Power the Transformation with Proportional Reasoning

1. Power the Transformation with Proportional Reasoning Begin by developing proportional reasoning with connections to equivalent fractions, coordinate graphing, function tables and geometric similarity. CMC Annual Conference 2013 Vicki Vierra vvierra@vcoe.org

2. Agenda • Welcome & Introductions • Algebra Estimation Jar • Is It Proportional? • Math Practice #4 Model with Mathematics • Morris 1, 2, 3, Boris and Doris • Proportional Reasoning Problem Types • Rectangle Ratios • Additional Resources

3. Take Off, Touch Down • If the description pertains to you, Take Off. • See who else shares your characteristic and Touch Down • K - 5th grade • 6th grade • 7th grade • 8th grade • HS • Advanced Learners • English Learners • Struggling problem solvers

4. Table Introductions Give each person a 20-second “spotlight” to share: • Name • Grade / Courses / Role • Location • Personal number

5. Algebra Estimation Jar • There are 3 different items in the jar: Marshmallows, Goldfish and Pretzels • There are three times as many pretzels as goldfish • There are two times as many marshmallows as goldfish. • Estimate how many of each item are in the jar. • Write your strategy and estimate on a piece of paper with your name.

6. Typical Textbook Treatment of Proportionality • Defines ratio, rate, and proportion • States that if a/b = c/d, then ad = bc, without proof or explanation • Provides some empirical evidence, e.g., 2/5 = 4/10 2 ∙ 10 = 4 ∙ 5 • Students do problems using a proportion (percent, maps, scale drawings, similar figures, mixtures) (Goldstein, 2008)

7. Proportional Reasoning includes: • Multiplication, Division and the inverse relation between the two; • The need for non-integer values (fraction, decimal, and percent representations); • Ratio and rate; • Proportion as a tool for solving problems; • Linear functions in the form y = kx, where k is the constant of proportionality (the graph of which is a line through the origin.) (Goldstein, 2008)

8. Never Tell An Answer Please remember the enormous responsibility we all have as learners not to spoil anybody else’s fun. The quickest way to spoil someone else’s fun is to tell them an answer before they have a chance to discover it themselves. Susan Pirie

9. Morris 1, 2, 3, Boris & Doris • Explore similarity by drawing figures using a coordinate system • Similarity is intuitively defined as being “the same shape.” • What attributes of similar figures are preserved when all the lengths are multiplied by a constant? • What happens to the area? • Plot Morris 1 on the grid paper, connecting the points in order • Following the rule, generate the points for Morris 2, 3, Boris and Doris. • Complete the table for Summary of Morris’s Noses

10. Proportional Reasoning Problem Types • 3 types of proportional reasoning problems • Missing value – given three values, find the fourth • Numerical comparison – Which of two given values represent more or less. • Qualitative comparison – Evaluate the effect on a ratio of a qualitative change on one or both of the quantities • Sort/label the proportion problems according to these three types. • Compare with a partner.

11. “Teaching concepts within a context has the advantages of: • Piquing students’ interest • Stimulating their imaginations • Giving functional mathematics knowledge useful in applications.” (Huetinck & Munshin, 2008)

12. Standards for Mathematical Practice – Habits of Mind Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning

13. Math Practice #4 Model with mathematics: • Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. • In early grades, … as simple as writing an addition equation to describe a situation. • In middle grades, … apply proportional reasoning to plan a school event or analyze a problem in the community. • By high school, … use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. • Comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. • Able to identify important quantities in a practical situation and map their relationships using such toolsas diagrams, two-way tables, graphs, flowcharts and formulas. • They can analyze those relationships mathematically to draw conclusions. • They routinely interpret their mathematical results in the context of the situation • Reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

14. Rectangle Ratios • An introduction to the proportionality of similar figures. • Sort the rectangles into “families” of similar shape and find the common characteristics of each family • Consider nesting, graphing, and finding equivalent ratios of sides. • Arrange each “family” from least to greatest • What patterns do you see within a “family”? • Stack each family of rectangles, sharing the lower left corner • Complete the length and width chart

15. Which Common Core Content Standards are embedded in: • Algebra Estimation Jar • Is It Proportional? • Morris 1, 2, 3, Boris & Doris • Proportional Reasoning Problem Types • Rectangle Ratios

16. 6th Domain: Ratios and Proportional Relationships • Understand ratio concepts and use ration reasoning to solve problems. • Understand the concept of a ratio and use ratio language to describe a ration relationship between two quantities. • Understand the concept of a unit rate • 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.

17. 7th Domain: Ratios and Proportional Relationships • Analyze proportional relationships and use them to solve real-world and mathematical problems. • Compute unit rates… • Recognize and represent proportional relationships between quantities. • Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent rations in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin • Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. • Represent proportional relationships by equations. • Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, …

18. 8th Domain: Expressions and Equations • Understand the connections between proportional relationships, lines, and linear equations. • Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. • Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane: derive the equation y = mx for a line through the origin ….

19. Additional Resources • Cat Food + Folded Square • Pizza Place + student work • Candy Jar + Bag of Marbles • Proportionality Problems Worksheet & Questions (Cookie Problem) • MS Math Short Tasks: Ratios and Proportional Relationships + Answers

20. “Properly presented, the concept of proportionality can truly serve to relate and clarify many other topics typically studied more or less separately in math courses: (Summac Forum, Dana Center )

21. Closing: Tell your elbow partner • How will you introduce proportional reasoning in your classroom? • How will you promote students’ access to powerful mathematics through student talk • Thank you for your participation! Please complete the evaluation for Session #752 • See you next year!