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.
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.
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.
Generalise this inequality involving integrals.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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.
An account of some magic squares and their properties and and how to construct them for yourself.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
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?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
A collection of games on the NIM theme
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.
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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?
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.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
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. . . .
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?”
Can you find the values at the vertices when you know the values on the edges?
Can you tangle yourself up and reach any fraction?
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.
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 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?
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 explain the surprising results Jo found when she calculated the difference between square numbers?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
What's the largest volume of box you can make from a square of paper?
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.
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Charlie has moved between countries and the average income of both has increased. How can this be so?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you use the diagram to prove the AM-GM inequality?
It would be nice to have a strategy for disentangling any tangled ropes...
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. . . .
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.
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
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?
What is the total number of squares that can be made on a 5 by 5 geoboard?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.