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# Reasoning, Convincing and Proving - Secondary Students

### The Number Jumbler

### More Less Is More

### Subtraction Surprise

### Statement Snap

### Missing Multipliers

### Your Number Is...

### Remainders

### Number Lines in Disguise

### Fruity Totals

### 5 by 5 Mathdokus

### Multiples Sudoku

### What Numbers Can We Make?

### American Billions

### Calendar Capers

### Crossed Ends

### Tilted Squares

### Olympic Measures

### Strange Bank Account

### Go Forth and Generalise

### Special Numbers

### Gabriel's Problem

### Place Your Orders

### Legs Eleven

### Triangle Numbers

### What Numbers Can We Make Now?

### Fibonacci Surprises

### Seven Squares - Group-worthy Task

### Impossibilities

### Overlaps

### Always a Multiple?

### Reversals

### Squares in Rectangles

### Think of Two Numbers

### Power Mad!

### All in a Jumble

### More Number Pyramids

### Tower of Hanoi

### Differences

### Consecutive Negative Numbers

### The Greedy Algorithm

### Reasoning, Convincing and Proving - Short Problems

### Arithmagons

### Cyclic Quadrilaterals

### Take Three from Five

### Marbles in a Box

### Pythagoras Proofs

### Product Sudoku

### Same Length

### Shopping Basket

### What Does it All Add up To?

### Curvy Areas

### A Little Light Thinking

### Speeding Boats

### Finding Factors

### How Old Am I?

### Factorising with Multilink

### Generating Triples

### Speed-time Problems at the Olympics

### Nutrition and Cycling

### Salinon

### Terminology

### Multiplication Square

### Multiplication Arithmagons

### CD Heaven

### Puzzling Place Value

### Painted Cube

### Picture Story

### Difference of Two Squares

### Why 24?

### LCM Sudoku

### Pythagoras Perimeters

### Difference Sudoku

### Harmonic Triangle

### Angle Trisection

### Latin Numbers

### 2-digit Square

### Mega Quadratic Equations

### Quad in Quad

### Kite in a Square

### Always Perfect

### Iff

### Direct Logic

### Integration Matcher

### Areas and Ratios

### Mind Your Ps and Qs

### Impossible Triangles?

Or search by topic

*Reasoning, Convincing and Proving is part of our Thinking Mathematically collection.*

Age 7 to 14

Challenge Level

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Age 7 to 14

Challenge Level

In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.

Age 7 to 14

Challenge Level

Try out some calculations. Are you surprised by the results?

Age 7 to 14

Challenge Level

You'll need to know your number properties to win a game of Statement Snap...

Age 7 to 14

Challenge Level

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Age 7 to 14

Challenge Level

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Age 7 to 14

Challenge Level

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Age 7 to 14

Challenge Level

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

Age 7 to 16

Challenge Level

In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

Age 7 to 16

Challenge Level

Can you use the clues to complete these 5 by 5 Mathematical Sudokus?

Age 11 to 14

Challenge Level

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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?

Age 11 to 14

Challenge Level

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Age 11 to 14

Challenge Level

Choose any three by three square of dates on a calendar page...

Age 11 to 14

Challenge Level

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

Age 11 to 14

Challenge Level

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Age 11 to 14

Challenge Level

These Olympic quantities have been jumbled up! Can you put them back together again?

Age 11 to 14

Challenge Level

Imagine a very strange bank account where you are only allowed to do two things...

Age 11 to 14

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.

Age 11 to 14

Challenge Level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Age 11 to 14

Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Age 11 to 14

Challenge Level

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Age 11 to 14

Challenge Level

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?

Age 11 to 14

Challenge Level

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Age 11 to 14

Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Age 11 to 14

Challenge Level

Play around with the Fibonacci sequence and discover some surprising results!

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?

Age 11 to 14

Challenge Level

Just because a problem is impossible doesn't mean it's difficult...

Age 11 to 14

Challenge Level

Can you find ways to put numbers in the overlaps so the rings have equal totals?

Age 11 to 14

Challenge Level

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Age 11 to 14

Challenge Level

Where should you start, if you want to finish back where you started?

Age 11 to 14

Challenge Level

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Age 11 to 14

Challenge Level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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.

Age 11 to 14

Challenge Level

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Age 11 to 14

Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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.

Age 11 to 14

Challenge Level

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Age 11 to 14

Challenge Level

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Age 11 to 14

Challenge Level

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

Age 11 to 16

A collection of short Stage 3 and 4 problems requiring Reasoning, Convincing and Proving.

Age 11 to 16

Challenge Level

Can you find the values at the vertices when you know the values on the edges?

Age 11 to 16

Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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?

Age 11 to 16

Challenge Level

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Age 11 to 16

Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

Age 11 to 16

Challenge Level

The clues for this Sudoku are the product of the numbers in adjacent squares.

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?

Age 11 to 16

Challenge Level

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

Age 11 to 18

Challenge Level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

Age 14 to 16

Challenge Level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Age 14 to 16

Challenge Level

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Age 14 to 16

Challenge Level

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Age 14 to 16

Challenge Level

Can you find the hidden factors which multiply together to produce each quadratic expression?

Age 14 to 16

Challenge Level

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?

Age 14 to 16

Challenge Level

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Age 14 to 16

Challenge Level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Age 14 to 16

Challenge Level

Have you ever wondered what it would be like to race against Usain Bolt?

Age 14 to 16

Challenge Level

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

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?

Age 14 to 16

Challenge Level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Age 14 to 16

Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Age 14 to 16

Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Age 14 to 16

Challenge Level

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at each price?

Age 14 to 16

Challenge Level

Can you explain what is going on in these puzzling number tricks?

Age 14 to 16

Challenge Level

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?

Age 14 to 16

Challenge Level

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Age 14 to 16

Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three?

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.

Age 14 to 16

Challenge Level

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

Age 14 to 16

Challenge Level

If you know the perimeter of a right angled triangle, what can you say about the area?

Age 14 to 16

Challenge Level

Use the differences to find the solution to this Sudoku.

Age 14 to 16

Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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.

Age 14 to 16

Challenge Level

Can you create a Latin Square from multiples of a six digit number?

Age 14 to 16

Challenge Level

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Age 14 to 18

Challenge Level

What do you get when you raise a quadratic to the power of a quadratic?

Age 14 to 18

Challenge Level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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?

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.

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?

Age 16 to 18

Challenge Level

Can you work through these direct proofs, using our interactive proof sorters?

Age 16 to 18

Challenge Level

Can you match the charts of these functions to the charts of their integrals?

Age 16 to 18

Challenge Level

Do you have enough information to work out the area of the shaded quadrilateral?

Age 16 to 18

Challenge Level

Sort these mathematical propositions into a series of 8 correct statements.

Age 16 to 18

Challenge Level

Which of these triangular jigsaws are impossible to finish?