Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

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.

Concrete Wheel

Age 11 to 14
Challenge Level

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

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?

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?

Konigsberg Plus

Age 11 to 14
Challenge Level

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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.

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?

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

Proof Sorter - Quadratic Equation

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.

Mediant Madness

Age 14 to 16
Challenge Level

Kyle and his teacher disagree about his test score - who is right?

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.

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?

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. . . .

The Triangle Game

Age 11 to 16
Challenge Level

Can you discover whether this is a fair game?

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.

Tessellating Hexagons

Age 11 to 14
Challenge Level

Which hexagons tessellate?

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?

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.

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?

More Number Pyramids

Age 11 to 14
Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

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?

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?

Folding Fractions

Age 14 to 16
Challenge Level

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

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?

Magic Squares II

Age 14 to 18

An article which gives an account of some properties of magic squares.

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.

What Numbers Can We Make?

Age 11 to 14
Challenge Level

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

A Long Time at the Till

Age 14 to 18
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?

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.

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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

The Frieze Tree

Age 11 to 16

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Iffy Logic

Age 14 to 18
Challenge Level

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

Yih or Luk Tsut K'i or Three Men's Morris

Age 11 to 18
Challenge Level

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

Königsberg

Age 11 to 14
Challenge Level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

What Numbers Can We Make Now?

Age 11 to 14
Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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?

Pythagorean Triples II

Age 11 to 16

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

Gabriel's Problem

Age 11 to 14
Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

More Number Sandwiches

Age 11 to 16
Challenge Level

When is it impossible to make number sandwiches?

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!

Take Three from Five

Age 11 to 16
Challenge Level

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

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. . . .

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. . . .

Pattern of Islands

Age 11 to 14
Challenge Level

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Paradoxes

Age 7 to 14

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.