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 Conference2013

Vicki Vierra vvierra@vcoe.org

Agenda

Welcome & IntroductionsAlgebra Estimation JarIs It Proportional? Math Practice #4 Model with MathematicsMorris 1, 2, 3, Boris and DorisProportional Reasoning Problem TypesRectangle RatiosAdditional Resources

Take Off, Touch DownIf the description pertains to you, Take Off.

See who else shares your characteristic and Touch Down

K - 5th grade6th grade7th grade8th gradeHSAdvanced LearnersEnglish LearnersStruggling problem solvers

Table Introductions

Give each person a 20-second “spotlight” to share:

NameGrade / Courses / RoleLocationPersonal number

Algebra Estimation JarThere are 3 different items in the jar:

Marshmallows, Goldfish and PretzelsThere are three times as many pretzels as

goldfishThere 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.

Typical Textbook Treatment of ProportionalityDefines 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)

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)

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

Morris 1, 2, 3, Boris & DorisExplore similarity by drawing figures using a

coordinate systemSimilarity 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 orderFollowing the rule, generate the points for Morris

2, 3, Boris and Doris. Complete the table for Summary of Morris’s Noses

Proportional Reasoning Problem Types3 types of proportional reasoning problems

Missing value – given three values, find the fourthNumerical 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.

“Teaching concepts within a context has the advantages of:

1) Piquing students’ interest2) Stimulating their imaginations3) Giving functional mathematics

knowledge useful in applications.” (Huetinck & Munshin, 2008)

Standards for

Mathematical Practice – Habits of Mind

12

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning

of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

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 situationReflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Math Practice #4

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 familyConsider nesting, graphing, and finding equivalent

ratios of sides. Arrange each “family” from least to greatestWhat patterns do you see within a “family”?Stack each family of rectangles, sharing the lower left

cornerComplete the length and width chart

Which Common Core Content Standards are embedded in: Algebra Estimation JarIs It Proportional?Morris 1, 2, 3, Boris & DorisProportional Reasoning Problem TypesRectangle Ratios

6th Domain: Ratios and Proportional RelationshipsUnderstand ratio concepts and use ration reasoning to solve problems.1.Understand the concept of a ratio and use ratio language to describe a ration relationship between two quantities.2.Understand the concept of a unit rate3.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.

7th Domain: Ratios and Proportional RelationshipsAnalyze proportional relationships and use them to solve real-world and mathematical problems. 1.Compute unit rates…2.Recognize and represent proportional relationships between quantities.

a. 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

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. d. Explain what a point (x, y) on the graph of a proportional

relationship means in terms of the situation, …

8th Domain: Expressions and EquationsUnderstand 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 ….

Additional ResourcesCat Food + Folded SquarePizza Place + student work Candy Jar + Bag of MarblesProportionality Problems Worksheet &

Questions (Cookie Problem)MS Math Short Tasks: Ratios and

Proportional Relationships + Answers

“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

Closing: Tell your elbow partnerHow 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!