THE MATHENAEUM

The Mathenæum (Mathenaeum, Matheneum) is a collection of mathematics games and activities for classroom learning

TRIG RATIO RACE

Trigonometry may seem arcane to a new learner, but it is a branch of mathematics rooted in ancient astronomy, and developed over many years by Babylonian, Egyptian (it helped them build the pyramids!), Greek, Indian, and Arab mathematicians before returning to Europe in the 13 th century.. This activity encourages a student to consider all possible side-length ratios in a triangle, and then AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. INTRODUCTION TO ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice building up expressions using Algebra Tiles, using zero pairs — i.e. a pair of opposite tiles that

TRIANGLE EXPLORER

Triangle Explorer. Manipulate and constrain triangles. Investigate the effects of constraining side lengths and angle sizes in triangles. Show side lengths Lock sides: a b c. Show angle sizes Lock angles: A B C. Constructions: Circumcircle A-b A-c B-a B-c C-a C-b a+ b+ c+. Overrides: None Translate Rotate Scale Flip Lock Constructions.

SPIROLATERALS

Spirolaterals are geometrical figures formed by the repetition of a simple rule. The base pattern is formed by drawing line segments of increasing length (in integer units) up to a particular size, turning a fixed angle after each segment (clockwise or anti-clockwise). Drawing continues in the same manner from the resulting position until FUNCTION VIEWER HELP To view the plot for some function of x, enter it as an expression into the text field, then hit enter or press the Plot button. The curve corresponding to the function will be added to the plot and table below. Many standard mathematical functions are supported. THE MATHEMATICS OF DOBBLE The formula for the number of points in a projective plane of order n is n2+n+1. You can use this formula to answer the following two questions: Verify that an order 2 projective plane has 7 points (as we saw with the Fano plane). Given that there are 57 symbols in Dobble, what is the order of the projective plane used to generate a full set

of

COLOUR SPINNER

Colour Spinner. Enter your choices and spin the wheel! Enter colours as a comma separated list, then press Update. All HTML colour names and hex values are supported. cyan,magenta,yellow,cyan,magenta,yellow.

MATCHSTICK PATTERNS

Matchstick Patterns. One good way to describe mathematics is that it provides a way to understand and describe the patterns and structures that we can see, and that we can imagine. It does this by revealing the underlying relationships , and how they arise and are changed by various transformations . For example, we can consider 2 × 3 as a

THE MATHENAEUM

The Mathenæum (Mathenaeum, Matheneum) is a collection of mathematics games and activities for classroom learning

TRIG RATIO RACE

Trigonometry may seem arcane to a new learner, but it is a branch of mathematics rooted in ancient astronomy, and developed over many years by Babylonian, Egyptian (it helped them build the pyramids!), Greek, Indian, and Arab mathematicians before returning to Europe in the 13 th century.. This activity encourages a student to consider all possible side-length ratios in a triangle, and then AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. INTRODUCTION TO ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice building up expressions using Algebra Tiles, using zero pairs — i.e. a pair of opposite tiles that

TRIANGLE EXPLORER

Triangle Explorer. Manipulate and constrain triangles. Investigate the effects of constraining side lengths and angle sizes in triangles. Show side lengths Lock sides: a b c. Show angle sizes Lock angles: A B C. Constructions: Circumcircle A-b A-c B-a B-c C-a C-b a+ b+ c+. Overrides: None Translate Rotate Scale Flip Lock Constructions.

SPIROLATERALS

Spirolaterals are geometrical figures formed by the repetition of a simple rule. The base pattern is formed by drawing line segments of increasing length (in integer units) up to a particular size, turning a fixed angle after each segment (clockwise or anti-clockwise). Drawing continues in the same manner from the resulting position until FUNCTION VIEWER HELP To view the plot for some function of x, enter it as an expression into the text field, then hit enter or press the Plot button. The curve corresponding to the function will be added to the plot and table below. Many standard mathematical functions are supported. THE MATHEMATICS OF DOBBLE The formula for the number of points in a projective plane of order n is n2+n+1. You can use this formula to answer the following two questions: Verify that an order 2 projective plane has 7 points (as we saw with the Fano plane). Given that there are 57 symbols in Dobble, what is the order of the projective plane used to generate a full set

of

COLOUR SPINNER

Colour Spinner. Enter your choices and spin the wheel! Enter colours as a comma separated list, then press Update. All HTML colour names and hex values are supported. cyan,magenta,yellow,cyan,magenta,yellow.

MATCHSTICK PATTERNS

Matchstick Patterns. One good way to describe mathematics is that it provides a way to understand and describe the patterns and structures that we can see, and that we can imagine. It does this by revealing the underlying relationships , and how they arise and are changed by various transformations . For example, we can consider 2 × 3 as a AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. THE MATHEMATICS OF DOBBLE The formula for the number of points in a projective plane of order n is n2+n+1. You can use this formula to answer the following two questions: Verify that an order 2 projective plane has 7 points (as we saw with the Fano plane). Given that there are 57 symbols in Dobble, what is the order of the projective plane used to generate a full set

of

FRACTAL ART FROM LINES Fractals Part II. This is the second of three activities that explore the construction and visualisation of fractals.. Although arising from simple processes, fractals exhibit infinite complexity, and exist at the nexus of mathematics, nature, and art. No matter how closely you look at a fractal, however much you zoom in, they remain equally complex (i.e. bumpy).

MATCHSTICK PATTERNS

Matchstick Patterns. One good way to describe mathematics is that it provides a way to understand and describe the patterns and structures that we can see, and that we can imagine. It does this by revealing the underlying relationships , and how they arise and are changed by various transformations . For example, we can consider 2 × 3 as a THE RULES OF DODGEBALL The rules of Dodgeball. The game below is played between two players, a Matcher and a Dodger. Each player has their own grid of tiles, and repeated clicks on a tile toggle its state between X and O. Each turn the Dodger turns a single tile, and the Matcher makes a complete row. The turn is finished when the ↩ button is pressed on the Matcher TILES - THEWESSENS.NET Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. Solving equations involves working with expressions on each side of an equals sign

simultaneously.

INTERSECTION DETECTION Intersection Detection. This activity is a game similar to the card game known as Dobble or Spot It! . It uses a set of cards, each showing some number of symbols. Overall, there are as many symbols as cards, and each card has the same number of symbols. They key feature is that for any chosen pair of cards, there will be exactly one symbol

in

DIRECTED NUMBER WITH ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice directed number multiplication and division by using the

unit Algebra Tiles.

TEMPLATE

Choose an activity below. A Counting Dot Machine. Numbers in any base using exploding dots. Exploding Dot Machines. Numbers in any base using exploding dots. Arithmetic with Exploding Dots. Performing arithmetic calculations with exploding dots. Polynomial Division. Dividing polynomials using exploding dots. DIRECTED NUMBER WITH ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice directed number addition and subtraction by using the

unit Algebra Tiles.

THE MATHENAEUM

The Mathenæum (Mathenaeum, Matheneum) is a collection of mathematics games and activities for classroom learning

TRIG RATIO RACE

Trigonometry may seem arcane to a new learner, but it is a branch of mathematics rooted in ancient astronomy, and developed over many years by Babylonian, Egyptian (it helped them build the pyramids!), Greek, Indian, and Arab mathematicians before returning to Europe in the 13 th century.. This activity encourages a student to consider all possible side-length ratios in a triangle, and then AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. PARABOLAS - THEWESSENS.NET Study parabolas in the number plane. Consider regions where the quadratic expression is positive, negative, or zero. and relate them to the equation of the parabola. INTRODUCTION TO ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice building up expressions using Algebra Tiles, using zero pairs — i.e. a pair of opposite tiles that

SPIROLATERALS

Spirolaterals. Spirolaterals are geometrical figures formed by the repetition of a simple rule. The base pattern is formed by drawing line segments of increasing length (in integer units) up to a particular size, turning a fixed angle after each segment (clockwise

or anti-clockwise).

FUNCTION VIEWER HELP To view the plot for some function of x, enter it as an expression into the text field, then hit enter or press the Plot button. The curve corresponding to the function will be added to the plot and table below. Many standard mathematical functions are supported.

TRIANGLE EXPLORER

Single Triangle Mode. Manipulate the triangle by dragging the vertices. Optionally lock any side or angle to its current size. Various constructions can be added for guidance:

COLOUR SPINNER

Enter colours as a comma separated list, then press Update.All HTML colour names and hex values are supported. THE MATHEMATICS OF DOBBLE Let's start by thinking about the Euclidean plane —a flat two dimensional space that extends without limit in all directions. We can consider this space as being made of infinitely many points, each of which has neither length nor breadth (that is they are zero dimensional), that may (somewhat magically) be connected to form straight lines—geometrical objects with length but no breath (and

THE MATHENAEUM

The Mathenæum (Mathenaeum, Matheneum) is a collection of mathematics games and activities for classroom learning

TRIG RATIO RACE

Trigonometry may seem arcane to a new learner, but it is a branch of mathematics rooted in ancient astronomy, and developed over many years by Babylonian, Egyptian (it helped them build the pyramids!), Greek, Indian, and Arab mathematicians before returning to Europe in the 13 th century.. This activity encourages a student to consider all possible side-length ratios in a triangle, and then AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. PARABOLAS - THEWESSENS.NET Study parabolas in the number plane. Consider regions where the quadratic expression is positive, negative, or zero. and relate them to the equation of the parabola. INTRODUCTION TO ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice building up expressions using Algebra Tiles, using zero pairs — i.e. a pair of opposite tiles that

SPIROLATERALS

Spirolaterals. Spirolaterals are geometrical figures formed by the repetition of a simple rule. The base pattern is formed by drawing line segments of increasing length (in integer units) up to a particular size, turning a fixed angle after each segment (clockwise

or anti-clockwise).

FUNCTION VIEWER HELP To view the plot for some function of x, enter it as an expression into the text field, then hit enter or press the Plot button. The curve corresponding to the function will be added to the plot and table below. Many standard mathematical functions are supported.

TRIANGLE EXPLORER

Single Triangle Mode. Manipulate the triangle by dragging the vertices. Optionally lock any side or angle to its current size. Various constructions can be added for guidance:

COLOUR SPINNER

Enter colours as a comma separated list, then press Update.All HTML colour names and hex values are supported. THE MATHEMATICS OF DOBBLE Let's start by thinking about the Euclidean plane —a flat two dimensional space that extends without limit in all directions. We can consider this space as being made of infinitely many points, each of which has neither length nor breadth (that is they are zero dimensional), that may (somewhat magically) be connected to form straight lines—geometrical objects with length but no breath (and AVANTI - THEWESSENS.NET About the activity This is the beta release of the Avanti Symbolic Calculator.It is designed to assist with the kinds of numerical and algebraic operations that are used in high school mathematics, while using proper mathematical notation. FRACTAL ART FROM LINES Fractals Part II. This is the second of three activities that explore the construction and visualisation of fractals.. Although arising from simple processes, fractals exhibit infinite complexity, and exist at the nexus of mathematics, nature, and art. No matter how closely you look at a fractal, however much you zoom in, they remain equally complex (i.e. bumpy). THE MATHEMATICS OF DOBBLE Let's start by thinking about the Euclidean plane —a flat two dimensional space that extends without limit in all directions. We can consider this space as being made of infinitely many points, each of which has neither length nor breadth (that is they are zero dimensional), that may (somewhat magically) be connected to form straight lines—geometrical objects with length but no breath (and TILES - THEWESSENS.NET Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. Solving equations involves working with expressions on each side of an equals sign

simultaneously.

THE RULES OF DODGEBALL Cantor's diagonal argument is a beautiful, simple, profound and historically significant part of mathematics, and the Dodgeball game is a wonderful way to make it accessible and intuitive.

MATCHSTICK PATTERNS

About the activity. This activity implements a common approach to learning algebra: start with a visual pattern, extract the associated numerical pattern (i.e. a sequence), realise the need for an efficient and general description, and construct such a description in algebraic

form.

INTERSECTION DETECTION The mathematical structures that underly this game are Finite Fields, and Projective Planes. These structures are interesting in themselves, and can be linked not only to Dobble-like games, but also to Graeco-Latin squares, Mobius strips and other fascinating algebraic, QUADRILATERAL EXPLORER Select a type of quadrilateral, and use the buttons below to optionally show the diagonals, and add markings to indicate the important angle, side and diagonal properties. DIRECTED NUMBER WITH ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice directed number multiplication and division by using the

unit Algebra Tiles.

DIRECTED NUMBER WITH ALGEBRA TILES Algebra tiles support the representation and manipulation of constant, linear and quadratic expressions. Expressions are built up from tiles corresponding to each of x 2, −x 2, x, −x, 1, and −1, where the shape and colour indicates the class of tile. The activities below help practice directed number addition and subtraction by using the

unit Algebra Tiles.

THE MATHENÆUM

MATHEMATICAL EXPLORATIONS, GAMES, AND LEARNING

* Home

* Number

* Geometry

* Models

* Algebra

* Chance & Data

* Calculus

* Utilities

* Guides

*

WELCOME TO THE MATHENÆUM An Athenæum is a place for reading and learning, such as a library, a museum, or scientific academy. The term comes from the temple of the goddess Athene in ancient Athens, which was used for teaching. Building on this tradition, you are now visiting my online Mathenæum — a website devoted to mathematical exploration, learning, and fun! Choose a topic from the menu above to begin your visit. (Follow @Mr_Wessen on Twitter to be notified

of updates.)

WHAT'S NEW?

Avanti — A symbolic calculator

*

Equivalent

Fraction Explorer

— Use a grid visualisation to explore equivalent fractions.

*

Wiggly Creature Constructor — Use mathematics to make amazing wiggly creatures!

*

Intersection Detection

— Find the common

symbol as fast as you can.

*

The Mathematics Of

Dobble — The

fascinating mathematics underlying Dobble and related games.

*

A

Counting Dot Machine — An Exploding Dot machine that just keeps counting.

*

Algebra

Tiles —

Multiple activities to model and solve problems involving directed number arithmetic and basic algebra.

*

Modelling

Word Problems

— Model

addition and multiplication word problems with Singapore Bar Models.

*

Fractions,

Decimals and Percentages

— Multiple

activities to model problems involving fractions, decimals and percentages with Singapore Bar Models.

*

Exploding Dots

— Multiple

activities to model arithmetic with Exploding Dots.

POPULAR

*

Equation Solver

— Generate and solve linear equations and inequations of increasing complexity.

*

Multiple Madness

— Identify multiples

as fast as you can.

*

Factor Champion

— Practice mental division

competitively.

*

Quadrilateral Explorer

— Explore angle,

side and symmetry properties of quadrilaterals.

*

Fibonacci Bamboozling

— Discover the

relationship between Fibonacci Numbers and the Golden Ratio by studying an impossible dissection

*

Problem of the Year

— Make expressions using the

digits of the year.

*

Sub Chase

— Track down the enemy submarine on a search grid.

*

Spirolaterals

— Explore shapes

generated by repeating simple drawing rules.

*

Conic Billiards

— Play and learn

about reflections and conic sections.

*

The Chaos Game

— Making fractals by transforming points.

*

Matchstick Patterns

— From patterns of matchsticks, to patterns in numbers, to algebra.

PAPERS

*   I have now published 4 articles in +plus magazine . My earlier article, Ping pong balls, infinity and superpowers

,

presents some paradoxes of infinity, and my new series of three articles, Not just a matter of time

,

discusses complexity and the limits of computation. See them all here

.

*   Here is a paper based on a talk I gave at the 2014 Mathematical Association of NSW conference describing this website: MANSW 2014 . *   Here is a paper based on a talk I gave about using paradoxes in maths teaching at the 2015 Mathematical Association of NSW conference: MANSW 2015

.

FEEDBACK

Click here to leave feedback, report bugs, make suggestions etc.

X

Name: * Email: * Subject: Message: * FULL INDEX Click here to show or hide a complete index of pages and activities.

NUMBER

Factor Explorer

Factor Champion

About Primes

About Factors

Figurate Numbers

One Hit Wonder

Make One

Equivalent Ratios

The Rate Escape

Function Pyramids

Patterns and Percentages

Four Numbers

Four Twos

Twenty Four

Problem of the Year

Dodgeball

Coding Number Patterns

Target Value

Multiple Madness

String Art

Multiplication Patterns Equivalent Fraction Explorer

GEOMETRY

Triangle Explorer

Triangle Area

Quadrilateral Explorer

Line Segments

Line Explorer

Parabola Explorer

Fibonacci Bamboozling

Circles on Circles

Trig Ratio Race

Sub Chase

Spirolaterals

Aristotle's Wheels

Ford Circles

Fractal Stamping

Fractal Line Art

The Chaos Game

Conic Billiards

Ancient Arcs

Intersection Detection The Mathematics Of Dobble

MODELS

* Algebra Tiles

Using Algebra Tiles

Directed Number: + and − Directed Number: × and ÷

Algebra with Tiles

Solving Equations

* Modelling Word Problems Addition Word Problems Multiplication Word Problems * Fractions, Decimals and Percentages Visualising Fractions

Comparing Fractions

Adding and Subtracting Fractions Multiplying Fractions

Dividing Fractions

Decimals

Percentages

Ratios

* Exploding Dots

A Counting Dot Machine Exploding Dot Machines Arithmetic with Exploding Dots

Polynomial Division

ALGEBRA

Algebra Squares

Sphinx Tiles

Equation Solver

Number Spirals

Quadratic Chaos

Matchstick Patterns

CALCULUS

The Learning Curve

Introducing the Derivative

Derivative Viewer

Introducing the Integral

CHANCE & DATA

Prisoner

Binomial Bomber

Understanding Outliers

Ants on a Stick

Higher or Lower

Computer Composer

Generating Gibberish

Crazy Dice

Notakto

Exploring Data

UTILITIES

Dice

Coins

Random

Choose

Colour Spinner

Function Viewer

GUIDES

A-maze-ing

Labyrinth

The heart curve

Seven Circles

Hungry Bugs

The Spiral of Theodorus Wiggly Creature Constructor WHAT IS MATHEMATICS?

1 × 1

11 × 11

111 × 111

1111 × 1111

11111 × 11111

111111 × 111111

1111111 × 1111111

11111111 × 11111111 111111111 × 111111111

= 1

= 121

= 12321

= 1234321

= 123454321

= 12345654321

= 1234567654321

= 123456787654321

= 12345678987654321

Mathematics is the search for order and explanation; a creative process of revealing and understanding hitherto hidden patterns and

relationships.

And mathematical understanding is powerful! By highlighting connectedness, symmetry and beauty in that which is around us and that in which we engage, it can change the way we see the world. A common misconception views mathematics as dry and technical: a collection of formulas, technical terms, and arcane processes. But that is to confuse aspects of mathematical presentation and expression with the core content. At its most fundamental level, mathematics is simply a playground of ideas — pure unencumbered ideas and abstractions, and the processes are those of imagination and

invention.

A learner approaching mathematics with this in mind will quickly find themselves immersed in a unique and creative domain full of discovery and reward. It is my hope that the activities available here will help equip and guide students on just such a journey of mathematical

learning.

So what are you waiting for? It's time to explore The Mathenæum!

Further reading:

–  A Mathematician's Lament –  Devlin's Angle –  Jo Boaler's blog at Youcubed –  Dan Finkel on Teaching and Mathematical Thinking

GUIDING PRINCIPLES

Each activity on this site has been developed on the basis of the following small set of guiding principles: * Interactive visualisation aids understanding and conceptual

development.

* Dynamic and exploratory activities generate engagement. * Historical background adds interest and helps provide context and connection for the mathematical material. * Good mathematical activities will promote group interaction. * Good mathematical activities naturally lead to discussion and

extension.

EXPLORING NUMBER

The activities in this section are designed to promote an appreciation of numbers as products, the concepts of factorisation and prime factorisation, and to practice working with fractions. Much use is made of graphical illustrations of number value to help visualise the properties being investigated.

EXPLORING GEOMETRY

The activities in this section are designed to help in the investigation of properties of triangles, lines, parabolas and general concepts in coordinate geometry. Graphical and algebraic representations are provided where appropriate. By supporting both free and constrained manipulation of curves and objects, families of shapes and curves can be constructed and explored.

EXPLORING ALGEBRA

The activities in this section are designed to help build an understanding of algebra both as a natural extension of number, and as an efficient way of describing things mathematically. EXPLORING CHANCE & DATA Here you will find activities relating to analysing randomness and strategies for games of strategy and chance, as well as those relating to statistics and the visualisation of data.

MODELS

Here you will find a variety of different models of arithmetic and algebra. These virtual manipulatives present a concrete visualisation for many arithmetical and algebraic tasks, and help develop understanding of the mathematics and processes underlying more formal mathematical representations.

EXPLORING CALCULUS

Here you will find activities that help with visualising and understanding how calculus deals with continuous functions.

UTILITIES

Some simple and useful utility applications.

GUIDES

Step by step instructions to guide you through various mathematical

constructions.

‹ ›

‹ ›

Explore factorisation as visualised by the rows and columns of rectangular grids with the Factor Explorer

, or use a similar

visualisation to explore Equivalent Fractions

.

Practice mental division competitively playing Factor Champion . There is some interesting mathematical and historical information along with related activities in the pages About Primes and About Factors . Extending the geometrical interpretation of numbers is the Figurate Numbers activity, for exploring natural sequences of numbers that arise from particular geometrical arrangements of points. The Multiple Madness activity also involves factorisation, providing a fun way to practice identifying multiples. A challenging activity is the game One Hit Wonder that presents a fun way to practice converting between fractions, decimals and percentages. Make One is a card game designed to practice basic operations on fractions. Equivalent ratios, including decimals and fractions, can be explored and systematically simplified using the Equivalent Ratios worksheet. Calculate and compare rates and escape the deadly rings when playing The Rate Escape Try and work out the secret functions by playing the Function Pyramid

activity.

Paint in coloured squares like Ellsworth Kelly using Patterns and Percentages , and analyse your artwork using fractions, decimals and percentages. A fun and creative way to play with numbers is to create different mathematical expressions from just 4 values, and this has been used in a number of games and puzzles over time. Four Numbers

, Four Twos

, Twenty Four

, and Problem of the Year are four such puzzles. Four Twos includes raising to a power among the available operations, and as a results the largest value that can be made is really quite amazingly large. See if you can find it. Target Value is a game with a similar basic idea, where you must arrange a set of numbers and arithmetic operations to result in a particular target value. Dodgeball is a game of strategy, and the winning strategy that teaches something amazing

about infinity!

Use coding to reveal Number Patterns as ASCII art, explore the patterns in multiplication tables with Multiplication Patterns , or create String Art from polygons using modular

arithmetic.

INDEX

* Factor Explorer

* Factor Champion

* About Primes

* About Factors

* Figurate Numbers

* One Hit Wonder

* Make One

* Equivalent Ratios

* The Rate Escape

* Function Pyramids

* Patterns and Percentages

* Four Numbers

* Four Twos

* Twenty Four

* Problem of the Year

* Dodgeball

* Coding Number Patterns

* Target Value

* Multiple Madness

* String Art

* Multiplication Patterns * Equivalent Fraction Explorer Explore triangle properties, congruency and related geometric constructions with the Triangle Explorer , and the relationship between the area of a triangle and the area of a rectangle with the same height and base with Triangle Area

.

Similarly, explore angle, side and symmetry properties of quadrilaterals using the Quadrilateral Explorer

.

Properties of line segments in the coordinate plane can be studied using Line Segments , and then extended to equations of lines with the Line Explorer , and parabolas with the

Parabola Explorer .

The Fibonacci Bamboozlement

and Coin Rolling

activities present two fun and somewhat counter-intuitive geometrical puzzles. Practice calculating _sin, cos_ and _tan_ from right angle triangles in the Trig Ratio Race

activity.

Track down the enemy submarine on a search grid playing Sub Chase

.

Explore Spirolaterals — shapes generated by repeating simple drawing rules. Be astounded by a paradox more than 2000 years old: do Aristotle's Wheels really show that all circles have the same circumference? Ford Circles illustrate an incredible fact about the distribution of the rational numbers — they are not actually that close to each other! The three activities Sierpiński Stamping , Fractal Art From Lines , and The Chaos Game allow you to explore some amazing aspects of fractal geometry, and how the most incredibly complex shapes can arise from simple processes. Play Conic Billiards to learn about reflections and conic sections. Ancient Arcs lets you explore some interesting shapes from antiquity made with circular

arcs.

Play the Intersection Detection game, and analyse it's fascinating mathematical basis with The Maths of Dobble

.

INDEX

* Triangle Explorer

* Triangle Area

* Quadrilateral Explorer

* Line Segments

* Line Explorer

* Parabola Explorer

* Fibonacci Bamboozling * Circles on Circles

* Trig Ratio Race

* Sub Chase

* Spirolaterals

* Aristotle's Wheels

* Ford Circles

* Fractal Stamping

* Fractal Line Art

* The Chaos Game

* Conic Billiards

* Ancient Arcs

* Intersection Detection * The Mathematics Of Dobble Move the tiles to perform algebraic operations and produce the target expression with the Algebra Squares game. Build larger and larger replicas of the basic _sphinx_ pattern with Sphinxes . Generate and solve linear equations and inequations of increasing complexity on the Equation

Solver , page.

Explore number patterns and algebra using Number Spirals

.

Iterate a simple quadratic function and generate infinite complexity with Quadratic Chaos . A common way to illustrate and the usefulness of algebra is through pattern recognition and description. The Matchstick Patterns

activity supports

exploring the link from a visual pattern to a sequence of numbers and then to an algebraic description.

INDEX

* Algebra Squares

* Sphinx Tiles

* Equation Solver

* Number Spirals

* Quadratic Chaos

* Matchstick Patterns What is the optimal arrangement for prisoners in the Prisoner

game?

Study the binomial distribution, or simply generate data to study in a frequency table with the Binomial Bomber

.

Play the Reaction Time Game and test your speed. Watch the distribution as you play and look for outliers. Why are they there? Is it fair that they be included in the results? The Exploring

Data activity

supports a more direct investigation of the connection between data values and summary values like the average and standard deviation. Ants on a Stick is a fun puzzle where a clever insight can instantly reduce mayhem and confusion to clarity and simplicity. Consider the probability for a card draw or dice roll, and try to win a prize playing Higher or Lower? The next two activities let you explore the statistics that underly music and language. First, use mathematics and randomness to describe and create music with the Computer Composer

. Then help the

Tralfamadorians decode English using a book by Charles Dickens while Generating Gibberish . Can you choose the winning die in this simple game? Crazy Dice

.

Applying mathematical reasoning to games and competition is becoming increasingly important. See how you go playing Notakto — a game of Noughts and Crosses without the noughts!

INDEX

* Prisoner

* Binomial Bomber

* Understanding Outliers

* Ants on a Stick

* Higher or Lower

* Computer Composer

* Generating Gibberish

* Crazy Dice

* Notakto

* Exploring Data

Model and solve problems involving directed number arithmetic, binomial expansion and factorisation, completing the square and linear equations using Algebra tiles

.

Singapore Bar Models support the pictorial modelling of problems involving the relationship of parts to the whole, or comparison of alternatives. One important use of these models is the representation

of Word Problems

. Another

important use is the representation of Fractions, Decimals and

Percentages ,

and their associated arithmetic. An alternative pictorial representation of arithmetic, that also naturally encompasses the concept of base and polynomial division is given by Exploding Dots

.

INDEX

* Algebra Tiles

* Using Algebra Tiles * Directed Number: + and − * Directed Number: × and ÷ * Algebra with Tiles

* Solving Equations

* Modelling Word Problems * Addition Word Problems * Multiplication Word Problems * Fractions, Decimals and Percentages * Visualising Fractions * Comparing Fractions * Adding and Subtracting Fractions * Multiplying Fractions * Dividing Fractions

* Decimals

* Percentages

* Ratios

* Exploding Dots

* A Counting Dot Machine * Exploding Dot Machines * Arithmetic with Exploding Dots * Polynomial Division The concept of a derivative is introduced and developed through The

Learning Curve ,

Introducing the Derivative , and the Derivative Viewer

.

Integration is shown in Introducing the Integral

.

INDEX

* The Learning Curve * Introducing the Derivative

* Derivative Viewer

* Introducing the Integral Roll some dice , flip some coins , generate some random numbers

, choose

from a list of entries, or spin for a random colour. Plot and compare functions using a variety of plot types with the

Function Viewer .

INDEX

* Dice

* Coins

* Random

* Choose

* Colour Spinner

* Function Viewer

Follow a simple algorithm to construct a maze

, or a labyrinth

.

Construct a Heart Curve , and other epicycloids using Clock Arithmetic. Construct a pattern with hexagonal symmetry using Seven Circles

.

Create and colour whirls from curves of pursuit using Hungry Bugs

.

Explore some maths from the 5th century BC by drawing The Spiral of

Theodorus .

See what interesting wiggly creatures you can create with the Wiggly Creature Constructor by cleverly combining circles.

INDEX

* A-maze-ing

* Labyrinth

* The heart curve

* Seven Circles

* Hungry Bugs

* The Spiral of Theodorus * Wiggly Creature Constructor The Mathenæum — © Ken Wessen (2014-19)