nrich.maths.org :: Mathematics Enrichment :: Articles for Secondary Teachers

Fibonacci's Three Wishes 2

Second of two articles about Fibonacci, written for students.

Extra Challenges from Madras

A few extra challenges set by some young NRICH members.

Going Places with Mathematicians

This article looks at the importance in mathematics of representing places and spaces mathematics. Many famous mathematicians have spent time working on problems that involve moving and mapping things.

All Is Number

Read all about Pythagoras' mathematical discoveries in this article written for students.

A-maze-ing

Did you know that ancient traditional mazes often tell a story? Remembering the story helps you to draw the maze.

Palindromes

Find out about palindromic numbers by reading this article.

An Introduction to Magic Squares

Find out about Magic Squares in this article written for students. Why are they magic?!

Going First

This article shows how abstract thinking and a little number theory throw light on the scoring in Go.

Pi, a Very Special Number

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

Arclets

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NFRICH website.

More on Mazes

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

Ratio or Proportion?

An article for teachers which discusses the differences between ratio and proportion, and invites readers to contribute their own thoughts.

Logic, and How it Should Influence Our Teaching

Providing opportunities for children to participate in group narrative in our classrooms is vital. Their contrasting views lead to a high level of revision and improvement, and through this process they become more aware of "thinking". This article looks at the way we handle these narratives.

Calendars

Calendars were one of the earliest calculating devices developed by civilisations. Find out about the Mayan calendar in this article.

Friezes

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Generating Number Patterns: an Email Conversation Amongst

This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.

Ever Had the Feeling You Are Going Round in Circles

This article teaches you how to draw cardiods, limacons, nephroids and ellipses - a lot easier than they sound! All you need is a pair of compasses and a pencil.

The Secret World of Codes and Code Breaking

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.

History of Morse

This short article, gives an outline of the origins of Morse code and its inventor and how the frequency of letters is reflected in the code they were given.

Pythagoras

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

Liethagoras' Theorem

Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him.

Got It! Article

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Mathematical Symbols

A brief article written for pupils about mathematical symbols.

Thinking 3D

How can we as teachers begin to introduce 3D ideas to young children? Where do they start? How can we lay the foundations for a later enthusiasm for working in three dimensions?

Can You Find a Perfect Number?

Can you find any perfect numbers? Read this article to find out more...

History of Measurement

This article tells you all about some early ways of measuring as well as methods of measuring tall objects we can still use today. You can even have a go at some yourself!

Algebra from Geometry

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Leonardo of Pisa and the Golden Rectangle

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.

Weigh to Go

This article for teachers recounts the history of measurement, encouraging it to be used as a spring board for cross-curricular study.

Thinking about Different Ways of Thinking

This article, the first in a series, discusses mathematical-logical intelligence as described by Howard Gardner.

Fibonacci's Three Wishes 1

First or two articles about Fibonacci, written for students.

Dominant Intelligences

The second in a series, this article looks at the possible opportunities for children who operate from different intelligences to be involved in "typical" maths problems.

When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

What Are Numbers?

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

The Cyclic Quadilateral

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

Latin Squares

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

Muggles, Logo and Gradients

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

Logic, Truth Tables and Switching Circuits Challenge

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record your findings.

A Knight's Journey

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Truth Tables and Electronic Circuits

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Ding Dong Bell

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Some Circuits in Graph or Network Theory

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Bands and Bridges: Bringing Topology Back

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

Roasting Old Chestnuts

Mainly for teachers. A discussion and examples of some of the school mathematics of yesteryear.

More Old Chestnuts

Mainly for teachers. More school mathematics of yesteryear.

Roasting Old Chestnuts 3

Mainly for teachers. More mathematics of yesteryear.

Roasting Old Chestnuts 4

For teachers. Yet more school maths from long ago-interest and percentages.

Euclid? No, but Carol, Yes

For teachers. About the teaching of geometry with some examples from school geometry of long ago.

Tangles

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

Shuffles Tutorials

Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.

Elastic Maths

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

Continued Fractions I

An article introducing continued fractions with some simple puzzles for the reader.

Curvature of Surfaces

How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins.

The 1999 Cricket World Cup: A Simulation Game for the Classroom

This article describes a simulation which can be played out in the classroom.

Divisibility Tests

Tim Rowland takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

Pythagorean Triples I

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

Pythagorean Triples II

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

Whole Number Dynamics II

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

Whole Number Dynamics IV

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

Euler's Formula and Topology

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the relationship between Euler's formula and angle deficiency of polyhedra.

Whole Number Dynamics V

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

Geometry and Gravity 1

This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.

Geometry and Gravity 2

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

Zooming in on the Squares

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Magical Maze - Lecture Summary

An introduction to Ian Stewart's RI Christmas Lectures on Mathematics and Nature with investigations and activities on mathematical patterns in cosmology, music, snowflakes, and flowers, animal movement, probability and risk, and patterns in the regularity and irregularity in nature.

Magical Maze - 35 Activities

Investigations and activities for you to enjoy on pattern in nature.

Magic Squares

An account of some magic squares and their properties and and how to construct them for yourself.

Magic Squares II

An article which gives an account of some properties of magic squares.

Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

Picturing Pythagorean Triples

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

An Investigation Based on Score

Class 2YP from Madras College was inspired by the problem in NRICH to work out in how many ways the number 1999 could be expressed as the sum of 3 odd numbers, and this is their solution.

Ways of Summing Odd Numbers

Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum of odd numbers.

Impossible Sandwiches

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

Magic Sums and Products

How to build your own magic squares.

The Codabar Check

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

Adding and Subtracting Positive and Negative Numbers

How can we help students make sense of addition and subtraction of negative numbers?

Magic Squares for Special Occasions

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Con Tricks

Here are some examples of 'cons', and see if you can figure out where the trick is.

Cricket Ratings

Like all sports rankings, the cricket ratings involve some maths. In this case, they use a mathematical technique known as exponential weighting. For those who want to know more, read on.

Squareo'scope Determines the Kind of Triangle

A description of some experiments in which you can make discoveries about mathematics.

Volume of a Pyramid and a Cone

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Mouhefanggai

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

What Is the Circle Scribe Disk Compass?

Introducing a geometrical instrument with 3 basic capabilities.

A Story about Absolutely Nothing

This article for the young and old talks about the origins of our number system and the important role zero has to play in it.

Bloom's Taxonomy

Bloom's taxonomy

Rich Tasks and Contexts

What are rich tasks and contexts and why do they matter?

The Best Card Trick Ever!

You will need an assistant, a witness and an ordinary deck of cards.

Educating the Gifted Mathematician

What is the classroom culture that you foster to support able learners?

Cultivating Creativity

Creativity in the mathematics classroom is not just about what pupils do but also what we do as teachers. If we are thinking creatively about the mathematical experiences we offer our pupils we can open up opportunities for them to be creative. Jennifer Piggott shares some of her thoughts on creative teaching, and how it can encourage creative learners.

MEI 2005

Presentation given at the MEI conference in Reading 2005

Sufficient but Not Necessary: Two Eyes and Seki in Go

The game of go has a simple mechanism. This discussion of the principle of two eyes in go has shown that the game does not depend on equally clear-cut concepts.

Round-robin Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Dancing with Maths

An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?

Paper Folding - Models of the Platonic Solids

A description of how to make the five Platonic solids out of paper.

Pattern Power

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Closing the Learning and Teaching Gap

This article discusses the findings of the 1995 TIMMS study how to use this information to close the performance gap that exists between nations.

Go Forth and Generalise

This article begins to look at what it means to generalise and the importance of looking beyond spotting patterns to understanding why the patterns are there.

Dice, Routes and Pathways

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to think mathematically, especially geometrically.

Games Related to Nim

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Sprouts

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought.

Problem Solving: Opening up Problems

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have greater potential.

Using Questioning to Stimulate Mathematical Thinking

Good questioning techniques have long being regarded as a fundamental tool of effective teachers. This article for teachers looks at different categories of questions that can promote mathematical thinking.

Using Questioning to Stimulate Mathematical Thinking: Addendum

In the process of working with some groups of teachers on using questions to promote mathematical thinking, the following table was developed. It provides examples of generic questions that can be used to guide children through a mathematical investigation, and at the same time prompt higher levels of thinking.

Learning Mathematics Through Games Series: 4. from Strategy Games

Basic strategy games are particularly suitable as starting points for investigations. Players instinctively try to discover a winning strategy, and usually the best way to do this is to analyse the outcomes of series of 'moves'. With a little encouragement from the teacher, a mathematical investigation is born.

Co-operative Problem Solving: Pieces of the Puzzle Approach

The content of this article is largely drawn from an Australian publication by Peter Gould that has been a source of many successful mathematics lessons for both children and student-teachers. It presents a style of problem-solving activity that has the potential to benefit ALL children in a class, both mathematically and socially, and is readily adaptable to most topics in mathematics curricula.

A Japanese Mathematics Lesson

Jenni Way describes her visit to a Japanese mathematics classroom.

Mathematical Patchwork

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

On Time?

This article explains how Greenwich Mean Time was established and in fact, why Greenwich in London was chosen as the standard.

History of Fractions

Who first used fractions? Were they always written in the same way? How did fractions reach us here? These are the sorts of questions which this article will answer for you.

What’s Inside/outside/under the Box?

This article describes investigations that offer opportunities for children to think differently, and pose their own questions, about shapes.

Eureka!

Read this article to find out about the discoveries and inventions of Archimedes.

Coordinates

Has it ever occurred to you how maps are made? Or perhaps who first thought of the idea of designing maps? We're here to answer these questions for you.

History of Money

If you would like a new CD you would probably go into a shop and buy one using coins or notes. (You might need to do a bit of saving first!) However, this way of paying for the things you want did not always exist. Find out more ...

Logic

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Adding with the Abacus

Nowadays the calculator is very familiar to many of us. What did people do to save time working out more difficult problems before the calculator existed?

Outside the Box

This article explores the links between maths, art and history, and suggests investigations that are enjoyable as well as challenging.

The Dangerous Ratio or to Be Male Is Odd

This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.

High Jumping

How high can a high jumper jump? How can a high jumper jump higher without jumping higher? Read on...

Where... over There...

This introduction to polar coordinates describes what is an effective way to specify position. This article explains how to convert between polar and cartesian coordinates and also encourages the creation of some attractive curves from some relatively easy equations.