Visualising
Treasure Hunt
Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?
Picturing Triangular Numbers
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Frogs
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Picturing Square Numbers
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Polygon Rings
Join pentagons together edge to edge. Will they form a ring?
Reflecting Squarely
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Polygon Pictures
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
Shady Symmetry
How many different symmetrical shapes can you make by shading triangles or squares?
Constructing Triangles
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Fence It
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Diminishing Returns
How much of the square is coloured blue? How will the pattern continue?
Hidden Squares
Can you find the squares hidden on these coordinate grids?
Isosceles Triangles
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?
Quadrilaterals in a Square
What's special about the area of quadrilaterals drawn in a square?
Square Coordinates
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?
On the Edge
If you move the tiles around, can you make squares with different coloured edges?
Shear Magic
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
Squares in Rectangles
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?
Tower of Hanoi
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Seven Squares - Group-worthy Task
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?
Route to Infinity
Can you describe this route to infinity? Where will the arrows take you next?
Coordinate Patterns
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
The Farmers' Field Boundary
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Rolling Around
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?
Seven Squares
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Cuboids
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?
Efficient Cutting
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.
Triangles to Tetrahedra
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Visualising - Short Problems
A collection of short Stage 3 and 4 problems on Visualising.
Semi-regular Tessellations
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Square It
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.
Diamond Collector
Collect as many diamonds as you can by drawing three straight lines.
Parallelogram It
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.
Cuboid Challenge
What's the largest volume of box you can make from a square of paper?
Marbles in a Box
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Rhombus It
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.
Nine Colours
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?
Steel Cables
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Speeding Boats
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Spotting the Loophole
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?
Doesn't Add Up
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Pick's Theorem
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.
Building Tetrahedra
Can you make a tetrahedron whose faces all have the same perimeter?
Attractive Tablecloths
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?
Mystic Rose
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Painted Cube
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?
Picture Story
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Partly Painted Cube
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?
The Spider and the Fly
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?
Bendy Quad
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.
Vector Journeys
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
Circles Ad Infinitum
A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?