Explore the triangles that can be made with seven sticks of the same length.
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?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Have a go at this 3D extension to the Pebbles problem.
What do these two triangles have in common? How are they related?
Can you create more models that follow these rules?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
These pictures show squares split into halves. Can you find other ways?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Can you find ways of joining cubes together so that 28 faces are visible?
How many models can you find which obey these rules?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
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?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
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?
How many triangles can you make on the 3 by 3 pegboard?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
Why does the tower look a different size in each of these pictures?
Explore one of these five pictures.
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?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.
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?
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.
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?
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 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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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.
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?
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?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.