TEACHER’S GUIDE
TEACHER’S GUIDE
TEACHER’S GUIDE
Extension Activity
Suggested Print Resources
Follow these steps below to learn more about the relationship of the
three sides of a right triangle and measurement.The Pythagorean
Theorem states that the sum of the squares of the two legs of a right triangle are equal to the hypotenuse squared.The formula for this is: a2 + b2
= c2 .
Materials needed: unlined white paper, metric ruler that has a millimeter
scale on it, calculator, protractor (to help you draw a right angle in the
right triangle).
• Gerdes, Paulus. Geometry from Africa: Mathematical and Educational
Explorations. The Mathematical Association of America, 1999.
• Johnson,Art. Famous Problems and Their Mathematicians. Teacher
Ideas Press, Portsmouth, NH; 2000.
• Lundy, Miranda. Sacred Geometry. Walker & Company, New York, NY;
2001.
1. Using a ruler, draw a right triangle. Make sure the right angle is exactly
90 degrees in measure.
The
Pythagorean Theorem
2. Label the right angle “C” and the other two angles “A” and “B.”
3.The hypotenuse is the longest line segment in your right triangle and it
should be labeled AB from step 2 above.
4. Using the metric ruler, measure the three sides of the triangle in millimeters.Try to be as accurate as possible. List the measurements of the
following line segments:AB = _____, BC = ______,AC = _____
Grades 8 & up
hese engaging programs complement traditional
lessons by encouraging mathematics discovery in
the real world. Using animated graphics, real-life
locales and vibrant young hosts, each program clearly
explains math concepts and presents students with
strategies to improve their problem-solving capabilities. Step-by-step examples of typical exam questions
are illustrated, along with common pitfalls to avoid.
T
5. Using your calculator, square (multiply the number times itself) each of
the three segments and write them below:
(AB)2 =_______, (BC)2 =_______, (AC)2 =______
6. Since AB is the hypotenuse, according to the Pythagorean Theorem:
(AB)2 = (BC)2 + (AC)2. Do the sides of your right triangle fit this
formula?
7. Remember that measurement is only APPROXIMATE so it is easy to be
“off” on your measurements. If the formula doesn’t work for your data,
try measuring more accurately or estimating to the nearest tenth of a
millimeter to get the Pythagorean Theorem to work for your right
triangle.
Suggested Internet Resources
Periodically, Internet Resources are updated on our Web site at
www.LibraryVideo.com
• www.cut-the-knot.org/pythagoras/index.shtml
This site contains over forty illustrated proofs of the Pythagorean
Theorem.
• www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/
TEACHER’S GUIDE
Paula J. Bense, M.Ed.
Curriculum Specialist, Schlessinger Media
COMPLETE LIST OF TITLES
• Area of Circles and
Composite Shapes
• Combined Probability
• Enlargement
Teacher’s Guides Included
and Available Online at:
• Loci
• The Pythagorean Theorem
• The Sine Ratio
• The Tangent Ratio
800-843-3620
Pythagoras.html
This site contains an interesting biography of Pythagoras and excellent
sources of information about math terms and symbols.
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Teacher’s Guide Copyright 2004 by Schlessinger Media,
a division of Library Video Company
P.O. Box 580, Wynnewood, PA 19096 • 800-843-3620
Program Copyright 2000 Channel Four Television Corporation
All rights reserved.
B2115
This guide is a supplement designed for teachers to
use when presenting this program and provides background information, vocabulary, practice questions
and answers, as well as Internet resources for students
and teachers to explore.
Please note that this series was produced in Great Britain,
where the decimal point is often drawn as a centered dot
(e.g., 3·1415) instead of a period (e.g., 3.1415) as is customary in the United States.
Background
Born on the Aegean island of Samos around 580 BC, Pythagoras traveled as a
young man to Egypt and Babylon. He eventually settled in Crotona, a Greek
colony in southern Italy.There he founded a brotherhood that studied religion, politics and philosophy as well as mathematics and science. Actually
Pythagoras’ famous ‘discovery’ was already known to the Babylonians, the
Chinese and the Egyptians. But the Pythagorean brotherhood may have provided the first general proof of the theorem.
Groups of three numbers (a,b,c) which satisfy a2 + b2 = c2 are called
Pythagorean triples. Here are the first 12 primitive triples (triples that are not
multiples of smaller triples):
3, 4, 5
Katie reveals that many civilizations, including the Chinese, Babylonians and
Egyptians, knew of and used the relationship whose ‘discovery’ is ascribed to
Pythagoras. She tells us about the work of Egyptian surveyors, known as
‘rope-stretchers,’ who designed the great pyramids.
Katie reviews the key features of right-angled triangles and looks in detail at
the special relationship between the sides that Pythagoras expressed in his
theorem.We see how the areas of the squares on the shorter sides fit exactly
into the area of the square on the hypotenuse, and how to express this
algebraically.
On a windswept site, Ben braves the elements to discover how the
Pythagorean Theorem is used by a team of engineers who are setting up a
wind farm.They use it to establish exactly where to construct anchor points
for steel guy ropes so that they remain taut.
Key Facts and Exam Tips
• In a right triangle, the longest side is called the hypotenuse. It is always
opposite the right angle.
• The Pythagorean Theorem states that in a right-angled triangle the area
of the square on the hypotenuse is equal to the sum of the areas of the
squares on the other two sides. For the triangle below the theorem can
be written as c2 = a2 + b2.
5, 12, 13
8, 15, 17
Learning Objectives
7, 24, 25
• Express the Pythagorean Theorem both algebraically and geometrically.
• Find the length of the hypotenuse of a right triangle.
• Work with squares and square roots.
• Use the converse of the theorem to prove that a triangle is right-angled.
• Rearrange the formula to change the subject of the equation.
• Calculate the length of a shorter side of a right triangle.
20, 21, 29
12, 35, 37
9, 40, 41
28, 45, 53
11, 60, 61
16, 63, 65
Vocabulary
33, 56, 65
Pythagorean Theorem — A theorem in geometry that states the square
of the length of the hypotenuse of a right triangle equals the sum of the
squares of the lengths of the other two sides.The formula is written
a2 + b 2 = c 2 .
adjacent — The triangle leg next to the angle being analyzed in a right
triangle.
degree — A unit equal to 1/360 of a circle.
hypotenuse — The longest side of a right triangle.
opposite — The triangle leg across from the angle being analyzed in a right
triangle.
right angle — An angle that is 90 degrees.
similar triangles — Triangles of different size but having the same angles.
trigonometry — The study of how the sides and angles of a triangle are
related to each other.
triangle — A geometric figure consisting of three points or vertices which
are connected with straight line segments called sides or legs.
square — To multiply a number by itself.
square root — A factor of a number that when squared gives the number.
48, 55, 73
Program Overview
The program shows how the Pythagorean Theorem can be used to find
unknown lengths in right-angled triangles and to prove that a triangle is
right-angled. It draws on the history of mathematics to show how this important theorem has been used for thousands of years to solve practical problems. Ben and Katie explore the relevance of the Pythagorean Theorem in our
world, witnessing the construction of an aerial slide by the army, the positioning of equipment at a wind farm, and measurement of a corner on a
soccer field.
At the army training camp, Ben shows how the length of the rope of an aerial
slide can be calculated because it forms the hypotenuse of a right triangle.
Ben does the calculation and introduces the idea of taking a square root. He
discusses how to find the square-root function on a calculator and how to
round an answer sensibly.With the dimensions of the slide established, Ben
glides down the hypotenuse!
(Continued)
2
3
• ‘Squaring’ a number means multiplying it by itself. So ‘c2’ means ‘c x c.’
Remember that c2 represents the area of a square with side c.
• When answering questions using the Pythagorean Theorem, make sure
you start with a clear diagram of the triangle. Label the right angle and
the sides of the triangle, then fill in the lengths that you know and identify the length you are going to find.
• When substituting the values into the formula, take care to replace the
letters with the lengths of the correct sides of the triangle.You must be
clear about which side is the hypotenuse before you can substitute
values. If the side you need to find is the hypotenuse, you can do the
arithmetic. If you need to find one of the shorter sides, you will need to
rearrange the expression first.
• At the end of the calculation, you need to take a square root. Make sure
you know how to do this on your calculator.
• You should know and be able to recognize the first 10 to 15 square
numbers, and also the squares of 20, 30, 40 and so on.
• You should be familiar with simple Pythagorean triples like (3,4,5),
(6,8,10), (5,12,13) and (7,24,25), and know that triangles with sides of
these lengths must be right-angled.
• When solving problems, remember to check that the size of your
answer is reasonable for the conditions given in the question. If you are
asked to find the hypotenuse, your answer should be the biggest of the
three sides.You can put all three values back into the formula and
check that they satisfy the Pythagorean rule. If they don’t, you know
you have made a mistake.
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