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May 2012, Volume 17, Issue 9

 FEATURES Functions and the Volume of VasesAnn C. McCoy, Rita H. Barger, Joann Barnett, and Emily CombsWhile filling vases with water and observing volume and height relationships, students learn the fundamentals of functions. Proportioning Cats and RatsKimberly A. MarkworthThis feline-rodent problem helps preservice teachers go beyond the cross-multiplication algorithm to think about proportional relationships. Second Look:Ratio and Proportion Reading Visual RepresentationsRheta N. Rubenstein and Denisse R. ThompsonA tool used in reading theory is adapted to help mathematics teachers ask good questions that help students interpret displays of information. Developing Formal Procedures through Sense MakingJeffrey M. Choppin, Carolyn B. Clancy, and Scott J. KochAllowing students to reason and communicate about integer operations, or any idea, before these ideas are formalized can be an important tool for fostering deep understanding. Talking about the Greek CrossStacy L. Reeder and George E. AbshireMeaningful mathematical discourse occurs when tasks are chosen carefully and the teacher steps back and allows students to move to the forefront of their own learning. Thanks from the Editorial PanelAn annual listing of all the volunteers who serve as reviewers and referees for MTMS. Classified Index, Volume 17, August 2011-May 2012- FREE PREVIEW!Online OnlyThe MTMS 2011-2012 index contains both an author index and a subject category.

 Departments On My MindOn My Mind: Looking at NAEP and the Standards through the Same Lens Informing PracticeInforming Practice: Examples as Tools for Constructing Justifications Cartoon CornerCartoon Corner: Fire Up the Grill Palette of Problems/Menu of ProblemsPalette of Problems - May 2012 Quick ReadsQuick Reads: Word Clouds in Math Classrooms Window on ResourcesWindow on Resources - May 2012 Math for RealMath for Real: What Drives Fuel Economy?

 Second Look - Ratio and Proportion Miniature Toys Introduce Ratio and Proportion with a Real-World ConnectionMiniature plastic toys are ideal models for introducing ratios and proportions. They concretely emphasize the real-world connection of the concepts to objects. Activity included. Problems that Encourage Proportion SenseSeveral problems promoting proportion sense, reasoning about quantities, and the various relationships among quantities in proportions. Using School Lunches to Study ProportionA project in which proportion, ratio, and percent are used to analyze the nutritional value of student lunches. Sample student work included. Illuminations Lesson: Understanding Rational Numbers and ProportionsIn this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions. Number and Operations Standard for Grades 6-8In grades 6–8, students should deepen their understanding of fractions, decimals, percents, and integers, and they should become proficient in using them to solve problems. By solving problems that require multiplicative comparisons (e.g., "How many times as many?" or "How many per?"), students will gain extensive experience with ratios, rates, and percents, which helps form a solid foundation for their understanding of, and facility with, proportionality. The study of rational numbers in the middle grades should build on students' prior knowledge of whole-number concepts and skills and their encounters with fractions, decimals, and percents in lower grades and in everyday life. Students' facility with rational numbers and proportionality can be developed in concert with their study of many topics in the middle-grades curriculum. For example, students can use fractions and decimals to report measurements, to compare survey responses from samples of unequal size, to express probabilities, to indicate scale factors for similarity, and to represent constant rate of change in a problem or slope in a graph of a linear function.