1. The Numbers Behind the Death Spiral Tetyana Berezovski (St.
Josephs University / Mathematics) Diana Cheng, EdD (Towson
University / Mathematics)
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. 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. 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. Movie https://www.youtube.com/watch?v=H3T5A4WsWCQ
6. Questions about the Death Spiral? What kinds of mathematical
investigations would you ask about the death spiral?
7. Death Spiral: 2D Projections z x y A O A O O M B BA
8. Death Spiral: Side View M
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. 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. Death Spiral: Solution ML = 9.00 in. BM = 6.00 in. L M B
Man head
12. Death Spiral (a): solution ML = 9.00 in. BM = 6.00 in. L M
B Man head
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. Death Spiral (b): Solution
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. Death Spiral:(b) Students Responses (IC)
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. 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. 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. 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. 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. Death Spiral: Birds Eye View Concentric circles O A B
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. 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. 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. 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. 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. 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. 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. 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!