Visualising and Representing is part of our Thinking Mathematically collection.
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Favourite
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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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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Method in multiplying madness?
Age
7 to 14
Challenge level
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
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Favourite
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?
list
Visualising and Representing - Short Problems
A collection of short Stage 3 and 4 problems on Visualising and Representing.
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Starting Fibonacci
Age
11 to 14
Challenge level
What is the first term of a Fibonacci sequence whose second term is 4 and fifth term is 22?
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Favourite
Parallelogram It
Age
11 to 16
Challenge level
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.
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Favourite
Your number was...
Age
11 to 14
Challenge level
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
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Perimeter Expressions
Age
11 to 14
Challenge level
Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
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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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Folded A4
Age
11 to 14
Challenge level
What shapes can be made by folding an A4 sheet of paper only once?
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Reflecting Lines
Age
11 to 14
Challenge level
Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.
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Picturing Triangular Numbers
Age
11 to 14
Challenge level
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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Picturing Square Numbers
Age
11 to 14
Challenge level
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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Favourite
Add to 200
Age
11 to 14
Challenge level
By selecting digits for an addition grid, what targets can you make?
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Days and Dates
Age
11 to 14
Challenge level
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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How far does it move?
Age
11 to 14
Challenge level
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects the distance it travels at each stage.
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Favourite
Translating Lines
Age
11 to 14
Challenge level
Investigate what happens to the equation of different lines when
you translate them. Try to predict what will happen. Explain your
findings.
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Favourite
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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Multiple Surprises
Age
11 to 16
Challenge level
Sequences of multiples keep cropping up...
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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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Growing Surprises
Age
11 to 14
Challenge level
Can you find the connections between linear and quadratic patterns?
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Diminishing Returns
Age
11 to 14
Challenge level
How much of the square is coloured blue? How will the pattern continue?
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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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Sieve of Eratosthenes
Age
11 to 14
Challenge level
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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Island Hopping
Age
11 to 14
Challenge level
What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?
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Favourite
Odds, Evens and More Evens
Age
11 to 14
Challenge level
Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...
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Favourite
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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Favourite
Square It
Age
11 to 16
Challenge level
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
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Favourite
Round and round and round
Age
11 to 14
Challenge level
Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?
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Squares in a Square
Age
11 to 14
Challenge level
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
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Favourite
Mixing Lemonade
Age
11 to 14
Challenge level
Can you work out which drink has the stronger flavour?
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Favourite
Fence it
Age
11 to 14
Challenge level
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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Favourite
Constructing Triangles
Age
11 to 14
Challenge level
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
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Daniel's Star
Age
11 to 14
Challenge level
A solid 'star' shape is created. How many faces does it have?
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Favourite
Isosceles Triangles
Age
11 to 14
Challenge level
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
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Favourite
Hidden Squares
Age
11 to 14
Challenge level
Can you find the squares hidden on these coordinate grids?
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Painted Octahedron
Age
11 to 14
Challenge level
What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?
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Favourite
Frogs
Age
11 to 14
Challenge level
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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Reflecting Squarely
Age
11 to 14
Challenge level
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
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Bishop's Paradise
Age
11 to 14
Challenge level
Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
Which of the statements about diagonals of polygons is false?
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Favourite
Fractions Rectangle
Age
11 to 14
Challenge level
The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?
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Favourite
Odds and Evens
Age
11 to 14
Challenge level
Are these games fair? How can you tell?
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Potatoes
Age
11 to 14
Challenge level
Weekly Problem 19 - 2009
When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?
When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?
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Printing Error
Age
11 to 14
Challenge level
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
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Favourite
Shady Symmetry
Age
11 to 14
Challenge level
How many different symmetrical shapes can you make by shading triangles or squares?
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Reflected Back
Age
11 to 14
Challenge level
Imagine reflecting the letter P in all three sides of a triangle in turn. What is the final result?
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Favourite
Searching for mean(ing)
Age
11 to 16
Challenge level
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
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Favourite
Rolling Around
Age
11 to 14
Challenge level
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
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Favourite
Quadratic Patterns
Age
11 to 14
Challenge level
Surprising numerical patterns can be explained using algebra and diagrams...
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Rhombus It
Age
11 to 16
Challenge level
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.
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Favourite
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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Favourite
An Unusual Shape
Age
11 to 14
Challenge level
Can you maximise the area available to a grazing goat?
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Cuboid challenge
Age
11 to 16
Challenge level
What's the largest volume of box you can make from a square of paper?
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Favourite
Quadrilaterals in a Square
Age
11 to 14
Challenge level
What's special about the area of quadrilaterals drawn in a square?
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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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Favourite
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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Favourite
Reversals
Age
11 to 14
Challenge level
Where should you start, if you want to finish back where you started?
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Favourite
Same Answer
Age
11 to 14
Challenge level
Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
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Favourite
Shear Magic
Age
11 to 14
Challenge level
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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Favourite
Opposite vertices
Age
11 to 14
Challenge level
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
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11x11 square
Age
11 to 16
Challenge level
Here's a neat trick you can do with an 11 by 11 square...
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Favourite
Speeding up, slowing down
Age
11 to 14
Challenge level
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its speed at each stage.
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Favourite
Pythagoras Proofs
Age
11 to 16
Challenge level
Can you make sense of these three proofs of Pythagoras' Theorem?
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Favourite
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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Favourite
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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Favourite
Counting Factors
Age
11 to 14
Challenge level
Is there an efficient way to work out how many factors a large number has?
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Coordinate Patterns
Age
11 to 14
Challenge level
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
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Favourite
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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Favourite
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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Favourite
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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Favourite
Route to infinity
Age
11 to 14
Challenge level
Can you describe this route to infinity? Where will the arrows take you next?
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Favourite
On the Edge
Age
11 to 14
Challenge level
If you move the tiles around, can you make squares with different coloured edges?
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Favourite
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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Favourite
Funny Factorisation
Age
11 to 16
Challenge level
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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Favourite
Triathlon and Fitness
Age
11 to 14
Challenge level
The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
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Doubly Symmetric
Age
11 to 14
Challenge level
What is the smallest number of additional squares that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?
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Don't Be Late
Age
11 to 14
Challenge level
Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.
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Favourite
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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Integral Polygons
Age
11 to 14
Challenge level
Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?
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Blockupied
Age
11 to 14
Challenge level
A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?
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Favourite
Hamiltonian Cube
Age
11 to 16
Challenge level
Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
Find the length along the shortest path passing through certain points on the cube.
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Favourite
Seven Squares
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?
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Same Face
Age
11 to 14
Challenge level
A cube is rolled on a plane, landing on the squares in the order shown. Which two positions had the same face of the cube touching the surface?
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Favourite
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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Favourite
Square coordinates
Age
11 to 14
Challenge level
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
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Favourite
Stars
Age
11 to 14
Challenge level
Can you work out what step size to take to ensure you visit all the dots on the circle?
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Night Watchmen
Age
11 to 14
Challenge level
Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?
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Turning N Over
Age
11 to 14
Challenge level
A card with the letter N on it is rotated through two different axes. What does the card look like at the end?
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Adjacent Factors
Age
11 to 14
Challenge level
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
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Favourite
The Farmers' Field Boundary
Age
11 to 14
Challenge level
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
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Favourite
Shapely pairs
Age
11 to 14
Challenge level
A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...
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Fifty Coins
Age
11 to 14
Challenge level
Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?
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Favourite
Alison's quilt
Age
11 to 14
Challenge level
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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Favourite
Mixing Paints
Age
11 to 14
Challenge level
Can you work out how to produce different shades of pink paint?
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Product and Sum
Age
11 to 14
Challenge level
When Jim rolled some dice, the scores had the same product and sum. How many dice did Jim roll?