Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Mathematical Thinking > Mathematical reasoning & proof

Proof Sorter - Geometric Series

Age 16 to 18 Challenge Level:

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

Proof Sorter - Sum of an AP

Age 16 to 18 Challenge Level:

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

Direct Logic

Age 16 to 18 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

Proof Sorter - Quadratic Equation

Age 14 to 18 Challenge Level:

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

Generally Geometric

Age 16 to 18 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Middle Man

Age 16 to 18 Challenge Level:

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

Yih or Luk Tsut K'i or Three Men's Morris

Age 11 to 18 Challenge Level:

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. . . .

Napoleon's Hat

Age 16 to 18 Challenge Level:

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?

Proof: A Brief Historical Survey

Age 14 to 18

If you think that mathematical proof is really clearcut and universal then you should read this article.

Dodgy Proofs

Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

Tree Graphs

Age 16 to 18 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

Iffy Logic

Age 14 to 18 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Euclid's Algorithm II

Age 16 to 18

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.

Thousand Words

Age 16 to 18 Challenge Level:

Here the diagram says it all. Can you find the diagram?

Multiplication Square

Age 14 to 16 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Sperner's Lemma

Age 16 to 18

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

To Prove or Not to Prove

Age 14 to 18

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Target Six

Age 16 to 18 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Dalmatians

Age 14 to 18 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Pair Squares

Age 16 to 18 Challenge Level:

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Impossible Triangles?

Age 16 to 18 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

Plus or Minus

Age 16 to 18 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Sprouts Explained

Age 7 to 18

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. . . .

Janine's Conjecture

Age 14 to 16 Challenge Level:

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Mind Your Ps and Qs

Age 16 to 18 Short Challenge Level:

Sort these mathematical propositions into a series of 8 correct statements.

Picture Story

Age 14 to 16 Challenge Level:

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

There's a Limit

Age 14 to 18 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Exhaustion

Age 16 to 18 Challenge Level:

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Polynomial Relations

Age 16 to 18 Challenge Level:

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Integral Inequality

Age 16 to 18 Challenge Level:

An inequality involving integrals of squares of functions.

Quadratic Harmony

Age 16 to 18 Challenge Level:

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Square Mean

Age 14 to 16 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Mechanical Integration

Age 16 to 18 Challenge Level:

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Contrary Logic

Age 16 to 18 Challenge Level:

Can you invert the logic to prove these statements?

Notty Logic

Age 16 to 18 Challenge Level:

Have a go at being mathematically negative, by negating these statements.

Some Circuits in Graph or Network Theory

Age 14 to 18

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

Rational Roots

Age 16 to 18 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Without Calculus

Age 16 to 18 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Road Maker 2

Age 16 to 18 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

Tetra Inequalities

Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

Road Maker

Age 14 to 18 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

On the Importance of Pedantry

Age 16 to 18

A introduction to how patterns can be deceiving, and what is and is not a proof.

Breaking the Equation ' Empirical Argument = Proof '

Age 7 to 18

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.

Magic Squares II

Age 14 to 18

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

Advent Calendar 2011 - Secondary

Age 11 to 18 Challenge Level:

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

An Introduction to Number Theory

Age 16 to 18

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

A Long Time at the Till

Age 14 to 18 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Calculating with Cosines

Age 14 to 18 Challenge Level:

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?