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

Three Ways

Age 16 to 18
Challenge Level

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Diverging

Age 16 to 18
Challenge Level

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.

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?

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?

Leonardo's Problem

Age 14 to 18
Challenge Level

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Always Perfect

Age 14 to 18
Challenge Level

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

Mechanical Integration

Age 16 to 18
Challenge Level

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

Look Before You Leap

Age 16 to 18
Challenge Level

Relate these algebraic expressions to geometrical diagrams.

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?

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.

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.

Pair Squares

Age 16 to 18
Challenge Level

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.

Plum Tree

Age 14 to 18
Challenge Level

Label this plum tree graph to make it totally magic!

How Many Solutions?

Age 16 to 18
Challenge Level

Find all the solutions to the this equation.

More Polynomial Equations

Age 16 to 18
Challenge Level

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.

Fibonacci Factors

Age 16 to 18
Challenge Level

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

Absurdity Again

Age 16 to 18
Challenge Level

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

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.

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

Quadratic Harmony

Age 16 to 18
Challenge Level

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.

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.

Particularly General

Age 16 to 18
Challenge Level

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

Polynomial Relations

Age 16 to 18
Challenge Level

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.

Really Mr. Bond

Age 14 to 16
Challenge Level

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

' 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?

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}.

Triangles Within Triangles

Age 14 to 16
Challenge Level

Can you find a rule which connects consecutive triangular numbers?

Triangles Within Squares

Age 14 to 16
Challenge Level

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

Reciprocals

Age 16 to 18
Challenge Level

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

Triangles Within Pentagons

Age 14 to 16
Challenge Level

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

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?

Magic W

Age 14 to 16
Challenge Level

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

Never Prime

Age 14 to 16
Challenge Level

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.

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. . . .

Attractive Tablecloths

Age 14 to 16
Challenge Level

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Magic Sums and Products

Age 11 to 16

How to build your own magic squares.

Interactive Number Patterns

Age 14 to 16
Challenge Level

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

Number Rules - OK

Age 14 to 16
Challenge Level

Can you produce convincing arguments that a selection of statements about numbers are true?

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.

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 ...?

Pythagoras Perimeters

Age 14 to 16
Challenge Level

If you know the perimeter of a right angled triangle, what can you say about the area?

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...

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. . . .

Janine's Conjecture

Age 14 to 16
Challenge Level

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. . . .

What's Possible?

Age 14 to 16
Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

System Speak

Age 16 to 18
Challenge Level

Five equations... five unknowns... can you solve the system?

Mediant Madness

Age 14 to 16
Challenge Level

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

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?

Series Sums

Age 14 to 16
Challenge Level

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.