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### Working mathematically

### For younger learners

### Advanced mathematics

# Visualising and Representing - Secondary Students

### Treasure Hunt

### Your Number Is...

### The Number Jumbler

### Subtraction Surprise

### Remainders

### Method in Multiplying Madness?

### Number Lines in Disguise

### Shady Symmetry

### Calendar Capers

### Your Number Was...

### Frogs

### Constructing Triangles

### Perimeter Expressions

### Fractions Rectangle

### Diminishing Returns

### Sieve of Eratosthenes

### Add to 200

### Reflecting Squarely

### Hidden Squares

### Polygon Rings

### Picturing Triangular Numbers

### Fence It

### Round and Round and Round

### Picturing Square Numbers

### Polygon Pictures

### Isosceles Triangles

### Quadrilaterals in a Square

### Always a Multiple?

### Think of Two Numbers

### Squares in Rectangles

### Square Coordinates

### Counting Factors

### The Farmers' Field Boundary

### Route to Infinity

### Shear Magic

### Tower of Hanoi

### Stars

### Rolling Around

### Impossibilities

### On the Edge

### Triangle Numbers

### Seven Squares

### Legs Eleven

### Alison's Quilt

### Seven Squares - Group-worthy Task

### Mixing Paints

### Reversals

### Quadratic Patterns

### Coordinate Patterns

### Triathlon and Fitness

### Same Answer

### More Number Pyramids

### Cuboids

### Efficient Cutting

### Triangles to Tetrahedra

### Visualising and Representing - Short Problems

### Diamond Collector

### Square It

### Semi-regular Tessellations

### Parallelogram It

### Rhombus It

### Marbles in a Box

### Searching for Mean(ing)

### Cuboid Challenge

### Funny Factorisation

### Nine Colours

### Factorising with Multilink

### Spotting the Loophole

### Speed-time Problems at the Olympics

### Which Is Cheaper?

### Hollow Squares

### Steel Cables

### Speeding Boats

### The Spider and the Fly

### Perfectly Square

### Pythagoras Perimeters

### Square Number Surprises

### Doesn't Add Up

### Puzzling Place Value

### Building Tetrahedra

### Mystic Rose

### Attractive Tablecloths

### A Question of Scale

### Terminology

### Plus Minus

### Which Is Bigger?

### Pick's TheoremLive

### Painted Cube

### Partly Painted Cube

### Picture Story

### Bendy Quad

Or search by topic

*Visualising and Representing is part of our Thinking Mathematically collection.*

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?

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

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

Age 7 to 14

Challenge Level

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

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

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

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 11 to 14

Challenge Level

How many different symmetrical shapes can you make by shading triangles or squares?

Age 11 to 14

Challenge Level

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

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?

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?

Age 11 to 14

Challenge Level

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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?

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?

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

Follow this recipe for sieving numbers and see what interesting patterns emerge.

Age 11 to 14

Challenge Level

By selecting digits for an addition grid, what targets can you make?

Age 11 to 14

Challenge Level

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Age 11 to 14

Challenge Level

Can you find the squares hidden on these coordinate grids?

Age 11 to 14

Challenge Level

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

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?

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?

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 รท 360. How did this help?

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?

Age 11 to 14

Challenge Level

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

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?

Age 11 to 14

Challenge Level

What's special about the area of quadrilaterals drawn in a square?

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

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

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

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?

Age 11 to 14

Challenge Level

Is there an efficient way to work out how many factors a large number has?

Age 11 to 14

Challenge Level

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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 work out what step size to take to ensure you visit all the dots on the circle?

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?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

If you move the tiles around, can you make squares with different coloured edges?

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

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

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

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

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

Can you work out how to produce different shades of pink paint?

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

Surprising numerical patterns can be explained using algebra and diagrams...

Age 11 to 14

Challenge Level

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Age 11 to 14

Challenge Level

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

Age 11 to 14

Challenge Level

Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?

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

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Age 11 to 14

Challenge Level

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Age 11 to 14

Challenge Level

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Age 11 to 16

A collection of short Stage 3 and 4 problems on Visualising and Representing.

Age 11 to 16

Challenge Level

Collect as many diamonds as you can by drawing three straight lines.

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.

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?

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.

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.

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

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?

Age 11 to 16

Challenge Level

What's the largest volume of box you can make from a square of paper?

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?

Age 11 to 16

Challenge Level

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

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

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

Age 14 to 16

Challenge Level

Which armies can be arranged in hollow square fighting formations?

Age 14 to 16

Challenge Level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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

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

Age 14 to 16

Challenge Level

There are unexpected discoveries to be made about square numbers...

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?

Age 14 to 16

Challenge Level

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

Age 14 to 16

Challenge Level

Can you make a tetrahedron whose faces all have the same perimeter?

Age 14 to 16

Challenge Level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Age 14 to 16

Challenge Level

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Age 14 to 16

Challenge Level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

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?

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

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.