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?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
I cut this square into two different shapes. What can you say about the relationship between them?
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?
Here are many ideas for you to investigate - all linked with the number 2000.
What do these two triangles have in common? How are they related?
Have a go at this 3D extension to the Pebbles problem.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
An investigation that gives you the opportunity to make and justify predictions.
How many tiles do we need to tile these patios?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
A follow-up activity to Tiles in the Garden.
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 triangles can you make on the 3 by 3 pegboard?
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 area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Explore one of these five pictures.
Investigate the number of faces you can see when you arrange three cubes in different ways.
A description of some experiments in which you can make discoveries about triangles.
In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?
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?
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?
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?
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!
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?
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?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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 challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
This challenge extends the Plants investigation so now four or more children are involved.
How many models can you find which obey these rules?
Can you create more models that follow these rules?
Why does the tower look a different size in each of these pictures?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?