About Course
Ratio and Proportional Relationships.
Ratios and proportional relationships are essential to mathematics and to all of science, and they are useful in daily life. They are a foundation for understanding rate of change, slope, linear-relationships, and other relationships in mathematics and science. In this chapter, we will motivate and define the concepts of ratio and proportional relationship. We will then see ways of reasoning to solve proportion problems with the aid of tables, graphs, double number lines, and strip diagrams. We will see how unit rates arise from ratios, how they connect ratios to fractions, and how they are behind the common cross-multiplying method for solving proportions. We will see that graphs of proportional relationships are of a special type and we will develop equations for proportional relationships by reasoning quantitatively. We will distinguish proportional relationships from other kinds of relationships, including inversely proportional relationships. Finally, we will study percent increase and decrease.
We focus on the following topics and practices within the Common Core State Standards for Mathematics.
Standards for Mathematical Content in the CCSSM
In the domain of Ratio and Proportional Relationships (Grades 6 and 7),students learn ratio concepts, including the concept of unit rate associated with a ratio, and they use ratio language, such as “3 cups flour to 2 cups water” and “3 cups flour for every 2 cups water.” They reason about ratios and rates with the aid of tables, double number lines, and strip diagrams to solve problems. They analyze and graph proportional relationships and they distinguish them from other kinds of relationships. They also use proportional relationships to solve multistep percent problems, such as problems involving percent increase and decrease.
Standards for Mathematical Practice in the CCSSM
Opportunities to engage in all eight of the Standards for Mathematical Practice described in the Common Core State Standards occur throughout the study of ratio and proportional relationships, although the following standards may be especially appropriate for emphasis:
• 2 Reason abstractly and quantitatively. Students engage in this practice when they use a ratio to describe a quality that mixtures or other related quantities have in common, and when they recognize that a single ratio can apply to both small and large amounts of the mixture or the related quantities.
• 4 Model with mathematics. Students engage in this practice when they use ratios and proportional relationships to model situations and when they examine relationships critically to determine if they are proportional or not and why or why not.
• 5 Use appropriate tools strategically. Students engage in this practice when they make and reason logically about ratio tables, double number lines, and strip diagrams as they solve problems involving ratios and proportional relationships.
(From Common Core Standards for Mathematical Practice. Published by Common Core Standards Initiative.)
