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
Areas of Squares
1
2
4
5
9
8
16
25
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
2
3
√5
√8
4
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
4 5
4 2
26
2 17
5
3 2
2 5
29
Lengths of Lines
Which of the following could be drawn by connecting dots
on the dotty paper?
•
17
• 2 3
•
2
• 2 2
•
6
• 3 5
•
13
• 2√5
•
11
• 2√7
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





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.
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.
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