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All About Surds A surd is a number which is written with a root sign and cannot be simplified e.g. 3 , or 17 Trying to write down the exact number as a decimal is impossible because surds are irrational numbers and therefore the decimal part ‘continues forever’, without repeating.

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All About Surds e.g = …

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Tilted Squares Using square dotty paper it is possible to draw squares by connecting 4 dots. They can be ‘straight on’ or ‘at an angle’

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Tilted Squares On a 6 by 6 dotty grid, how many different sized squares can you find? You might need several copies of the grids.

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**Tilted Squares This resource from nrich may help you to find them.**

Once you think you have them all, find the area of each one.

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Possible Squares

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**How might you find the area of this square?**

Areas of Squares How might you find the area of this square?

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Areas of Squares

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Areas of Squares 1 2 4 5 9 8 16 25 10 17 13

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**Side Lengths of Squares**

Knowing the areas, can you find the length of the side of each square?

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**Side Lengths of Squares**

1 √2 2 √ 5 3 √ 8 4 5 √ 10 √ 17 √ 13

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**Side Lengths of Squares**

One of these expressions can be simplified. You might notice that the larger square is an enlargement of the smaller one – twice the side length (although 4 times the area). If the side length of the smaller one is √2, the larger one must be 2√2

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**Side Lengths of Squares**

This means that 8 =2 2 8 = 4×2 8 = 4 × 2 8 =2 2

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Lengths of Lines Can you find the length of each line on the next slide? Simplify where possible. Hint: think about each line as the hypotenuse of a right-angled triangle

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4 5 4 2 26 2 17 5 3 2 2 5 29

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Lengths of Lines Which of the following could be drawn by connecting dots on the dotty paper? 17 2 6 13 11 2 3 2 2 3 5 2√5 2√7

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**Lengths of Lines 17 2 6 13 11 2 3 2 2 3 5 2√5 2√7 **

Which of the following could be drawn by connecting dots on the dotty paper? 17 2 6 13 11 2√5 2√7

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Angry Surds Having learned about surd lengths in right angled triangles, you might like to play Angry Surds.

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Teacher notes In this edition, the focus is on surds and familiarisation with lengths of sides in right angled triangles involving surds, culminating in playing a new mathematical game. Students should have previously used Pythagoras’ theorem. Some parts of the activity are suitable for Foundation GCSE students, others for Higher GCSE or AS students Tilted squares Slide 4 The class will need to decide whether position and orientation are ‘important’ in this task. With rotations, translations and reflections of a square considered to be ‘the same’ there are 11 different squares that can be found.

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Teacher notes Areas of squares Slides 8 & 9 Students can show their methods for finding the area using the ‘ink annotation’ tool (which becomes visible when the pointer is allowed to hover over the bottom left of the PPT slide). It is worth flagging up Pythagoras theorem if no-one comes up with it. You might ask which method is the most time efficient.

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Teacher notes Slide 9 Students can show their methods for finding the area using the ‘ink annotation’ tool (which becomes visible when the pointer is allowed to hover over the bottom left of the PPT slide). They might divide the square up into smaller shapes, enclose the square or use Pythagoras’ theorem.

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Teacher notes Side Lengths of squares Slides 11& 12 Students should realise that once they have the area of the square, the side length is simple the square root of it. Slide 13 Simplifying roots. Lengths of lines Slides 16 & 17 It’s helpful to point out where lines pass through dots, linking to the idea of enlargement i.e. the dots on g split the line into 3, making it 3 times bigger than a 2 line.

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Acknowledgements Square Root of 2 to places Accessed 29/10/14 nrich Tilted Squares ‘checker’

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