion 300 000 by Nova Development, purchased by the author.

Reviews

The following are some comments people have made about M1 Maths, mainly on Facebook groups. They are included here with permission. Further comments, complimentary or critical, are always welcome. (See the 'Contact' section above.)

This website is absolutely AMAZING and we use it regularly! Thank you for such a wonderful resource!

Michelle Pacey, Middle School Math Facebook Group

Thank you so much! So much effort but blessing so many especially me! The topics and levels are well-laid out, the font, size and spacing are reader-friendly. Love the interesting bits of background info that makes self-learning easy. Suitable for WA OLNA test revision, Primary Maths Extension (which I'll be using) and also good recap for Year 11/12 Essential/General Maths. I'd also be using the Maths Games! If you print/publish it into loose leaf folders, I'd buy it! For the love of Maths and learners, thank you!

Anna Tan, Middle School Math Facebook Group

These are so very well done David! This will save me so much time going ahead with online teaching, especially since I've always wanted to trial flipped classroom learning.

Piyumi Fernando, QLD Teachers Facebook Group

I recommend to all of you!!! Please go there, you will just love mathematics.

BroCharleskwaku Odoom De Optimistic, Math and Beyond Facebook Group

Thank you, David! This is such an amazing living legacy project and you are so generous to share it. It will be a huge help to us.

Alexandra Edgar, QLD Teachers Facebook Group

Thanks for sharing! I've just started my Maths teaching career so this really helps.

Yvette Millan, Mathematics Teachers of NSW Facebook Group

This is so incredibly generous of you. Having taught for 30+ years myself I know the work that would have gone into this. There's a special place in Heaven for you.

Jenny Cawood, Australian Secondary Mathematics Teachers 7-12 Facebook Group

This is amazing, David! I’m definitely using them and will let you know how I go. Thank you!

Neelam Naidu, Queensland Senior Mathematics Teachers Facebook Group

Thank you for sharing this TERRIFIC resource David Ilsley. We need more veteran teachers like yourself who are willing to help the next generation. May you be richly blessed for all your hard work. Thanks again mate.

Nile Sheep, Australian Secondary Mathematics Teachers 7-12 Facebook Group

I love the Settler/Zoomer resources.

Callum Day, Australian Secondary Mathematics Teachers 7-12 Facebook Group

I’ve used the lessons on modelling and they were fantastic.

Soph Em, Queensland Senior Mathematics Teachers Facebook Group

I would like to thank you very much for creating this website. It is such a great website. It is user friendly and has resources to both help in extending the high achievers and reinforcing concepts for the slow learners. Thank you so much for putting in so much effort to provide us teachers with such great resources.

Indra Kumar, Email communication

M1Maths.com banner

MODULES

M1Maths

Number

Level 1       Level 2       Level 3       Level 4       Level 5       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

The Number strand is treated in a fairly conventional way. The emphasis is on students developing number sense and a range of methods and short cuts for computation.

Visual conception of fractions is promoted before symbolic manipulation. The visual approach is more amenable to students developing their own methods and short cuts and, once concepts have been gained, they are less likely to be forgotten.

N1:     Number - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

N1-1 pdf     docx

Whole numbers

  • whole numbers to trillions
  • the terms: whole number, counting number, multiple, factor, even, odd, composite, prime, >, <
  • divisibility rules, products of prime factors
4 h
N1-2 pdf     docx

Fraction Meanings

  • understanding common fractions, decimal fractions and percentages
3 h
N1-3 pdf     docx

Fraction Conversions

  • converting between equivalent common fractions, between mixed numbers and improper fractions and between common fractions, decimal fractions and percentages
3 h N1-2
N1-3 pdf     docx

Fraction Conversions

Alternative 'Learn' Section: Learning by Thinking

4 h N1-2
N1-4 pdf     docx

Negatives

  • understanding negative numbers and placing them on a number line
1 h
N1-5 pdf     docx

Calculators

  • using calculators to add, subtract, multiply and divide whole numbers, decimal fractions, common fractions and negative numbers
1 h N1-3 N1-4
N1-6 pdf     docx

Powers

  • whole number powers and square roots
  • expressing counting numbers as products of prime factors in power form
2 h N1-1 N1-5
N1-7 pdf     docx

Order of Operations

  • order of operations conventions
3 h
N1-8 pdf     docx

Common Fraction Operations 1

  • mental methods to perform simple operations on common fractions
  • approximation to perform more difficult operations
2 h N1-3
N1-9 pdf     docx

Decimal Operations 1

  • exact and approximate mental/written methods to add and subtract whole numbers and decimal fractions and multiply and divide whole numbers
3 h N1-3
N1-10 pdf     docx

Rounding and Approximation

  • rounding of numbers
  • approximate calculations
3 h N1-5 N1-9

N2:     Number - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

N2-1 pdf     docx

Number Sets

  • the various sets of numbers
2 h
N2-2 pdf     docx

Fractions of Numbers

  • finding fractions (common, decimal and percent) of numbers
  • adding and subtracting percentages to/from numbers, including by multiplying
2 h
N2-3 pdf     docx

Rates

  • rates
4 h
N2-4 pdf     docx

Ratios

  • ratios
2 h N2-3
N2-5 pdf     docx

Decimal Operations 2

  • exact and approximate mental/written methods to multiply and divide whole numbers and decimal fractions
  • predicting when decimal fractions will terminate, recur and continue without recurrence
3 h N2-2
N2-6 pdf     docx

Negative Operations

  • mental/written methods to perform operations on positive and negative numbers
3 h
N2-7 pdf     docx

Common Fraction Operations 2

  • mental/written methods to add, subtract, multiply and divide common fractions
3 h

N3:     Number - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

N3-1 pdf     docx

Scientific Notation

  • scientific notation
2 h
N3-2 pdf     docx

Simple Interest

  • simple interest
3 h
N3-2 pdf     docx

Simple Interest

Alternative 'Learn' Section: Learning by Thinking

5 h
N3-3 pdf     docx

Proportion

  • direct proportion
  • inverse proportion
2 h

N4:     Number - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

N4-1 pdf     docx

Compound Interest

  • compound interest
3 h
N4-1 pdf     docx

Compound Interest

Alternative 'Learn' Section: Learning by Thinking

4 h

N5:     Number - Level 5

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 5 modules.

N5-1 pdf     docx

Simplifying Surds

  • simplifying expressions containing surds
2 h A5-2

N6:     Number - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

N6-1 pdf     docx

Complex Numbers

  • plotting complex numbers on an Argand diagram
  • converting between Cartesian, polar and exponential form
  • adding, subtracting, multiplying and dividing complex numbers and finding powers and roots
  • application of complex numbers to trigonometric identities and fractals
9 h
N6-2 pdf     docx

Vectors

  • scalars and geometric vectors in 2 and 3 dimensions
  • addition, subtraction, multiplication by a scalar
  • scalar and vector products
  • applications of geometric vectors
  • storage vectors
10 h
N6-3 pdf     docx

Matrices

  • types of matrix
  • matrix operations
  • applications of matrices
5 h

Algebra

Level 1       Level 2       Level 3       Level 4       Level 5       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

The learning sequence for Algebra is somewhat unconventional. Algebra is treated as the study and use of relations, a relation being information which allows us to find the value of one quantity if we know the value of another quantity. As such, all new knowledge is presented as a technique for solving real-life problems: nothing is introduced as an abstract idea or as something that will be needed later. This avoids the common lament of 'When are we ever going to use this?'

By the time a student reaches the end of Level 4 or 5, s/he will know evrything a student taught traditionally will know, but the knowledge will seem more coherent, meaningful and useful.

A1:     Algebra - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

A1-1 pdf     docx

Relations 1

  • using relations given as tables, as graphs and as sets of ordered pairs
3 h
A1-2 pdf     docx

Relations 2

  • the terms 'variable', 'independent' and 'dependent'
  • converting between different forms of a relation
3 h A1-1
A1-3 pdf     docx

Patterns

  • recognising patterns in sequences of numbers
  • determining whether a relation has a pattern (from both tables and graphs)
  • knowing that relations with patterns can be expressed as formulae
3 h A1-2
A1-4 pdf     docx

Discrete vs Continuous Relations

  • distinguishing discrete and continuous relations
  • presenting them appropriately as tables and graphs
2 h A1-3
A1-5 pdf     docx

Substitution

  • substituting for the independent variable in a formula and doing the arithmetic to find the value of the dependent variable
  • converting from formulae to other forms of a relation
  • substituting into formulae with more than one independent variable
3 h A1-4
A1-6 pdf     docx

Equations

  • substituting for the dependent variable in a formula and solving the resulting equation to find the value of the independent variable
  • algebraic shorthand
7 h A1-5

A2:     Algebra - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

A2-1 pdf     docx

Writing Equations

  • solving problems by modelling them with equations
3 h
A2-2 pdf     docx

Collecting Terms

  • collecting terms to solve equations
2 h A2-1
A2-3 pdf     docx

Expanding

  • expanding brackets to solve equations
3 h A2-2

A3:     Algebra - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

A3-1 pdf     docx

Undoing the Unknown

  • undoing the unknown to solve equations
3 h
A3-2 pdf     docx

Choosing the Unknown

  • writing and solving equations where the choice of unknown is not obvious
2 h A3-1
A3-3 pdf     docx

Squares and Fractions

  • solving equations with squares and fractions
4 h A3-2
A3-4 pdf     docx

Rearranging Formulae

  • changing the subject of a formula
2 h A3-3
A3-5 pdf     docx

Solving by Graphing

  • using a graphics calculator or computer graphing program
  • graphing relations and solving equations using a graphics calculator
2 h
A3-6 pdf     docx Domain and Range
  • domain and range
2 h
A3-7 pdf     docx

Functions

  • functions and the language of functions
3 h A3-6
A3-8 pdf     docx

Linear Functions

  • general form and graph shape
  • using y=mx+c to convert between formulas, tables and graphs for linear functions
  • finding the formula for a linear function from the gradient and points on the line
  • equation solution methods
  • applications
  • finding equations of lines on the Cartesian plane
  • parallel and perpendicular lines
  • mid-point of a line segment
5 h A3-7 M3-3
A3-9 pdf     docx

Reciprocal Functions<

  • general form and graph shape
  • equation solution methods<
  • applications
2 h A3-8
A3-10 pdf     docx

Index Laws 1-5

  • using the index laws for natural number powers
4 h
A3-10 pdf     docx

Index Laws 1-5

Alternative 'Learn' Section: Learning by Thinking

5 h

A4:     Algebra - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

A4-1 pdf     docx

Factorising

  • factorising with circles
  • expanding and factorising quadratics
5 h
A4-2 pdf     docx

Quadratic Functions

  • general form and graph shape
  • equation solution methods
  • applications
  • writing and solving quadratic equations
6 h A4-1
A4-3 pdf     docx

Simultaneous Equations - Linear

  • solving pairs of linear simultaneous equations by equating, graphing, substitution, elimination and calculator
6 h
A4-4 pdf     docx

Inequalities

  • solving single-unknown inequalities and graphing the solutions on a number line
1 h
A4-5 pdf     docx

Algebraic Fractions

  • manipulating algebraic fractions
3 h A4-1
A4-6 pdf     docx

Algebraic Proofs

  • algebraic proofs
2 h

A5:     Algebra - Level 5

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 5 modules.

A5-1 pdf     docx

Polynomial Functions

  • general form and graph shape
  • equation solution methods
  • applications
3 h <
A5-2 pdf     docx

Index Laws 6-10

  • zero, negative and fractional indices
  • converting between fractional indices and surds
3 h
A5-2 pdf     docx

Index Laws 6-10

Alternative 'Learn' Section: Learning by Thinking

4 h
A5-3 pdf     docx

Power Functions

  • general form and graph shape
  • equation solution methods
  • applications
2 h A5-2
A5-4 pdf     docx

Exponential Functions and Logs

  • general form and graph shape
  • equation solution methods and logs
  • applications
5 h A5-2
A5-5 pdf     docx

Simultaneous Equations - General

  • solving pairs of general simultaneous equations by equating, graphing and substitution
  • solving larger sets of simultaneous equations
6 h A5-3 A5-4
A5-6 pdf     docx

Finding Formulae for Functions

  • finding formulae for functions from known value pairs
3 h A5-5
A5-7 pdf     docx

Model Functions

  • finding model functions for data using technology
3 h A5-6
A5-8 pdf     docx

Calculator Equation Solving

  • the equation solving function of a graphics calculator
1 h A5-1
A5-9 pdf     docx

Algebraic Transformations

  • how changes to formulae affect their graphs and vice versa
4 h
A5-10 pdf     docx

Further Relations

  • composition of functions
  • inverse functions and y^2 = x
  • discontinuities
  • rational functions
  • piecewise and step functions
  • absolute value functions
  • circles
5 h A5-1
A5-11 pdf     docx

Trigonometric Functions

  • general form and graph shape of sinusoidal functions
  • applications of sinusoidal functions
  • drawing graphs of sinusoidal functions from the formulas and vice versa
  • non-sinusoidal trigonometric functions
5 h M5-1
A5-12 pdf     docx

Trigonometric Equations

  • solving equations derived from trigonometric functions
5 h A5-11 M5-1
A5-13 pdf     docx

Logs

  • the meaning of logs
  • logarithmic functions
  • equations from logarithmic functions
  • the log laws
  • logarithmic scales
7 h A5-4

A6:     Algebra - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

A6-1 pdf     docx

Combinations and the Binomial Expansion

  • multiplication principle
  • permutations and combinations
  • expansion of (x+y)^n
  • Pascal’s triangle
6 h
A6-2 pdf     docx

Arithmetic Sequences

  • recursive and explicit formulae
  • sum to n terms
4 h
A6-3 pdf     docx

Geometric Sequences

  • recursive and explicit formulae
  • sum to n terms
  • sum to infinity
3 h A6-2
A6-4 pdf     docx

Linear Programming

  • graphing formula inequalities on the Cartesian plane
  • solving linear programming problems
3 h
A6-5 pdf     docx

Further Methods of Proof

  • proof by deduction
  • proof by exhaustion
  • proof by contradiction
  • proof by induction
3 h
A6-6 pdf     docx

Loci and Conic Sections

  • loci
  • conic sections
4 h

Measurement

Level 1       Level 2       Level 3       Level 4       Level 5       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

The Measurement strand is treated fairly conventionally except that students are encouraged to be able to find perimeters, areas, volumes etc. through common sense methods before being introduced to the formulae. This hopefully will avoid the tendency of students to rely on memorising formulae rather than understanding the mathematical concepts.

M1:     Measurement - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

M1-1 pdf     docx

Dimensions, Size and Mass

  • dimensions and measures of size in 1D, 2D and 3D
  • measuring and estimating length, area, volume and mass
5 h
M1-2 pdf     docx

Time 1

  • reading and recording times to the nearest minute on digital and analogue clocks
  • the order of the months and how many days in each
  • calendars
  • simple timetables
2 h
M1-3 pdf     docx

Unit Conversion

  • converting between units for mass and length, between L, mL and cm3 and between units for time
2 h M1-1 M1-2
M1-4 pdf     docx

Length, Area and Volume 1

  • calculating perimeters of polygons
  • calculating areas of rectangles and volumes of rectangular prisms
2 h M1-1 M1-3
M1-4 pdf     docx

Length, Area and Volume 1

Alternative 'Learn' Section: Learning by Thinking

3 h M1-1 M1-3

M2:     Measurement - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

M2-1 pdf     docx

Graduated Scales

  • reading graduated scales
2 h
M2-2 pdf     docx

Time 2

  • determining start time, finish time and duration of events given the other two
  • measuring and estimating duration of events
  • converting between 12h and 24h time
  • at a given local time, finding the time in other parts of the country in winter and in summer
  • estimating times in other parts of the world
  • more difficult timetables
5 h
M2-3 pdf     docx

Length, Area and Volume 2

  • perimeters of circles and sectors of circles
  • approximating areas of non-rectangular shapes using an enveloping rectangle
  • areas of triangles, circles, parallelograms, trapeziums and sectors of circles
  • surface areas (flat faces only)
  • volumes of prisms and cylinders
6 h N2-2 G2-4
M2-3 pdf     docx

Length, Area and Volume 2

Alternative 'Learn' Section: Learning by Thinking

8 h N2-2 G2-4

M3:     Measurement - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

M3-1 pdf     docx

Pythagoras

  • using Pythagoras theorem to find any side of a right-angle triangle in 2- and 3-dimensional situations
3 h
M3-2 pdf     docx

Trigonometry

  • using tangent, sine and cosine ratios to find side lengths and angles in right-angle triangles
  • solving problems using trigonometry including those involving direction and angles of elevation/depression
6 h M3-1
M3-3 pdf     docx

Slope

  • slope as an angle
  • gradient
2 h M3-2
M3-4 pdf     docx

Length, Area and Volume 3

  • perimeters, areas and volumes of compound shapes
  • distinguishing length, area and volume formulae by inspection
  • the effect on length, area and volume of multiplying the dimensions of a shape
8 h M3-1
M3-4 pdf     docx

Length, Area and Volume 3

Alternative 'Learn' Section: Learning by Thinking

8 h M3-1

M4:     Measurement - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

M4-1 pdf     docx

Length, Area and Volume 4

  • converting between different units for area and volume
  • volumes of pyramids, cones and spheres
  • surface areas of cylinders, cones and spheres
  • approximating volumes of irregular 3D shapes
4 h
M4-1 pdf     docx

Length, Area and Volume 4

Alternative 'Learn' Section: Learning by Thinking

5 h

M5:     Measurement - Level 5

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 5 modules.

M5-1 pdf     docx

Unit Circle and Trig Identities

  • sines, cosines and tangents in terms of the unit circle
  • identities: tan theta = sin theta/cos theta, Pythagorean identity, sin theta = cos (90 - theta)
4 h
M5-2 pdf     docx

Solving Triangles

  • cosine and sine rules
  • area formulae
5 h M5-1

M6:     Measurement - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

M6-1 pdf     docx

Radians

  • radians, conversion between radians and degrees and the radian values of common angles
5 h
M6-2 pdf     docx

Exact Trig Values

  • exact values of the trig functions of multiples of 30 and 45 degrees
3 h M6-1

Geometry

Level 1       Level 2       Level 3       Level 4       . . . . .       . . . . .

Level 1 is Years 5-7, Level 6 Years 11-12.

The Geometry strand is treated fairly conventionally, though experiences are designed to encourage the development of knowledge of theorems etc. through discovery rather than rote learning.

G1:     Geometry - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

G1-1 pdf     docx

Drawings

  • plans and elevations, perspective drawings, cross sections and nets
3 h M1-1
G1-2 pdf     docx

Angles

  • estimating, measuring and constructing angles
  • the meaning of turn and angle
2 h
G1-3 pdf     docx

Position

  • describing position using eight compass points, distance and bearing, coordinates and latitude and longitude
3 h G1-2

G2:     Geometry - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

G2-1 pdf     docx

Maps and Scales

  • making and reading maps and plans with legends and scales
2 h
G2-2 pdf     docx

Geometric Figures

  • conventions for labelling geometric figures
  • geometric theorems
5 h
G2-3 pdf     docx

Properties of Polygons

  • polygon names
  • internal angles
  • types of triangles and their properties
  • types of quadrilateral and their properties
5 h
G2-4 pdf     docx

Geometric Vocabulary

  • words used in geometry
2 h
G2-5 pdf     docx

Transformations and Symmetry

  • translations, reflections, rotations and dilations of points and shapes
  • reflectional and rotational symmetry
3 h G2-2
G2-6 pdf     docx

Congruence

  • the meaning of congruence
  • tests for congruence of triangles
3 h G2-2

G3:     Geometry - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

G3-1 pdf     docx

Similarity

  • the meaning of similarity
  • tests for similarity
  • applications of similar triangles
3 h

G4:     Geometry - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

G4-1 pdf     docx

Geometric Proofs

  • proving geometric statements using chains of reasoning
  • circle theorems
4 h
G4-2 pdf     docx

Networks

  • graphs, vertices, edges, traversability
  • weighted graphs, shortest paths
3 h

Statistics

Level 1       Level 2       Level 3       Level 4       Level 5       . . . . .

Level 1 is Years 5-7, Level 6 Years 11-12.

The main use students will have in later life for the statistics they learn is in interpreting data presentations and critically assessing data presentations that make a point. These are thus the primary thrusts of the strand, though other skills like presenting data (especially with technology), calculating statistical measures etc. are covered.

Most data presentations in the media are not one of the 'standard' types normally learnt at school. Thus, time is devoted to learning to understand, interpret and read unfamiliar data presentations. With this skill, reading the 'standard' types comes fairly naturally.

S1:     Statistics - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules

S1-1 pdf     docx

Data Displays 1

  • reading varied data displays
  • reading and drawing tables, picture graphs, dot plots, bar graphs, scatter graphs and line graphs
6 h
S1-2 pdf     docx

Data Summary

  • mean, median, mode and range
2 h

S2:     Statistics - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

S2-1 pdf     docx

Data Collection

  • data and surveys
  • extracting data
2 h

S3:     Statistics - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

S3-1 pdf     docx

xlsx

Spreadsheets

  • using spreadsheets to store data, to calculate and to produce graphs
4 h
S3-2 pdf     docx

Data Displays 2

  • reading varied data displays
  • reading and drawing grouped tables, compound bar graphs, histograms, stem and leaf plots and pie charts
5 h
S3-3 pdf     docx

Critiquing

  • critiquing data collection, data representation and conclusions drawn from data
3 h S3-2
S3-4 pdf     docx

Data Types

  • numerical, categorical and ordinal data
  • univariate, bivariate and multivariate data
  • time series
1 h
S3-5 pdf     docx

Data Distributions

  • distribution shape
  • unimodal and bimodal distributions
  • normal distributions
  • skewed distributions
2 h S3-2

S4:     Statistics - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

S4-1 pdf     docx

Quantiles and Spread

  • quantiles and ogives
  • five-number summaries, box plots
  • inter-quartile range
  • standard deviation
5 h
S4-2 pdf     docx

Linear Regression

  • trends in data, regression lines, interpolation and extrapolation
  • correlation
  • causality
3 h

S5:     Statistics - Level 5

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 5 modules.

S5-1 pdf     docx

Grouped Data

  • calculating statistics from grouped data
3 h

Probability

Level 1       Level 2       Level 3       Level 4       . . . . .       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

Many textbooks treat probability very poorly. One common error is to define probability as the number of outcomes in the event divided by the number of outcomes in the sample space. This depends on the probability of all outcomes in the sample space being equal, something often not mentioned. The definition is also circular in that the definition of probability depends on the idea that the probabilities of the outcomes are equal and thus on knowing what probability means. A more appropriate definition of probability is the fraction of times that something will happen in the long run and this is the way it is introduced here.

Probabilities can be determined using data, symmetry or guesswork, guesswork being required for untestable and unrepeatable events like finding there is an afterlife if you die. By the end of the course, student will be able to do all the usual calculations, but will have a more solidly grounded concept of what it is all about.

P1:     Probability - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

P1-1 pdf     docx

Probability

  • the meaning of probability
  • finding approximate probabilities using data (from experiment and pre-existing) and understanding the reliability of the approximations
  • estimating probabilities by guesswork
  • finding exact probability using indifference, deciding when it applies
  • calculating expected frequency
8 h N1-3

P2:     Probability - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

P2-1 pdf     docx

Compound Events

  • addition and multiplication rules
5 h
P2-2 pdf     docx

Two-way Tables

  • using two-way tables to determine probabilities
1 h P2-1
P2-3 pdf     docx

Venn Diagrams

  • using Venn diagrams to determine numbers and probabilities
3 h

P3:     Probability - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

P3-1 pdf     docx

Tree Diagrams

  • using tree diagrams to determine probabilities
3 h

P4:     Probability - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

P4-1 pdf     docx

Complex Probabilities

  • using the addition and multiplication rules in combination
4 h

P6:     Probability - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

P6-1 pdf     docx

Sets

  • set terminology and notation
2 h
P6-2 pdf     docx

Conditional Probability

  • conditional probability
3 h
P6-3 pdf     docx

Discrete Random Variables

  • discrete random variables
  • discrete probability distributions
  • expected value and variation
5 h
P6-4 pdf     docx

Binomial Distributions

  • binomial probabilities
5 h P6-3 A6-1
P6-5 pdf     docx

Continuous Random Variables

  • probability density functions
  • expected value, quantiles and spread
  • cumulative distribution functions
  • mean and variance of aX + b
5 h P6-3 C6-10
P6-6 pdf     docx

Normal Distributions

  • finding probabilities from values
  • finding values from probabilities
  • z-scores
5 h P6-5
P6-7 pdf     docx

Confidence Intervals for Proportions

  • confidence intervals for proportions
4 h P6-4
P6-8 pdf     docx

Confidence Intervals for Means

  • confidence intervals for means
3 h P6-7

Calculus

. . . . .       . . . . .       . . . . .       . . . . .       . . . . .       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

Calculus is treated fairly conventionally, though time is spent developing some of the basic concepts like how rates of change appear on graphs in practical situations.

Also, differentials are given individual meaning, rather than just their ratio. This makes things like the chain rule easier to understand.

C6:     Calculus - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

C6-1 pdf     docx

Velocity Graphically

  • displacement vs time graphs: high/low, rising/falling, steeper/flatter
  • gradient = velocity
  • gradient of secant = average velocity
  • gradient of tangent = instantaneous velocity
4 h
C6-2 pdf     docx

Velocity Algebraically

  • velocity at a given time using reducing intervals
  • velocity at given time using an unspecified interval, dt
  • velocity at any time
6 h C6-1
C6-3 pdf     docx

Velocity by Rule

  • differentiating at^n and sums of such terms by rule
6 h C6-2
C6-4 pdf     docx

Other Relations

  • calculus with relations other than between time, displacement and velocity
  • generic x-y relations
4 h C6-3
C6-5 pdf     docx

Applications of Derivatives

  • small increments
  • tangents and normals
  • finding a point with a given gradient
  • optimisation
  • turning points and curve sketching
8 h C6-4
C6-6 pdf     docx

Chain, Product and Quotient Rules

  • chain, product and quotient rules
7 h C6-5
C6-7 pdf     docx

Other Derivatives

  • the derivatives of sin x, cos x, a^x, e^x, ln x
5 h C6-6
C6-8 pdf     docx

Mastering Differentiation

  • differentiating all functions fluently
12 h C6-7
C6-9 pdf     docx

Differential Equations

  • differential equations
10 h C6-8
C6-10 pdf     docx

Integration

  • definite integrals and the fundamental theorem of calculus
  • solving problems with integration
  • area under and between curves
  • trapezoidal rule
  • indefinite integrals
14 h C6-9
C6-11 pdf     docx

Pert

  • differential equations of the form dA/dt = rA
4 h C6-10
C6-12 pdf     docx

Higher-order Derivatives

  • second and higher-order derivatives
3 h C6-10
C6-13 pdf     docx

Graph Sketching

  • sketching graphs of polynomial and rational functions using knowledge of function shape, extreme behaviour, discontinuities, axis intercepts, stationary points and spot values
4 h C6-8
C6-14 pdf     docx

Optimisation

  • finding the value of a control quantity which gives the optimum value of an objective quantity
6 h C6-8
C6-15 pdf     docx

Simpson's Rule

  • finding the value of a control quantity which gives the optimum value of an objective quantity
2 h C6-10

Skills

Mental Arithmetic       Problem Solving       Investigating       Communicating

It might be argued that, for many students, skills like mental arithmetic and problem solving will be more important to them in life than being able to do things like solve quadratic equations, use trigonometry and find the mid-point of a line segment. Thus it is hoped that significant time will be devoted to skills development. Often, curricula and programs are so tightly packed with content knowledge that the teaching of skills tends to be squeezed out. However, in the longer run, having the skills can allow students to develop the content knowledge more quickly. Not only this, the learning will be more soundly based and thus better retained. This idea is elaborated here.

Devoting a quarter to a third of learning time to the skills is probably appropriate. To this end, a lot of material is provided in the modules on Mental Arithmetic, Problem Solving, Investigating and Communicating.

Investigating is a skill ignored in many programs, but it can be helpful in convincing students that mathematics is something that they can develop themselves in response to a need rather than something handed down from on high in immutable and unquestionable form and without which no progress can be made.

Unlike with the knowledge modules, each skill is presented as a single module divided into sections for the different levels.

Skills: Mental Arithmetic

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

pdf     docx
  • know the addition and multiplication facts to 9+9 and 9x9 and the corresponding subtraction and division facts
  • develop, choose and use a flexible range of methods for performing arithmetic calculations mentally and on paper
  • have a sense of the size of numbers; be able to estimate the result of a calculation before performing it and check the reasonableness of an answer after calculating it

Skills: Problem Solving

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

pdf     docx
  • attack problems by understanding them, trying anything, and working from both ends
  • show persistence in solving problems
  • use the problem solving strategies: operation, guess and check, picture, list, pattern, equation
  • solve Fermi problems
  • present solutions appropriately

Skills: Investigating

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

pdf     docx
  • investigate using the approach: data, pattern, show, extend

Skills: Communicating

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

pdf     docx
  • communicate mathematical ideas effectively and in conventional style
  • prove mathematical statements

M1Maths.com banner

FOR STUDENTS

M1Maths

Free Maths Websites

This is a list of good free websites for students of secondary maths under the six categories below.

Websites designed more for teachers are listed in the Extras for Teachers section.

Explanations     Practice & Problems     Games     Tools     Interest     Other

  Explanations

  Site   Content   Comments

YouTube

Video lessons on most secondary maths topics

Just search for the topic. Generally, many options available.

Khan Academy

Systemmatic lessons on maths topics from Year 1 to University

A vast site. Lessons are spoken and simultaneously hand-written/drawn.

Math Is Fun

Facts and explanations covering Years 7-10 maths

A good reference where one can easily look up mathematical facts. Well organised.

Math Planet

Written and video explanations of maths ideas, Years 7-10

American site. No practice.

BBC Bitesize

Explanations and examples covering secondary maths

Comprehensive coverage of secondary maths in the UK.

MATHhelp

Video Maths lessons for middle school to college maths in the US

Purplemath

Video lessons for Years 5 to 10 and higher-level algebra.

American site connected to MATHhelp

Eddie Woo

Video'd lessons on secondary maths

Eddie Woo is a teacher in Sydney. This is his Youtube channel.

AMSI

Text explanations and some examples and exercises covering the Australian curriculum to Year 10 and Maths Methods

Organised as modules.

Virtual Nerd

All seconday maths

Includes 1500 video lessons.

Weird Number

Equivalent Fractions

An entertaining animated story about natural numbers, rational numbers and equivalent fractions.

PatrickJMT

About 1000 videos covering secondary maths and some tertiary maths

Good explanations of how to do things, though not always why the methods work or applications of the methods.

Practice and Problems

  Site   Content   Comments

AMSI

Includes modules on all Australian Curriculum topics up to Year 10 and Maths Methods

Text explanations and some examples and exercises.

What's going on in this graph?

Practice at interpreting and reading graphs

Various graphs from the New York Times containing real data.

Worksheet Generator

Arithmetic and algebra worksheets

Generated printable worksheets in arithmetica and basic algebra. Questions can be randomised so they are different each time.

Kuta Software

Worksheets for many Years 7-10 topics

Printable worksheets. Answers available. A pay site, but with a bit of free material avaliable.

Transum

A collection of questions, problems, puzzles, activities, instructional videos etc.

An extensive site produced by one person in the UK.

Project Euler

Several hundred mathematical problems suitable for solution by writing computer programs.

You can register your solutions. Each problem has been solved by hundreds to hundreds of thousands of users.

Games

  Site   Content   Comments

Kahoot

A site where one can make up on-line quizzes on any topic. Multiple-choice questions are projected on a screen and students answer on an Internet-capable device and compete to get the top score.

Widely used in schools. Students enjoy it.

Cool Maths Games

Problem solving

Large collection of games, though many aren't terribly mathematical.

Prodigy

Game based on Years 1-8 Maths

Students have to solve maths problems to make progress.

Jefferson Lab

Arithmetic, Years 3-11

Site includes a small collection of maths games.

Arithmetic Four

Arithmetic, Years 2-8

A game that practises arithmetic.

Figure This

Arithmetic, Years 2-8

An NCTM site containing maths games and challenges. Designed to encourage families to do maths together.

Tools

  Site   Content   Comments

Geogebra

A graphing package for functions and geometry with various tools and other teaching/learning resources.

A site widely used in schools.

Desmos

Many facilities, including graphing functions, plotting data, evaluating equations, exploring transformations.

A site widely used in schools.

Wolfram

A calculator which will do symbolic manipulations like solving equations, finding indefinite integrals etc.

Turtle Graphics

Online coding in Logo

Site that allows coding in Logo and turtle graphics

Interest

  Site   Content   Comments

Cut The Knot

Interactive Mathematics Miscellany and Puzzles.

Good for generating interest and spurring investigations.

Funbrain

Various, Years K-8

Games, books and videos involving maths

I Will Derive

Amusing song about calculus

Clever take-off of 'I Will Survive'. Worth showing to calculus students.

Mandelbrot Zoom

10^275 times zoom into the Mandelbrot Set

Trippy patterns of artistic merit as well as mathematical interest.

Mandelbrot Zoom

Video and song

A fairly shallow zoom into the Mandelbrot Set, but with a catchy song about the fractal. The song contains the f-word, though it's not very obvious.

Fractal Zoom

Zoom into various fractals

Artistic merit as well as mathematical interest.

Pendulum Waves

Video of a set of swinging balls

A set of balls with slightly differing frequencies produce various patterns.

Mathematical Quotations Server

Thousands of mathematical quotations

Fractal Art<

Collection of fractal art images

Artistic application of maths. Have to log in to Pinterest to see all images.

Maths Tricks

A large collection of mathematical tricks as well as other materials

Part of www.pedagonet.com

Other Sites

  Site   Content   Comments

Maths.scot

All secondary maths with an emphasis on worked examples

A site which would complement M1Maths. It covers fairly much the same topics but with more worked examples.

Astronomy

The link below is to the website of an astronomy club. The site contains astronomy information, particularly under the buttons: 'Space Facts', 'Tidbits', 'Glossary' and 'Internet Resources'.

Dark Side Astronomy

M1Maths.com banner

FOR TEACHERS

M1Maths

Learning and Assessment Programs

Many different maths curricula are specified around the world and many different school programs are written in accordance with these. However, up to about Year 10, these curricula and programs all tend to contain basically the same maths topics, just arranged in a different sequence and maybe assigned to different year levels.

Furthermore, school programs up to Year 10 are in most cases not required to match the state-specified timings exactly, as long as students have covered the required material by the end of the period concerned.

For schools wishing to use the M1Maths materials as their primary student text, a number of options for learning and assessment programs are described below, but hybrids of these and completely different approaches are of course possible.

Option 1 - Fitting the M1Maths Modules to the Existing Program

For a school program up to Year 10, it should be possible to assign the corresponding M1Maths module(s) to each topic in the program and then to work through those modules accordingly.

This is basically the same as using a commercial textbook with the school program.

Option 2 - Using the M1Maths Levels as the School Program

This is a suggestion for a curriculum-paced school program based on M1 Maths. A schedule of modules could be decided upon, assigning the Level 1 modules to Year 7, the Level 2 modules to Year 8 and so on. The following is one possible sequence. Note that Level 5 is for students aspiring to the higher-level maths in senior.

Year 7: Level 1

M1-1..4

A1-1..3

N1-1..5

G1-1..3

S1-1..2

N1-6..10

A1-4..6

P1-1

Year 8: Level 2

M2-1..3

N2-1..4

G2-1..6

A2-1..3

P2-1..3

S2-1..2

 

 

Year 9: Level 3

N3-1..3

A3-1..5

G3-1

P3-1

S3-1..3

M3-1..4

A3-6..10

 

Year 10: Level 4

N4-1

A4-1..3

M4-1

P4-1

A4-4..6

G4-1

 

 

Year 10: Level 5

A5-1..8

N5-1

M5-1..2

A5-9..13

 

 

 

 

The skills would need to be developed in parallel with the knowledge. Knowledge and skills might be assessed once a term or at other intervals. The level tests below are one way to do this. Students could sit say the Level 2 test at the end of each term in Year 8 and see their progress as the year goes on and they learn more. Any students who work ahead of the class would be suitably rewarded.

Option 3 - Student-Paced Program

In a student-paced program, students work through the learning at a speed that suits their individual background, ability, aspirations and dedication, mastering more basic concepts before being expected to use them to develop more advanced concepts. This type of program is conducive to the develoment of a growth mindset in the students.

Many education authorities, though, specify what should be taught in each year level to all students regardless of their present state of knowledge. This can produce a lock-step progression through the material which might suit average to slightly above-average students, but which restricts the more able and enthusiastic students and leaves those students who have not yet mastered the pre-requisite knowledge to get lost, to get disillusioned and to get further and further behind.

In many cases, however, this requirement is not externally monitored and thus it is possible to use a program which allows all students to progress at their own pace and thus as far as suits their particular circumstances.

M1Maths is particularly suited to such a student-paced program. This is why the material is divided into Levels 1 to 6 rather than year-levels 7 to 12. The student-paced program links below are to a document which discusses the rationale for such an approach and gives a detailed account of such a program of learning and assessment. The Progress Tracker link is to a spreadsheet that can be used to record the students' progress within this program.

Student-paced program:   pdf     docx                     Progress Tracker:   xlsx

Click on the required format. Word documents and spreadsheets need to be downloaded and opened in Word or Excel to display properly.

Level Tests

The links below are to Knowledge tests and Skills test for each level from Level 1 to Level 5. The tests were produced by writing a question or two from each of the modules at that level, plus a scattering of questions from earlier levels. There will eventually be four versions of each test. However, not all have been written to date.

These tests could be used as school tests. A version should be picked at random and minor changes made (e.g. changing some of the numbers, maybe changing the problems and the investigation). Students can then still use the versions below for test preparation / practice without invalidating the tests. The tests won't be high stakes after all. Alternatively, schools might decide to write their own tests from scratch. In that case, the tests below could be used as models.

Click on the required format. Word documents need to be downloaded and opened in Word to display properly.

  Version A Level 1 Level 2 Level 3 Level 4 Level 5
  Knowledge pdf     docx pdf     docx pdf     docx pdf     docx pdf     docx
  Skills pdf     docx pdf     docx pdf     docx pdf     docx pdf     docx
  Version B Level 1 Level 2 Level 3 Level 4 Level 5
  Knowledge pdf     docx pdf     docx pdf     docx        
  Skills pdf     docx pdf     docx pdf     docx        
  Version C Level 1 Level 2 Level 3 Level 4 Level 5
  Knowledge pdf     docx pdf     docx            
  Skills pdf     docx pdf     docx pdf     docx        
  Version D Level 1 Level 2 Level 3 Level 4 Level 5
  Knowledge pdf     docx pdf     docx            
  Skills pdf     docx pdf     docx pdf     docx        

Curriculum Correlation

M1Maths covers the maths generally taught in secondary school (or middle and high school) up to and including calculus.

The learning sequence broadly matches that used in most curricula, the ideas being assigned to levels in a way that makes for a logical progression which matches typical mathematical development and in which all pre-requisite ideas have been met when they are needed.

If the sequence in M1Maths is used, students should develop the appropriate and required learnings by the critical stages in their education and thus, learning should meet the requirements of authority curricula. Suggested detailed module sequences are available under Learning and Assessment Programs.

However, because curricula vary in detail between authorities, the M1Maths sequence may not match any authority curriculum or school program exactly. If it is necessary to match learning to an authority curriculum on a year-to year (or shorter-term basis), using M1Maths as a regular text for learning will require the modules to be matched to the elements of the curriculum or program in use.

Being Australian, I have done this for the Australian Years 7 to 10 syllabus and the Australian Years 11-12 Mathematical Methods syllabus. Being a Queenslander, I have done the same for the Queensland Years 11-12 syllabuses. The links below will take you to the various tables. These tables can also be used as templates for correlation with other curricula.

I'm afraid I haven't done the same for other places. But, if anyone would like to produce them for their country or state, I would gladly include them here.

Word documents need to be downloaded and opened in Word to display properly.

Australian Curriculum

Years 7-10 Maths v8.4   (Implementation up to 2023)       pdf       docx

Years 7-10 Maths v9.0   (Implementation 2024)       pdf       docx

A summary of the essential points of the Australian Years 7-10 Maths Curriculum is provided here.

The full Australian curriculum can be viewed at v9.australiancurriculum.edu.au.

Years 11-12 Maths Methods   pdf       docx

Queensland Syllabuses

Years 11-12 Essential Maths  pdf       docx

Years 11-12 Maths Methods   pdf       docx

Years 11-12 General Maths   pdf       docx

Years 11-12 Specialist Maths   pdf       docx

Fun and Games

These are fun activities and games which can be played as a class or by small groups. They help develop or reinforce the concepts addressed in the learning modules. Click the preferred format in the first column to download instructions and materials for that activity. Word documents need to be downloaded and opened in Word to display properly. The instructions are designed primarily for teachers to allow them to conduct the activity with a class.

Activity

Format

Relevance

Description

Bingo

pdf     docx

Mental arithmetic (all levels)

A game of bingo where students have to perform simple mental computations to work out the numbers called

Goodies and Baddies

pdf     docx

General

A quiz game between one half of the class and the other. The questions can be made up to suit the needs of the class

Group Problem Solving

pdf     docx

Problem solving – persistence and cooperation

A competition in which students work in groups to try to solve more problems correctly than the other groups

12 Question Challenge

pdf     docx

Problem solving – persistence and cooperation

A competition in which students work in groups to try to solve more problems correctly than the other groups

2 by 5 Challenge

pdf     docx

Problem solving – persistence and cooperation

A competition in which students work in groups to try to solve more problems correctly than the other groups

Relay

pdf     docx

Problem solving – persistence and cooperation

A competition in which students work in groups to try to solve a series of problems more quickly than the other groups

Number Facts Race

pdf     docx

Number Facts Fluency

A race between groups in which students have to recall and state number facts

Mind Reading

pdf     docx

Problem solving

A simple magic trick for students to work out

Twenty

pdf     docx

Problem solving

A strategy game for students to work out so they can succeed in the challenge of beating the teacher

Last One Standing

pdf     docx

General

A quick class game that rehearses recollection of facts and mental arithmetic skills. Can be used as a break and leg stretch

4-Corner Quiz

pdf     docx

General

This is a fun quiz involving movement that can be used to revise and reinforce any topic

Target

pdf     docx

Number sense, mental arithmetic,estimation, order of operations

A card game in which students use numbers to make an expression as close in value as possible to a given target number

Back to Back

pdf     docx

Geometry, Communication

An activity where students have to give clear and precise mathematical instructions

Protractor Golf

pdf     docx

Estimating and measuring angles and distances

A game for two in which students aim for holes on a paper golf course by estimating then measuring direction and distance

Walk the Plank

pdf     docx

xlsx

Adding and subtracting positive and negative numbers

A simulation of walking the plank using addition and subtraction of positive and negative numbers to decide which direction and how far to go

Fraction Line

pdf     docx

Fraction meanings and fractions of numbers

A game where one half of the class competes against the other half to best estimate given fractions of the way along a line

Fraction Dominoes

pdf     docx

Fraction conversion

A game of dominoes to develop fluency with recognising different expressions of the same fraction and with fraction conversion

Millionaire

pdf     docx

pdf     docx

Fractions of numbers

A board game that rehearses fractions among other things

Stomp

pdf     docx

Number facts and mental arithmetic

A board game for two

Greedy Pig

pdf     docx

Probability

A game mostly for fun, but which helps develop intuitive ideas of probability. It can be used with all levels

Professional Development Articles

Professional Development might be a bit of a grandiose title for this section, which consists of a bunch of random articles written over the years by David Ilsley. The articles bear some relation to mathematics education, though not all are terribly serious.

Some of the articles have been published in association journals; some have been used in face-to-face professional development; and some have just been written for lack of anything better to do.

Word and Excel documents need to be downloaded and opened in Word or Excel to display properly.

Title

Format

Summary

Polar Art

pdf

Some artistic polar function graphs

A Mathematician's Glossary of Psychological Disorders

pdf

A bit of a joke

Algebra as the Study of Relations

pdf

The philosophy behind the approach to algebra used in M1Maths.

Bad Language in the Maths Classroom

pdf

A not-too-serious article about the misuse of certain words in maths teaching

Mathematical Quotes

pdf     docx

Quotes about mathematics - some humorous, some more serious.

Avoiding Confusion in Probability

pdf

Teaching programs and textbooks often cause students to develop misconceptions in elementary probability. This article suggests a way to avoid that. The article explains the philosophy behind the approach to probability used in M1Maths.

A Single Area Formula for All 2D Shapes

pdf

A suggestion that students can learn the single area formula A = l x w x f for all 2D shapes

Setting Up a School Orienteering Course

pdf

Some tips on setting up a course

Is 9 a Random Number?

pdf

Some thoughts on what makes numbers random

Numerical Solution of Differential Equations Using Spreadsheets

pdf     xlsx

Using spreadsheets to solve differential equations numerically. The method is analogous to the numerical methods of finding definite integrals, but it can be applied to a much broader range of problems including those which lead to second- and higher-order equations and those which lead to equations in several variables which have to be solved simultaneously. Download the spreadsheet in Excel so it renders properly.

The Envelope Paradox

pdf

An interesting paradox

The Envelope Paradox - Resolution

pdf

A resolution to the paradox above.

The Unexpected Exam Paradox

pdf

An interesting paradox whose resolution is beyond me

All Whole Numbers Can Be Expressed in Eleven Words or Less

pdf

Another paradox

Fluid Resistance: Proportional to v or to v squared?

pdf

It can be either. An explanation of what determines which is the case in physical situations.

Learning Mensuration Formulae

pdf

A suggetion for making mensuration formulae easier to remember and more difficult to confuse.

Collective Nouns for Teachers

pdf

Little-known collective nouns for groups of teachers

Programming on the TI83 Calculator

pdf

Three sample programs to illustrate the programming potential of graphics calculators

Bad Maths

pdf

Extending Pascal's Triangle to make Pascal's Hexagon.

Finding Angles

pdf

Two methods for finding angles between intersectiong lines, one using geomety theorems, one using the idea of bearings

Deviant Polynomials

pdf

A non-standard general form for polynomials can be easier to get one's head around and can make curve sketching more intuitive.

Entropy with Combinations

pdf

Using combinations to get a feel for entropy

The Hairy Dog Theorem

pdf

My favourite mathematical theorem

The Ham and Cheese Sandwich Theorem

pdf

An application of the concept of 'degrees of freedom'

Round Things

pdf

An easy way to remember length, area and volume formulae for circles and spheres

Piracy

pdf

A practical application of complex numbers

Serious Problems 1-10

pdf     docx

Challenging problems that teachers and top-gun students might enjoy

Serious Problems 1-10 - Solutions

pdf     docx

Worked solutions to serious problems

Serious Problems 11-20

pdf     docx

Challenging problems that teachers and top-gun students might enjoy

Serious Problems 11-20 - Solutions

pdf     docx

Worked solutions to serious problems

Serious Problems 21-30

pdf     docx

Challenging problems that teachers and top-gun students might enjoy

Serious Problems 21-30 - Solutions

pdf     docx

Worked solutions to serious problems

Serious Problems 31-34

pdf     docx

Challenging problems that teachers and top-gun students might enjoy

Serious Problems 31-34 - Solutions

pdf     docx

Worked solutions to serious problems

Shifting the Emphasis from Memorising to Thinking

pdf     docx

pptx

Queensland Association of Maths Teachers
Annual Conference 2023 Presentation

Free Maths Websites

This is a list of good free websites for teachers of secondary maths.

Sites designed more for students are included in the Extras for Students section.

Resources for Teachers

  Site   Content   Comments

Illuminations

Various resources for maths teachers.

The NCTM public site.

nRich

Mathematics resources for children, parents and teachers

The Nrich Maths Project Cambridge, England. Designed to enrich learning.

Scootle

Lesson resources for many topics P-10.

ACARA site.

MathsBot

Online manipulatives, worksheets etc for primary and secondary maths

A lot of stuff to browse.

National Library of Virtual Manipulatives

Online manipulatives for many topics

Large collection. Requires Java. Can be a problem with some web browsers.

youcubed

Jo Boaler's Mindset approach to Maths Eduction

American site for teachers

Topmarks

Database of resources for teachers including many for secondary maths

Searchable by subject and level. UK-based.

Background image: https://pixabay.com/illustrations/beach-background-sea-ocean-3892386/ (cropped) (Creative Commons licence)
Student image: https://www.flickr.com/photos/83633410@N07/7658219802 (cropped) (Creative Commons licence)