A Comparison of Textbooks' Presentation of Fractions

In the United States, fractions are an important part of the middle school curriculum, yet many middle school students struggle with fraction concepts. Teachers also have difficulty with the conceptual understanding needed to teach fractions and rely on textbooks when making instructional decisions. This reliance on textbooks, the idea that teaching and learning of fractions is a complex process, and that fraction understanding is the foundation for later topics such as proportionality, algebra, and probability, makes it important to examine the variation in presentation of fraction concepts in U.S. textbooks, especially the difference between traditional and standards‐based curricula. The purpose of this study is to determine if differences exist in the presentation of fractions in conventional and standards‐based textbooks and how these differences align with the recommendations of National Council of Teachers of Mathematics, Common Core State Standards, and the research on the teaching and learning of fractions.     

Frequency-stabilized cavity ring-down spectrometer for high-sensitivity measurements of water vapor concentration

We present a portable spectrometer that uses the frequency-stabilized cavity ring-down spectroscopy technique capable of high-precision measurements of trace water vapor concentration. Measuring one of the strongest rovibrational transitions in the ν₁+ν₃ water vapor combination band near ˜ν=7181.156 cm^-1, we compare spectroscopic and thermodynamic determinations of trace water vapor in N₂, and find systematic differences attributable to water vapor background effects and/or uncertainties in line intensities. We also compare the frequency-stabilized ring-down method with other cavity ring-down approaches that are based on unstabilized probe lasers and unstabilized ring-down cavities. We show that for the determination of water vapor concentration, the frequency-stabilized cavity ring-down method has the minimum measurement uncertainty of these techniques. The minimum noise-equivalent absorption coefficient of the spectrometer was 1.2×10^-10 cm^-1 Hz^-1/2, which further corresponds to a minimum detectable water vapor mole fraction equal to 0.7×10^-9 for an absorption spectrum of 10 minutes duration. PACS 33.20.-t; 33.70.Jg; 33.70.Fd; 42.62.Fi     

Sample shape dependent EPR and the measurement of Van Vleck paramagnetism

The dependence of the EPR resonance field on sample shape has been measured for Fe³⁺ and Yb³⁺ ions substituted into thullium gallium garnet. The results enable the second order paramagnetism of the lowest singlet Tm³⁺ energy level to be found.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

Ketones derived from phyllocladene

Phyllocladene has been converted into some 15- and 16-oxo derivatives. The rotary dispersion curves of these indicate that XV is the most likely structure for mirene.     

First-order Kondo-frustated antiferromagnetic ordering in the heavy electron compound YbAs

¹⁷⁰Yb Mössbauer measurements show that the heavy electron compound YbAs undergoes a first order paramagnetic-antiferromagnetic transition centered at T_N=0.58K. The saturated 4f-shell magnetic moments are reduced by 47% relative to the value predicted by a crystal field model. A study of the line shapes below T_N in the presence of an external field shows that the RKKY exchange energy is about 20 times bigger than k_B T_N. These results show that YbAs presents a magnetic order frustrated by the Kondo effect.supported by CNPq