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

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

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

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

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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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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.

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What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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

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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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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?

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

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

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We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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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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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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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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How many faces can you see when you arrange these three cubes in different ways?

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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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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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Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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How many models can you find which obey these rules?

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

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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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Can you find ways of joining cubes together so that 28 faces are visible?

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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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How many ways can you find of tiling the square patio, using square tiles of different sizes?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

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

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