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

Polynomial Interpolation

Age 16 to 18
Challenge Level

Can you fit polynomials through these points?

System Speak

Age 16 to 18
Challenge Level

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

' 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?

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.

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.

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.

How Many Solutions?

Age 16 to 18
Challenge Level

Find all the solutions to the this equation.

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.

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.

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.

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.

Look Before You Leap

Age 16 to 18
Challenge Level

Relate these algebraic expressions to geometrical diagrams.

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.

Sums of Squares

Age 16 to 18
Challenge Level

Can you prove that twice the sum of two squares always gives the sum of two squares?

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.

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.

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.

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.

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.

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. . . .

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?

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?

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.

Consecutive Squares

Age 14 to 16
Challenge Level

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

Number Rules - OK

Age 14 to 16
Challenge Level

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

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?

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?

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?

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?

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.

Magic Sums and Products

Age 11 to 16

How to build your own magic squares.

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

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.

Square Number Surprises

Age 14 to 16
Challenge Level

There are unexpected discoveries to be made about square numbers...

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?

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?

Particularly General

Age 16 to 18
Challenge Level

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

Puzzling Place Value

Age 14 to 16
Challenge Level

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

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.

Algebra Match

Age 11 to 16
Challenge Level

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

Enriching Experience

Age 14 to 16
Challenge Level

Find the five distinct digits N, R, I, C and H in the following nomogram

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?

Diophantine N-tuples

Age 14 to 16
Challenge Level

Can you explain why a sequence of operations always gives you perfect squares?

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?

Mediant Madness

Age 14 to 16
Challenge Level

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

Back to Basics

Age 14 to 16
Challenge Level

Find b where 3723(base 10) = 123(base b).

Fair Shares?

Age 14 to 16
Challenge Level

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Plum Tree

Age 14 to 18
Challenge Level

Label this plum tree graph to make it totally magic!

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. . . .