Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

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.

Volume of a Pyramid and a Cone

Age 11 to 14

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Unit Fractions

Age 11 to 14
Challenge Level

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

Rhombus in Rectangle

Age 14 to 16
Challenge Level

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Gift of Gems

Age 14 to 16
Challenge Level

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

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

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?

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!

Folding Fractions

Age 14 to 16
Challenge Level

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

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.

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?

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.

A Chordingly

Age 11 to 14
Challenge Level

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Ratty

Age 11 to 14
Challenge Level

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Fitting In

Age 14 to 16
Challenge Level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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?

Pent

Age 14 to 18
Challenge Level

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Towering Trapeziums

Age 14 to 16
Challenge Level

Can you find the areas of the trapezia in this sequence?

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.

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.

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.

Three Balls

Age 14 to 16
Challenge Level

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Lens Angle

Age 14 to 16
Challenge Level

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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?

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?

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?

More Number Sandwiches

Age 11 to 16
Challenge Level

When is it impossible to make number sandwiches?

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?

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?

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.

Chameleons

Age 11 to 14
Challenge Level

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

Pythagorean Triples II

Age 11 to 16

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

More Mathematical Mysteries

Age 11 to 14
Challenge Level

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

Logic

Age 7 to 14

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Convex Polygons

Age 11 to 14
Challenge Level

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Con Tricks

Age 11 to 14

Here are some examples of 'cons', and see if you can figure out where the trick is.

Cross-country Race

Age 14 to 16
Challenge Level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

Tourism

Age 11 to 14
Challenge Level

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Disappearing Square

Age 11 to 14
Challenge Level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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?

Diophantine N-tuples

Age 14 to 16
Challenge Level

Can you explain why a sequence of operations always gives you perfect squares?

Go Forth and Generalise

Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Round and Round

Age 14 to 16
Challenge Level

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Coins on a Plate

Age 11 to 14
Challenge Level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Concrete Wheel

Age 11 to 14
Challenge Level

A huge wheel is rolling past your window. What do you see?

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?

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.

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?

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. . . .