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With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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What fraction of this square is shaded?

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Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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If you move the tiles around, can you make squares with different coloured edges?

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It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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Can you find the squares hidden on these coordinate grids?

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What fractions can you divide the diagonal of a square into by simple folding?

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What can you see? What do you notice? What questions can you ask?

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On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

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Can you work out the area of the inner square and give an explanation of how you did it?

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What is the minimum number of squares a 13 by 13 square can be dissected into?

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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Can you recreate squares and rhombuses if you are only given a side or a diagonal?

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A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

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A Short introduction to using Logo. This is the first in a twelve part series.

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Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

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Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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Look at how the pattern is built up - in that way you will know how to break the final pattern down into more manageable pieces.

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Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

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Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?