Proportions

Research says...

 * If you start with cross products, students never really get a sense of what proportions are.
 * Using cross products before learning scaling up and scaling down can be detrimental.
 * Expose students to such a range of problem situations that they become very flexible and willing to persevere.

=Facts about Ratios and Proportions...= =Resources= NCTM Book - Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning (Grades 6-8) AMSTI Units OGAP Proportional Reasoning Framework and Item Bank Tape Diagrams Video - The real meaning of MPH
 * What distinguishes a proportional relationship from a non-proportional one is that it is multiplicative throughout.
 * A fraction is one quantity that has a unique place on the number line. Fractions are always part to whole. A ratio is a comparison of two or more quantities (part to whole, part to part, part to part to part)
 * Start a ratios teaching unit by comparing to fractions and how they are different. Avoid writing ratios as fractions.
 * Use a double number line to display two different quantities.
 * Comparing and Scaling
 * Stretching and Shrinking
 * Go to OGAP website, click on OGAP Projects, scroll down to General Information to download Framework

6th Grade
1.Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities. [6-RP1] 2.Understand the concept of a unit rate associated with a ratio //a//://b// with //b// ≠ 0, and use rate language in the context of a ratio relationship. [6-RP2] 3.Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. [6-RP3]
 * Understand ratio concepts and use ratio reasoning to solve problems. **
 * a.Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. [6-RP3a]
 * b.Solve unit rate problems including those involving unit pricing and constant speed. [6-RP3b]
 *  c.Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means times the quantity); solve problems involving finding the whole, given a part and the percent. [6-RP3c]
 * d.Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. [6-RP3d]

7th Grade
1.Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. [7-RP1] 2.Recognize and represent proportional relationships between quantities. [7-RP2]  3.Use proportional relationships to solve multistep ratio and percent problems. [7-RP3]
 * Analyze proportional relationships and use them to solve real-world and mathematical problems. **
 * a.Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. [7-RP2a]
 * b.Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. [7-RP2b]
 * c.Represent proportional relationships by equations. [7-RP2c]
 * <span style="color: #000000; font-family: 'Times New Roman','serif'; font-size: 14.6667px;"> d.Explain what a point (//x, y//)on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, //r//)where //r// is the unit rate. [7-RP2d]