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Resources tagged with Mathematical reasoning & proof similar to Pericut:

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

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Rotating Triangle

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Coins on a Plate

Stage: 3 Challenge Level: Challenge Level:1

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.

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Round and Round

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Pythagoras Proofs

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Appearing Square

Stage: 3 Challenge Level: Challenge Level:1

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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Disappearing Square

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Proof Sorter - Quadratic Equation

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Three Balls

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Zig Zag

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Cyclic Quad Jigsaw

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Cyclic Quadrilaterals

Stage: 3 Challenge Level: Challenge Level:1

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

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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!

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Go Forth and Generalise

Stage: 3

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.

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A Chordingly

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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The Pillar of Chios

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

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The Triangle Game

Stage: 3 and 4 Challenge Level: Challenge Level:1

Can you discover whether this is a fair game?

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Picturing Pythagorean Triples

Stage: 4 and 5

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.

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Encircling

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Pythagorean Triples II

Stage: 3 and 4

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

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

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?

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Quads

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?

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Rolling Coins

Stage: 4 Challenge Level: Challenge Level:1

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Some Circuits in Graph or Network Theory

Stage: 4 and 5

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

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Angle Trisection

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Postage

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Seven Squares - Group-worthy Task

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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Ratty

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Road Maker

Stage: 4 and 5 Challenge Level: Challenge Level:1

Which of these roads will satisfy a Munchkin builder?

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Converse

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Happy Numbers

Stage: 3 Challenge Level: Challenge Level:1

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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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.

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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.

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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.

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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.

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Always Perfect

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Sixational

Stage: 4 and 5 Challenge Level: Challenge Level:1

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. . . .

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The Great Weights Puzzle

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

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

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Cosines Rule

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Iffy Logic

Stage: 4 and 5 Challenge Level: Challenge Level:1

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

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Gift of Gems

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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There's a Limit

Stage: 4 and 5 Challenge Level: Challenge Level:1

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?

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Multiplication Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Fitting In

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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A Biggy

Stage: 4 Challenge Level: Challenge Level:1

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.

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Shuffle Shriek

Stage: 3 Challenge Level: Challenge Level:1

Can you find all the 4-ball shuffles?

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Long Short

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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DOTS Division

Stage: 4 Challenge Level: Challenge Level:1

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}.

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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?