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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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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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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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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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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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A follow-up activity to Tiles in the Garden.

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

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

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

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

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

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

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

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

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In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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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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What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

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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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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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How many models can you find which obey these rules?

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

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This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.