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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?

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Can you make the birds from the egg tangram?

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You have a set of the digits from 0 â€“ 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

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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?

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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?

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Here is a version of the game 'Happy Families' for you to make and play.

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This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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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?

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What do these two triangles have in common? How are they related?

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What is the greatest number of squares you can make by overlapping three squares?

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Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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An activity making various patterns with 2 x 1 rectangular tiles.

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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?

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This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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These pictures show squares split into halves. Can you find other ways?

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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?

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Explore the triangles that can be made with seven sticks of the same length.

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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?

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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?

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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?

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These practical challenges are all about making a 'tray' and covering it with paper.

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Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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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?

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These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

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Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

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How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

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Exploring and predicting folding, cutting and punching holes and making spirals.

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Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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Can you deduce the pattern that has been used to lay out these bottle tops?

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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?

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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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Can you split each of the shapes below in half so that the two parts are exactly the same?