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

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 DIFFERENT quadrilaterals can be made by joining the dots on the 8-point 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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Can you find all the different triangles on these peg boards, and find their angles?

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

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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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What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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

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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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Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to 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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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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

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

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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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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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Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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

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

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Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

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Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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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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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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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 long does it take to brush your teeth? Can you find the matching length of time?

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

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First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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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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How many different rectangles can you make using this set of rods?

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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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Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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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.

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

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In this game for two players, take it in turns to shade one petal, or two petals next to each other. Is it better to go first or second?

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