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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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Can you draw a square in which the perimeter is numerically equal to the area?

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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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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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This activity investigates how you might make squares and pentominoes from Polydron.

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What is the best way to shunt these carriages so that each train can continue its journey?

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These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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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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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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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These practical challenges are all about making a 'tray' and covering it with paper.

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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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How many different triangles can you make on a circular pegboard that has nine pegs?

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How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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If you had 36 cubes, what different cuboids could you make?

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My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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Design an arrangement of display boards in the school hall which fits the requirements of different people.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

This task challenges you to create symmetrical U shapes out of rods and find their areas.

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Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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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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When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

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

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Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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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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Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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Can you find all the different triangles on these peg boards, and find their angles?