Reasoning, Convincing and Proving - Secondary Students

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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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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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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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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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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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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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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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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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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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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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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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Odds and Evens

Age

11 to 14

Challenge level

Are these games fair? How can you tell?

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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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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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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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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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Cryptarithms

Age

11 to 14

Challenge level

Can you crack these cryptarithms?

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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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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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Multiple Surprises

Age

11 to 16

Challenge level

Sequences of multiples keep cropping up...

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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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Mixing Lemonade

Age

11 to 14

Challenge level

Can you work out which drink has the stronger flavour?

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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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Polygon Pictures

Age

11 to 14

Challenge level

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

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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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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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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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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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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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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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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 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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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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Thousands and Millions

Age

11 to 14

Challenge level

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

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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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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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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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Reversals

Age

11 to 14

Challenge level

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

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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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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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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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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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Pythagoras Proofs

Age

11 to 16

Challenge level

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 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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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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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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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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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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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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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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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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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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Repetitiously

Age

14 to 16

Challenge level

Can you express every recurring decimal as a fraction?

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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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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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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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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?

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Latin Numbers

Age

14 to 16

Challenge level

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

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Harmonic Triangle

Age

14 to 16

Challenge level

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

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2-Digit Square

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?