Other tags that relate to Cocked Hat
Quadratic equations. Families of Graphs. Surds. Symmetry. Differentiation of parametric and implicit functions. Physics. Turning points. Graph sketching. Golden ratio. Mathematical reasoning & proof.There are 174 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof
Target Six
Age 16 to 18 Challenge Level:
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Continued Fractions II
Age 16 to 18
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Golden Eggs
Age 16 to 18 Challenge Level:
Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
Plus or Minus
Age 16 to 18 Challenge Level:
Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
Power Quady
Age 16 to 18 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Folding Fractions
Age 14 to 16 Challenge Level:
What fractions can you divide the diagonal of a square into by simple folding?
Proof Sorter - Quadratic Equation
Age 14 to 18 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.
Square Pair Circles
Age 16 to 18 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.
Pent
Age 14 to 18 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Folding Squares
Age 14 to 16 Challenge Level:
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
Encircling
Age 14 to 16 Challenge Level:
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
The Golden Ratio, Fibonacci Numbers and Continued Fractions.
Age 14 to 16
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
Matter of Scale
Age 14 to 16 Challenge Level:
Prove Pythagoras' Theorem using enlargements and scale factors.
Square Mean
Age 14 to 16 Challenge Level:
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
More Dicey Decisions
Age 16 to 18 Challenge Level:
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
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.
Pythagorean Golden Means
Age 16 to 18 Challenge Level:
Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
Calculating with Cosines
Age 14 to 18 Challenge Level:
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
The Frieze Tree
Age 11 to 16
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Picturing Pythagorean Triples
Age 14 to 18
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.
Kite in a Square
Age 14 to 16 Challenge Level:
Can you make sense of the three methods to work out the area of the kite in the square?
Where Do We Get Our Feet Wet?
Age 16 to 18
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Pythagorean Triples II
Age 11 to 16
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Pythagorean Triples I
Age 11 to 16
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
Age 16 to 18 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?
Angle Trisection
Age 14 to 16 Challenge Level:
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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.
Quadratic Harmony
Age 16 to 18 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.
Mechanical Integration
Age 16 to 18 Challenge Level:
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Pair Squares
Age 16 to 18 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.
More Number Sandwiches
Age 11 to 16 Challenge Level:
When is it impossible to make number sandwiches?
Big, Bigger, Biggest
Age 16 to 18 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
Fitting In
Age 14 to 16 Challenge Level:
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
Long Short
Age 14 to 16 Challenge Level:
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Our Ages
Age 14 to 16 Challenge Level:
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
There's a Limit
Age 14 to 18 Challenge Level:
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
Impossible Sandwiches
Age 11 to 18
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 Clue Is in the Question
Age 16 to 18 Challenge Level:
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
Classifying Solids Using Angle Deficiency
Age 11 to 16 Challenge Level:
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Salinon
Age 14 to 16 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Pythagoras Proofs
Age 14 to 16 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
Areas and Ratios
Age 16 to 18 Challenge Level:
Do you have enough information to work out the area of the shaded quadrilateral?
Direct Logic
Age 16 to 18 Challenge Level:
Can you work through these direct proofs, using our interactive proof sorters?
Mind Your Ps and Qs
Age 16 to 18 Short Challenge Level:
Sort these mathematical propositions into a series of 8 correct statements.
Iffy Logic
Age 14 to 18 Challenge Level:
Can you rearrange the cards to make a series of correct mathematical statements?
Cyclic Quadrilaterals
Age 11 to 16 Challenge Level:
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Geometric Parabola
Age 14 to 16 Challenge Level:
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
NRICH is part of the family of activities in the Millennium Mathematics Project.