Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.
You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
This article by Alex Goodwin, age 18 of Madras College, St Andrews
describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n
An account of some magic squares and their properties and and how to construct them for yourself.
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Bricks are 20cm long and 10cm high. How high could an arch be built
without mortar on a flat horizontal surface, to overhang by 1
metre? How big an overhang is it possible to make like this?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you find the area of a parallelogram defined by two vectors?
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
A game for 2 players
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
A collection of games on the NIM theme
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Can you find the values at the vertices when you know the values on the edges?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Make and prove a conjecture about the cyclic quadrilateral
inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
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.
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. . . .
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
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?”
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Equal touching circles have centres on a line. From a point of this
line on a circle, a tangent is drawn to the farthest circle. Find
the lengths of chords where the line cuts the other circles.
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
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 ...?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
An article which gives an account of some properties of magic squares.