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# Reasoning, Justifying, Convincing and Proof

*Reasoning, Justifying, Convincing and Proof is part of our Working Mathematically collection.*

### More Less Is More

### Fruity Totals

### 5 by 5 Mathdokus

### Tilted Squares

### Summing Consecutive Numbers

### Elevenses

### Magic Letters

### Go Forth and Generalise

### Gabriel's Problem

### Number Pyramids

### Calendar Capers

### Diminishing Returns

### Largest Product

### What Numbers Can We Make?

### Sticky Numbers

### Legs Eleven

### Route to Infinity

### Tower of Hanoi

### Power Mad!

### Seven Squares

### More Number Pyramids

### What Numbers Can We Make Now?

### 1 Step 2 Step

### Always a Multiple?

### Seven Squares - Group-worthy Task

### Consecutive Negative Numbers

### Reasoning, Justifying, Convincing and Proof - Short Problems

### Cyclic Quadrilaterals

### What's it Worth?

### Arithmagons

### Take Three from Five

### Marbles in a Box

### Pythagoras Proofs

### Same Length

### What Does it All Add up To?

### Factorising with Multilink

### How Old Am I?

### A Little Light Thinking

### Salinon

### Curvy Areas

### Painted Cube

### Attractive Tablecloths

### Perfectly Square

### Picture Story

### What's Possible?

### Multiplication Square

### Why 24?

### Angle Trisection

### Latin Numbers

### Iff

### Always Perfect

### Quad in Quad

### Kite in a Square

### Integration Matcher

### Direct Logic

### Mind Your Ps and Qs

### Areas and Ratios

### Impossible Triangles?

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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 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

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

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?

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?

Age 11 to 14

Challenge Level

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

Age 11 to 14

Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

How much of the square is coloured blue? How will the pattern continue?

Age 11 to 14

Challenge Level

Which set of numbers that add to 100 have the largest product?

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

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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

Can you describe this route to infinity? Where will the arrows take you next?

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

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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

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?

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

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

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

Age 11 to 16

A collection of short Stage 3 and 4 problems on Reasoning, Justifying, Convincing and Proof.

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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

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

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

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

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

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

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

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?

Age 14 to 16

Challenge Level

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

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

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 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 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

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 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

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

Age 16 to 18

Challenge Level

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

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

Which of these triangular jigsaws are impossible to finish?