Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

On the solid state structure of 4-iodobenzoic acid

The solid-state structure of 4-iodobenzoic acid has been confirmed by variable temperature X-ray diffraction, variable temperature solid-state NMR and differential scanning calorimetry. 4-iodobenzoic acid crystallizes in the space group P2(1)/n, and dimerizes in the solid state about a center of inversion. Using extensive X-ray crystallographic data collections, the placement of the carboxylate H atoms from the residual electron density in difference Fourier maps was determined. The position of the electron density associated with the proton is found to vary with temperature in that the population of the disordered sites changes with varying temperature. Determination of the crystal structure between the temperatures of 248 and 198 K was not possible due to a phase transition, an endothermic event occurring at 230.77 K. The phase transition is also indicated by a change in the relaxation time of the ring carbon atoms in the solid-state NMR data. Though the dominating force in the dimeric unit in the solid state is the presence of strong hydrogen bonds, there are also van der Waals forces present between the iodine atoms. In the layered structure, the iodine-iodine distance is within the van der Waals contact radii, an interaction which causes a deformation in the electron density of the iodine atoms.     

Novel carbohydrate-appended metal complexes for potential use in molecular imaging

Seven discrete sugar-pendant diamines were complexed to the {M(CO)(3)}(+) ((99m)Tc/Re) core: 1,3-diamino-2-propyl beta-D-glucopyranoside (L(1)), 1,3-diamino-2-propyl beta-D-xylopyranoside (L(2)), 1,3-diamino-2-propyl alpha-D-mannopyranoside (L(3)), 1,3-diamino-2-propyl alpha-D-galactopyranoside (L(4)), 1,3-diamino-2-propyl beta-D-galactopyranoside (L(5)), 1,3-diamino-2-propyl beta-(alpha-D-glucopyranosyl-(1,4)-D-glucopyranoside) (L(6)), and bis(aminomethyl)bis[(beta-D-glucopyranosyloxy)methyl]methane (L(7)). The Re complexes [Re(L(1)-L(7))(Br)(CO)(3)] were characterized by (1)H and (13)C 1D/2D NMR spectroscopy which confirmed the pendant nature of the carbohydrate moieties in solution. Additional characterization was provided by IR spectroscopy, elemental analysis, and mass spectrometry. Two analogues, [Re(L(2))(CO)(3)Br] and [Re(L(3))(CO)(3)Br], were characterized in the solid state by X-ray crystallography and represent the first reported structures of Re organometallic carbohydrate compounds. Conductivity measurements in H(2)O established that the complexes exist as [Re(L(1)-L(7))(H(2)O)(CO)(3)]Br in aqueous conditions. Radiolabelling of L(1)-L(7) with [(99m)Tc(H(2)O)(3)(CO)(3)](+) afforded in high yield compounds of identical character to the Re analogues. The radiolabelled compounds were determined to exhibit high in vitro stability towards ligand exchange in the presence of an excess of either cysteine or histidine over a 24 h period.     

Faster GC analyses performed by flow programming in short capillary columns

The technique of programming the carrier gas flow rate in gas chromatography, especially in connection with the use of capillary columns shorter than 10 m can significantly accelerate GC analyses. Equations for calculation of the parameters of the exponential flow function and retention data are described. The effects of flow programming in a short capillary column are shown in a few chromatograms. Different programming rates are tested and compared with temperature programming.     

Sulfur K-edge XAS and DFT calculations on [Fe4S4]2+ clusters: effects of H-bonding and structural distortion on covalency and spin topology

Sulfur K-edge X-ray absorption spectroscopy of a hydrogen-bonded elongated [Fe4S4]2+ cube is reported. The data show that this synthetic cube is less covalent than a normal compressed cube with no hydrogen bonding. DFT calculations reveal that the observed difference in electronic structure has significant contributions from both the cluster distortion and from hydrogen bonding. The elongated and compressed Fe4S4 structures are found to have different spin topologies (i.e., orientation of the delocalized Fe2S2 subclusters which are antiferromagnetically coupled to each other). It is suggested that the H-bonding interaction with the counterion does not contribute to the cluster elongation. A magneto-structural correlation is developed for the Fe4S4 cube that is used to identify the redox-active Fe2S2 subclusters in active sites of HiPIP and ferredoxin proteins involving these clusters.     

A novel germanate, Cu2Fe2Ge4O13, with a four tetrahedra oligomer

The structure of Cu₂Fe₂Ge₄O₁₃, previously thought to be CuFeGe₂O₆, has been determined from single-crystal X-ray diffraction data to be monoclinic, P2₁/m, a=12.1050(6), b=8.5073(4), c=4.8736(2) Å, β=96.145(1)°, Z=2, with R1=0.0231 and wR2=0.0605. The unique structure has an oligomer of four germanate tetrahedra, cross-linked laterally by square-planar copper ions, joined end-to-end by a zigzag chain of edge-sharing iron oxide octahedra. Running along the a-direction the metal oxide chain consists of alternating Cu-Cu and Fe-Fe dimers. A hypothetical series of homologous structures (Cu_n−2Fe₂Ge_nO_3n+1 with n=3,4,…,∞) with different length germanate oligomers is proposed, where as n increases, the infinite chain of the CuGeO₃ is approached. In this context, Cu₂Fe₂Ge₄O₁₃ is viewed as being built from blocks of CuGeO₃ and the Fe oxide chains. This material has significance to the study of low-dimensional mixed-spin systems.     

An unprecedented uranyl phosphate framework in the structure of [(UO2)3(PO4)O(OH)(H2O)2](H2O)

The new uranyl phosphate [(UO2)3(PO4)O(OH)(H2O)2](H2O) (1) with an unprecedented framework structure has been synthesized at 150 and 185 degrees C. The structure (tetragonal, P4(2)/mbc, a = 14.015(1) A, c = 13.083(2) A, V = 2575.6(4) A(3), Z = 8) contains uranyl phosphate chains composed of uranyl pentagonal and hexagonal bipyramids and phosphate tetrahedra linked by sharing of polyhedral edges. The uranyl phosphate chains are aligned both along [100] and [010] and are linked into a novel framework structure involving channels along [001]. Topologically identical chains occur linked into sheets in more than a dozen uranyl phosphate minerals, but these chains have never been observed in opposing orientations and linked into a framework as in 1.