Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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 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?
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 dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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?
How many triangles can you make on the 3 by 3 pegboard?
How many models can you find which obey these rules?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
These practical challenges are all about making a 'tray' and covering it with paper.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
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?
Here is a version of the game 'Happy Families' for you to make and play.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
An activity making various patterns with 2 x 1 rectangular tiles.
How is it possible to predict the card?
What do these two triangles have in common? How are they related?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
Can you create more models that follow these rules?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
How do you know if your set of dominoes is complete?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Here are some ideas to try in the classroom for using counters to investigate number patterns.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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 activity investigates how you might make squares and pentominoes from Polydron.
Exploring and predicting folding, cutting and punching holes and making spirals.
This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?
What is the greatest number of squares you can make by overlapping three squares?
Can you visualise what shape this piece of paper will make when it is folded?
Make a flower design using the same shape made out of different sizes of paper.
Can you deduce the pattern that has been used to lay out these bottle tops?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Reasoning about the number of matches needed to build squares that share their sides.
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?