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Visualising and Representing is part of our Developing Mathematical Thinking collection.
The Number Jumbler can always work out your chosen symbol. Can you work out how?
Try out some calculations. Are you surprised by the results?
Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Choose any three by three square of dates on a calendar page...
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
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?
How much of the square is coloured blue? How will the pattern continue?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
By selecting digits for an addition grid, what targets can you make?
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Can you find the squares hidden on these coordinate grids?
Join pentagons together edge to edge. Will they form a ring?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What's special about the area of quadrilaterals drawn in a square?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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?
Is there an efficient way to work out how many factors a large number has?
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Can you work out what step size to take to ensure you visit all the dots on the circle?
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?
Just because a problem is impossible doesn't mean it's difficult...
If you move the tiles around, can you make squares with different coloured edges?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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?
Can you work out how to produce different shades of pink paint?
Where should you start, if you want to finish back where you started?
Surprising numerical patterns can be explained using algebra and diagrams...
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
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.
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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?
A collection of short Stage 3 and 4 problems on Visualising and Representing.
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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.
Collect as many diamonds as you can by drawing three straight lines.
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.
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
What's the largest volume of box you can make from a square of paper?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
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?
Have you ever wondered what it would be like to race against Usain Bolt?
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?
Which armies can be arranged in hollow square fighting formations?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
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.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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?
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?
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
If you know the perimeter of a right angled triangle, what can you say about the area?
There are unexpected discoveries to be made about square numbers...
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Can you make a tetrahedron whose faces all have the same perimeter?
Can you explain what is going on in these puzzling number tricks?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
If a sum invested gains 10% each year how long before it has doubled its value?