This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes 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?
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.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
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.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
How many triangles can you make on the 3 by 3 pegboard?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
How many models can you find which obey these rules?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
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 order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Make a flower design using the same shape made out of different sizes of paper.
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?
What shape is made when you fold using this crease pattern? Can you make a ring design?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What do these two triangles have in common? How are they related?
Can you visualise what shape this piece of paper will make when it is folded?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
Exploring and predicting folding, cutting and punching holes and making spirals.
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?
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 ...
What is the greatest number of squares you can make by overlapping three squares?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
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.
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?