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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.

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

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

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

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

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An investigation that gives you the opportunity to make and justify predictions.

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

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

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

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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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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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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?

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