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Visualising and Representing is part of our Thinking Mathematically collection.
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?
Try out some calculations. Are you surprised by the results?
The Number Jumbler can always work out your chosen symbol. Can you work out how?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
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.
Choose any three by three square of dates on a calendar page...
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.
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?
Can you find the connections between linear and quadratic patterns?
Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...
By selecting digits for an addition grid, what targets can you 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?
Can you find the squares hidden on these coordinate grids?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.
Join pentagons together edge to edge. Will they form a ring?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
In how many ways can you fit all three pieces together to make shapes with line symmetry?
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?
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?
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Just because a problem is impossible doesn't mean it's difficult...
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
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?
Where should you start, if you want to finish back where you started?
Can you work out what step size to take to ensure you visit all the dots on the circle?
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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?
Surprising numerical patterns can be explained using algebra and diagrams...
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?
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?
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?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
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?
What's special about the area of quadrilaterals drawn in a square?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
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.
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 square.
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.
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.
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
Have you ever wondered what it would be like to race against Usain Bolt?
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?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
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?
There are unexpected discoveries to be made about square numbers...
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you explain what is going on in these puzzling number tricks?
Can you make a tetrahedron whose faces all have the same perimeter?
Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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?
I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?
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?
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?