Student errors are springboards for analyzing, reasoning, and justifying. The mathematics education community recognizes the value of student errors, noting that "mistakes are seen not as dead ends but rather as potential avenues for learning." To induce specific errors and help students learn, choose tasks that might produce mistakes.…

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…

One challenge that middle school teachers commonly face is finding insightful, hands-on applications when teaching basic mathematical concepts. One concept that is a foundation of middle school mathematics is the notion of "linear functions." Although a variety of models can be used for linear equations, such as temperature conversions,…

Presents solutions at several different levels of sophistication for a problem involving 10 stools arranged in a row at a diner: how many different ways can the ten stools be occupied, discounting which of three persons sit on any stool, so that at least one stool is between two people? (AIM)

Describes a series of lessons in which 6th grade students explore notions of rates of change and their effect on the shapes of graphs. Addresses aspects of the algebra content standard for the middle grades. (YDS)

Explores the mathematics of flight. Also contains a student worksheet. (YDS)

Describes the use of mathematics skills in daily life using a home renovation business as an example. (YDS)

Presents activity pages containing world population data for students to use to predict future population figures and Consumer Price Index (CPI) data to track changes in the costs of the goods. Includes answer key. (KHR)

Explains the approach of integrating notions of algebra in problem-solving experiences and the design of instructional activities that integrate nonroutine, nontraditional problems. Presents some problems and students' interpretations, conjectures, and generalizations about the patterns they discovered. (AIM)

Explores why it is impossible to make a square from six tangram pieces and, in the process, develops insight into the nature of learning mathematics as well as strategies for teaching. (YDS)

Describes middle school students' efforts to investigate and fairly compensate the three bears, who incurred some losses at the hands of Goldilocks. Problems include investigating replacement costs of the damaged items, the cost to house Goldilocks in a state school, and the value of appropriate types of community service before determining…

Discusses mathematical thinking and problems posed by seventh- grade students as part of a unit on the study of algebraic ideas. The unit involves showing how variables cause changes and also incorporates the solving of systems of equations by constructing line-graph drawings and combination charts. (AIM)

Examines the definition of basic mathematics skills in a discussion of meeting the needs of mathematics students in a changing society. One basic skill is understanding formulas that are used in real life. Challenges teachers to teach students to reason mathematically, improve communication, and build mathematical connections with real-life…

Presents an activity through which students develop mathematical concepts connected to area and perimeter in a rich, real-life setting and use tables, graphs, diagrams, and spreadsheets to explore the problem, see relationships, and make and test conjectures about those relationships. Encourages algebraic thinking. (SYB)

Describes a Food-Choice Unit that includes the following activities: drawing circle graphs, graphing dietary content, understanding heart attacks, reducing dietary fat, charting fat content of restaurant foods, and calculating percent of fat. Includes extension activities. (CSCO)