problem solving tactics amt pdf

High School Problem Solving

A website about solving mathematical problems, mathematics problem solving resources.

“This is your last chance. After this, there is no turning back. You take the blue pill – the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill – you stay in Wonderland and I show you how deep the rabbit-hole goes.” The Matrix

Over time I have accumulated many different materials and resources that I have found helpful in learning and practicing mathematics, ranging from online practice environments to handouts and books. Here I present a collection of some of my favorite ones (many of which are completely free):

How To Write A Math Solution (AoPS) Covers general tips on writing and explaining proofs in mathematics – a worthwhile read, even though it doesn’t cover actual topics.

Some of my favorite websites and links for mathematical enrichment:

• Art of Problem Solving This website has a vast collection of problems from mathematics competitions around the world for all levels. In addition, they have an active community with numerous resources in various topics. And if you’re interested further, they also have classes and textbooks going from basic pre-algebra all the way through to Olympiad Geometry and Calculus – this is where I learnt and practiced a lot of my skills starting out.
• AwesomeMath Similarly to AoPS, this website has a wide range of high-quality books on different math olympiad topics, as well as offering classes.
• Australian Maths Trust – The AMT runs the math olympiads and competitions in Australia, and has a variety of useful books in their shop
• Mathematical Association of America – Similar to the AMT, the MAA runs the math olympiad program in America, and has several publications in their shop as well as general information.

And if you want more, this wiki page has a huge list of links to peruse for all levels.

Some books I found particularly helpful are

• Euclidean Geometry in Mathematical Olympiads (EGMO). ( Amazon Link ). Written by IMO medallist and Olympiad enthusiast Evan Chen, this book is a comprehensive text on most of what you’ll see in Olympiad Geometry. It covers topics from the basics like similarity and congruence all the way through to more advanced synthetic techniques, and contains plenty of practice problems with hints and selected solutions.
• Problem Solving Tactics by the AMT is a great book that covers both basic and more advanced topics in all areas of Olympiad mathematics – algebra, geometry, combinatorics and number theory, with numerous worked examples and questions to practice on (although some of these practice problems don’t have solutions provided). Bear in mind that it is a little pricey, so get it if you are enthusiastic.
• A Beautiful Journey Through Olympiad Geometry (Free pdf). This book is a free alternative to EGMO, and covers similar topics going from the fundamentals of Geometry to the advanced.
• Number Theory (AwesomeMath). This book is a great introduction to number theory, going from basic results to more advanced topics with plenty of worked examples as well as practice problems to do. AwesomeMath also has many other books on different concepts such as Combinatorial Problems

These resources have many excellent free handouts on all kinds of mathematical topics:

• Evan Chen’s Webpage has many free high-quality resources with great explanations and problems to practice, although many of the topics are not very beginner-friendly.
• Yufei Zhao’s Olympiad page (Canada IMO) has many handouts from his lectures at Olympiad camps – some of these can get quite advanced, but they are a great resource nonetheless.

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Resources for the classroom

Jump into a class of problem solving with these interesting and thought-provoking lesson cards, which offer multiple points of interest and ability for students in years 2 through 10.

Lesson Cards

Get your students exploring problem-solving tactics by basing a lesson on one of these cards. Each one features a problem and follow up challenges, and covers various levels of difficulty to suit a range of abilities in your classroom. 

You will also find the complete cards with solutions and links to the ACARA website (for full information on how each activity is linked to the Australian Curriculum).

Suitable years:

Animal, Vegetable, Mineral is suitable from years 2–12. This question is versatile in the sense that teachers can ask for either a simple answer or the intricate details within each answer.

Lesson Areas:

We have a rundown of the learning areas and descriptors taken from the Australian Curriculum so you’ll easily be able to drop these lesson cards into your schedule without deviating from the curriculum.

• Year 2, ACMMG046 Identify and describe half and quarter turns
• Year 3, ACMMG066 Identify symmetry in the environment
• Year 4, ACMMG088 Compare and describe two dimensional shapes that result from combining and splitting common shapes
• Year 4, ACMMG091 Create symmetrical patterns, pictures and shapes
• Year 5, ACMMG114 Describe translations, reflections and rotations of two-dimensional shapes, and identify line and rotational symmetries
• Year 6, ACMMG142 Investigate combinations of translations, reflections and rotations
• Year 8, ACMMG200 Define congruence of plane shapes using transformations
• Year 10, ACMMG244 Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes

Additional Resources:

We have created an interactive tool on GeoGebra as a visual aid to help. Click on the links below and walk your students through our visual aid.

Solutions Card:

Download the solutions card . The card contains all the items on the lesson card as well as the solutions to each question.

Magic Dice is suitable from years 2–6. This question is versatile in the sense that teachers can ask for either a simple answer or the intricate details within each answer.

Learning areas:

Below is a rundown of the learning areas and descriptors are taken from the Australian Curriculum so you’ll easily be able to drop these lesson cards into your schedule without deviating from your class schedule.

• Year 2, ACMNA029 Explore the connection between addition and subtraction
• Year 3, ACMNA060 Describe, continue and create number patterns resulting from performing addition or subtraction
• Year 4, ACMNA083 Find unknown quantities in number sentences involving addition and subtraction and identify equivalent number sentences involving addition and subtraction
• Year 5, ACMNA121 Find unknown quantities in number sentences involving multiplication and division and identify equivalent number sentences involving multiplication and division
• Year 6, ACMNA134 Explore the use of brackets and order of operations to write number sentences 
• Dice Templates

Solution Card:

Download the Solution card which contains all the items on the lesson card as well as the solutions to each question.

Suitable Years:

The Frobenius coin problem is suitable from years 7–12.

Below is a rundown of the learning areas and descriptors taken from the Australian Curriculum so you’ll easily be able to drop these lesson cards into your schedule without deviating from the curriculum.

• Year 7, ACMNA149  Investigate index notation and represent whole numbers as products of powers of prime numbers
• Year 8, ACMNA183  Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies
• Year 9, ACMNA208  Plot linear relationships on the Cartesian plane with and without the use of digital technologies
• Year 10, ACMNA235  Solve problems involving linear equations, including those derived from formulas

Resources for the problem:

• Sliding graph

Supplementary Lesson Card:

Have a look at the  Supplementary Lesson Card . It Provides you with extended questions for advanced students in your classroom.

Download the solutions.  The card provides all the items on the lesson card as well as the solutions to each question.

Scaly Tiles is suitable from years 5–12. This question is a playful investigation of how shapes fit together which leads to the sophisticated concepts of self-similarity.

The following is not intended to be an exhaustive list but indicates how the above activity aligns with various stages of the mathematics curriculum. Follow the links to the ACARA website for elaborations.

• Year 5, ACMMG114 Describe translations, reflections and rotations of two-dimensional shapes, and identify line and rotational symmetries
• Year 5, ACMMG115 Apply the enlargement transformation to familiar two dimensional shapes and explore the properties of the resulting image compared with the original
• Year 6, ACMMG142 Investigate combinations of translations, reflections and rotations
• Year 7 ACMMG161 Draw different views of prisms and solids formed from combinations of prisms
• Year 7 ACMMG163 Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal
• Year 7 ACMMG166 Demonstrate that the angle sum of a triangle is 180◦ and use this to find the angle sum of a quadrilateral
• Year 7 ACMMG181 Describe translations, reflections in an axis and rotations of multiples of 90◦ on the Cartesian plane using coordinates
• Year 8, ACMMG200 Define congruence of plane shapes using transformations
• Year 8, ACMMG201 Develop the conditions for congruence of triangles
• Year 9, ACMMG220 Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar
• Year 9, ACMMG221 Solve problems using ratio and scale factors in similar figures
• Year 9, ACMMG222 Investigate Pythagoras’ Theorem and its application to solving simple problems involving right-angled triangles
• Year 10, ACMMG243 Formulate proofs involving congruent triangles and angle properties
• Grid paper – isometric- small
• Grid paper – isometric- large
• Grid paper – square- small
• Grid paper -square- large

Solution Card

Download the Solution Card. The card contains the above information as well as detailed solutions for all questions. 

Problemo Student Portal

Problemo Student  is a free problem-solving platform that allows students to test their skill in mathematics and informatics. It’s packed with problems covering a range of topics and year levels. Jump in a try one question or work your way through a series based on your chosen topic.  

AMC Practice Problems Online

• 2019 AMC Practice Problems – All Levels
• 2019 AMC Practice Problem Solutions – All Levels

Choose your year level (3-12) and prepare for the AMC on our new platform. Users don’t need to log in to participate. Solutions for each question are supplied.

AMC Posters

• AMT Pathway Poster (A3)
• Upper Primary AMC Poster (A4)
• Junior AMC Poster (A4)
• Senior AMC Poster (A4)

AMC Teacher's Handbook

• AMC Online Version
• AMC Paper Version

Your guide to running the AMC. A step by step of instructions for the day and what to do afterwards.

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Problemo – making maths meaningful

Problemo – powered by The Australian Maths Trust (AMT) – is an online maths problem-solving resource designed to help teachers engage students and encourage mathematical thinking and creative problem-solving.

The platform has a library of more than 400 maths problems designed by AMT’s expert team of mathematicians and teachers, and each one is mapped to the Australian Curriculum. Problemo allows teachers of years 3-10 to build rich learning experiences by using a powerful search function to create problem sets, real-time student quizzes and lessons plans in a few easy steps.

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Problem Solving Tactics PDF

Download Problem Solving Tactics PDF

Description

Table of contents.

About this book......Page 6 Problems......Page 14 Converse......Page 18 If and only if......Page 19 Contrapositive......Page 20 Proof by contradiction......Page 21 Proof by induction......Page 22 Strong induction......Page 24 Proof by exhaustion (case bashing)......Page 25 Pigeonhole principle......Page 26 Advanced pigeonhole principle......Page 28 Extremal principle......Page 29 Telescoping......Page 31 Problems......Page 32 Fundamental theorem of arithmetic......Page 37 Dealing with digits......Page 38 Floor function......Page 39 Square roots and conjugates......Page 40 Powers of two......Page 41 Euclid's algorithm......Page 42 Integers base-n......Page 44 Construction problems......Page 45 Chinese remainder theorem......Page 47 From Fermat to Euler......Page 48 Existence of a generator......Page 50 Problems......Page 52 Monotonicity......Page 56 Bounding arguments......Page 57 Polynomial modulus......Page 58 Modular arithmetic......Page 59 Divisibility and gcds......Page 60 Reduction of variables......Page 61 Infinite descent......Page 62 Vieta jumping......Page 63 Cyclotomic recognition......Page 64 Plane geometry......Page 66 Problems......Page 67 Cyclic quadrilaterals......Page 72 One step at a time......Page 73 Triangle centres......Page 75 Constructions......Page 77 Extend to the circumcircle......Page 79 Reverse reconstruction......Page 81 Trigonometry......Page 82 Areas......Page 83 Relate to known diagrams......Page 85 Create beautiful pictures......Page 86 Important configurations in geometry......Page 88 Angle bisector and perpendicular bisector......Page 89 Pivot theorem......Page 90 Radical axis theorem......Page 91 Similar switch......Page 92 Radical axis bisects common tangent......Page 93 Perpendicularity......Page 94 Alternate segment switch......Page 95 Ratios for collinearity......Page 96 Points of contact of incircle and excircle......Page 97 Circumcircle, incentre and excentre......Page 98 Simson line......Page 99 Pascal's theorem......Page 100 Desargues' theorem......Page 101 Quadrilateral and incircle......Page 102 Nine-point circle......Page 103 Euler line......Page 104 Four lines and four circles......Page 105 Newton–Gauss line......Page 106 Alternative characterisation of symmedian......Page 107 Convex cyclic hexagon and diagonals......Page 108 Quadrilateral, triangles and incircles......Page 109 Median, inradius and chord of incircle......Page 110 Incentre and midpoints......Page 111 Incentre, excentre, midpoint and contact points......Page 112 Incentre and chord of incircle......Page 113 Harmonic quadrilateral......Page 114 Incentre and mixtilinear incircle......Page 115 Butterfly theorem......Page 116 Problems......Page 118 Collinear points......Page 121 Menelaus' theorem......Page 122 Concurrent lines......Page 123 Ceva's theorem......Page 124 Concyclic points......Page 126 Power of a point......Page 130 Radical axes......Page 131 Ellipses......Page 132 Pascal's theorem......Page 135 Transformation geometry......Page 138 Problems......Page 139 Rotations......Page 143 Dilations......Page 144 Spiral symmetries......Page 146 Affine transformations......Page 147 Problems......Page 150 Addition ideas......Page 153 Angles......Page 154 Multiplication ideas......Page 155 Similarity ideas......Page 157 Roots of unity......Page 159 Polynomials......Page 162 Problems......Page 163 Division algorithm......Page 167 Fundamental theorem of algebra......Page 168 Vieta's formulas......Page 169 Integer polynomials......Page 171 Algebraic trickery......Page 173 Irreducibility......Page 175 Polynomials modulo p (upstairs–downstairs)......Page 176 Lagrange interpolation......Page 179 Root focus......Page 180 Problems......Page 182 Cauchy's functional equation......Page 186 Guess and hope......Page 187 Substitutions......Page 188 Injective, surjective and bijective......Page 189 The associative trick......Page 191 Involutions......Page 192 Fixed points......Page 193 Somewhere versus everywhere......Page 195 Completely multiplicative functions......Page 196 Well-ordering of N+......Page 197 Problems......Page 200 Squares are non-negative......Page 203 Rearrangement inequality......Page 204 Cauchy–Schwarz inequality......Page 206 Power means inequality......Page 208 Jensen's inequality......Page 209 Substitutions......Page 210 Addition and multiplication of inequalities......Page 211 Expand and conquer......Page 213 Homogeneous inequalities......Page 214 Muirhead's inequality......Page 215 Weighted inequalities......Page 216 Problems......Page 220 Reflection principle......Page 223 Transformations......Page 226 Trigonometry......Page 227 Parametrisation......Page 228 Ptolemy's inequality......Page 229 Locus and tangency......Page 232 Isoperimetric inequalities......Page 234 Triangle formulas......Page 235 Problems......Page 238 Subtraction and division......Page 241 Binomial identities......Page 242 Bijections......Page 244 The supermarket principle......Page 245 Pigeonhole principle......Page 246 Principle of inclusion–exclusion......Page 247 Double counting......Page 248 Injections......Page 250 Recursion......Page 251 Double counting via tables......Page 253 Combinatorial reciprocal principle......Page 254 Graph theory......Page 256 Problems......Page 257 Degree......Page 260 Connected graphs, cycles and trees......Page 261 Pigeonhole principle......Page 263 Euler trails......Page 264 Paths......Page 266 Extremal principle......Page 267 Count and count again......Page 268 Planar graphs......Page 269 Polyhedra......Page 270 Graph theory and inequalities......Page 271 Problems......Page 274 Number invariants......Page 280 Parity......Page 281 Modular arithmetic invariants......Page 282 Colouring invariants......Page 283 Monovariants......Page 285 Invariants as cost......Page 286 Permutation parity......Page 287 Combinatorial games......Page 288 Position analysis......Page 289 Pairing strategies......Page 290 Strategy stealing......Page 291 Problems......Page 294 Extremal principle......Page 298 Perturbation......Page 300 Discrete intermediate value theorem......Page 301 Convex hull......Page 302 Euler's formula......Page 303 Pigeonhole principle......Page 304 Colouring......Page 305 How do complex numbers work?......Page 306 Function notation......Page 309 Directed angles......Page 310 Some useful triangle formulas......Page 311 Index......Page 312

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Problemo – an online platform helping teachers make maths more meaningful for students

The Australian Maths Trust (AMT) has today launched Problemo – a unique online platform that has the potential to change the way mathematical problem solving is taught in Australian and New Zealand classrooms. 

Problem solving is a central proficiency in the Australian Curriculum and has been identified as one of the top skills needed for future employability, and the Australian Maths Trust is leading the way in developing this crucial skill in the education sector.

With its development led by teachers, Problemo has been carefully designed to support student learning regardless of the context – in the classroom or learning remotely at home.

“Problemo is the culmination of the AMT’s 40 years of expertise and passion for helping young Australians experience the inspiration that comes with problem solving,” said Nathan Ford, AMT CEO.

“Mathematical problem solving is at the heart of our modern world – its challenges, technology, obstacles and opportunities – and we look forward to helping teachers and students explore and master this world as we continue to develop this resource for schools across the country.”

The platform contains a library of more than 400 maths problems for years 3-10 aligned to the Australian Curriculum and AMT will add problems and content areas to Problemo each term to make sure the platform addresses the needs of teachers in all learning environments.

Each problem is classified using strands, topics, difficulty, keywords, ACARA codes, and strategies for powerful yet easy searching.

Additionally, each problem comes with enabling and extending prompts, and solutions which help both students and teachers expand their maths problem solving capacity.

Teachers can also access official lesson cards and can create their own to help with lesson preparation.

Problemo caters for in-class and remote teaching with the ability to:

• create PDFs of problems to download
• display problems digitally for classroom use
• create online quizzes for students, with results displaying in the teacher platform.

The AMT is a national charity dedicated to ensuring all young Australians can realise their potential in problem solving through maths and algorithmics.  It has delivered the Australian Mathematics Competition, one of Australia’s largest educational events, for more than 40 years and to more than 15 million students.

101 Problems in Algebra from the Training of the USA IMO Team

By titu andreescu and zuming feng, promotional material (tbs), what the reviewers say.

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Australian Mathematical Olympiads Book 2 1996-2011 PDF

This book is a complete collection of all Australian Mathematical Olympiad papers from the first paper in 1996 to 2011. Solutions to all of the problems are included and in a number of cases alternative solutions are also offered.

The material covered is recommended for senior and advanced students.

Authors: H Lausch, A Di Pasquale, DC Hunt & PJ Taylor

This book is a digital download. After purchase, it will be delivered to you automatically via a download link in an email.

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