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
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
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
Table Introductions
Give each person a 20-second “spotlight” to share:
 Name
 Grade / Courses / Role
 Location
 Personal number
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.
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)
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 & Doris
 Explore similarity by drawing figures using a coordinate

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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
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.
“Teaching concepts within a context has the
advantages of:
1) Piquing students’ interest
2) Stimulating their imaginations
3) Giving functional mathematics knowledge
useful in applications.”
(Huetinck & Munshin, 2008)
Standards for
Mathematical Practice – Habits of Mind
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning
of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
12
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.
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
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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
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
6th Domain: Ratios and Proportional Relationships
 Understand 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 rate
3. 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 Relationships
 Analyze proportional relationships and use them
to solve real-world and mathematical problems.
Compute unit rates…
2. Recognize and represent proportional relationships
between quantities.
1.
a.
b.
c.
d.
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, …
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 nonvertical line in the coordinate plane: derive the
equation y = mx for a line through the origin ….
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
“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:
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
(Summac Forum, Dana Center )
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!