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This problem is intended to get children to look really hard at something they will see many times in the next few months.

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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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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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The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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Investigate these hexagons drawn from different sized equilateral triangles.

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

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

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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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I cut this square into two different shapes. What can you say about the relationship between them?

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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?

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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?

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

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

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

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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?

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Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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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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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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Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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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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Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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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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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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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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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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Sort the houses in my street into different groups. Can you do it in any other ways?

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Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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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.

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Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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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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Investigate the different ways you could split up these rooms so that you have double the number.

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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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What is the largest cuboid you can wrap in an A3 sheet of paper?

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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?

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A follow-up activity to Tiles in the Garden.

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A challenging activity focusing on finding all possible ways of stacking rods.

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In how many ways can you stack these rods, following the rules?

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This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.