Resources tagged with: Creating and manipulating expressions and formulae

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Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae

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.

Sitting Pretty

Age 14 to 16 Challenge Level:

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Just Touching

Age 16 to 18 Challenge Level:

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Circles in Circles

Age 16 to 18 Challenge Level:

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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?

Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

The Medieval Octagon

Age 14 to 16 Challenge Level:

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Three Four Five

Age 14 to 16 Challenge Level:

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Incircles

Age 16 to 18 Challenge Level:

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Semi-square

Age 14 to 16 Challenge Level:

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Nicely Similar

Age 14 to 16 Challenge Level:

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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.

Gutter

Age 14 to 16 Challenge Level:

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Ball Bearings

Age 16 to 18 Challenge Level:

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Terminology

Age 14 to 16 Challenge Level:

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Screen Shot

Age 14 to 16 Challenge Level:

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Generating Triples

Age 14 to 16 Challenge Level:

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Pareq Calc

Age 14 to 16 Challenge Level:

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Diophantine N-tuples

Age 14 to 16 Challenge Level:

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

Pinned Squares

Age 14 to 16 Challenge Level:

What is the total number of squares that can be made on a 5 by 5 geoboard?

Areas of Parallelograms

Age 14 to 16 Challenge Level:

Can you find the area of a parallelogram defined by two vectors?

Complex Partial Fractions

Age 16 to 18 Challenge Level:

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Training Schedule

Age 14 to 16 Challenge Level:

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Always Perfect

Age 14 to 16 Challenge Level:

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

Marbles in a Box

Age 11 to 16 Challenge Level:

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Square Pizza

Age 14 to 16 Challenge Level:

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Puzzling Place Value

Age 14 to 16 Challenge Level:

Can you explain what is going on in these puzzling number tricks?

Algebra from Geometry

Age 11 to 16 Challenge Level:

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

One and Three

Age 14 to 16 Challenge Level:

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Hand Swap

Age 14 to 16 Challenge Level:

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

AMGM

Age 14 to 16 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

Pick's Theorem

Age 14 to 16 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

A Tilted Square

Age 14 to 16 Challenge Level:

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Triangles Within Triangles

Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

Robert's Spreadsheet

Age 14 to 16 Challenge Level:

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number.

Partly Painted Cube

Age 14 to 16 Challenge Level:

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Little and Large

Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Around and Back

Age 14 to 16 Challenge Level:

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Algebra Match

Age 11 to 16 Challenge Level:

A task which depends on members of the group noticing the needs of others and responding.

Magic Sums and Products

Age 11 to 16

How to build your own magic squares.

Mediant Madness

Age 14 to 16 Challenge Level:

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

Multiplication Square

Age 14 to 16 Challenge Level:

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

Simplifying Doughnut

Age 14 to 18 Challenge Level:

An algebra task which depends on members of the group noticing the needs of others and responding.

Dating Made Easier

Age 14 to 16 Challenge Level:

If a sum invested gains 10% each year how long before it has doubled its value?

Interactive Number Patterns

Age 14 to 16 Challenge Level:

How good are you at finding the formula for a number pattern ?

Reaction Rates!

Age 16 to 18

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

Sweeping Satellite

Age 16 to 18 Challenge Level:

Derive an equation which describes satellite dynamics.

Triangles Within Squares

Age 14 to 16 Challenge Level:

Can you find a rule which relates triangular numbers to square numbers?

Painted Cube

Age 14 to 16 Challenge Level:

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?