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Resources tagged with Mathematical reasoning & proof similar to Squ-areas:

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

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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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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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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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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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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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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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Rhombus in Rectangle

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

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.

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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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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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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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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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Dicing with Numbers

Stage: 3 Challenge Level: Challenge Level:1

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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Aba

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

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

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Not Necessarily in That Order

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

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

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The Genie in the Jar

Stage: 3 Challenge Level: Challenge Level:1

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

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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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Con Tricks

Stage: 3

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

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Logic

Stage: 2 and 3

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

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Online

Stage: 3 Challenge Level: Challenge Level:1

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

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Matter of Scale

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

Prove Pythagoras Theorem using enlargements and scale factors.

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Cycle It

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

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

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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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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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Children at Large

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

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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Paradoxes

Stage: 2 and 3

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.

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Thirty Nine, Seventy Five

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Volume of a Pyramid and a Cone

Stage: 3

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.

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Cross-country Race

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

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

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More Mathematical Mysteries

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Circle Box

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

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?

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Marbles

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

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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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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Tri-colour

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

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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Winning Team

Stage: 3 Challenge Level: Challenge Level:1

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

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More Marbles

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

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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Pyramids

Stage: 3 Challenge Level: Challenge Level:1

What are the missing numbers in the pyramids?

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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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Growing Ls

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

Can you fit Ls together to make larger versions of themselves?

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Kite in a Square

Stage: 4 Challenge Level: Challenge Level:1

Can you make sense of the three methods to work out the area of the kite in the square?

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Areas and Ratios

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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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Greetings

Stage: 3 Challenge Level: Challenge Level:1

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

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Cyclic Quadrilaterals

Stage: 3 Challenge Level: Challenge Level:1

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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Multiplication Square

Stage: 3 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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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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Impossible Sandwiches

Stage: 3, 4 and 5

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.

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The Frieze Tree

Stage: 3 and 4

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

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Magic Squares II

Stage: 4 and 5

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

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What Numbers Can We Make Now?

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.