What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
Can you find ways of joining cubes together so that 28 faces are visible?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.
Explore the triangles that can be made with seven sticks of the same length.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Can you create more models that follow these rules?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
An activity making various patterns with 2 x 1 rectangular tiles.
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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.
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?
What do these two triangles have in common? How are they related?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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 can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
How many models can you find which obey these rules?
In how many ways can you stack these rods, following the rules?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Investigate the different ways you could split up these rooms so that you have double the number.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
How many triangles can you make on the 3 by 3 pegboard?