GOAL:
The purpose of these lessons is to develop, analyze, explain and
use methods for solving proportions while solving problems in
a variety of situations.WHY: Standard 7, "Computation and Estimations," from the National Council of Teachers of Mathematics Curriculum and Evaluation Standards (1989) states: In grades 5-8, the mathematics curriculum should develop the concepts underlying computation and estimation in various contexts so that students can develop, analyze, and explain methods for solving proportions; select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods; use computation, estimation, and proportions to solve problems; use estimation to check the reasonableness of results. ( pg. 94)The Addenda Series publication "Understanding Rational Numbers and Proportions" (NCTM, 1994) addresses the issue more: A proportion is a mathematical statement expressing the equality of two ratios. Many mathematics students equate the study of proportions with the cross multiply rule. However, the ability to apply this rule is not an accurate indicator of students' understanding of proportions or their ability to recognize a proportional relationship. Understanding proportional relationships requires a deep, complex system of knowledge about a particular quantitative relationship that permeates the world around us. As a result, students' ability to reason using proportional relationships develops over time.... Presenting meaningful, alternative methods to the cross multiply rule strengthens connections between equivalent fractions, unit rates, and proportions. (pg. 3-4)The importance of students developing a strong rational-number-and-proportion foundation for maximizing career options cannot be overstated. Carefully planned instruction that builds on students' natural language and informal notions of fractions, ratios, decimals, percents, and proportions will help learners become quantitatively literate citizens as they apply concepts of rational numbers and proportions in their daily living. (pg. 1)
WHAT: Proportional thinking involves the ability to understand and compare ratios, and to predict and produce equivalent ratios. It requires comparisons between quantities and also between the relationships between quantities. It involves quantitative thinking as well as qualitative thinking. It is not dependent on a skill with a mechanical or algorithmic procedure. Therefore, this material does not simply show the cross multiply method and assume students can do proportional thinking. Time is spent exploring proportions from a variety of perspectives: equivalent fractions, multiplication by unit rates, graphing, as well as the cross multiply method. A feature of proportional situations is the multiplicative relationship among the quantities. The relationship is explored with tables, graphs, and equivalent fractions.
HOW: This topic consists of six lessons, a 14 minute video, and an extension activity. The last page of the teacher section provides examples of proportional reduction of situations with large numbers into situations with more comprehensible numbers. The video should be shown several times: after doing Lesson 1 to motivate students and introduce the material, throughout the lessons to examine individual segments, and for closure or assessment. All pages of the print do not need to be copied for all students. Many lessons can be limited to one classroom set or one per group if cooperative groups are used. Some lesson sheets can be used as classroom guides. Key terms are in bold type within the context of the lessons. Many of the lessons involve measurements which are better done with teams of students. This set of lessons, "Proportions: Expressing Relationships," is the second in a set of four from the Math Vantage Proportional Reasoning Unit. The first topic, "Rates and Ratios: Comparisons," focuses on creating and using rates and ratios. The third set of lessons in the Proportional Reasoning Unit also addresses proportions; however, it is concerned more with proportions used in scale modeling. The fourth set of lessons focuses on percents. Lesson 1 of this set of lessons builds on students' knowledge about equivalent fractions and their multiplicative sense for proportions. Proportions are defined. Lesson 2 continues to approach proportional reasoning using statements of equivalent fractions and multiplication by unitary rates. Lesson 3 uses student interest in bicycle gear ratios to practice creating and using ratios and proportions. In Lesson 4, students are asked to represent proportional situations in diagrams. Lesson 5 introduces the cross multiplication procedure and sampling techniques. It is important that students realize not all number patterns represent proportions. Lesson 6 asks students to gather data from different situations and describe them as proportional or not proportional. The extension activity compares direct and inverse relationships and also presents inverse relationships that are inverse proportions.