1. 1. The Numbers Behind the Death Spiral Tetyana Berezovski (St. Josephs University / Mathematics) Diana Cheng, EdD (Towson University / Mathematics)
2. 2. Abstract In light of the most recent Winter Olympic Games, mathematical modeling problems involving algebra, geometry, trigonometry & calculus are presented via dynamic geometry software in the context of pairs figure skating. An aesthetically pleasing & athletically demanding pairs figure skating element, the death spiral, is discussed. Activities related to the pairs death spiral which are suitable for middle & high school students are provided in this workshop. Teachers work on these problems are analyzed & discussed.
3. 3. CCSS - Modeling Modeling links classroom mathematics & statistics to everyday life, work, & decision-making. Modeling is the process of choosing & using appropriate mathematics and statistics to analyze empirical situations, to understand them better, & to improve decisions. Quantities & their relationships in physical, economic, public policy, social, & everyday situations can be modeled using mathematical & statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, & comparing predictions with data.
4. 4. Death Spiral Required element of a pairs routine The man pivots in a circle with both of his skates at a fixed point on the ice, while the lady moves around him in a circular path with only one of her blades touching the ice (Kerrigan & Spencer 2003) The ladys torso is low and can be, at times, almost parallel to the ice Soviets Ludmila Belousova & Oleg Protopopov, pairs figure skating champions of the 1964 & 1968 Winter Olympic Games, were credited with making death spirals famous (US Figure Skating Association 1998)
5. 5. Movie https://www.youtube.com/watch?v=H3T5A4WsWCQ
6. 6. Questions about the Death Spiral? What kinds of mathematical investigations would you ask about the death spiral?
7. 7. Death Spiral: 2D Projections z x y A O A O O M B BA
8. 8. Death Spiral: Side View M
9. 9. Dearth Spiral: Related Rates In the following activity we investigate the Death Spiral taking in consideration the simultaneous motion of both partners. During the entrance stage of the Death Spiral, the man lowers his body vertically, allowing the lady to slide farther away horizontally. It is important to notice that while in the element, partners hands form a straight line, and a straight-hands distance ML is constant (9 feet), while vertical distance MB and horizontal distance BL are varying over the time. MLB is the angle between the straight- hands distance AB and the ice surface. It is known that by the time the distance MB is 6 feet, the lady is sliding horizontally at the rate 1 foot per second.
10. 10. Death Spiral: Student Questions a. At what rate does the man lower his body vertically? Will the partners position change at the same or different rate? Explain. b. Determine how fast MLB changes. c. If the distance MB decreases as the man lowers his body, does MLB increase or decrease? How do you know? Draw a graph illustrating the relationship between the mans height and MLB.
11. 11. Death Spiral: Solution ML = 9.00 in. BM = 6.00 in. L M B Man head
12. 12. Death Spiral (a): solution ML = 9.00 in. BM = 6.00 in. L M B Man head
13. 13. Death Spiral (a): students responses Students had to determine the correct equation, the Pythagorean theorem, for which the derivative would provide information on the mans rate of change. Some students forgot that the straight-arms distance ML , is constant over time. Students had trouble visualizing how the hypotenuse of a right triangle stays constant while the vertical and horizontal dimensions changed.
14. 14. Death Spiral (b): Solution
15. 15. Death Spiral:(b) Students Responses Some students correctly wrote out the expression for d(MLB)/ dt using the Quotient Rule then made substitution errors to arrive at an incorrect solution Even when correct substitutions are made, students can make conceptual errors (see next)
16. 16. Death Spiral:(b) Students Responses (IC)
17. 17. Death Spiral (c): Students Responses Students stated that would decrease because there was a direct relationship between the angle and the mans height, without further explanation Students concluded that a direct relationship exists, however their graphs did not indicate proportional relationships
18. 18. Death Spiral (c): Students Responses A student found the original angle MLB value to be 41.81o, and then plugged in a lower MB to see if is greater than or less than 41.81o. This happens to produce the correct response that decreases due to the approximately linear relationship between MB and angle MLB (see Table 2). However, the student had only checked two data points and generalized based on this limited data.
19. 19. Death Spiral (c): Students Responses A student found the original value to be 41.81o, and then plugged in a lower to see if is greater than or less than 41.81o. This happens to produce the correct response that decreases due to the approximately linear relationship between and (see Table 2). However, the student had only checked two data points and generalized based on this limited data. None of the students took into consideration the possibility that a nonlinear or trigonometric relationship could exist.
20. 20. Death Spiral (c): Students Responses Some students showed lines with rays extending on and other students showed distinct data points Continuous graph: Discrete graph:
21. 21. Dearth Spiral (c): relationship between the mans height & MLB y = 8.0973x - 6.6787 R = 0.999 0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 AngleMLB(degrees) MB height in feet
22. 22. Death Spiral: Birds Eye View Concentric circles O A B
23. 23. Concentric Circle Tracings Question 1: Does Keauna or Rockne travel faster during the death spiral? Explain how you determined your answer. Keauna travels faster as she is moving on bigger circle. Rockne is hardly moving. (why does traveling on a larger circle connect with faster rate?) Keauna. It takes both of them the same time to make one complete rotation, but the distance travelled in that time is greater for Keauna because her circle has a larger radius, which means a larger circumference. Since speed = distance / time, Keaunas greater distance implies greater speed.
24. 24. Concentric Circle Tracings (2a) If OA = 10 feet, and OB = 2 feet, answer the following questions: How much further does Keauna travel than Rockne? Explain. Circumference of a circle = 2 where r is the radius of the circle. Keauna distance = 2 10 = 20 62.83 Rockne distance = 2 2 = 4 12.56 Keauna travels 16 further or approx. 50.27 ft further. 20 4 = 16 62.83 12.56 = 50.27
25. 25. GSP Animation: Concentric Circles c1 c2 Length arc SA( ) Length arc S'B( ) = 21.21 cm Length arc S'B = 5.33 cm Length arc SA = 26.55 cm Animate Segment AB S' O A B S
26. 26. Concentric Circles (2b) Keauna 20 8 = 5 2 7.85 Rockne 4 8 = 2 1.57 The differences in speeds are 5 2 2 = 2 6.28 . If the death spiral took 8 seconds to complete, what is the difference in speeds of Keauna and Rockne? Explain.
27. 27. Concentric Circle (2c) Specific solution Keauna and Rocknes friend, Lisa, is trying to answer previous two questions and she first calculates the difference between the lengths of OA and OB. Is this a valid first step to solve the above two questions? If so, how could she use it, and if not, explain why not. So Lisa finds the difference to be 8 feet, which would aide her in finding the difference of their distance traveled which will still result in = 28 = 16 50.27 It will also aide in calculating the difference of their 2 speeds using D= RT 16 8 = 2 6.28 Using Lisas method allows finding the solutions much quicker. Although using this method you do not know how far each traveled or the speed which may be necessary information.
28. 28. Concentric Circles Question 2c (general solution) Lisa could use the difference between OA and OB to find Keauna and Rocknes speed. The distance they each travel create two circles. The distance they travel can be calculated by finding the circumference of each circle. The formula = 2 can be used to find the circumference. Keaunas distance is 2 and Rocknes distance is 2 when = and = . To find the difference in their distance, subtract their formulas: 2 2 2( ) 2( ) If Lisa finds the differences between the radii of the circles, she could multiply the difference by 2 which would give her the differences between the distances they travel during the death spiral. Finally, if she divides the difference between their distances by 8, time, this
29. 29. Concentric Circles (2c) Geometric solution Subtracting the circumferences of two circles is equivalent to finding the circumference of one circle whose radius was the difference of the two circles radii (Posamentier, 2003). c3 c2 c1 OB = 0.33 cm OA = 7.09 cm Circumference c3 = 44.52 cm Circumference c1( ) Circumference c2( ) = 44.52 cm Circumference c2 = 2.10 cm Circumference c1 = 46.62 cm Animate Point B A B O
30. 30. Conclusion The ideas introduced and discussed today could be used to enhance the instruction of applications of derivatives & trigonometry, distance / rate / time, circle circumferences, and algebra. Consistent with the ideas of modeling explained by the Common Core State Standards Standards for Practice (CCSI 2010), we believe that teachers should be able to comprehend mathematical content at much deeper level, making connections between real life & various topics in mathematics; such connections will help challenging mathematical ideas to be understood and to stick!
31. 31. References: