# 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 ### Operating Machines

##### Age 16 to 18 Challenge Level:

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT? ### Calculus Countdown

##### Age 16 to 18 Challenge Level:

Can you hit the target functions using a set of input functions and a little calculus and algebra? ### How Do You React?

##### Age 14 to 16 Challenge Level:

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling... ### Sweeping Satellite

##### Age 16 to 18 Challenge Level:

Derive an equation which describes satellite dynamics. ### 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... ### Snookered

##### Age 14 to 18 Challenge Level:

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion? ### 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. ### 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. ### Lap Times

##### Age 14 to 16 Challenge Level:

Can you find the lap times of the two cyclists travelling at constant speeds? ### 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? ### How Many Solutions?

##### Age 16 to 18 Challenge Level:

Find all the solutions to the this equation. ### 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. ### ' Tis Whole

##### Age 14 to 18 Challenge Level:

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed? ### Always Two

##### Age 14 to 18 Challenge Level:

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2. ### W Mates

##### Age 16 to 18 Challenge Level:

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T. ### 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. ### Polynomial Interpolation

##### Age 16 to 18 Challenge Level:

Can you fit polynomials through these points? ### Plum Tree

##### Age 14 to 18 Challenge Level:

Label this plum tree graph to make it totally magic! ### Chocolate 2010

##### Age 14 to 16 Challenge Level:

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2... ### Interpolating Polynomials

##### Age 16 to 18 Challenge Level:

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials. ### And So on - and on -and On

##### Age 16 to 18 Challenge Level:

Can you find the value of this function involving algebraic fractions for x=2000? ### 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? ### 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? ### DOTS Division

##### Age 14 to 16 Challenge Level:

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}. ### Harmonic Triangle

##### Age 14 to 16 Challenge Level:

Can you see how to build a harmonic triangle? Can you work out the next two rows? ### Perfectly Square

##### Age 14 to 16 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why? ### Triangles Within Squares

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? ### 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. ### 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? ### Unit Interval

##### Age 14 to 18 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product? ### Lower Bound

##### Age 14 to 16 Challenge Level:

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = ### 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. ### 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. . . . ### Triangles Within Triangles

##### Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers? ### Binomial

##### Age 16 to 18 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn ### Triangles Within Pentagons

##### Age 14 to 16 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number. ##### 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? ### 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. ### Telescoping Functions

##### Age 16 to 18

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra. ### Puzzling Place Value

##### Age 14 to 16 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases. ### Steel Cables

##### Age 14 to 16 Challenge Level:

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? ### 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? ### Poly Fibs

##### Age 16 to 18 Challenge Level:

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys. ### 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? ### Difference of Two Squares

##### Age 14 to 16 Challenge Level:

What is special about the difference between squares of numbers adjacent to multiples of three? ### Square Number Surprises

##### Age 14 to 16 Challenge Level: 