Other tags that relate to Polynomial Interpolation
Graphs. Inequalities. Generalising. Polynomial functions and their roots. Expanding and factorising quadratics. Simultaneous equations. Factors and multiples. Creating and manipulating expressions and formulae. Working systematically. Mathematical reasoning & proof.There are 76 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising
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. . . .
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.
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?
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?
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?”
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?
Pinned Squares
Age 14 to 16 Challenge Level:
What is the total number of squares that can be made on a 5 by 5 geoboard?
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?
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?
Lower Bound
Age 14 to 16 Challenge Level:
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Rational Roots
Age 16 to 18 Challenge Level:
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
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?
Shape and Territory
Age 16 to 18 Challenge Level:
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Can it Be
Age 16 to 18 Challenge Level:
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Odd Differences
Age 14 to 16 Challenge Level:
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.
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?
Square Pizza
Age 14 to 16 Challenge Level:
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?
Multiplication Arithmagons
Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Arithmagons
Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges?
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?
Take Three from Five
Age 14 to 16 Challenge Level:
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?
Games Related to Nim
Age 5 to 16
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.
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?
Pick's Theorem
Age 14 to 16 Challenge Level:
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.
Pareq Calc
Age 14 to 16 Challenge Level:
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. . . .
Plus Minus
Age 14 to 16 Challenge Level:
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Generally Geometric
Age 16 to 18 Challenge Level:
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Mystic Rose
Age 14 to 16 Challenge Level:
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
More Twisting and Turning
Age 11 to 16 Challenge Level:
It would be nice to have a strategy for disentangling any tangled ropes...
Winning Lines
Age 7 to 16
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Nim
Age 14 to 16 Challenge Level:
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.
Problem Solving, Using and Applying and Functional Mathematics
Age 5 to 18 Challenge Level:
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.
One, Three, Five, Seven
Age 11 to 16 Challenge Level:
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.
Hypotenuse Lattice Points
Age 14 to 16 Challenge Level:
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Pentanim
Age 7 to 16 Challenge Level:
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.
Magic Squares
Age 14 to 18
An account of some magic squares and their properties and and how to construct them for yourself.
Fractional Calculus III
Age 16 to 18
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Polycircles
Age 14 to 16 Challenge Level:
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?
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?
Why Stop at Three by One
Age 16 to 18
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
Converging Means
Age 14 to 16 Challenge Level:
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. . . .
Areas of Parallelograms
Age 14 to 16 Challenge Level:
Can you find the area of a parallelogram defined by two vectors?
Fractional Calculus II
Age 16 to 18
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.
Generating Triples
Age 14 to 16 Challenge Level:
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?
Irrational Arithmagons
Age 16 to 18 Challenge Level:
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
NRICH is part of the family of activities in the Millennium Mathematics Project.