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

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

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

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

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

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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 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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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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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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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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In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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

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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 thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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

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

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This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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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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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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Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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How many trapeziums, of various sizes, are hidden in this picture?

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Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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

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Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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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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Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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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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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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Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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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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Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?