Anne Collins & Linda Dacey 
It’s All Relative [PDF ebook] 
Key Ideas and Common Misconceptions about Ratio and Proportion, Grades 6-7

For years, students have been taught to cross multiply when presented with proportion problems. Few understood what they were doing or why they were doing it. Rarely were students invited to think about various representations to determine the relationship between the two ratios being compared. The ideas and activities in this guide will help you provide students with the conceptual understanding of ratios and proportional thinking that is essential for solving real-world problems and achieving success in higher-level mathematics.It’s All Relative includes thirty modules, which are aligned to specific Common Core State Standards. Each module in this accessible, spiral-bound flipchart addresses specific mathematical practices and common misconceptions; features timely classroom examples; includes proven, ready-to-use activities; and provides ideas for adapting activities to meet the needs of individual learners. The authors particularly emphasize how to use visual models-tables, tape diagrams, graphs, and double number lines-to represent ratios and proportional thinking. These visual models, which may be new to many teachers, enhance students’ conceptual understanding and improve their ability to solve a wide range of proportional reasoning problems.