Proof
AS Level
- Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including proof by deduction, proof by exhaustion
- Disproof by counter example
A Level
- Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs)
- ALL
- Presentation
- Interactive resource
- Activity sheet
- External link
Presentation
Triominoes
Carom Maths provides extension materials for teachers and students of A Level mathematics.
This presentation shows how, when placing triominoes onto a chessboard, there is always one empty square. An algebraic proof is developed to show that the empty square will always appear in the same location, or in one of its rotations, using a visual approach.
The activity is designed to explore aspects of the subject which may not normally be encountered, to encourage new ways to approach a problem mathematically and to broaden the range of tools that an A Level mathematician can call upon.
Interactive resource
Proof
This interactive excel resource illustrates a number of proofs. It explores properties of odd, even and consecutive numbers- both numerically and algebraically- and also covers properties of the sum, difference and product of these numbers, again these are explored numerically and algebraically.
The final twelve sheets each deal with a proof relating to the properties of odd, even, square, triangular and cubic numbers. Each line of algebra can be revealed leading to a complete proof of each of the statements.
Activity sheet
The Language of Mathematics and Proof
This RISP activity is ideal for introducing, consolidating or revising the idea of proof using a mathematical argument and appropriate use of logical deduction.
Students are asked to choose two triangular numbers and find when the difference is a prime number. Students should then be encouraged to attempt to prove their conjecture. The teacher notes suggest a pictorial proof and an algebraic proof.
Suggested extensions include investigating when the difference of two squares is prime and when the difference of two cubes is prime.
Risps for A2 Level Core: Proof 2
Two Repeats covers the revision of algebraic topics including changing the subject of a formula, graphical solution of equations, solving simultaneous and quadratic equations and manipulating surds.
Given a simple starting premise, students have to solve the puzzle by solving a number of different kinds of equations, working logically and systematically to arrive at the solution.
External link
NRICH Proof Sorter - the Square Root of 2 Is Irrational
In this interactive NRICH activity, the steps of the proof have to be arranged into the correct order.
This resource is one of four Proof Slider activities, that also cover geometric series, quadratic equations and the sum of an arithmetic series.
NRICH: Dodgy Proofs
This resource from NRICH includes thirteen 'proofs' that give obviously incorrect results, such as 3 = 0. Students are asked to find for each of the 'proofs' the exact point at which the arguement breaks down.
