Reasoning, Justifying, Convincing and Proof
More Less Is More
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
Fruity Totals
In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?
5 by 5 Mathdokus
Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
Largest Product
Which set of numbers that add to 100 have the largest product?
What Numbers Can We Make?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Sticky Numbers
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Summing Consecutive Numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Elevenses
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Magic Letters
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Go Forth and Generalise
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Gabriel's Problem
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Number Pyramids
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Calendar Capers
Choose any three by three square of dates on a calendar page...
Diminishing Returns
How much of the square is coloured blue? How will the pattern continue?
Seven Squares - Group-worthy Task
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?
Legs Eleven
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
Route to Infinity
Can you describe this route to infinity? Where will the arrows take you next?
Tower of Hanoi
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Power Mad!
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Seven Squares
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
More Number Pyramids
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What Numbers Can We Make Now?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
1 Step 2 Step
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?
Always a Multiple?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Consecutive Negative Numbers
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Reasoning, Justifying, Convincing and Proof - Short Problems
A collection of short Stage 3 and 4 problems on Reasoning, Justifying, Convincing and Proof.
Cyclic Quadrilaterals
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
What's it Worth?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Arithmagons
Can you find the values at the vertices when you know the values on the edges?
Pythagoras Proofs
Can you make sense of these three proofs of Pythagoras' Theorem?
Same Length
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Take Three from Five
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?
Marbles in a Box
How many winning lines can you make in a three-dimensional version of noughts and crosses?
What Does it All Add up To?
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
Factorising with Multilink
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
How Old Am I?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
A Little Light Thinking
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Salinon
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?
Curvy Areas
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What's Possible?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Multiplication Square
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Why 24?
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.
Painted Cube
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Attractive Tablecloths
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Perfectly Square
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Picture Story
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Latin Numbers
Can you create a Latin Square from multiples of a six digit number?
Angle Trisection
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Iff
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Always Perfect
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Quad in Quad
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Kite in a Square
Can you make sense of the three methods to work out what fraction of the total area is shaded?
Direct Logic
Can you work through these direct proofs, using our interactive proof sorters?
Integration Matcher
Can you match the charts of these functions to the charts of their integrals?
Mind Your Ps and Qs
Sort these mathematical propositions into a series of 8 correct statements.
Areas and Ratios
Do you have enough information to work out the area of the shaded quadrilateral?
Impossible Triangles?
Which of these triangular jigsaws are impossible to finish?
