Pythagoras

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Pythagoras

  1. 1. Lesson Planning Pythagoras’ Theorem The first method The second method The third method Presentation Internet Coursework assessment End Hisham Hanfy
  2. 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
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  14. 14. see text book page 59 Number: 1,2,3 and 4 End Main Hisham Hanfy
  15. 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. 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 End Main Hisham Hanfy
  17. 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. End Main Hisham Hanfy
  18. 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 End Main Hisham Hanfy
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  20. 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. 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 End Main Hisham Hanfy
  22. 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. 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 End Main Hisham Hanfy
  24. 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). End Main Hisham Hanfy
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  26. 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. End Main Hisham Hanfy
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  30. 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. End Main Hisham Hanfy
  31. 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 End Main Hisham Hanfy
  32. 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 End Main Hisham Hanfy
  33. 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 End Main Hisham Hanfy
  34. 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 End Main Hisham Hanfy
  35. 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 End Main Hisham Hanfy
  36. 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 End Main Hisham Hanfy Main slide
  37. 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 End Main Hisham Hanfy
  38. 38. assessment End Hisham Hanfy
  39. 39. The length of the side CB is 5 cm 4 cm 8 cm End Main Hisham Hanfy
  40. 40. The length of the side AC is 5 cm 4 cm 8 cm End Main Hisham Hanfy
  41. 41. The length of the side CB is 8 cm 5cm 6cm End Main Hisham Hanfy
  42. 42. Wrong Try Again Back Hisham Hanfy
  43. 43. Wrong Try Again Back Hisham Hanfy
  44. 44. Wrong Try Again Back Hisham Hanfy
  45. 45. Back to assessment Hisham Hanfy

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