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?
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?
Here is a version of the game 'Happy Families' for you to make and play.
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?
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.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?
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.
How many models can you find which obey these rules?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
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?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
How many triangles can you make on the 3 by 3 pegboard?
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?
Can you each work out the number on your card? What do you notice? How could you sort the cards?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Factors and Multiples game for an adult and child. How can you make sure you win this game?
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?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
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?
You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?
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.
Here are some ideas to try in the classroom for using counters to investigate number patterns.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
How do you know if your set of dominoes is complete?
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
What do these two triangles have in common? How are they related?
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?
Explore the triangles that can be made with seven sticks of the same length.
Can you deduce the pattern that has been used to lay out these bottle tops?
Can you make the birds from the egg tangram?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
An activity making various patterns with 2 x 1 rectangular tiles.
Can you create more models that follow these rules?
Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?