Activities for Students: Trigonometry through a Ferris Wheel

Gail Kaplan

doi:10.5951/mt.102.2.0138

Activities for Students: Trigonometry through a Ferris Wheel

Mathematics Teacher ◽

10.5951/mt.102.2.0138 ◽

2008 ◽

Vol 102 (2) ◽

pp. 138-143

Author(s):

Gail Kaplan

Keyword(s):

Phase Shifts ◽

Generalized Functions ◽

Study Phase ◽

Trigonometric Functions ◽

Ferris Wheel

In the traditional study of trigonometry, students graph the basic trigonometric functions. They study phase shifts, horizontal and vertical translations, and changes in period so that they can sketch the graph of generalized functions such as f(x) = acosb(x − c) + d by recognizing the information provided by the constants a, b, c, and d. Far too often, students master this material by memorizing it and thus have little comprehension of why and how each value in an equation affects the graph.

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Circadian clock resetting by arousal in Syrian hamsters: the role of stress and activity

AJP Regulatory Integrative and Comparative Physiology ◽

10.1152/ajpregu.00222.2003 ◽

2003 ◽

Vol 285 (4) ◽

pp. R917-R925 ◽

Cited By ~ 54

Author(s):

R. E. Mistlberger ◽

M. C. Antle ◽

I. C. Webb ◽

M. Jones ◽

J. Weinberg ◽

...

Keyword(s):

Wheel Running ◽

Physical Restraint ◽

Rest Period ◽

Phase Shifts ◽

Study Phase ◽

Mild Stress ◽

Synthesis Inhibitor ◽

Apparent Paradox ◽

Large Phase ◽

Cortisol Release

Circadian rhythms in the Syrian hamster can be markedly phase shifted by 3 h of wheel running or arousal stimulation during their usual daily rest period (“subjective day”). Continuous wheel running is predictive but not necessary for phase shifts of this “nonphotic” type; hamsters aroused by gentle handling without running can also show maximal shifts. By contrast, physical restraint, a standard stress procedure and thus presumably arousing, is ineffective. To resolve this apparent paradox, phase-shifting effects of 3-h sessions of restraint or other stress procedures were assessed. In a preliminary study, phase shifts to arousal by gentle handling were significantly potentiated by the cortisol synthesis inhibitor metyrapone, suggesting that stress-related cortisol release may inhibit phase shifts to arousal. Next, it was confirmed that restraint in the subjective day does not induce phase shifts, but behavioral observations revealed that it also does not sustain arousal. Restraint combined with noxious compressed air blasts did sustain arousal and induced a significant cortisol response compared with arousal by gentle handling but did not induce shifts. Restraint combined with continuous horizontal rotation was also ineffective, as was EEG-validated arousal via confinement to a pedestal over water. However, 3 h of resident-intruder interactions (an intense psychosocial stress) or exposure to an open field (a mild stress) did induce large shifts that were positively correlated with indexes of forward locomotion. The results indicate that large phase shifts associated with arousal in the usual sleep period are neither induced nor prevented by stress per se, but are dependent on the expression of at least low levels of locomotor activity. Sustained arousal alone is not sufficient.

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Students' Conceptions of Sine and Cosine Functions When Representing Periodic Motion in a Visual Programming Environment

Journal for Research in Mathematics Education ◽

10.5951/jresematheduc.49.4.0390 ◽

2018 ◽

Vol 49 (4) ◽

pp. 390-423 ◽

Cited By ~ 4

Author(s):

Anna F. DeJarnette

Keyword(s):

Prior Knowledge ◽

Secondary Students ◽

Periodic Motion ◽

Visual Programming ◽

Programming Environment ◽

Trigonometric Functions ◽

Sine And Cosine Functions ◽

Cosine Functions ◽

Ferris Wheel

In support of efforts to foreground functions as central objects of study in algebra, this study provides evidence of how secondary students use trigonometric functions in contextual tasks. The author examined secondary students' work on a problem involving modeling the periodic motion of a Ferris wheel through the use of a visual programming environment. This study illustrates the range of prior knowledge and resources that students may draw on in their use of trigonometric functions as well as how the goals of students' work inform their reasoning about trigonometric functions.

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Limitations to the Spacing Effect

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie) ◽

10.1027/1618-3169.52.4.257 ◽

2005 ◽

Vol 52 (4) ◽

pp. 257-263 ◽

Cited By ~ 22

Author(s):

Peter P. J. L. Verkoeijen ◽

Remy M. J. P. Rikers ◽

Henk G. Schmidt

Keyword(s):

Free Recall ◽

Phase Retrieval ◽

Incidental Learning ◽

Factor Model ◽

Recall Performance ◽

Spacing Effect ◽

Word List ◽

Intentional Learning ◽

Study Phase ◽

Learning Condition

Abstract. The spacing effect refers to the finding that memory for repeated items improves when the interrepetition interval increases. To explain the spacing effect in free-recall tasks, a two-factor model has been put forward that combines mechanisms of contextual variability and study-phase retrieval (e.g., Raaijmakers, 2003 ; Verkoeijen, Rikers, & Schmidt, 2004 ). An important, yet untested, implication of this model is that free recall of repetitions should follow an inverted u-shaped relationship with interrepetition spacing. To demonstrate the suggested relationship an experiment was conducted. Participants studied a word list, consisting of items repeated at different interrepetition intervals, either under incidental or under intentional learning instructions. Subsequently, participants received a free-recall test. The results revealed an inverted u-shaped relationship between free recall and interrepetition spacing in both the incidental-learning condition and the intentional-learning condition. Moreover, for intentionally learned repetitions, the maximum free-recall performance was located at a longer interrepetition interval than for incidentally learned repetitions. These findings are interpreted in terms of the two-factor model of spacing effects in free-recall tasks.

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