Resources tagged with Mathematical reasoning & proof similar to A Method of Defining Coefficients in the Equations of Chemical Reactions:
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Working systematically. Real world. Manipulating algebraic expressions/formulae. Linear equations. Simultaneous equations. Experimental probability. Theoretical probability. Mathematical reasoning & proof. Long problems. Mathematical modelling.There are 164 results
Problem Solving, Using and Applying and Functional Mathematics
Stage: 1, 2, 3, 4 and 5 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.
Leonardo's Problem
Stage: 4 and 5 Challenge Level:
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
Stone Henge
Stage: 5 Challenge Level:
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Proof Sorter - Quadratic Equation
Stage: 4 and 5 Challenge Level:
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
Circle Box
Stage: 4 Challenge Level:
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
Pythagorean Triples II
Stage: 4 and 5
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Square Mean
Stage: 4 Challenge Level:
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Zig Zag
Stage: 4 Challenge Level:
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
Rhombus in Rectangle
Stage: 4 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Square Pair Circles
Stage: 5 Challenge Level:
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
Gift of Gems
Stage: 4 Challenge Level:
Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .
Picturing Pythagorean Triples
Stage: 4 and 5
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Exhaustion
Stage: 4 Challenge Level:
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
The Great Weights Puzzle
Stage: 4 Challenge Level:
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Round and Round
Stage: 4 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Pythagoras Proofs
Stage: 4 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
Quadratic Harmony
Stage: 5 Challenge Level:
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Logic, Truth Tables and Switching Circuits Challenge
Stage: 2, 3, 4 and 5
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to. . . .
Pair Squares
Stage: 5 Challenge Level:
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
Pythagorean Triples I
Stage: 4 and 5
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Little and Large
Stage: 5 Challenge Level:
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
Truth Tables and Electronic Circuits
Stage: 2, 3 and 4
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
Power Quady
Stage: 4 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Modulus Arithmetic and a Solution to Differences
Stage: 5
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
Modulus Arithmetic and a Solution to Dirisibly Yours
Stage: 5
Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
Continued Fractions II
Stage: 5
In this article we show that every whole number can be written as a continued fraction of the form k/[1+k/[1+k/etc.]].
Euclid's Algorithm II
Stage: 5
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Impossible Sandwiches
Stage: 3, 4 and 5
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
The Frieze Tree
Stage: 4
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Recent Developments on S.P. Numbers
Stage: 5
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
Transitivity
Stage: 5
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
Sums of Squares and Sums of Cubes
Stage: 5
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.
Try to Win
Stage: 5
Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
Can it Be
Stage: 5 Challenge Level:
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Whole Number Dynamics I
Stage: 3, 4 and 5
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Why 24?
Stage: 4 Challenge Level:
Take any prime number greater than 3, square it, subtract one and divide by 24. Make a statement about what you notice about dividing by 24 (a conjecture) and prove that what you say is always true.
Modular Fractions
Stage: 5 Challenge Level:
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
Angle Trisection
Stage: 4 Challenge Level:
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Whole Number Dynamics II
Stage: 3, 4 and 5
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
Whole Number Dynamics III
Stage: 5
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
Where Do We Get Our Feet Wet?
Stage: 5
Jenny Piggott chose this article. Professor Körner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Yih or Luk Tsut K'i or Three Men's Morris
Stage: 3, 4 and 5 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Telescoping Functions
Stage: 5
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
Whole Number Dynamics V
Stage: 4 and 5
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
A Knight's Journey
Stage: 4 and 5
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Whole Number Dynamics IV
Stage: 4 and 5
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
Magic Squares II
Stage: 4 and 5
An article which gives an account of some properties of magic squares.
Euler's Formula and Topology
Stage: 4 and 5
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
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