What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What do these two triangles have in common? How are they related?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
A follow-up activity to Tiles in the Garden.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
I cut this square into two different shapes. What can you say about the relationship between them?
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Have a go at this 3D extension to the Pebbles problem.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Explore one of these five pictures.
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?
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?
In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
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?
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?
A challenging activity focusing on finding all possible ways of stacking rods.
Here are many ideas for you to investigate - all linked with the number 2000.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
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?
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.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
How many tiles do we need to tile these patios?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
Can you create more models that follow these rules?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This challenge extends the Plants investigation so now four or more children are involved.
How many models can you find which obey these rules?
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?
Can you find ways of joining cubes together so that 28 faces are visible?
In how many ways can you stack these rods, following the rules?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
An investigation that gives you the opportunity to make and justify predictions.