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

Plum Tree

Age 14 to 18 Challenge Level:

Label this plum tree graph to make it totally magic!

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.

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.

Interactive Number Patterns

Age 14 to 16 Challenge Level:

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

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.

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?

Look Before You Leap

Age 16 to 18 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

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.

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.

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.

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?

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?

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?

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?

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.

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?

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

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.

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?

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.

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

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

Triangles Within Squares

Age 14 to 16 Challenge Level:

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

Triangles Within Triangles

Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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?

AMGM

Age 14 to 16 Challenge Level:

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

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?

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.

Sweeping Satellite

Age 16 to 18 Challenge Level:

Derive an equation which describes satellite dynamics.

Sums of Pairs

Age 11 to 16 Challenge Level:

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?”

Hike and Hitch

Age 14 to 16 Challenge Level:

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

Matchless

Age 14 to 16 Challenge 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 ?

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?

Sixational

Age 14 to 18 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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.

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.

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?

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. . . .

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?

System Speak

Age 16 to 18 Challenge Level:

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

Lap Times

Age 14 to 16 Challenge Level:

Can you find the lap times of the two cyclists travelling at constant speeds?

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...

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?

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.

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.

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?

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?

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}.