In many textbooks of chemical-engineering thermodynamics, a gas mixture obeying the fundamental law pV[subscript m] = RT is most often called ideal-gas mixture (in some rare cases, the term perfect-gas mixture can be found). These textbooks also define the fundamental concept of ideal solution which in theory, can be applied indifferently to…

In classroom science laboratories, unlike a real science laboratory, the teacher can guide students away from potential dead ends and toward data that are most likely to result in accurate conclusions. Sometimes, though, allowing students to pursue dead ends and to collect "bad" data can provide especially rich learning opportunities.…

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

The desire to persuade students to avoid strictly memorizing formulas is a recurring theme throughout discussions of curriculum and problem solving. In combinatorics, a branch of discrete mathematics, problems can be easy to write--identify a few categories, add a few restrictions, specify an outcome--yet extremely challenging to solve. A lesson…

Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…

Mr. Perez and Mrs. Peterson co-teach a ninth-grade algebra class. Perez and Peterson's class includes four students with individualized education programs (IEPs). In response to legislation, such as the No Child Left Behind (NCLB) Act (2001) and the Individuals with Disabilities Education Improvement Act (2006), an increasing number of students…

By the time they reach middle school, all students have been taught to add fractions. However, not all have "learned" to add fractions. The common mistake in adding fractions is to report that a/b + c/d is equal to (a + c)/(b + d). It is certainly necessary to correct this mistake when a student makes it. However, this occasion also…

What if you could challenge your 11th graders to figure out the best response to a partial meltdown at a nuclear reactor in fictional Gammatown, USA? With this volume in the "STEM Road Map Curriculum Series," you can! "Radioactivity" outlines a journey that will steer your students toward authentic problem solving while…

Automated theorem proving (ATP) for Propositional Classical Logic is an algorithm to check the validity of a formula. It is a very well-known problem which is decidable but co-NP-complete. There are many algorithms for this problem. In this paper, an educationally oriented implementation of Semantic Tableaux method is described. The program has…

Many students have trouble grasping algebra. In this book, bestselling authors Judith, Gary, and Erin Muschla offer help for math teachers who must instruct their students (even those who are struggling) about the complexities of algebra. In simple terms, the authors outline 150 classroom-tested lessons, focused on those concepts often most…

"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

The Chebyshev polynomials named after a Russian mathematician, Pafnuty Lvovich Chebyshev, have various mathematical applications. A process for obtaining Chebyshev polynomials, and a mathematical inquiry into the patterns they generate, is presented.

Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…

In light of evidence indicating that errors in making accurate pie charts are prevalent, an alternative method for constructing pie charts is proposed. It involves the use of a simple trigonometric formula to compute a factor of the length of the chord that defines a given percentage sector. This factor is multiplied by the radius of the circle to…

Presents calculations after discussing exponential growth that deal with the determination of the time of exhaustion at different rates of consumption. (PR)