Exploring Special Lines (Pappus, Desargues, Pascal s Mystic Hexagram)

Transcription

1 Exploring Special Lines (Pappus, Desargues, Pascal s Mystic Hexagram) Introduction These three lab activities focus on some of the discoveries made by famous mathematicians by investigating lines. The first activity focuses on the work by Desargues. Desargues ( ) was from a wealthy family and had unlimited access to books and attended great educational institutions. Although recognized for his designs of a spiral staircase and other inventions, he is best known for his work in geometry. He is considered the inventor of projected geometry. Pappus Line is the focus of the second lab. Pappus ( ) lived in Alexandria and was one of the famous Greek geometers. He wrote Mathematical Collection and is considered the father of projective geometry. The third lab allows students to investigate Pascal s mystic hexagram. Frenchman Blaise Pascal ( ) was most famous for the Pascal triangle and his work on the cycloid. He discovered his mystic hexagram at the age of 16 and read Euclid s Elements at the age of 12. Key Words: Lines, modern geometry Ohio State Model Curriculum Objectives 1) Students will be able to compare, order, and determine equivalence of real numbers. 2) Students will be able to write inequalities for various triangular relationships. Learning Objectives 1) Students will complete basic constructions with a series of lines. 2) Students will make conjectures and test their results using geometry software. 3) Students will discuss collinear and non-collinear points. Materials Computers or calculators with Cabri geometry installed Exploring Special Lines lab worksheet Procedures Discuss the mathematicians Pappus, Desargues, and Pascal. Divide the students into groups of no more than three (two is preferable) Have students complete the lab worksheet. Monitor students progress during the activity. Use results from the lab activity as assessment. Review findings prior to the conclusion of class.

2 Exploring Special Line - Desargues Line Lab #1 Worksheet Team members: File Name: Date: Lab Goals Students will investigate a famous result discovered by Desargues through his work with lines. Desargues ( ) was from a wealthy family and was welleducated. Although recognized for his designs of a spiral staircase and other inventions, he is best known for his work in geometry. He is considered the inventor of projected geometry. Procedures 1) Draw point A. (use point tool) 2) Construct three different lines BCD and EFG. Do not put much space between the different points. (use line tool) 3) Place points E, F, and G on lines AB, AC, and AC, respectively. (use point on object tool) 4) Construct triangle BCD and EFG. (use triangle tool and attribute tool) Make the triangle thick to stand out. 5) Construct lines BC, BD, CD, EF, FG, EG. (use line tool) 6) Find the intersection of lines BC and EF. Label the point H. Next, find the intersection of lines BD and EG. Label this point I. Finally, find the intersection

3 of lines CD and FG. Label this point J. (use intersection points tool and attribute tool) 7) What do you notice about the points H, I, and J? 8) Test your result from #7. Was your conjecture correct? 9) Now grab a vertex of triangle BCD and move it around. Record your observations below. 10) Repeat #9 using triangle EFG. Record your observations. Triangles BCD and EFG are defined as being homological (or in perspective). Since all three lines pass through point A, point A is described as the homological center. The line that contains points H, I, and J is the homological axis. Extension Draw a line connecting H, I, and J. Hide everything except point A, the two triangles and the homological axis (line HJ ). Experiment by moving the vertices of the triangles and making generalizations about the effects of moving each vertex. Record your thoughts. Also, describe why the triangles are described as being in perspective. Discuss why the terms homological axis and homological center are used.

4 Exploring Special Lines - Pappus Line Lab #2 Worksheet Team members: File Name: Date: Lab Goals Students will investigate a famous result discovered by Pappus through his work with lines. Pappus Line is the focus of this lab. Pappus ( ) lived in Alexandria and was one of the famous Greek geometers. He wrote Mathematical Collection and is considered the founder of projective geometry. It is actually a special case of Pascal s Mystic Hexagram, which is the subject of the next lab. Procedure 1) Construct line AB. Place point C on the line. 2) Draw line DE. Place F on the line. (use line tool and use objects on line tool) (use line tool and use objects on line tool) 3) Draw lines AD, and EC. Label the intersection of those lines point G. 4) Draw lines BE and AF. Label the intersection of those lines point H. 5) Draw lines CF and DB. Label the intersection of these lines point I. (use line tool and use intersection tool)

5 6) Record your observations below. Focus on the points G, H, and I. What do you notice? 7) Test your results from above. Is your conjecture from #7 correct? 8) Grab the different points and move them around. Record any other observations that you see. Extension Connect the points G, H, and I. Hide the lines and watch the points as they move. Describe the locus of points as they move around the screen. Record any additional observations below.

6 Exploring Special Lines Pascal s Mystic Hexagram Lab #3 Worksheet Team members: File Name: Date: Lab Goals Students will investigate a famous discovery of Pascal through his work with lines - the mystic hexagram. Frenchman Blaise Pascal ( ) was most famous for the Pascal triangle and his work on the cycloid. He discovered his mystic hexagram at the age of 16 and read Euclid s Elements at the age of 12. The Pappus line in lab #2 is a special case of Pascal s work. Procedure 1) Construct a circle of any size. Place six points on the circle in the following order: A, B, C, D, E, and F. (use circle tool and use points on object tool) 2) Draw lines DA and EB. Label the intersection of these lines point G. 3) Draw lines AC and BF. Label the intersection of these lines points H.

7 4) Draw lines CE and FD. Label the intersection of these points I. 5) Record your observations below. Focus on the points G, H, and I. What do you notice? 6) Test your conjecture from #7. Is your conjecture from #7 correct? 7) Grab the different points and move them around. Record any other observations that you see. The line HG is known as the Pascal line. Extension Connect the points G, H, and I. Hide the lines and watch the points as they move. Describe the locus of points as they move around the screen. Record any additional observations below. Draw different lines to see if you can find other Pascal lines.