The standard for measuring fuel efficiency in the U.S. has been miles per gallon (mpg). However, the Environmental Protection Agency's (EPA) switch in rating fuel efficiency from miles per gallon to gallons per hundred miles with the 2013 model-year cars leads to interesting and relevant mathematics with real-world connections. By modeling…

Of the most universal applications in integral calculus are those involved with finding volumes of solids of revolution. These profound problems are typically taught with traditional approaches of the disk and shell methods, after which most calculus curriculums will additionally cover arc length and surfaces of revolution. Even in these visibly…

A well-known lesson taught by many upper-elementary and early-middle-school teachers at the beginning of the school year asks students to compare how arm span relates to height. Students measure their height and arm span and compare their measurements to those of their classmates. This lesson gets students measuring, graphing, and practicing…

This article describes how sixth-grade students developed concrete models to solve division of fractions story problems. Students developed separate two-step procedures to solve measurement and partitive problems, drawing on invented procedures for division of whole numbers. Errors also tended to be specific to the type of division problem…

The statistician has n independent estimates of a parameter he knows is positive, but, as is the case in components-of-variance problems, some of the estimates may be negative. If the n estimates are to be combined into a single number, we compare the obvious rule, that of averaging the n values and taking the positive part of the result, with…

The inadequacies in present measurement models are indicated and a description is given of how tree theory, a theory-generative model, overcomes these inadequacies. Among the weaknesses cited in many measurement models are their untested assumptions of linear order and unidimensionality and their inability to generate non-associational…

TO DEVELOP A BETTER MATHEMATICAL BASIS FOR FACTOR ANALYSIS, A MATHEMATICAL APPROACH WAS FORMULATED FOR FIXING COMMUNALITIES AND THEIR COMPLEMENTS, UNIQUENESSES, AS WELL. WHILE THE INVESTIGATOR WAS UNABLE TO PROVE THAT MINIMUM-TRACE CRITERIA FIX COMMUNALITIES UNIQUELY, A SET OF IDENTITIES AND INEQUALITIES WAS LOCATED AMONG THE COMMUNALITIES AND,…

This paper proposes a general model for free-response data collected for measuring a specified unidimensional psychological process; systematizes situations which vary with respect to the scoring level of items; and finds out general conditions for the operating characteristic of an item response category to provide a unique maximum likelihood…

THIS PAPER DEALS WITH SOME TESTS OF HYPOTHESIS FREQUENTLY ENCOUNTERED IN THE ANALYSIS OF MULTIVARIATE DATA. THE TYPE OF HYPOTHESIS CONSIDERED IS THAT WHICH THE STATISTICIAN CAN ANSWER IN THE NEGATIVE OR AFFIRMATIVE. THE DOOLITTLE METHOD MAKES IT POSSIBLE TO EVALUATE THE DETERMINANT OF A MATRIX OF HIGH ORDER, TO SOLVE A MATRIX EQUATION, OR TO…

Discusses the classification of binary oxides as acidic, basic, or amphoteric. Demonstrates how a numerical scale for acidity/basicity of binary oxides can be constructed using thermochemical data for oxoacid salts. Presents the calculations derived from the data that provide the numeric scale values. (TW)