These rectangles have been torn. How many squares did each one have inside it before it was ripped?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
How many triangles can you make on the 3 by 3 pegboard?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
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?
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
Can you draw a square in which the perimeter is numerically equal to the area?
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?
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?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
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?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
These practical challenges are all about making a 'tray' and covering it with paper.
This activity investigates how you might make squares and pentominoes from Polydron.
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?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?
How many different triangles can you make on a circular pegboard that has nine pegs?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Investigate the different ways you could split up these rooms so that you have double the number.
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
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?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
My coat has three buttons. How many ways can you find to do up all the buttons?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.