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

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

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

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Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

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

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

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How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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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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In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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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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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 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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If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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

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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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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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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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This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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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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How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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

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

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Can you fill in the empty boxes in the grid with the right shape and colour?

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

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How many models can you find which obey these rules?

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

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Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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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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Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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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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Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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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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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 to have an ordered approach.

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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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When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.