Fraction Rules Research Articles

GENERALIZED FRACTION RULES FOR MONOTONICITY WITH HIGHER ANTIDERIVATIVES AND DERIVATIVES

We first introduce the generic versions of the fraction rules for monotonicity, i.e., the one that involves integrals known as Gromov’s theorem and the other that involves derivatives known as L’Hôpital’s rule for monotonicity, which we then extend to high-order antiderivatives and derivatives, respectively.

The dynamic nature of children’s strategy use after receiving accuracy feedback in decimal comparisons

When students start learning decimals, they may incorrectly apply features of their prior numerical knowledge (e.g., whole-number or fraction rules). However, because whole numbers, fractions, and decimals all have their own unique features, these whole-number and fraction strategies do not always lead to correct solutions. We examined whether receiving immediate accuracy feedback while comparing decimal pairs that were either congruent with whole-number rules (e.g., decimals with more digits were larger in magnitude) or incongruent with whole-number rules (e.g., decimals with fewer digits were larger in magnitude) would lead students to change their decimal comparison strategies. We also examined whether students’ potential improvement after feedback would generalize to decimal comparisons involving different numbers of digits. We found that sixth- to eighth-grade students’ use of the whole-number strategy declined and their use of the normative decimal strategy increased over the course of receiving feedback, whereas no significant strategy change was observed among students who did not receive any feedback. Students who received feedback were also less likely to use a whole-number strategy and more likely to use a decimal strategy in different decimal comparisons in an immediate posttest and a 2-week delayed posttest. Our exploratory analyses found that students’ improvement on decimal comparisons did not transfer to decimal arithmetic. Moreover, students’ inhibitory control also predicted strategy use in immediate and delayed posttests. Our study provides insights into the mechanisms of rapid strategy change and has implications for designing interventions to improve children’s understanding of decimal magnitudes.

Eye gaze patterns reflect how young fraction learners approach numerical comparisons

Learning fractions is notoriously difficult, yet critically important to mathematical and general academic achievement. Eye-tracking studies are beginning to characterize the strategies that adults use when comparing fractions, but we know relatively little about the strategies used by children. We used eye-tracking to analyze how novice children and mathematically-proficient adults approached a well-studied fraction comparison paradigm. Specifically, eye-tracking can provide insights into the nature of differences: whether they are quantitative—reflecting differences in efficiency—or qualitative—reflecting a fundamentally different approach. We found that children who had acquired the basic fraction rules made more eye movements than did either adults or less proficient children, suggesting a thorough but inefficient problem solving approach. Additionally, correct responses were associated with normative gaze patterns, regardless of age or proficiency levels. However, children paid more attention to irrelevant numerical relationships on conditions that were conceptually difficult. An exploratory analysis points to the possibility that children on the verge of making a conceptual leap attend to the relevant relationships even when they respond incorrectly. These findings indicate the potential of eye-tracking methodology to better characterize the behavior associated with different levels of fraction proficiency, as well as to provide insights for educators regarding how to best support novices at different levels of conceptual development.

Some results of bottom sediment monitoring in oil production area in the southeastern Baltic Sea

Monitoring studies in the area of Kravtsovskoe oil field in the southeastern Baltic Sea have been carried out since 2003. Terrigenous bottom sediments (0–5 cm layer) in the area (23 stations) are represented by all grain size types. Down to the depth of 30–50 m, these are mostly quartz sands; at depths of 60–110 m, these are terrigenous mud with Corg content of 1–3%. The concentration and distribution of toxic elements (Cu, Co, Cd, Pb, Ba, and Hg) over sediment types and bed area are the same as in the Baltic Sea as a whole: Cu, Co, Cd, Pb, and, partially, Hg are distributed in accordance with the “fraction rules” (with minimums in sands and coarse aleurites and maximums in mud); Ba distribution is independent of the sediment type. The concentrations of toxic elements tend to increase with time: the concentrations of such elements in samples (at the same stations) somewhat increase compared with their concentrations before the construction of D-6 oil platform, during its construction, and during oil production. This trend in the concentrations of toxic elements in sediments can be due to the steelwork of D-6 platform. The increase in Ba concentration is likely due to drill mud spills during exploration drilling. None of the elements shows sediment pollution in excess of MAC anywhere in the area.

Prediction of Remaining Useful Life for Used Gas Turbine Blades

Used turbine blades are replaced by new once based on manufacturers recommended useful life. However, depending on the turbine operating conditions the blades might have different creep service life. Thus, the aim of this study was to predict the remaining creep life and investigating the microstructure of used gas turbine blades. This was done by using Larson-Miller parameter and Robinson life fraction rules under certain conditions of stress and temperature. The change in microstructure of the materials was analyzed by using the Field Emission Scanning Electron Microscope (FESEM). The result shows that turbine blades suffered from several microstructure changes based on their service life. The method used to predict the remaining useful life of used turbines could be an input to a decision to repair or replace used turbine blades. Keywords: Larson-Miller parameter,life fraction rule, microstructure, creep life, turbine blades.

Creep-Fatigue Evaluation by Hysteresis Energy in Modified 9Cr-1Mo Steel

Researchers proposed the methods of creep-fatigue evaluation, such as time fraction rules or ductility exhaustion rules. However, the microstructure change during creep-fatigue should not be directly considered in these methods. The hysteresis energy contributes to the microstructure change before the crack initiation and the crack initiation and propagation. The creep-fatigue has evaluated by the hysteresis energy in modified 9Cr-1Mo steel which is a candidate for structural material in Fast Breeder Reactor (FBR) plant. Creep-fatigue and fatigue tests were carried out at 723-873K in air. The hysteresis energy per hour at the middle of life (Nf / 2, Nf is the number of cycles to failure) has been evaluated. It is clear that the relationship between this parameter and the time to failure can be expressed by the power-law function. The creep-fatigue life can be evaluated based on the hysteresis energy an hour at Nf / 2 using this relation.

Learning Mathematical Rules with Reasoning

This research focused on students learning of mathematical rules with reasoning. A small group of students (age 11-12 yrs) was observed closely by the first author as she taught them fraction rules. The area of focus was fractions and activities were designed pertaining to the four rules of fractions: addition, subtraction, multiplication, and division. The study was undertaken from a constructivist perspective of learning according to which students’ learn through active participation in the construction of knowledge (Glasersfeld, 1995). This was significant in the context of mathematics classrooms in Pakistan which usually subscribe to the objectivist epistemology i.e. knowledge of the ultimate reality is possible. An implication of adhering to this epistemology is the knowledge transmission view of teaching and learning (Halai, 2000). Key findings of the study were that there were two significant factors that enabled students to learn rules with reasoning were: Teachers questions and opportunities for students to explain thinking; and opportunity for students to engage with concrete and semi concrete materials. The study also provides some useful insights into the sequence of teaching the fraction rules and raises implications for mathematics teaching and teacher education.

Use of life and strain fraction rules for creep life prediction of pressurized pipe components undergoing geometry change

The results of creep failure life predictions of pressurized plain pipes, pipe bends, and a thick-walled pipe with a circumferential weld are used to demonstrate the applicability of a stress-based life fraction rule (LFR) and a strain-based strain fraction rule (SFR), under conditions of geometric non-linearity (GNL). The material properties used are related to a CrMoV pipe, at 640°C. Both the LFR and the SFR predictions are based on the stress and strain solutions obtained from the finite element calculations using a Norton creep law. The results obtained were compared with those obtained from corresponding creep damage analyses. For the cases investigated, it has been shown that, compared with damage results, conservative failure lives with consistent failure positions were obtained for the pipe weld, using both the LFR and the SFR methods. For plain pipes and pipe bends, the SFR prediction produces conservative results. However, the LFR prediction overestimates the failure life for the range of pipe diameter ratios investigated. The results obtained indicate that the LFR and SFR methods have potential for predicting creep failure lives in cases when the effect of geometry change is significant and hence the method based on the steady state peak stresses obtained from small deformation analysis is not applicable.

Creep fracture parameters of functionally graded coating

Functionally graded coating (FGC) has been widely used in severe conditions for its excellent mechanical characteristics compared with pure coating material. The graded component in the FGC can be described by specified local volume fraction rules of coating material. In this paper an inhomogeneous finite element method is used to investigate the crack behavior in FGC at elevated temperature with the powerlaw constitutive equation. The path‐ independent integral value C* characterizes amplitude of the crack tip stress fields at steady stage. Different volume fractions of coating material, creep coefficients and creep indices are analyzed by systematic computation to obtain the effects of these factors on crack growth and obtain the knowledge to minimize crack damage in the FGC. In addition, strain energy density rate factor, ?, is proposed as a material parameter for measuring crack resistance capability in creep conditions. The preliminary analysis with ? is also applied in FGC and the numerical results showed that ? can also characterize the coating load capacity.