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Resources tagged with Mathematical reasoning & proof similar to Algebra Match:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

Leonardo's Problem

Stage: 4 and 5 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?

Common Divisor

Stage: 4 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.

Janine's Conjecture

Stage: 4 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. . . .

Perfectly Square

Stage: 4 Challenge Level:

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

Multiplication Square

Stage: 4 Challenge Level:

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

Number Rules - OK

Stage: 4 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...

Composite Notions

Stage: 4 Challenge Level:

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

Always Perfect

Stage: 4 Challenge Level:

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

Never Prime

Stage: 4 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.

Unit Interval

Stage: 4 and 5 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?

AMGM

Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

Archimedes and Numerical Roots

Stage: 4 Challenge Level:

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Pythagoras Proofs

Stage: 4 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?

Mediant

Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Geometric Parabola

Stage: 4 Challenge Level:

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

Stage: 4 Challenge Level:

Four jewellers share their stock. Can you work out the relative values of their gems?

A Knight's Journey

Stage: 4 and 5

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

Pythagorean Triples I

Stage: 3 and 4

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!

Whole Number Dynamics IV

Stage: 4 and 5

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?

Whole Number Dynamics III

Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

Whole Number Dynamics II

Stage: 4 and 5

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 I

Stage: 4 and 5

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

Why 24?

Stage: 4 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Pythagorean Triples II

Stage: 3 and 4

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

Sixational

Stage: 4 and 5 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. . . .

Natural Sum

Stage: 4 Challenge Level:

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Cosines Rule

Stage: 4 Challenge Level:

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Picture Story

Stage: 4 Challenge Level:

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

Proof: A Brief Historical Survey

Stage: 4 and 5

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

Rotating Triangle

Stage: 3 and 4 Challenge Level:

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Square Mean

Stage: 4 Challenge Level:

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

There's a Limit

Stage: 4 and 5 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?

A Biggy

Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Postage

Stage: 4 Challenge Level:

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Long Short

Stage: 4 Challenge Level:

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

DOTS Division

Stage: 4 Challenge Level:

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Our Ages

Stage: 4 Challenge Level:

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Angle Trisection

Stage: 4 Challenge Level:

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Proofs with Pictures

Stage: 4 and 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Breaking the Equation ' Empirical Argument = Proof '

Stage: 2, 3, 4 and 5

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.

Stage: 3 and 4 Challenge Level:

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Folding Fractions

Stage: 4 Challenge Level:

What fractions can you divide the diagonal of a square into by simple folding?

Iffy Logic

Stage: 4 and 5 Challenge Level:

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

The Great Weights Puzzle

Stage: 4 Challenge Level:

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

L-triominoes

Stage: 4 Challenge Level:

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

A Long Time at the Till

Stage: 4 and 5 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?

Kite in a Square

Stage: 4 Challenge Level:

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

More Number Sandwiches

Stage: 3 and 4 Challenge Level:

When is it impossible to make number sandwiches?

Calculating with Cosines

Stage: 4 and 5 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?

Same Length

Stage: 3 and 4 Challenge Level:

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