1. Lesson Planning
Pythagoras’ Theorem
The first method The second method The third method
Presentation Internet Coursework
assessment
End Hisham Hanfy
2. To enable the students to 1) Board marker
1) Know Pythagoras’ theorem 2) Computer Lap top
3) projector – pointer
2) Solve problems on Pythagoras’ theorem
Click Here
I’m using my lap top
and projector in the
class room and didn’t
Click Here access to computer
room
Click Here
End assessment Hisham Hanfy
14. see text book page 59
Number: 1,2,3 and 4
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15. To enable the students to. 1) Board marker
Know some information about Pythagoras
2) Computer room
Know Pythagoras’ Theorem.
Understanding some Proofs of Pythagoras’ 3) projector – pointer
Theorem.
Solving Problems on Pythagoras’ Theorem.
In this lesson I access to
computer room and using
internet connection
Click Here
End assessment Hisham Hanfy
16. Subject : Mathematics
Age Range : Year 10
Timescale : Lesson
Scope : Individual student
Topic : Pythagoras' Theorem
Materials : Computer room with an Internet access available for students.
Objectives : To enable the students to.
Know some information about Pythagoras
Know Pythagoras’ Theorem.
Understanding some Proofs of Pythagoras’ Theorem.
Solving Problems on Pythagoras’ Theorem.
Beginning : Some information about Pythagoras
To know some information about Pythagoras go to web page
http://www.mathsnet.net/dynamic/pythagoras/index.html
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17. Procedures :
Students are required to open the following web pages:
1) to Know Pythagoras' Theorem go to this web page
http://www.mathsnet.net/dynamic/pythagoras/theorem.html
in this page you will see Pythagoras' Theorem. Pythagoras's Theorem is all
about right-angled triangles. If squares are constructed on the three sides of
a right-angled triangle, then these three squares have a very simply but
important connection.
2) Proofs of Pythagoras's Theorem:
to proof this theorem, we have different proofs to discuses this proofs go
to this pages :
The first proof:
http://www.mathsnet.net/dynamic/pythagoras/proof01.html
In the diagram below the areas of the yellow and green squares add up to
the area of the blue square. The two diagrams to the right illustrate how this
must be so. Use your mouse to move any of the red points.
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18. The second proof:
http://www.mathsnet.net/dynamic/pythagoras/proof02.html
In the diagram below use your mouse to slide the colored shapes from
the large square into the smaller squares
The third proof:
http://www.mathsnet.net/dynamic/pythagoras/proof03.htm
Use your mouse to move any of the blue points. Observe how the larger
square is made up of the same pieces as the two smaller squares
combined.
3) Example:
ask students to open the following web pages
http://www.mathsnet.net/dynamic/pythagoras/problem01.html
http://www.mathsnet.net/dynamic/pythagoras/problem02.html
and discuses how to solve this problems by using different ways
4) Home work:
I will ask students to open the following web pages web page and try to
solve the problems by them selves
http://www.mathsnet.net/dynamic/pythagoras/problem03.html
http://www.mathsnet.net/dynamic/pythagoras/problem04.html
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20. Click on the following to see
My coursework sheet
Planning document
Information about Pythagoras
Types of triangles
Drawing triangles
Right angled triangle
Pythagoras’ theorem
Calculate right angled triangle
Mathematical proof
Some proofs of Pythagoras’ theorem
Links on the world wide web
Hisham Hanfy
End assessment
21. My coursework sheet
Writing an introduction
The grade : 10h
My investigation is called : Pythagoras' theorem
I have been set the task of : collecting information from the internet about
some information about Pythagoras
some information about types triangles
what is right angled triangle
what is Pythagoras' theorem
what is mathematical proof
Some proofs of Pythagoras' theorem
Writing a method
To start with I will divide my pupils into 3 groups
The first group search for
some information about Pythagoras
some information about types triangles
The second group search for
what is right angled triangle
what is Pythagoras' theorem
The third group search for
what is mathematical proof
Some proofs of Pythagoras' theorem
The method I plan to use will be search in world wide web
When I have done this I will show my result using powerPoint presentation
Using results:
From the result I have noticed that : the relation between the three sides is (AC)2 = (AB)2 + (BC)2 where AC is the hypotenuse and AB , BC
are the other sides
I can use this to predict that : I can find the length of any side if I know the other sides
And the measure of any angle in the right angled triangle
I will cheek to see if my prediction is right by measuring the lengths of the three sides in any right angled triangle and then satisfy the
relation .
Now test my prediction on new Data. Data I have not already used to produce the rule
My rule for Pythagoras' theorem is (AC)2 = (AB)2 + (BC)2 where AC is the hypotenuse and AB , BC are the other sides
My rule works because we satisfy it in many right angled triangle
Conclusion:
From my results I know 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 sides "
I have found a rule which always works. This rule is AC)2 = (AB)2 + (BC)2 where AC is the hypotenuse and AB , BC are the other sides
My rule works because we satisfy it in many right angled triangle and found many proofs for this rule
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22. Specify and planning Minimum requirement Teacher's Notes
My investigation is called : Pythagoras' The pupils can, with help understand
theorem a simple task and produce
I have been set the task of working out : 1.some information about Pythagoras
1)some information about Pythagoras 2.some information about triangles
2)some information about types triangles 3.the Pythagoras' theorem
3)what is right angled triangle 4- The proof of the theorem
4)what is Pythagoras' theorem
5)what is mathematical proof
6) Some proofs of Pythagoras' theorem
Collect, Process and represent Minimum requirement Teacher's Notes
I will divide my students to three groups
The first group search for
1) some information about Pythagoras
2) some information about types triangles
The second group search for
1) what is right angled triangle The pupils collect / sample largely data
2) what is Pythagoras' theorem
The third group search for
1) what is mathematical proof
2) Some proofs of Pythagoras' theorem
Interpret and Discuss Minimum requirement Teacher's Notes
In this step I will discuss with my pupils Pupils comment on their data and
all the results of their search and then I then
will told them the main points which pupils must be know
pupils must be know 1.types of angles
2.Pythagoras' theorem
End Main 3- Proof of Pythagoras' theorem
Hisham Hanfy
23. Born on the island of Samos, Pythagoras was instructed in the teachings of the
early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras
is said to have been driven from Samos by his disgust for the tyranny of
Polycrates. About 530 bc Pythagoras settled in Crotona, a Greek colony in
southern Italy, where he founded a movement with religious, political, and
philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras
is known only through the work of his disciples
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24. Types of Triangles
Triangles are classified in terms of their sides and angles. Scalene triangles have no equal sides (fig.
1), isosceles triangles have two equal sides (fig. 4), and equilateral triangles have three equal sides
(fig. 5). In acute triangles, all the angles are less than 90 (fig. 1). In right triangles, one angle is equal
to 90 (fig. 3). In obtuse triangles, one angle is more than 90 (fig. 2). A line is called an altitude if it is
drawn from a vertex perpendicular to the opposite side (fig. 6). A line is called a median if it is drawn
from a vertex to the midpoint of the opposite side (fig. 7). A line is called an angle bisector if it divides
an angle into two equal angles (fig. 8). A line is called a perpendicular bisector if it is drawn
perpendicular to a side through its midpoint (fig. 9). A triangle drawn on the surface of a sphere is
called a spherical triangle (fig. 10). A figure with three arbitrary curves is sometimes called a triangle
(fig. 11).
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26. Right-angled triangles
Here is a triangle. There are three angles inside it: one we've called "theta"; one is left empty and one has a small square drawn in it. This
small square indicates that that angle is a right-angle, which is 900 or p/2 in radians. The triangle is therefore known as a right-angled
triangle.
The longest side of a right-angled triangle is called the "hypotenuse" as shown. The other two sides are labelled in terms of their position
to the angle q. (So if we were considering the other angle, rather than q, their labels would be swapped over).
We are going to define two new quantities called sine and cosine. They depend on the angle q: if q changes the they will also change.
The sine of q, written as sin(q), is defined as the length of the opposite side divided by the length of the hypotenuse.
Similarly, the cosine of q, written as cos(q), is defined as the length of the adjacent side divided by the length of the hypotenuse.
So
These are simple definitions, so let's see what they give us for sine and cosine for various angles.
First, if we imagine squashing the triangle down until the angle q reaches zero, then the opposite side will be zero
too and the adjacent side will be the same as the hypotenuse. (Our triangle has really just turned into a horizontal
line now).
This means that by the definitions above, sin(0)=0 and cos(0)=1.
Next, let's go to the other extreme: imagine increasing the angle q until the adjacent side shrinks to zero,
and now the opposite is the same as the hypotenuse. The angle q is now 900, or p/2.(The triangle is now
just a vertical line).
From our definitions of sine and cosine this tells us that sin(900)=1 and cos(900)=0.
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30. An example of a mathematical proof is the following argument, which proves that the Pythagorean theorem is true.
Figure 1 and figure 2 demonstrate that the relationship A2 + B2 = C2 holds in a right-angled triangle with sides A and B
and hypotenuse C. Figure 1 shows that a square of side A + B can be divided into four of the right-angled triangles, a
square of side A, and a square of side B. Figure 2 shows that a square of side A + B can also be dissected into four of
the right-angled triangles and a square of side C. Since the two squares of side A + B have the same area, they must
still have the same area once the four triangles are removed from each of them. The total area of the squares that
remain on the left side is A 2 + B2, and the area of the square remaining on the right side is C2. Thus A2 + B2 = C2.
The Greek mathematician Euclid laid down some of the conventions central to modern mathematical proofs. His book
The Elements, written about 300 bc, contains many proofs in the fields of geometry and algebra. This book illustrates
the Greek practice of writing mathematical proofs by first clearly identifying the initial assumptions and then reasoning
from them in a logical way in order to obtain a desired conclusion. As part of such an argument, Euclid used results
that had already been shown to be true, called theorems, or statements that were explicitly acknowledged to be self-
evident, called axioms; this practice continues today.
In the 20th century, proofs have been written that are so complex that no one person understands every argument
used in them. In 1976 a computer was used to complete the proof of the four-color theorem. This theorem states that
four colors are sufficient to color any map in such a way that regions with a common boundary line have different
colors. The use of a computer in this proof inspired considerable debate in the mathematical community. At issue
was whether a theorem can be considered proven if human beings have not actually checked every detail of the
proof.
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31. Mathematical Proof: Figures 1 and 2
These diagrams can be used to prove the Pythagorean theorem, which states
that if a right triangle has sides of length A and B, and a hypotenuse of length C,
then A2 + B2 = C2. Figure 1 and Figure 2 each have four right triangles with
sides of length A and B, and a hypotenuse of length C. Since the Figure 1 and
Figure 2 both have the same area, removing the four triangles from Figure 1
leaves a region that must have the same area as the region that is left when the
four triangles are removed from Figure 2. The area of the region left in Figure 1
is A2 + B2, and the area of the region left in Figure 2 is C2. Thus A2 + B2 = C2,
proving the Pythagorean
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32. Pythagoras' Proof to the Theorem
In the diagram above, we can see that
Area of the outer larger square = Area of the inner smaller square + Area of the four triangles
=> ( a + b ) 2 = c2 + 4 ( a.b/2)
Expanding gives:
=> a2 + 2a.b + b2 = c2 + 2a.b
Rearranging gives:
=> a2 + b2 = c2
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33. Proof Using Similar Triangl
In the above diagram, AD and BC are perpendicular. The triangles ABD and ACD are similar since ABD, BAD and ADB are equal
to CAD, ACD and ADC respectively.
Therefore, using ratios:
AB/BC = BD/AB
and
AC/BC = DC/AC.
Multiplying appropriately to get rid of the fractions gives:
AB·AB = BD·BC
and
AC·AC = DC·BC
Adding them together gives:
AB2 + AC2 = BD·BC + DC·BC
Factorising gives:
AB2 + AC2 = BC ( BD + DC )
Substituting (BD + DC) for BC gives:
AB2 AC+ 2 CB= 2
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34. President Garfield's Proof
Similar to Pythagroas' Proof, in the diagram above, we can see that:
Area of the Trapezium = Area of the inner smaller square + Area of the four triangles
=> (a + b).(a + b) /2 = (a + b) /2 + (a + b c.c) / 2
Expanding and multipyling by 2 gives:
a2 + 2ab + b2 = 2ab + c2
Reducing gives:
a2 + b2 = c2
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35. A Geometrical Proof to the Theorem
Consider the two triangles ABD and ACI, we know that, since ABHI and ACED are squares, that:
AI=AB, AC=AD and IAC=DAB=pic/2 + BAC
This means that the triangles ABD and ACI are congruent since they both have two of the same lengths and one
same angle. Therefore, their areas are equal and since the area of the square ABHI and that of the rectangle
ADJK are twice those of the triangles ACI and ABK respectively. Hence, ABHI and ADJK have the same area.
Similarly, with the triangles BCE and CFAm we can conclude that the areas BCFG and CEJK are equal.
Since,
Area ACED = Area ADJK + Area CEJK,
Replacing ADJK with ABHI and CEJK with BCFG:
=>Area ACED = Area ABHI + Area BCFG,
Therefore:
AC2 + BC2 =AB 2
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36. Another Proof to the Theorem
Similarly, in the diagram above, we can see that:
Area of the outer larger square = Area of the inner smaller square + Area of the four triangles
=> c2 = ( b - a )2 + 4( a.b / 2)
Expanding gives:
c2 = b2 - 2ab + a2 + 2ab
Reducing gives:
c2 = a2 + b2
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Main slide
37. 1) More on Pythagoras' Life
A short biography
http://www.andrews.edu/~calkins/math/biograph/biopytha.htm
A Longer Version with Mathematics in mind
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
Just as long with nice simple website
http://www.age-of-the-sage.org/greek/philosopher/pythagoras_biography.html
2) Proof
An Interactive Proof of Pythagoras' Theorem
http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html
This Website has 40 over proofs of his Pythagoras' theorem
http://www.cut-the-knot.org/pythagoras/index.shtml
More proofs
http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html
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