Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

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?

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?

Cyclic Quad Jigsaw

Age 14 to 16
Challenge Level

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

Towering Trapeziums

Age 14 to 16
Challenge Level

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

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?

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.

Pythagorean Triples II

Age 11 to 16

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

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?

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.

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.

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!

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?

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.

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.

Encircling

Age 14 to 16
Challenge Level

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Folding Fractions

Age 14 to 16
Challenge Level

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

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?

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.

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.

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.

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?

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. . . .

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

Thirty Nine, Seventy Five

Age 11 to 14
Challenge Level

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

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. . . .

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.

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.

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?

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?

Con Tricks

Age 11 to 14

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

More Number Sandwiches

Age 11 to 16
Challenge Level

When is it impossible to make number sandwiches?

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.

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?

Concrete Wheel

Age 11 to 14
Challenge Level

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

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?

Cyclic Quadrilaterals

Age 11 to 16
Challenge Level

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

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.

Gift of Gems

Age 14 to 16
Challenge Level

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

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.

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.

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.

Triangle Incircle Iteration

Age 14 to 16
Challenge Level

Keep constructing triangles in the incircle of the previous triangle. What happens?

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?

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?

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.

Proximity

Age 14 to 16
Challenge Level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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. . . .

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.