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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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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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How many models can you find which obey these rules?

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

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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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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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A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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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.

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

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

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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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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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Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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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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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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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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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?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

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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?

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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In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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

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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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How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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A follow-up activity to Tiles in the Garden.

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

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Why does the tower look a different size in each of these pictures?

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Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

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