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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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Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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

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

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

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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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Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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

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

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

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

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

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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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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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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 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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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.

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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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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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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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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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Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

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