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

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 = 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
TIME: 7 minutes (for Slides 44 to 46)
INTENT: To understand the Standards for Mathematical Practice as outlined in the CCSS
___________________________________________________________________________________
Have participants locate the CCSS for Mathematics handout and turn to the Standards for Mathematical Practice found on pp.1-2. Briefly discuss each of the Standards for Mathematical Practice.
HANDOUTS:
CCSS for Mathematics
California’s Common Core State Standards: Toolkit | Overview

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 tools as 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. units/unitary analysis
“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 )
ratio
proportion
rate
units/unitary analysis
“per”
percent
scale
similarity
slope
parts of a whole
interest
enlargements/ reductions
probability/odds
frequency distributions
motion and speed
comparison

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!