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Resources tagged with Manipulating algebraic expressions/formulae similar to Fractional Calculus II:

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Chocolate 2004

Stage: 4 Challenge Level: Challenge Level:1

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

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Ball Bearings

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Poly Fibs

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Incircles

Stage: 5 Challenge Level: Challenge Level:1

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 ...?

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More Polynomial Equations

Stage: 5 Challenge Level: Challenge Level:1

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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Complex Partial Fractions

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Calculus Countdown

Stage: 5 Challenge Level: Challenge Level:1

Can you hit the target functions using a set of input functions and a little calculus and algebra?

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Operating Machines

Stage: 5 Challenge Level: Challenge Level:1

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

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Sums of Pairs

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

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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Janine's Conjecture

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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Fibonacci Factors

Stage: 5 Challenge Level: Challenge Level:1

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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Absurdity Again

Stage: 5 Challenge Level: Challenge Level:1

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

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System Speak

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

Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6

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Diverging

Stage: 5 Challenge Level: Challenge Level:1

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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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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Telescoping Functions

Stage: 5

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.

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Mechanical Integration

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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There and Back

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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Pair Squares

Stage: 5 Challenge Level: Challenge Level:1

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

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Unusual Long Division - Square Roots Before Calculators

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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Squares

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

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Orbiting Billiard Balls

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

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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And So on - and on -and On

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Sitting Pretty

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

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?

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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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Always Two

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

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.

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Triangles Within Squares

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

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

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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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Magic Sums and Products

Stage: 3, 4 and 5

How to build your own magic squares.

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Sweeping Satellite

Stage: 5 Challenge Level: Challenge Level:1

Derive an equation which describes satellite dynamics.

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2008 January Poster

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

Pair Products

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Dating Made Easier

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

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

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Lap Times

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

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

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Look Before You Leap

Stage: 5 Challenge Level: Challenge Level:1

Relate these algebraic expressions to geometrical diagrams.

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Cocked Hat

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A new solution to a Tough Nut problem. Aleksander has drawn graphs for members of the family of functions given by the implicit equation (x^2 + 2ay -a^2)^2 = y^2(a^2 - x^2) corresponding to different. . . .

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Polynomial Relations

Stage: 5 Challenge Level: Challenge Level:1

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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Root to Poly

Stage: 4 Challenge Level: Challenge Level:1

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is 1+sqrt2+sqrt3.

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Quadratic Harmony

Stage: 5 Challenge Level: Challenge Level:1

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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Reciprocals

Stage: 5 Challenge Level: Challenge Level:1

Prove that for any positive numbers x1,x2,...,xn (x1 + x2 + ... + xn)([1/(x1)] + [1/(x2)] + ... + [1/(xn)]) is greater than or equal to n^2.

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Binomial

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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' Tis Whole

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

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?

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Graphic Biology

Stage: 5 Challenge Level: Challenge Level:1

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

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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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The Medieval Octagon

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

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

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Three Ways

Stage: 5 Challenge Level: Challenge Level:1

Given that x + y = -1 find the largest value of xy (a) by coordinate geometry (b) by calculus (c) by algebra.

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How Many?

Stage: 5 Challenge Level: Challenge Level:1

Find all the solutions to the this equation.

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Sums of Squares

Stage: 5 Challenge Level: Challenge Level:1

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

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Nicely Similar

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

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?

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Matchless

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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Never Prime

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.