Where is the rate in the rule?

A well-developed understanding of rate is foundational to conceptual understanding of introductory calculus. Many students achieve procedural competence with the application of rules for differentiation without developing an awareness of the connection between derivative and rate. In addition, rate-related reasoning is needed to make informed decisions in many everyday applications of rate. This paper reports on additional data collected during interviews for a project investigating the different ways rate may be experienced by pre-calculus students. Many researchers (for example Kaput, 1999) have suggested that the conceptual understanding of function may be enhanced through the presentation and exploration of multiple representations of a variety of functions. In this paper, one section of each interview is considered in detail to evaluate the participants’ understanding in a specific rate context. Participants were asked to discuss a dynamic geometry simulation of a blind on two different windows one rectangular and the other not. Detailed analysis of the video-record of each participant’s interview provides insights into their perceptions of rate in several different representations. In the sections below, the conceptual framework is described; details of the interviews and the computer-based simulation are provided; and the analysis of the data is discussed.

Windows on practice : investigating equity in technology based mathematics classrooms

The strengths and weaknesses of using ethnographic research to investigate equity in a study of a grade 9 class that used a dynamic geometry program with laptop computers will be presented. It will be argued that research approaches that involve “windows on practice” provide understanding of not only who is advantaged and disadvantaged in technology-mediated classrooms but how this occurs. The way that other paradigms such as reflexive methods may enhance qualitative research will be proposed. Studies that involve “windows on equitable practice” will provide mathematics educators with models for advancing equity in mathematics learning when teaching with technology.

Comparison of static and dynamic engine models on the transient performance of a passenger vehicle powertrain

This paper presents experimental and computational results obtained on the Ford Barra 190 4.0 litres I6 gasoline engine and on the Ford Falcon car equipped with this engine. Measurements of steady engine performance, fuel consumption and exhaust emissions were first collected using an automated test facility for a wide range of cam and spark timings vs. throttle position and engine speed. Simulations were performed for a significant number of measured operating points at full and part load by using a coupled Gamma Technologies GT-POWER/GT-COOL engine model for gas exchange, combustion and heat transfer. The fluid model was made up of intake and exhaust systems, oil circuit, coolant circuit and radiator cooling air circuit. The thermal model was made up of finite element components for cylinder head, cylinder, piston, valves and ports and wall thermal masses for pipes. The model was validated versus measured steady state air and fuel flow rates, cylinder pressure parameters, indicated and brake mean effective pressures, and temperature of metal, oil and coolant in selected locations. Computational results agree well with experiments, demonstrating the ability of the approach to produce fairly accurate steady state maps of BMEP and BSFC, as well as to optimize engine operation changing geometry, throttle position, cam and spark timing. Measurements of the transient performance and fuel consumption of the full vehicle were then collected over the NEDC cycle. Simulations were performed by using a coupled Gamma Technologies GT-POWER/GT-COOL/GT-DRIVE model for instantaneous engine gas exchange, combustion and heat transfer and vehicle motion. The full vehicle model is made up of transmission, driveshaft, axles, and car components and the previous engine model. The model was validated with measured fuel flow rates through the engine, engine throttle position, and engine speed and oil and coolant temperatures in selected locations. Instantaneous engine states following a time dependent demand for torque and speed differ from those obtained by interpolating steady state maps of BSFC vs. BMEP and speed. Computational results agree well with experiments, demonstrating the utility of the approach in providing a more accurate prediction of the fuel consumption over test cycles.

Analysis of dynamic strains in tibia during human locomotion based on flexible multibody approach integrated with magnetic resonance imaging technique

Bone is known to adapt to the prevalent strain environment while the variation in strains, e.g., due to mechanical loading, modulates bone remodeling, and modeling. Dynamic strains rather than static strains provide the primary stimulus of bone functional adaptation. The finite element method can be generally used for estimating bone strains, but it may be limited to the static analysis of bone strains since the dynamic analysis requires expensive computation. Direct in vivo strain measurement, in turn, is an invasive procedure, limited to certain superficial bone sites, and requires surgical implementation of strain gauges and thus involves risks (e.g., infection). Therefore, to overcome difficulties associated with the finite element method and the in vivo strain measurements, the flexible multibody simulation approach has been recently introduced as a feasible method to estimate dynamic bone strains during physical activity. The purpose of the present study is to further strengthen the idea of using the flexible multibody approach for the analysis of dynamic bone strains. Besides discussing the background theory, magnetic resonance imaging is integrated into the flexible multibody approach framework so that the actual bone geometry could be better accounted for and the accuracy of prediction improved.