# Resources tagged with: Creating and manipulating expressions and formulae

Filter by: Content type:
Age range:
Challenge level:

### There are 114 results

Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae ### Mechanical Integration

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge 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. ### Unit Interval

##### Age 14 to 18Challenge Level

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

##### Age 14 to 16Challenge Level

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

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

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

##### Age 14 to 16Challenge 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. ### Triangles Within Squares

##### Age 14 to 16Challenge Level

Can you find a rule which relates triangular numbers to square numbers? ### Janine's Conjecture

##### Age 14 to 16Challenge 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. . . . ### Leonardo's Problem

##### Age 14 to 18Challenge 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? ### And So on - and on -and On

##### Age 16 to 18Challenge Level

Can you find the value of this function involving algebraic fractions for x=2000? ### Look Before You Leap

##### Age 16 to 18Challenge Level

Relate these algebraic expressions to geometrical diagrams. ##### Age 16 to 18Challenge 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. ### Diverging

##### Age 16 to 18Challenge 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. ### Pair Squares

##### Age 16 to 18Challenge 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. ### DOTS Division

##### Age 14 to 16Challenge 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}. ### Multiplication Square

##### Age 14 to 16Challenge Level

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

##### Age 16 to 18Challenge Level

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

##### Age 14 to 16Challenge Level

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 ? ### A Tilted Square

##### Age 14 to 16Challenge Level

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

##### Age 16 to 18Challenge 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. ### Chocolate 2010

##### Age 14 to 16Challenge Level

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

##### Age 11 to 16

How to build your own magic squares. ### W Mates

##### Age 16 to 18Challenge 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. ### Fibonacci Factors

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

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

##### Age 14 to 18Challenge Level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. ### Magic Squares for Special Occasions

##### Age 11 to 16

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line. ### Harmonic Triangle

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge 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? ### Pythagoras Perimeters

##### Age 14 to 16Challenge Level

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

##### Age 14 to 18Challenge Level

Label this plum tree graph to make it totally magic! ### 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. ### Steel Cables

##### Age 14 to 16Challenge Level

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

##### Age 11 to 16Challenge Level

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

##### Age 16 to 18Challenge 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. ### What's Possible?

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

Can you explain what is going on in these puzzling number tricks? ### Difference of Two Squares

##### Age 14 to 16Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three? ### Series Sums

##### Age 14 to 16Challenge Level

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

##### Age 11 to 16Challenge Level

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

##### Age 16 to 18Challenge Level

Five equations... five unknowns... can you solve the system? ##### Age 14 to 16Challenge Level

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue? ##### Age 14 to 16Challenge Level

Kyle and his teacher disagree about his test score - who is right? ### Square Number Surprises

##### Age 14 to 16Challenge Level ### Particularly General

##### Age 16 to 18Challenge Level

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

##### Age 14 to 18Challenge Level

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