There are 162 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Three Neighbours
Age 7 to 14
Challenge Level
Take three consecutive numbers and add them together. What do you notice?
Tower of Hanoi
Age 11 to 14
Challenge Level
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
Pythagoras Proofs
Age 11 to 16
Challenge Level
Can you make sense of these three proofs of Pythagoras' Theorem?
Power Mad!
Age 11 to 14
Challenge Level
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Elevenses
Age 11 to 14
Challenge Level
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Iffy Logic
Age 14 to 18
Challenge Level
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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?
Seven Squares - Group-worthy Task
Age 11 to 14
Challenge Level
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Perfectly Square
Age 14 to 16
Challenge Level
The sums of the squares of three related numbers is also a perfect square - can you explain why?
More Number Pyramids
Age 11 to 14
Challenge Level
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Always Perfect
Age 14 to 18
Challenge Level
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Children at Large
Age 11 to 14
Challenge Level
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Leonardo's Problem
Age 14 to 18
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?
Take Three from Five
Age 11 to 16
Challenge Level
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Largest Product
Age 11 to 14
Challenge Level
Which set of numbers that add to 10 have the largest product?
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.
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
1 Step 2 Step
Age 11 to 14
Challenge Level
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Number Rules - OK
Age 14 to 16
Challenge Level
Can you produce convincing arguments that a selection of statements about numbers are true?
Lens Angle
Age 14 to 16
Challenge Level
Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
Iff
Age 14 to 18
Challenge Level
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Why 24?
Age 14 to 16
Challenge Level
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
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.
Zig Zag
Age 14 to 16
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?
Unit Interval
Age 14 to 18
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?
Common Divisor
Age 14 to 16
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.
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?
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?
Cosines Rule
Age 14 to 16
Challenge Level
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
Picture Story
Age 14 to 16
Challenge Level
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Neighbourly Addition
Age 7 to 14
Challenge Level
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Triangular Intersection
Age 14 to 16 Short
Challenge Level
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Gabriel's Problem
Age 11 to 14
Challenge Level
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Towering Trapeziums
Age 14 to 16
Challenge Level
Can you find the areas of the trapezia in this sequence?
Same Length
Age 11 to 16
Challenge Level
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Kite in a Square
Age 14 to 18
Challenge Level
Can you make sense of the three methods to work out what fraction of the total area is shaded?
What Numbers Can We Make Now?
Age 11 to 14
Challenge Level
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Advent Calendar 2011 - Secondary
Age 11 to 18
Challenge Level
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Binomial Coefficients
Age 14 to 18
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
What Numbers Can We Make?
Age 11 to 14
Challenge Level
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
L-triominoes
Age 14 to 16
Challenge Level
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
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.
Breaking the Equation ' Empirical Argument = Proof '
Age 7 to 18
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.
Sticky Numbers
Age 11 to 14
Challenge Level
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Folding Fractions
Age 14 to 16
Challenge Level
What fractions can you divide the diagonal of a square into by simple folding?
Air Nets
Age 7 to 18
Challenge Level
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
NRICH is part of the family of activities in the Millennium Mathematics Project.