Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Mathematical Thinking > Mathematical reasoning & proof

Truth Tables and Electronic Circuits

Age 11 to 18

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

Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Stonehenge

Age 16 to 18 Challenge Level:

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Air Nets

Age 7 to 18 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Fractional Calculus III

Age 16 to 18

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Exponential Intersection

Age 16 to 18 Challenge Level:

Can the pdfs and cdfs of an exponential distribution intersect?

Telescoping Functions

Age 16 to 18

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Common Divisor

Age 14 to 16 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

And So on - and on -and On

Age 16 to 18 Challenge Level:

Can you find the value of this function involving algebraic fractions for x=2000?

Cube Net

Age 16 to 18 Challenge Level:

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

Perfectly Square

Age 14 to 16 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Leonardo's Problem

Age 14 to 18 Challenge Level:

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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. . . .

Unit Interval

Age 14 to 18 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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.

Composite Notions

Age 14 to 16 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Binomial

Age 16 to 18 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Number Rules - OK

Age 14 to 16 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Mouhefanggai

Age 14 to 16

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.

Three Ways

Age 16 to 18 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Sixational

Age 14 to 18 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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.

Flexi Quad Tan

Age 16 to 18 Challenge Level:

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

Little and Large

Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Look Before You Leap

Age 16 to 18 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

A Knight's Journey

Age 14 to 18

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

Particularly General

Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases.

Geometric Parabola

Age 14 to 16 Challenge Level:

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

How Many Solutions?

Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

Interpolating Polynomials

Age 16 to 18 Challenge Level:

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Never Prime

Age 14 to 16 Challenge Level:

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Diverging

Age 16 to 18 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

A Computer Program to Find Magic Squares

Age 16 to 18

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Dodgy Proofs

Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

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.

Euler's Formula and Topology

Age 16 to 18

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. . . .

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

The Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

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.

Classifying Solids Using Angle Deficiency

Age 11 to 16 Challenge Level:

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

Same Length

Age 11 to 16 Challenge Level:

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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?

Impossible Sandwiches

Age 11 to 18

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.

Mind Your Ps and Qs

Age 16 to 18 Short Challenge Level:

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

More Dicey Decisions

Age 16 to 18 Challenge Level:

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

Kite in a Square

Age 14 to 16 Challenge Level:

Can you make sense of the three methods to work out the area of the kite in the square?

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.

Continued Fractions II

Age 16 to 18

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).