All About Surds

A surd is a number which is written with a root sign and cannot be simplified e.g. 3, 5 2 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.

All About Surds

e.g. 2 = 1.414213562373095048801688724209698078569671875376

94807317667973799073247846210703885038753432764157 27350138462309122970249248360558507372126441214970 99935831413222665927505592755799950501152782060571 47010955997160597027453459686201472851741864088919 86095523292304843087143214508397626036279952514079 89687253396546331808829640620615258352395054745750 28775996172983557522033753185701135437460340849884 71603868999706990048150305440277903164542478230684 92936918621580578463111596668…

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’

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.

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.

Possible Squares

Areas of Squares

How might you find the area of this square?

Areas of Squares

1 4 9 16 25

Areas of Squares

2 8 5 10 17 13

Side Lengths of Squares

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

Side Lengths of Squares

1 2 3 4 √2 √ 8 √ 5 5 √ 10 √ 17 √ 13

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

Side Lengths of Squares

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

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

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

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

Lengths of Lines

Which of the following could be drawn by connecting dots on the dotty paper?

• • • • • 17 2 6 11    13   • 2 3 • 2 2 • 3 5 • 2√5     • 2√7 

Angry Surds

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

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.

S tudents 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.

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.

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.

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.

Acknowledgements

Square Root of 2 to 1 000 000 places http://apod.nasa.gov/htmltest/gifcity/sqrt2.1mil

Accessed 29/10/14

nrich Tilted Squares ‘checker’

http://nrich.maths.org/content/id/2293/squareAnimation3.swf

Accessed 29/10/14