Resources tagged with Mathematical reasoning & proof similar to The Frieze Tree:
Other tags that relate to The Frieze Tree
Inequality/inequalities. Translations. Mathematical reasoning & proof. Rotations. Symmetry. Compound transformations. Glide reflections. Reflections. Number theory. Manipulating algebraic expressions/formulae.There are 167 results
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?
Encircling
Stage: 4 Challenge Level: 
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
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.
Always Perfect
Stage: 4 Challenge Level:
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Prime AP
Stage: 4 Challenge Level:
Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?
Sixational
Stage: 4 Challenge Level:
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
Fitting In
Stage: 4 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. . . .
A Biggy
Stage: 4 Challenge Level:
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
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. . . .
Power Quady
Stage: 4 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Pythagorean Triples II
Stage: 4 and 5
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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.
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!
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.
Common Divisor
Stage: 4 Challenge Level:
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
Postage
Stage: 4 Challenge Level:
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
Long Short
Stage: 4 Challenge Level:
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
Proof: A Brief Historical Survey
Stage: 4 and 5
If you think that mathematical proof is really clearcut and universal then you should read this article.
Picture Story
Stage: 4 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Exhaustion
Stage: 4 Challenge Level:
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Rotating Triangle
Stage: 3 and 4 Challenge Level:
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Janine's Conjecture
Stage: 4 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. . . .
Quads
Stage: 4 and 5 Challenge Level:
The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?
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?
Natural Sum
Stage: 4 Challenge Level:
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
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. . . .
Our Ages
Stage: 4 Challenge Level:
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .
DOTS Division
Stage: 4 Challenge Level:
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
There's a Limit
Stage: 4 and 5 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?
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?
Triangle Incircle Iteration
Stage: 4 Challenge Level:
Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .
N000ughty Thoughts
Stage: 4 Challenge Level:
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
Mod 3
Stage: 4 Challenge Level:
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
Sprouts
Stage: 2, 3, 4 and 5
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians. . . .
Some Circuits in Graph or Network Theory
Stage: 4 and 5
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Salinon
Stage: 4 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?
Three Frogs
Stage: 4 Challenge Level:
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
Perfectly Square
Stage: 4 Challenge Level:
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Unit Interval
Stage: 4 and 5 Challenge Level:
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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?
The Golden Ratio, Fibonacci Numbers and Continued Fractions.
Stage: 4
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.
Symmetric Tangles
Stage: 4
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Pythagoras Proofs
Stage: 4 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
Breaking the Equation ' Empirical Argument = Proof '
Stage: 2, 3, 4 and 5
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
Take a Square II
Stage: 4 Challenge Level:
What fractions can you divide the diagonal of a square into by simple folding?
Iffy Logic
Stage: 4 Short Challenge Level:
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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