M1MATHS

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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

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

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

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

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

N5-1

pdf docx

Simplifying Surds

simplifying expressions containing surds

2 h

A5-2

N6: Number - Level 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

G3-1

pdf docx

Similarity

the meaning of similarity
tests for similarity
applications of similar triangles

3 h

G4: Geometry - Level 4

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

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

S2-1

pdf docx

Data Collection

data and surveys
extracting data

2 h

S3: Statistics - Level 3

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

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

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

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

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

P3-1

pdf docx

Tree Diagrams

using tree diagrams to determine probabilities

3 h

P4: Probability - Level 4

P4-1

pdf docx

Complex Probabilities

using the addition and multiplication rules in combination

4 h

P6: Probability - Level 6

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

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

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

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

pdf docx

investigate using the approach: data, pattern, show, extend

Skills: Communicating

pdf docx

communicate mathematical ideas effectively and in conventional style
prove mathematical statements

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

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)