Reasoning, Convincing and Proving - Secondary Students

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Statement snap

Age

7 to 14

Challenge level

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

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Fruity totals

Age

7 to 16

Challenge level

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

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Missing multipliers

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?

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The Number Jumbler

Age

7 to 14

Challenge level

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

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5 by 5 Mathdokus

Age

7 to 16

Challenge level

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

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More less is more

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.

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Xavi's t-shirt

Age

7 to 16

Challenge level

How much can you read into a t-shirt?

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Treasure hunt

Age

7 to 14

Challenge level

Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

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Subtraction surprise

Age

7 to 14

Challenge level

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

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Your number is...

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?

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Number lines in disguise

Age

7 to 14

Challenge level

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

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Remainders

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?

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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?

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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?

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American billions

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...

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Place your orders

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?

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Crossed ends

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?

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Olympic measures

Age

11 to 14

Challenge level

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

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Strange bank account

Age

11 to 14

Challenge level

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

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Special numbers

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?

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Multiples Sudoku

Age

11 to 14

Challenge level

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

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Cryptarithms

Age

11 to 14

Challenge level

Can you crack these cryptarithms?

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Tilted squares

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?

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Calendar capers

Age

11 to 14

Challenge level

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

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Arithmagons

Age

11 to 16

Challenge level

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

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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?

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Multiple surprises

Age

11 to 16

Challenge level

Sequences of multiples keep cropping up...

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Triangles in circles

Age

11 to 14

Challenge level

Can you find triangles on a 9-point circle? Can you work out their angles?

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Mixing lemonade

Age

11 to 14

Challenge level

Can you work out which drink has the stronger flavour?

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Odds and evens

Age

11 to 14

Challenge level

Are these games fair? How can you tell?

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Polygon rings

Age

11 to 14

Challenge level

Join pentagons together edge to edge. Will they form a ring?

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Semi-regular tessellations

Age

11 to 16

Challenge level

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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Star polygons

Age

11 to 14

Challenge level

Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?

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Legs eleven

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?

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Squares in rectangles

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?

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Product Sudoku

Age

11 to 16

Challenge level

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

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Always a multiple?

Age

11 to 14

Challenge level

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

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Triangle numbers

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?

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What does it all add up to?

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?

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Fill me up

Age

11 to 14

Challenge level

Can you sketch graphs to show how the height of water changes in different containers as they are filled?

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All in a jumble

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.

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Marbles in a box

Age

11 to 16

Challenge level

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

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Thousands and millions

Age

11 to 14

Challenge level

Here's a chance to work with large numbers...

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Think of two numbers

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?

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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?

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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?

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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.

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More number pyramids

Age

11 to 14

Challenge level

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

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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?

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Impossibilities

Age

11 to 14

Challenge level

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

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Overlaps

Age

11 to 14

Challenge level

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

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Reversals

Age

11 to 14

Challenge level

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

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Pythagoras proofs

Age

11 to 16

Challenge level

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

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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?

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Fibonacci surprises

Age

11 to 14

Challenge level

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

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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.

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Differences

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?

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Which solids can we make?

Age

11 to 14

Challenge level

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

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Consecutive negative numbers

Age

11 to 14

Challenge level

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

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Shopping basket

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?

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The greedy algorithm

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.

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Common divisor

Age

14 to 18

Challenge level

Can you find out what numbers divide these expressions? Can you prove that they are always divisors?

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Generating triples

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?

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How old am I?

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?

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Speed-time problems at the Olympics

Age

14 to 16

Challenge level

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

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Finding factors

Age

14 to 16

Challenge level

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

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Factorising with multilink

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?

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Nutrition and cycling

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?

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Double trouble

Age

14 to 16

Challenge level

Simple additions can lead to intriguing results...

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Making sixty

Age

14 to 16

Challenge level

Why does this fold create an angle of sixty degrees?

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Mega quadratic equations

Age

14 to 18

Challenge level

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

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Curvy areas

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?

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Speeding boats

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?

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Circles in quadrilaterals

Age

14 to 16

Challenge level

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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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?

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A little light thinking

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?

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Isosceles seven

Age

14 to 16

Challenge level

Is it possible to find the angles in this rather special isosceles triangle?

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Quad in quad

Age

14 to 18

Challenge level

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

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Nicely similar

Age

14 to 16

Challenge level

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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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.

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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?

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Multiplication arithmagons

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?

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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?

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LCM Sudoku

Age

14 to 16

Challenge level

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

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Repetitiously

Age

14 to 16

Challenge level

Can you express every recurring decimal as a fraction?

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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?

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Terminology

Age

14 to 16

Challenge level

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

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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.

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Pythagoras perimeters

Age

14 to 16

Challenge level

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

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Back fitter

Age

14 to 18

Challenge level

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

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Painted cube

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?

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Difference of two squares

Age

14 to 16

Challenge level

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

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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?

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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?

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Puzzling place value

Age

14 to 16

Challenge level

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

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Doesn't add up

Age

14 to 16

Challenge level

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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Partly painted cube

Age

14 to 16

Challenge level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?