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

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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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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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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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Sort the houses in my street into different groups. Can you do it in any other ways?

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

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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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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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Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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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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In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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

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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.

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

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

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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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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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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 different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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