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

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

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

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

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

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

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

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

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

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

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

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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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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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Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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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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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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

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

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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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Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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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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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.

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Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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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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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.