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.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
An account of some magic squares and their properties and and how to construct them for yourself.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
Can you find the values at the vertices when you know the values on the edges?
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?
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.
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 terms.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
It would be nice to have a strategy for disentangling any tangled ropes...
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
A game for 2 players with similarities 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.
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?”
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 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.
Generalise this inequality involving integrals.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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?
A game for 2 players
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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.
A collection of games on the NIM theme
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?
Can you tangle yourself up and reach any fraction?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Charlie has moved between countries and the average income of both has increased. How can this be so?
What's the largest volume of box you can make from a square of paper?
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 moves.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?
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.
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 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.
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. . . .
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
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?