Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
This activity investigates how you might make squares and pentominoes from Polydron.
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
These practical challenges are all about making a 'tray' and covering it with paper.
Can you cover the camel with these pieces?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
An activity making various patterns with 2 x 1 rectangular tiles.
Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
How many triangles can you make on the 3 by 3 pegboard?
How many models can you find which obey these rules?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
What is the best way to shunt these carriages so that each train can continue its journey?
The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
If you had 36 cubes, what different cuboids could you make?
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
How many different triangles can you make on a circular pegboard that has nine pegs?
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
This train line has two tracks which cross at different points. Can you find all the routes that end at Cheston?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.