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Breaking the equation 'empirical argument = proof '
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
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Sprouts explained
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Air nets
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Geometry and gravity 2
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Impossible sandwiches
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Yih or Luk tsut k'i or Three Men's Morris
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and knot arithmetic.
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The bridges of Konigsberg
Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
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What does it all add up to?
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An introduction to proof by contradiction
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Proof: a brief historical survey
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Binomial coefficients
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A knight's journey
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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.
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To prove or not to prove
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Some circuits in graph or network theory
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Common divisor
Can you find out what numbers divide these expressions? Can you prove that they are always divisors?
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Network trees
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Summing geometric progressions
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
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Unit interval
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There's a limit
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Mega quadratic equations
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Dalmatians
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Proof sorter - quadratic equation
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Always perfect
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Curve fitter
This problem challenges you to find cubic equations which satisfy different conditions.
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Back fitter
10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?
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Calculating with cosines
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Always two
Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
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Quad in quad
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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Kite in a square
Can you make sense of the three methods to work out what fraction of the total area is shaded?
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Sixational
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Impossible sums
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Iff
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Difference of odd squares
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Pent
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The converse of Pythagoras
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?
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Leonardo's problem
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A long time at the till
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An introduction to number theory
An introduction to some beautiful results in Number Theory.
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On the importance of pedantry
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Telescoping functions
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Where do we get our feet wet?
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Why stop at three by one
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Modulus arithmetic and a solution to differences
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Sums of squares and sums of cubes
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Transitivity
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Modulus arithmetic and a solution to dirisibly yours
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Continued fractions II
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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Fractional calculus III
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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Sperner's lemma
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Euler's formula and topology
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A computer program to find magic squares
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An alphanumeric
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Powerful properties
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The kth sum of n numbers
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Euclid's algorithm II
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
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Big and small numbers in physics - group task
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Sixty-seven squared
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Proof sorter - geometric sequence
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Proof sorter - the square root of 2 is irrational
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Proof sorter - sum of an arithmetic sequence
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Stats statements
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Fixing it
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Integration matcher
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Fibonacci factors
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Prime sequences
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Flexi quad tan
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Napoleon's hat
Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
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Summats clear
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Diverging
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Polynomial relations
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Tetra inequalities
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Stonehenge
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More dicey decisions
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Staircase
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Code to zero
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Quadratic harmony
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Without calculus
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Pythagorean golden means
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Middle man
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Three ways
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Big, bigger, biggest
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Trig rules OK
Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...
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Magic W wrap up
Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.