Resources tagged with: Mathematical reasoning & proof

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

Zig Zag

Age 14 to 16 Challenge Level:

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Cosines Rule

Age 14 to 16 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.

Pythagorean Triples II

Age 11 to 16

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

Picturing Pythagorean Triples

Age 14 to 18

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.

Pythagorean Triples I

Age 11 to 16

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!

L-triominoes

Age 14 to 16 Challenge Level:

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

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.

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?

More Number Sandwiches

Age 11 to 16 Challenge Level:

When is it impossible to make number sandwiches?

No Right Angle Here

Age 14 to 16 Challenge Level:

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Iffy Logic

Age 14 to 18 Challenge Level:

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

Pareq Exists

Age 14 to 16 Challenge Level:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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.

Converse

Age 14 to 16 Challenge Level:

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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.

Triangular Intersection

Age 14 to 16 Short Challenge Level:

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Folding Squares

Age 14 to 16 Challenge Level:

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

Age 11 to 16 Challenge Level:

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

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?

Folding Fractions

Age 14 to 16 Challenge Level:

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

The Great Weights Puzzle

Age 14 to 16 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?

Long Short

Age 14 to 16 Challenge Level:

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

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.

Angle Trisection

Age 14 to 16 Challenge Level:

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

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?

Find the Fake

Age 14 to 16 Challenge Level:

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

Postage

Age 14 to 16 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. . . .

Parallel Universe

Age 14 to 16 Challenge Level:

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Age 14 to 18 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

Russian Cubes

Age 14 to 16 Challenge Level:

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Age 11 to 18 Challenge Level:

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

Our Ages

Age 14 to 16 Challenge Level:

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

Circle Box

Age 14 to 16 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

The Pillar of Chios

Age 14 to 16 Challenge Level:

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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.

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?

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.

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.

Ordered Sums

Age 14 to 16 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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. . . .

Natural Sum

Age 14 to 16 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. . . .

Similarly So

Age 14 to 16 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

The Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

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.

Knight Defeated

Age 14 to 16 Challenge Level:

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

Doodles

Age 14 to 16 Challenge Level:

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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.

Salinon

Age 14 to 16 Challenge Level:

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?