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Saltire Software, home of Geometry Expressions and GXWeb

Symbolic computations on these pages use Nerdamer Symbolic JavaScript to complement the in-built computer algebra system of GXWeb.

Introduction

 

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Jigsaws and Quizzes

GXWeb Jigsaws and Quizzes: Fancy some mathematical fun? Use jigsaws and GXWeb to explore some mathematical gems drawn from geometry, number theory and elementary algebra - and then follow the links provided to see some of their many and varied applications!   

 

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Check My Steps

GXWeb Check My Steps: Check My Steps offers an open-ended, easy to use web-based graphing calculator, based upon Nerdamer Symbolic JavaScript and the GXWeb web App. It supports the learning of algebra through easy access to a simple Computer Algebra System (CAS) and powerful graphing tools.   

 

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Meeting a Friend

GXWeb Meaningful Algebra Collection 1: Two friends decide to meet during their lunch hour: if they agree to wait for x minutes, what is their chance of meeting? 

  • Area of a triangle

  • Linear inequalities

 
 

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The Beach Race

GXWeb Meaningful Algebra Collection 2: A beach race begins at a point 4 km offshore and ends at the end of 10 km beach. If I swim at 4 km/h and run at 10 km/h, at what point on the beach should I aim to come ashore to minimise my race time? 

  • Pythagoras' Theorem

  • Linear functions

  • Calculus extension

 
 

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PaperFold Problem

GXWeb Meaningful Algebra Collection 3: I fold the top left corner of an A4 page to a point on the opposite side. Where should I fold to maximise the area of the triangle formed? 

  • Area of a triangle

  • Pythagoras' Theorem

  • Calculus extension

 
 

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The Diminishing Square

GXWeb Meaningful Algebra Collection 4: A small square is constructed within a larger unit square as shown. Can you find the algebraic model for the area of the smaller square as the point X changes? 

  • Area of a triangle

  • Pythagoras' Theorem

  • Similar triangles

 
 

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Birthday Buddies

GXWeb Meaningful Algebra Collection 5: What is the chance of someone else in your class sharing the same birthday with you? How big would your group need to be for this to be better than 50% - actually smaller than you might think! 

  • Chance and probability

  • Factorial functions

 
 

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The Falling Ladder

GXWeb Meaningful Algebra Collection 6: I am at the top of a ladder, overhanging a wall, when the ladder begins to slide away from the base of the wall. At what point am I falling fastest? 

  • Pythagoras' Theorem

  • Similar Triangles

  • Calculus extension

 
 

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Algebra Tiles

GXWeb Meaningful Algebra Collection 7: Enjoy some mathematical fun and challenges with GXWeb Jigsaws and Quizzes

  • Tak Tiles

  • Algebra Tiles

  • Fractured Fractions

  • Fraction Trees

  • Kissing Circles

 
 

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TakTiles

GXWeb Meaningful Algebra Collection 8: Build Algebraic expressions using TakTile Jigsaws 

  • Meaningful Algebra

 
 

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Harmonic Mathematics

Every once in a while in Mathematics we come across problems in which the numbers just do not seem to behave themselves. They appear simple enough at the outset, but we quickly discover that our mathematical intuition fails us, and the solution - if it comes at all - arrives as the result of a long and tortuous process. 

  • Means and Sequences

  • Geometric Constructions

 
 

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Exploring Bezier Curves

Most of the curves you see on a computer screen or printed page - everything from text fonts to animations - are generated mathematically using Bezier Curves (sometimes called Bezier Splines). 

  • Recursive Functions

  • Parametric Equations

 
 

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What are the Chances?

Suppose you have two samples from a population (for example, before and after samples from a medical treatment or exam results on a common test for two parallel class groups) and you wish to know if there is a significant difference between these - or whether any differences might just result from chance or error. We can use probability distributions to decide! 

  • Chance and probability

  • Probability Distributions

  • Hypothesis Testing

 
 

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Kissing Curves

At exactly what values of a and x do the curves y = ax and y = loga(x) kiss? 

  • Exponential and Logarithmic functions

 
 

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Kissing Circles Challenge

GXWeb Kissing Circles Collection 1: Imagine two circles - kissing! Now add a third and fourth circle to kiss both of these - each can only be placed in one of THREE positions, but could have many different sizes. Once the fourth kissing circle is added, however, everything changes! 

  • Geometric Constructions

  • Kissing Circles

 
 

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Arbelos: the Shoemaker's Knife

GXWeb Kissing Circles Collection 2: Arbelos is an old Greek name for a shoemaker's knife, and is made up of three kissing semi-circles. 

  • Perimeters and areas of circles

  • Geometric Constructions

 
 

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Farey Numbers, Fraction Trees and Kissing Circles

GXWeb Kissing Circles Collection 3: Suppose you wanted to make a list of all the rational numbers between 0 and 1. How would you start? 

  • Rational numbers and infinite series

 
 

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Fraction Tree Explorer

Did you know that fractions grow on trees? Or that (fraction) trees can grow from (continued) fractions? 

  • Rational numbers and infinite series

  • Continued fractions

  • Kissing circles

 
 

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Descartes Kissing Circles Theorem

GXWeb Kissing Circles Collection 4: Rene Descartes was not just the Father of Modern Philosophy, but also the Father of Analytic Geometry, building the bridge between Algebra and Geometry. This gem of a theorem describes the nature of Kissing Circles. 

  • Geometric Constructions

  • Kissing Circles

  • Quadratic Functions

 
 

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Apollonian Bubbles

GXWeb Kissing Circles Collection 5: Kissing Circles take on a whole new look when they become bubbles in an Apollonian Gasket! Apollonios of Perga studied these mathematical bubbles in the third century BC, but it took a couple of modern mathematicians in the 1990s to show how, once again, Descartes Theorem can help to produce the coolest screensavers. 

  • Geometric Constructions

  • Kissing Circles

  • Quadratic Functions

 
 

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Dragons, Folds and Fractions

Take a piece of paper and fold it in half, from right to left. If, after folding, you open the page out again, you will observe the paperfold sequence - a series of downward and upward folds that may catch you by surprise! Multiple folds result in fractions of the page jumbled within the folds, and the shape of the folds themselves lead to the Dragon Curve! 

  • Dragons and Fractions

  • Sequences and Fractal Curves

 
 

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PaperFolds and Points in a Fold

Take a piece of paper and fold it in half, from right to left. The fraction 1/2 of the page lies at the fold. Repeat and 1/2 will now lie on the left and 1/4 and 3/4 at the fold. Multiple folds result in fractions of the page jumbled within the folds, and the shape of the folds themselves lead to the Dragon Curve! 

  • Rational numbers and infinite series

  • Orders of fractions

  • Sequences and Fractal Curves

 
 

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Fractured Fractions

GXWeb Continued Fraction Collection 1: Every real number, rational and irrational, can be represented as a continued fraction. While normal fractions can only represent rational numbers, continued fractions are different. Rational numbers produce finite continued fractions, while irrationals become infinite continued fractions. 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Extending Stern-Brocot and Continued Fractions

GXWeb Continued Fraction Collection 2: Here I describe some discoveries I have come across recently which have not only extended my thinking and understanding of Stern-Brocot, but of continued fractions in general, potentially opening new doorways for further exploration. 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Fractured Functions

GXWeb Continued Fraction Collection 3: Most of the continued fractions you will come across are likely to be of the simple variety, but the many generalised forms offer so many interesting patterns and possibilities for further exploration! Try a few and see what you discover... 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Continued Fraction Arithmetic

GXWeb Continued Fraction Collection 4: Contrary to everybody, this self contained paper will show that continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic. (Bill Gosper, 1972) 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Continued Logarithms

GXWeb Continued Fraction Collection 5: In the newly dawning computer age of the 1970s, Bill Gosper recognised both the limitations of continued fractions in dealing with very large and very small numbers, and the possibilities for a new type of binary representation, reducing such numbers to highly accurate arrays of zeros and ones. Gosper described it as a 'sort of recursive version of scientific notation' (1972) 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Golden Fractions

GXWeb Continued Fraction Collection 6: The Golden Ratio is probably one of history's most interesting numbers, but it is only through continued fractions that we understand why it is also the most irrational of all numbers! 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Bessel Functions

GXWeb Continued Fraction Collection 7: What might appear to be the simplest of continued fractions turns out to be anything but simple - and related to a family of functions with important applications to physics - welcome to the Bessel Function Family! 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Gamma Function

GXWeb Continued Fraction Collection 8:

The factorial function is well known - \(3! = 3 \cdot 2 \cdot 1 = 6, 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 12,\)
\( n! = n \cdot (n-1) \cdot (n-2)\)...
But what happens in between? The Gamma function includes all values, not just the integers, and has some interesting properties. 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Chaos Theory

GXWeb Continued Fraction Collection 9:

Imagine a population, say of fish in a pond. If the pond is fixed in size and limited in the amount of food which it can provide, then the population of fish cannot grow unbounded. But what happens over the next few years becomes the basis for chaos theory. 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Musical Fractions

GXWeb Continued Fraction Collection 10: The modern musical scale is based upon a geometric sequence - each successive note is produced my multiplying by the twelfth root of 2 (\( \sqrt [12] 2 \) or \( 2^{1 \over 12} \) - approximately 1.059463). Now what if you could turn any continued fraction into a melody? 

  • Rational and Irrational Numbers

  • Approximations

 
 

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Accelerometer and Step Counter

GXWeb STEM Collection: Can you use the 3-D accelerometer of your mobile device to build a pedometer? 

  • STEM application (for mobile devices)

 
 

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Accelerometer-controlled Robot

GXWeb STEM Collection: Can you use the 3-D accelerometer of your mobile device to control a BLE Robot? 

  • STEM application (for mobile devices)

Build and program your Arduino or TI LaunchPad robot?

 
 

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Magnetometer and Compass

GXWeb STEM Collection: Can you use the 3-D magnetometer of your mobile device to build a compass? 

  • STEM application (for mobile devices)

 
 

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Magnetometer and GPS Navigation

GXWeb STEM Collection: Can you use the 3-D magnetometer and GPS of your mobile device to help you navigate? 

  • STEM application (for mobile devices)

 
 

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Learn How...

For some more detailed ideas about how to use such wonderful tools, please feel free to read:

If you have any questions or would like to share your experience with these tasks, please drop me an email!

 

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