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.     

Radiation-induced grafting of hexafluoroacetone to polyethylene

Although hexafluoroacetone is not polymerized by ionizing radiation, it is shown that γ-irradiation of hexafluoroacetone dissolved in polyethylene films produces a graft with a G value of 500 and, therefore, a kinetic chain length of 200. The effects of dose rate (0.021–3.55 Mrad/hr), temperature (21–53°C), and pressure (1.5–6.2 atm) on the graft rates have been measured. Also the effect of temperature (21–53°C) on the postirradiation grafting reaction and on the physical properties of the grafted films have been investigated. Together with solubility, diffusivity, infrared, and EPR data, the results lead to the following mechanism: The first step represents production of secondary alkyl radicals in the polyethylene by irradiation of the polymer–monomer system. The second step involves the linkage of the monomer to the radical site to form the alkoxy radical. Since it cannot add to another monomer unit, this radical abstracts a hydrogen atom from an adjacent polyethylene chain in the third step. Radical R· can then continue the kinetic chain. Radical combination and radical–impurity reactions terminate the chain. The graft may be unique in that it is the only one we have found in which a pendant group containing only one monomer unit is attached by a chain reaction. At dose rates up to 0.215 Mrad/hr, the grafting was linear with time and proportional to the 0.73 power of the dose rate at 21°C and to the 0.81 power at 53°C. The reaction is insensitive to increases in dose rate above 0.215 Mrad/hr where diffusivity measurements show the reaction to be diffusion-controlled. The rate of reaction increased 10% when the temperature was increased from 21 to 53°C. While there was significant postirradiation grafting reaction at 21°C, there was none at 53°C. The results do not fit the equations of reaction-controlled steady-state graft-polymerization kinetics. The deviations arise from an observed increase in monomer solubility in the film with increasing graft combined with low diffusivity of the monomer in polyethylene, and the presence of a radical-scavenging impurity which terminates the kinetic chain with the appearance of a relatively stable radical. EPR data suggests that the impurity is a trace of oxygen which may be produced radiolytically.     

Hypoxanthine,xanthine and theobromine complexes with palladium(II) and platinum(IV) chlorides

Summary Complexes of the general types Pd(L)(LH)Cl (LH=hxH, xnH, or tbH) and Pt(L)(LH)Cl₃ (LH=hxH, or xnH) are formed by boiling under reflux 21 molar mixtures of hypoxanthine (hxH), xanthine (xnH) or theobromine (tbH) and PdCl₂ or PtCl₄ in ethanol-triethyl orthoformate. These complexes appear to be linear chain polymeric species, characterized by single monoanionic L^– ligands bridging between adjacent Pd²⁺ or Pt⁴⁺ ions. Inclusion of one terminal neutral LH and one terminal chloro-ligand completes the coordination sphere in the square-planar Pd²⁺ complexes, while the Pt⁴⁺ complexes aretrans-octahedral, involving three terminal chloro and one terminal LH ligand per platinum. The possible binding sites of the bidentate bridging L^– and the unidentate terminal LH are discussed.     

Synthesis and antiinflammatory activity of the sulfoxides of 4-[3-(dimethylamino)propyl] - 3,4- dihydro- 2-(1-hydroxyethyl)-3-phenyl-2H-1,4-benzothiazine

Oxidation of 4-[3-(dimethylamino)propyl]-3,4-dihydro-2-(1-hydroxyethyl)-3-phenyl-2H-1,4-benzothiazine, hydrochloride (I) with hydrogen peroxide yielded a mixture of two sulfoxides (II). Since this mixture exhibited antiinflammatory activity, the two components (Isomers A and B) were prepared in purified form by oxidation of I with N-chlorosuccinimide and hydrogen peroxide, respectively. Isomer A was more potent than Isomer B in the carrageenin-induced edema test.     

Engineered protein cages for nanomaterial synthesis

Self-assembled particles of genetically engineered human L subunit ferritin expressing a silver-binding peptide were used as nanocontainers for the synthesis of silver nanoparticles. The inner cavity of the self-assembled protein cage displays a dodecapeptide that is capable of reducing silver ions to metallic silver. This chimeric protein cage when incubated in the presence of silver nitrate exhibits the growth of a silver nanocrystal within its cavity. Our studies indicate that it is possible to design chimeric cages, using specific peptide templates, for the growth of other inorganic nanoparticles.     

Diethyl benzoylphosphonate as a ligand

Summary Complexes (2 : 1) of diethyl benzoylphosphonate (debp) with 3d metal perchlorates were synthesized and characterized by means of i.r. and electronic spectral, magnetic susceptibility and conductance measurements. In new complexes of the types [M(debp)₂(OClO₃)(OH₂)](ClO₄) (M = Fe, Co, Zn) and [Fe(debp)₂(OClO₃)(OH₂)](ClO₄)₂, both debp ligands function as bidentate chelating agents, coordinating through the P=O and C=O oxygens. In contrast, in the manganese(II) and nickel(II) complexes, which are of the [M(debp)₂(OClO₃)(OH₂)₂](ClO₄) type, one debp acts as a bidentate chelating ligand, while the second debp is unidentate, coordinating only through the P=O oxygen. Hexacoordination in the new cationic complexes is completed by coordination of aqua and unidentate perchlorato ligands, which are in competition for sites in the inner coordination sphere of the central metal ion with the weak debp ligand. On the other hand, debp, owing to its bulkiness, and especially the presence of the benzoyl substituent, introduces sufficiently severe steric hindrance during coordination. As a result of this, the formation of [M(debp)₃]ⁿ⁺ tris-chelate cationic complexes with the 3 d metal ions under study does not seem to be possible.     

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.     

Comparison by capillary SFC and SFC-MS of soxhlet and supercritical fluid extraction of hamster feces

Components of hamster feces ranging from low molecular weight fatty acids through the expected range of triglycerides have been eluted in a single SFC run with simultaneous pressure and temperature programming. Temperature programming from 140°C to 240°C was required to provide optimum conditions for separation of the fatty acids and to move the elution region of the sterol esters away from that of the triglycerides. Data from chemical ionization and electron impact mass spectrometry of compounds separated by SFC were used to confirm identities suggested by retention measurements and to provide tentative identities of unknown compounds. SFC with flame ionization detection was used to compare Soxhlet extraction, off-line supercritical fluid extraction (SFE), and on-line SFE of the feces. Although samples obtained by Soxhlet extraction and SFE produced very similar chromatograms, SFE required far less time and consumed much smaller quantities of organic solvent.     

Effect of phase ratio on van't Hoff analysis in reversed-phase liquid chromatography,and phase-ratio-independent estimation of transfer enthalpy

In analysis of the thermodynamics of the transfer of a solute from the mobile phase to the stationary phase in reversed-phase liquid chromatography, it is nearly always assumed that the phase ratio is constant. This type of analysis is typically performed by applying a form of the van't Hoff equation, which relates the retention factor to temperature via the enthalpy and entropy of transfer. When non-linear van't Hoff plots are observed, it is often assumed that the enthalpy and entropy of transfer change with temperature. However, when the possibility of a change in the phase ratio is considered, it becomes apparent that non-linear van't Hoff behavior may or may not be due to changes in enthalpy or entropy. In this work, we present mathematical evidence that phase ratio changes, if they occur, can cause deviations from linearity in a van't Hoff plot. We also show that the phase ratio influence can be eliminated by considering the molecular difference between two solutes instead of the solutes themselves. The resulting selectivity van't Hoff plots may be linear, even when the van't Hoff plots of the two solutes are non-linear. In such cases, temperature-dependent phase ratio changes, and not necessarily changes in the transfer enthalpy, may be responsible for the curved van't Hoff plots of the individual solutes. In addition, we present chromatographic evidence that different solutes may "see" different thermodynamic phase ratios. It is clear that the concept of a phase ratio in reversed-phase chromatography is not nearly as well defined as a phase ratio in a bulk system like a liquid-liquid extraction.