Utilize CAS to close the gap between two viewpoints on parabolas

2. Utilize CAS to close the gap between two viewpoints on parabolas Nurit Zehavi MathComp Project The Weizmann Institute of Science, Israel Nurit.Zehavi@weizmann.ac.il Columbus, Ohio 2001

4. Principles • When student work, CAS performance, and student reflection are intertwined – the ‘mathematical assistant’ (as Derive is called) serves as a learning environment. • The teaching techniques need to consider both the up-to-date technology and modern approaches to student learning (e.g., the balance between conveying of knowledge by the teacher and its construction by the student). • Since the role of the teacher who teaches with modern technology is so complex, the structure and options of a computer-based learning unit should be initially clear to the teacher at a global level.

5. Types of tasks • Tasks that are designed to overcome identifieddifficulties in a specific topic. We chose to treat those difficulties for which we believe that the technology has the potential to help students to better understand the topic. • New tasks that highlight mathematical ideas within a learned topic. Such tasks could not be dealt with, practically or effectively, without CAS. • Tasks that extend the topic at hand by connecting it to previously learned topics or to topics that will be learned at a later stage.

6. Two Widows and One Parabola • Rational and task description • Formative development - • Classroom experience • Teacher workshop • Discussion of curricular episodes - • Background • Initiation • Follow-up • Implications • Summary

7. Two main limitations of the traditional presentation of parabolas in the high school curriculum: • The difficulties students have in expressing the relationship between the parameter ‘a’ of the quadratic function y = ax2 and the shape of its graph. • The gap between the two viewpoints on the parabola in the traditional curriculum: the algebraic view of the graph of the quadratic equation, and the analytic-geometry view of loci.

8. Vertical Stretching

9. Horizontal shrinking

10. Slopes

11. A Special Chord

13. Two alternative approaches • “ A special chord of the parabola y = ax2 is the chord that connects two symmetric points on the parabola’s arms, where the distance between the points is 1/|a| .” • “A special chord of a parabola y = ax2 is the chord that connects two symmetric points on the parabola’s arms, where the slopes are 1 or –1.”

14. The parabola is a collection of points for which the distance between a point P and the Focus point F is equal to the sum of the Y- coordinates of P and F.

15. LOCI (t,t2)

16. Classroom experience • Two teachers volunteered to implement their preferred version of the unit in their Grade 10 classes. Class I (n = 25) used the first version, and Class II (n = 32) used the second version. Each class worked on the unit for 5 periods. • It became apparent that students in both classes were encountering conceptual difficulties. In class I (40%) felt at a certain stage that they just “discovered” that the length of the special chord is 1/a . • In class II we realized that the students needed a lot of help because they did not adequately master,at this stage of learning, the concepts of slope and derivative.

17. Problem 1 Fitting special chords(a) Plot the graph of y = 0.5x2and its special chord [[ , ],[ , ]].(b) Multiply by 2 the coordinates of the end-points of the special chord and plot the new segment 2*[[ , ],[ , ]].

18. Plot a parabola ______ where the new segment is its special chord.(c)By what number do we have to multiply the coordinates of the end-points of the special chord in y = 0.5x2to obtain this of y = -1.5x2.

19. Reflection(curricular episode) “If you want to fit the special chord of a parabola y = ax2to the parabola y = bx2, you should multiply the coordinates bya/b.”

20. Discussion of curricular episodes - • Background • Initiation • Follow-up • Implications R

21. Webs of meaningsimilarity mapping [x, y] [t*x, t*y] Expanding/contracting a parabola

22. Which is which?

23. Problem 2: Fitting pointsThe mid-point of the special chord in a parabola y = ax2 is (0, 0.75) Suggest at least two methods to determine which of the points A(2, 2.25) B(1.5, 0.75) C(-3, 3) D(-4, 3.75) belong to the parabola.

24. Webs of meaningProve that FQ < QQ’ + FO(curricular episode)

25. Webs of meaning G

26. more curricular episodes Eccentricity etc.

27. Principles and Standards for School Mathematics - NCTM 2000 The Technology Principle • Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. • Technology cannot replace the mathematics teacher.

28. Teacher Workshops • Since the role of the teacher who teaches with modern technology is so complex, the structure and options of a computer-based learning unit should be initially clear to the teacher at a global level. • Teaching techniques: tasks designed to achieve specific goals for which the individual student is responsible, gradually turn into open-ended tasks for which the group and the teacher share the responsibility. • The teachers decide on the nature of their role on the basis of the following components: the cognitive aspects of learning revealed by observing individual student work, the interaction between the participants, and curricular episodes.

29. Introductionof the special chord(an integrated approach) What are the features of the marked chord?

30. Identify a special chord Suggest a definition of a special chord so that its length could serve as a measure for the shape of the parabola

31. The general parabola

33. Summary • The advent of graphing technology in the last two decades introduced a curricular change that made the function a unifying concept. Attempts were made to put under the same roof calculus and analytic geometry (in a diluted form), with little emphasis on the concept of geometrical loci. Even the ellipse became a result of transforming a circle.

34. Summary • Then CAS entered the mathematical environment. We realized that it might be possible to introduce the parabola in an innovative way that combines the two viewpoints. To be concrete, in the learning unit, “Two windows, one parabola” students see from the beginning that the graph of a quadratic function is a parabola in the geometric sense, but not the other way around.

35. Summary • The curricular change implied by this approach includes geometric transformations like translation, rotation, and expansion, which are performed easily using CAS. This is an integrated approach to curriculum, in which the old traditional concepts are embedded in new webs of meaning.

36. Historical note Hippocrates, 5th BCE Menaechamus, 4th BCE Euclid, 3rd BCE Archimedes, 3rd BCE

37. Archimedes’ treatise: The Method Let ABC be a segment of a parabola bounded by the straight line AC and the parabola ABC, and let D be the middle point of AC. Draw the straight line DBE parallel to the axis of the parabola and join AB,BC. Then shall the segment ABC be 4/3 of the triangle ABC. Proposition 1

38. Historical note Apollonius, 2-3rd BCE Diocles, 2nd B.C.E. Pappus, 4th C.E. Kepler, 16th C.E. R

39. Optical property of the parabola Diocles On Burning Mirrors A Focus F Directrix O M ‘A

40. http://www.hoxie.org/Math/algebra/conics.htm G

41. Old traditional concepts are embedded in new Webs of meaning.

42. http://ltsn.mathstore.ac.uk/came/ Computer Algebra in Mathematics Education This is the CAME web site for Computer Algebra in Mathematics Education, an open, international organization for those interested in the use of computer algebra software in mathematics education. G