1. Ratios & Proportional Relationships
Progressions Highlights

2. Overview Changes from current practice
Connections to previous and subsequent grades
Surprises
Conclusion

3. Why should you listen to our presentation?

4. Changes: Not solving proportional equations in 6th grade
In sixth grade, the focus is on equivalent ratios, not proportions and solving proportions. The latter is best saved as the culmination of a lot of productive struggle in the sixth grade.

5. Changes: Not Cross-multiplying until seventh grade
Timmy: “Mommy, remember how last year the bullies in seventh grade kicked sand in my face and told me I couldn’t cross-multiply? Well, this year I can cross-multiply AND represent equivalent ratios as equations in two variables with a constant of proportionality! Also, today is stand up for gingers day so everybody was nice to me!”

6. Visual Models: Tape Diagram
This diagram represents a mixture of apple and grape juice with a ratio of 3 to 2.

7. Visual Models: Equivalent Ratios Vs. Equivalent Fractions

8. Visual Models: Double Number Line

9. How the ideas grow from grade to grade: Where they are coming from
Use the multiplication table to focus on situations that involve rows and columns. Focus on the rows and columns of a ratio table as multiples of each other
Skip counting to extend tables and fill in missing entries
Fraction bar representation introduced in elementary school

10. How the ideas grow from grade to grade: Where they are going
Direct variation extends into 8th grade
Apply proportions in context for scale drawings
Foundation for study of functions, specifically linear functions
Extend understanding of quantity to new units like miles per hour instead of 3 miles in one hour

11. Surprises, Part I

12. Surprises Part II
Introduce constant of proportionality in 7th grade (as opposed to 8th)
The idea of a ratio is distinguished from a fraction, all terms have precise definitions which may be different from definitions used by certain textbooks
Tape diagrams and double number line - new terminology and alternate representations of ratios

13. Conclusion
The focus for teaching ratios and proportionss in 6th grade is visual and intuitive, rather than computational.
The focus in 7th grade is to represent proportional relationships with equations with visual represen- tations of graphs and tables.
These strategies seem practical for students and help them to develop number sense and reasoning skills more than the current strategies used.