Other tags that relate to Polynomial Interpolation
Generalising. Graph sketching. Polynomial functions and their roots. Transformation of functions. Mathematical reasoning & proof. Simultaneous equations. Creating and manipulating expressions and formulae. Inequalities. Factors and multiples. Expanding and factorising quadratics.There are 77 results
Broad Topics > Thinking Mathematically > Generalising
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?
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.
Arithmagons
Age 11 to 16
Challenge Level
Can you find the values at the vertices when you know the values on the edges?
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?
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?
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?
Plus Minus
Age 14 to 16
Challenge Level
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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?
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. . . .
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?
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.
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.
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?
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 =
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. . . .
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. . . .
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.
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.
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.
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?
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?”
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?
More Twisting and Turning
Age 11 to 16
Challenge Level
It would be nice to have a strategy for disentangling any tangled ropes...
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?
Harmonic Triangle
Age 14 to 16
Challenge Level
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Sliding Puzzle
Age 11 to 16
Challenge Level
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.
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?
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?
Train Spotters' Paradise
Age 11 to 16
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
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.
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.
Beelines
Age 14 to 16
Challenge Level
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Gnomon Dimensions
Age 14 to 16
Challenge Level
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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?
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.
Chocolate 2010
Age 14 to 16
Challenge Level
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Of All the Areas
Age 14 to 16
Challenge Level
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Loopy
Age 14 to 16
Challenge Level
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
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?
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.
Jam
Age 14 to 16
Challenge Level
To avoid losing think of another very well known game where the patterns of play are similar.
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?
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.