Grade 3 students’ mathematization through modeling: Situation models and solution models with mutli-digit subtraction problem solving

In considering mathematics problem solving as a model-eliciting activity ( [Lesh and Doerr, 2003], [Lesh and Harel, 2003] and [Lesh and Zawojewski, 2008]), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic.     

Ab initio molecular dynamical investigation of the finite temperature behavior of the tetrahedral Au19 and Au20 clusters

Density functional molecular dynamics simulations have been carried out to understand the finite temperature behavior of Au19 and Au20 clusters. Au20 has been reported to be a unique molecule having tetrahedral geometry, a large HOMO-LUMO energy gap, and an atomic packing similar to that of the bulk gold (Li, J.; et al. Science 2003, 299, 864). Our results show that the geometry of Au19 is exactly identical with that of Au20 with one missing corner atom (called a vacancy). Surprisingly, our calculated heat capacities for this nearly identical pair of gold clusters exhibit dramatic differences. Au20 undergoes a clear and distinct solid-like to liquid-like transition with a sharp peak in the heat capacity curve around 770 K. On the other hand, Au19 has a broad and flat heat capacity curve with continuous melting transition. This continuous melting transition turns out to be a consequence of a process involving a series of atomic rearrangements along the surface to fill in the missing corner atom. This results in a restricted diffusive motion of atoms along the surface of Au19 between 650 to 900 K during which the shape of the ground state geometry is retained. In contrast, the tetrahedral structure of Au20 is destroyed around 800 K, and the cluster is clearly in a liquid-like state above 1000 K. Thus, this work clearly demonstrates that (i) the gold clusters exhibit size sensitive variations in the heat capacity curves and (ii) the broad and continuous melting transition in a cluster, a feature that has so far been attributed to the disorder or absence of symmetry in the system, can also be a consequence of a defect (absence of a cap atom) in the structure.     

Sodium borohydride efficiently removes copper from amino acid-copper complexes

Sodium borohydride very efficiently removed copper from amino acid-copper complexes. The copper in the amino acid-copper complex was reduced to insoluble copper(I) oxide and the free amino acid was released in pure form. This method is rapid, nontoxic and inexpensive compared to the currently used methods.     

Synthesis, structure, UV-Vis-IR spectra, magnetism and theoretical studies on Cu[(2-aminomethyl)pyridine](thiocyanate)2 and comparisons with an analogous Cu complex

The title complex was synthesized under self-assembly conditions using Cu(acetate)₂·H₂O, 2-amp (= 2-aminomethylpyridine) and KSCN, and was characterized by IR, elemental analysis and single crystal structural analysis, and its spectral and RT magnetic properties were investigated. The asymmetric unit consists of a square planar Cu(II) center, with two ligand N atoms and two anionic Ns forming the square plane. In the unit cell, the monomeric complex assembles into 2-D layers through very weak non-bonded interactions between anionic S and Cu²⁺. Further, the structure was satisfactorily modeled by calculations based on Density Functional Theory (DFT), and the UV-Vis and IR spectra are analyzed in depth with the help of Time Dependent DFT (TDDFT). The results indicate that the absorption maxima are at relatively high energy and are mainly assigned to π → π^∗ transitions (in pyridine), with a fair contribution of metal to ligand charge transfer (MLCT) and ligand to ligand charge transfer (LLCT) transitions. All the low lying transitions are categorized as mixed MLCT/LLCT. A very weak but broad band in the higher wavelength region has been detected and identified as a d-d transition band. Also, it has been found that when the ligand ratio is modified, the formation of Cu(2-amp)₂(SCN)₂ takes place under the same self-assembly conditions, whose structure only has been recently reported. Structural, spectral and theoretical comparisons are presented for both complexes.     

Effect of nonlinear absorption in estimation of order of nonlinear optical process by fluorescence intensity

We report the effect of nonlinear absorption on the fluorescence signal. Attenuation of the incident beam intensity (I) in the medium due to nonlinear absorption reduces the fluorescence signal. We show that fluorescence signal proportional to I² is not always indicative of two-photon absorption process but such a dependence can be obtained for three or four-photon absorption process due to attenuation of the pump beam by nonlinear absorption. This reduction in the fluorescence signal due to nonlinear absorption is found to be more sensitive in the higher order nonlinear optical processes.     

Transmission grating spectrograph with on-line recording of XUV soft X-ray spectra from laser produced plasmas

The setting up and on-line operation of a transmission grating spectrograph in the XUV soft X-ray region is described. The detector was a microchannel plate–image intensifier combination, and the output of the image intensifier was coupled to a CCD camera–frame grabber system through an imaging lens. The spectrograph could be operated in the 5–20 Å range with 0.6 Å spectral resolution and in the 5–50 Å range with 1 Å resolution, respectively. The high sensitivity of the detector enables single shot operation, which is useful for several laser plasma interaction studies.     

Rotational analysis of the 0-0 band of the A3 ∏ inv-X3 ∑ − system of ND

The 0-0 band of the A³ ∏ _inv?X³ ∑ ^? system of ND was excited in an electrodeless microwave oscillator (2450 Mc./s.) discharge and photographed on a 6·6 meter concave grating spectrograph in the second order at a dispersion of 0·56 A/mm. Twenty-five out of the predicted twenty-seven branches have been identified in the rotational structure of the 0-0 band. From a rotational analysis, the following rotational constants have been determined: (i)X ³ ∑ ^? state—

     

Density functional analysis of the structural evolution of Gan (n=30-55) clusters and its influence on the melting characteristics

Recent experimental results have reported surprising variations in the shapes of the heat capacity curves and melting temperatures of gallium clusters in the size range of 30-55 atoms [G. A. Breaux et al., J. Am. Chem. Soc. 126, 8628 (2004)]. In the present work, we have carried out an extensive density functional investigation on ten selected clusters in the above mentioned size range. In particular, we have analyzed the ground state geometry and the nature of bonding in these clusters using electron localization function. We demonstrate that the existence or otherwise of a large island of atoms bonded with similar strength (i.e., the local order) in the ground state geometry is responsible for the variation in the shape of the heat capacity curve. We attribute the observed higher melting temperatures of some of the clusters (viz., Ga45-Ga48) to the presence of a distinct core and strong covalent bonds between the core and surface atoms. The present work clearly demonstrates that it is possible to understand the general trends observed in the heat capacity curves across the entire series on the basis of the analysis of their ground state.     

Synthesis and structure of coordination polymers of Ag(I) with isomeric (aminomethyl)pyridines. Formation of a novel circular helicate and 2-D networks via Ag...Ag contacts and coordination shell expansion under anion control

Reaction between Ag(I) salts and the three isomers of (aminomethyl)pyridines, viz., 2-amp, 3-amp, and 4-amp, lead to either discrete or polymeric (1-D and 2-D) structures influenced by anions and closed shell Ag.Ag contacts. Characterization data for Ag(2-amp)BF(4) (1) follow: monoclinic, space group C2/c, with a = 16.788(2) A, b = 11.5719(6) A, c = 11.3864(7) A, beta = 123.671(8) degrees, and Z = 8. For Ag(2)(2-amp)(3)(PF(6))(2) (2): monoclinic, space group P2(1)/a, with a = 10.029(7) A, b = 20.291(12) A, c = 13.907(6) A, beta = 95.38(5) degrees, and Z = 4. For Ag(2)(3-amp)(3)(PF(6))(2) (4): triclinic, space group P1, with a = 10.4482(7) A, b = 11.1468(9) A, c = 12.2720(11) A, alpha = 81.018(7) degrees, beta = 80.668(6) degrees, gamma = 80.977(6) degrees, and Z = 2. For Ag(4-amp)BF(4).0.75CH(3)CN (5): orthorhombic, space group C222(1), with a = 9.272(2) A, b = 16.164(12) A, c = 27.851(2) A, and Z = 8. For Ag(4-amp)PF(6) (6): monoclinic, space group P2(1)/m, with a = 5.2089(7) A, b = 14.3950(17) A, c = 7.0149(14) A, beta = 96.538(14) degrees, and Z = 2. While Ag(I) is 2-coordinate in 1, 5 and 6, it shows 3-coordination in 2 and 4. Compound 1 consists of a 1-D polymeric cation chain with interchain Ag...Ag contacts and the anions sitting on the edges of the chains. The dication in 2 is held in the form of a circular helicate by closed shell Ag...Ag interactions. Compound 4 generates a 2-D network with channels big enough to accommodate the anions. Compound 5 is a 2-D chiral network of chains connected by Ag...Ag contacts. Compound 6 shows a simple 1-D chain structure with an alternating arrangement of cationic chains and anions.