Ratio and Proportion PD Template

RATIO, PROPORTION, AND PROPORTIONAL REASONING
Essential understandings for teachers and students.
ProfessionalDevelopment Guide

E S S E NTIA L U NDE R S TA ND INGS F OR TE A CH E R S A N D S TU DE NTS :
Ratio, Proportion, and Proportional
Reasoning
Northbrook Academy Mathematics Department

Table of Contents
Introduction: Goals, background
information & context i
S E S S I O N 1
Introduction 1
Additive Versus Multiplicative Thinking 2
Composing the Unit 3
References 4
S E S S I O N 2
Ratios, Fractions and Proportions 5
Complexity of Proportional Reasoning 6
References 7
S E S S I O N 3
Steepness of a Ramp. Rates 8
Evaluating Teaching. Superficial Cues 9
Closing 10
REFLECTION 11
EVALUATION 13
REFERENCES 16
APPENDIX: 18
I C O N K E Y
 Multimedia
 Discussion
 Analyze Pupil Data
 Problem Solving
 Record Responses
Key Ideas

Introduction
Ratio, proportion, and proportional reasoning
are mathematics concepts that link a wide
array of mathematical topics together. Some
topics are those already encountered in earlier
grades, such as expanding (writing fractions in
higher terms) and reducing fractions and basic
fraction/decimal operations, while other
topics are those that will be encountered later
in mathematics, such as slope, linear
equations, direct variation, trigonometry, and
similarity in geometry. The use of
proportional reasoning is a form of
comparison that requires multiplicative
thinking. These comparisons are expressed as
ratios and rates (Chapin & Johnson, 2006).
Many everyday situations require rates and
ratios: The train is traveling at 70 mph,
pepperoni costs $4.69/lb, 4 out of 5 dentists
recommend trident sugarless gum,
chance of rain is 35%.
It is estimated that over half the adult
population do not reason proportionally
(Lamon, 1999).
Furthermore, it is estimated that over 90% of
students entering high school do not reason
well enough to learn high school mathematics
and science with understanding (Lamon,
1999). With respect to the middle school
grades, a short unit on ratios in grade 6 is
insufficient for sound learning. Students need
to be able to explore multiplicative thinking
throughout elementary grades and work on
developing the concept of multiplicative
thinking in the middleschool years (Chapin &
Johnson, 2006). Furthermore, the concept of
proportionality “is an important integrative
thread that connects many of the mathematics
topics studied in grades 6 – 8” (NCTM, 2000,
p. 217).
Goals
The main goal of the professional
development program will focus on teacher
learning by providing “opportunities for
teachers to build their content and
pedagogical content knowledge and examine
[their] practice” (Loucks-Horsley, Love, Stiles,
Mundry, & Hewson, 2003). Much of the
literature on ratio, proportion, and
proportional reasoning has demonstrated the

need for teaching of more multiplicative
thinking throughout the middle grades
(Lobato et al, 2010; Lo et al, 2004; Chapin &
Johnson, 2006; NCTM, 2000). There has
been much debate over rates versus ratios,
particularly with respect to whether
considering same or different units
(Thompson, 1994; Lobato et al, 2010). Many
questions are raised with respect to methods
used to solve proportions and the extent to
which they show a deep understanding of
ratio and proportion (Kaput & West, 1994;
Chapin & Johnson, 2006; Frudenthal, 1973,
1978 as cited by Lamon, 2007; Lobato et al,
2010). It is through this program that
teachers will have an opportunityto gain a
profound understanding of foundational
mathematics (Ma, 1999) with respect to the
topics of ratio, proportional, and proportional
reasoning. Many concepts that are connected
with these big ideas connect from content in
the elementary grades as well as connect to
content in the secondary grades beyond
middle school.
It is the goal of this professional
development program to provide teachers of
mathematics with the framework
proposed by Loucks-Horsley et al (2010).
The program will harbor a commitment to
vision and standards, the opportunity to
analyze student learning and other data, and
to set instructional goals in their teaching of
ratio, proportion and proportional reasoning.
It is my hope that the teachers will be able to
plan lessons based on some activities in the
program, carry them out in their own classes,
and evaluate the results for future planning.
Background
Information
The content will focus on ten
essential understandings of ratio, proportion,
and proportional reasoning (Lobato et al,
2010). The essential understandings are as
follows: 1.) Reasoning with ratios involves
attending to and coordinating two quantities;
2.) A ratio is a multiplicative comparison of
two quantities, or it is a joining of two
quantities in a composed unit; 3.) forming a
ratio as a measure of a real-world attribute
involves isolating that attribute from other
attributes and understanding the effect of
changing each quantity on the attribute of
interest; 4.) A number of mathematical

connections link ratios and fractions; 5.)
ratios can be meaningfully reinterpreted as
quotients; 6.) A proportion is a relationship
of equality between two ratios. In proportion,
the ratio of two quantities remains constant as
the corresponding values of the quantities
change; 7.) proportional reasoning is complex
and a.) involves understanding that equivalent
ratios can be created by iterating and/or
partitioning a composed unit; b.) If one
quantity in a ratio is multiplied or divided by a
particular factor, then the other quantity must
be multiplied or divided by the same factor to
maintain the proportional relationship; c.) The
two types of ratios – composed units and
multiplicative comparisons – are related; 8.) a
rate is a set of infinitely many equivalent
ratios; 9.) there are several ways of reasoning,
all grounded in sense making, can be
generalized into algorithms for solving
proportion problems; 10.) superficial
cues present in the context of a problem do
not provide sufficient evidence of
proportional relationships between quantities.
Participants will engage in discourse and
problem solving as they work problems out
and see for themselves which ideas
correspond to the essential understandings.
Participants will examine different methods of
working with ratios, rates and proportions
that extend beyond some of the simple
standard methods (Kaput & West, 1994;
Chapin & Johnson, 2006; Thompson, 1994).
There will be also discussion of student work
and thinking on students' problem solving and
their level of understanding of ratios and
proportion (e.g. Canada et al, 2008; Roberge
& Cooper, 2010).
Proportionality is urged to be used as a
major middle school theme (Lanius &
Williams, 2003). First of all, proportionality is
“complex and requires significant time to
master” (p. 395). Secondly, proportionality
“involves topics with which students have
great difficulty” (p. 395). Finally,
proportionality “provides a framework for
studying topics in algebra, geometry,
measurement, and probability and statistics –
all of which are important topics in the middle
grades” (p. 395).
Context
As Loucks-Horsley et al. (2003)
state, “Professional development does not

come in one-size-fits-all. It needs to be
tailored to fit the context in which teachers
teach and their students learn.” Therefore,
this PD program will take the students,
teachers, and curriculum into consideration.
The professional development program is
aimed at teachers of mathematics or teachers
who teach mathematics in alternative settings.
This program has been designed to meet the
needs of teachers, regardless of their
certification in mathematics. For teachers
who address learning needs of students, this
program takes an important mathematical idea
which is heavily connected to other areas of
mathematics and science. As many teachers
are teaching for conceptual understanding in
addition to procedural fluency, there are many
activities in this program which aim at
developing conceptual understanding.
Carpenter and Hiebert (1992) state: “A
mathematical idea or procedure or fact is
understood if it is part of an internal
network.” In other words, the mathematics is
understood if its mental representation is part
of a network of representations. The degree
of understanding depends upon the number
of connections and the strength of each
connection. Thorough understanding of a
mathematical idea, procedure or fact is
achieved if it is linked to existing with
stronger or more numerous connections
(Carpenter & Hiebert, 1992). This program
will certainly develop teachers' connections
among the essential understandings of ratio,
proportion, and proportional reasoning.
Measurement
Quantities and
Covariation
Relative
Thinking
Reasoning Up
and Down
Rational Number
Interpretations
Sharing and
Comparing
Unitizing

T I T L E
1
Interpreting Ratios
Overview
n thissession, participantswilldevelopmeaningofthetermratio. Inparticular,participants willexamine bothadditive
andmultiplicativethinking. Participantswillalsoconsider ratioasamultiplicativecomparison oftwoquantitiesorthe
joiningoftwoquantitiesinacomposed unit. Finally,participants willcompleteataskwhichwillhavethemapplysome
ofwhattheyhavelearnedandtothinkabouthowthetaskmightrevealcertainstudentmisconceptions ofratio.
Part 1: Introduction
Goals:
 Access teachers’ ideas about the meaning of ratio.
 Introduce participants to the purposes of the PD
Time: 15 minutes
Materials:
 Overhead Projector
 Paper
 Pencils
 Calculators
Directions:
*Power Point will accompany the entire session. 
1. Introductions (facilitator and participants share a little about themselves)
2. Provide teachers with a brief overview of the session schedule and goals.
3. Ratio will be formally defined. Ask teachers:
- How would you define ratio to your students?
- What are some other examples of ratio encountered in daily life?
- How can students develop a strong concept of ratio?
4.  Record responses onto a transparency or chart paper.
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5.  Quickly discuss some key ideas from the list and assure participants that many of
the topics will be covered in the PD and choose one that connects well as a transition
to thinking multiplicatively vs. additively.
Part 2: Additive Versus Multiplicative Thinking
Goals:
 To introduce teachers to the two types of comparison thinking.
 To recognize that additive thinking is not always sufficient.
 To understand the power of multiplicative thinking.
 To identify strategies and tools necessary to help students develop multiplicative thinking.
Time: 1 hour
Materials:
 Papers
 Pencils
 Calculators
Directions:
1. Participants will be presented with a problem for thought and discussion. This
problem is aimed at illustrating the two main ways of thinking of a problem: Additive
and multiplicative.
2.  Discuss different ways of interpreting the problem. Also discuss how to help
students realize that there is another way to think about the information.
3. The first essential understanding of ratios will be formalized: Reasoning with
Ratios involves attending to and coordinating two quantities (Lobato et al, 2010).
4. Ask teachers:
- What errors might students make when comparing two quantities?
- How can we as teachers help students think through the problem correctly?
6. Participants will reflect on what is needed to understand the power of multiplicative
thinking.
7.  Discuss ways of thinking multiplicatively with different ratios presented on slide
show.
8. Present a pizza problem (adapted from some ratio lessons developed by Cynthia
Lanius). Participants will consider visualand pictorial representations for developing
multiplicativethinking in creating ratios.
9. The second essential understanding will be formalized: Ratios are often
meaningfully interpreted as quotients.

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Part 3: Composing the Unit
Goals:
 To introduce teachers to ratio as a composed unit.
 To show different ways of composing a unit (iterating or partitioning)
 To evaluate and examine student thinking on ratio formation.
Time: 1 hour
Materials:
 Paper
 Pencils
 Calculators
 Handout: Which Tastes More Juicy?
Directions:
1. Discussion will center around the second part of the third essential understanding
involving ratios: A ratio is a multiplicative comparison of two quantities, or it is a
joining of two quantities in a composed unit.
2.  A problem will be presented for discussion where different solution strategies are
explored and discussed. Participantswill be asked to describe their thinking for the
different parts of the problem.
3. Participant responses will be recorded onto a transparency or chart paper.
4. Participant responses will be used to formally define and illustrate some of the
thinking, such as partitioning, quotients, and iterating.
5. Participants will complete the worksheet: Which Tastes More Juicy?
6.  Discuss student misconceptions about which tastes more juicy. Also think and
discuss about some other ways students can be helped to solve mixture problems
using some of the strategies discussed in this session (e.g. multiplicative thinking,
composing a unit, ratio as quotient).
Part 4: Closing
Goals:
 Summarize various teaching strategies and essential ideas of understanding needed for students
Time: 15 minutes
Directions:
1. At this time, we shall summarize the three essential understandings encountered thus
far and think about what we as teachers can do in our own classrooms to better
provide our own students with these essential understandings.

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REFERENCES:
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math You Teach, Grades
6 – 8, 2nd
Ed. Math Solutions Publications, Sausilito, CA.
Lamon, S. (1999). Teaching Fractions and Ratios for Understanding: Essential Content
Knowledge and Strategies for Teachers. Mahwah, NJ. Lawrence Erlbaum Associates, Inc
Lanius, C.S. & Williams, S.A. (2003). Proportionality: A Unifying Theme for the Middle
Grades. Mathematics Teaching in the Middle School, 8(8), 392 – 396.
Lobato, J., Ellis, A.B., Charles, R.I., and Zbiek, R.M. (2010). Developing Essential
Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching
Mathematics in Grades 6 – 8. National Council of Teachers of Mathematics, Reston, VA.

T I T L E
5
Ratios and Proportions
Overview
his session, participants will be able to differentiate between ratios and fractions.
Participants will also explore the meaning of a proportion as well as a number of
methods usedto solvemissing values inproportions. There will be somediscussion and
analysis of student work and thinking as well.
Part 1: Ratios, Fractions and an Introduction to
Proportions.
Goals:
 Participants will compare and contrast ratios and fractions.
 As we introduce proportions, participants will use prior knowledge from the prior session to
determine what is equal in a proportion.
Time: 1 hour 15 minutes
Materials:
 Copy of Handout: Proportions: What is Equal?
Directions:
1. Brief recap of the three key understandings from the previous session.
2. Provide teachers with a brief overview of the session schedule and the goal.
3. Participants will be presented with some examples of ratios and fractions.
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4.  Differences between ratios and fractions will be discussed.
6. Some notable differences between ratios and fractions will be discussed:
-ratios may be written in three ways (colon, fraction form, use of word “to”)
-ratios can show part – part relations while fractions can only show part – whole
-ratios can be compared to zero while fractions cannot
-ratios can compare noninteger values while fractions cannot
7. A number of mathematical connections link ratios and fractions (Lobato et al,
2010).
8.  Discussion of the fraction interpretation of ratios will lead into the development of
proportions.
9. Participants will now work on the handout, Proportions: What is Equal?
10.  Discussion will focus around equality of such ideas as equal composed units or
equal quotients.
11. A proportion is a relationship of equality between two ratios. In a proportion, the
ratio of two quantities remains constant as the corresponding values of the quantities
change (Lobato et al, 2010).
12.  Discussion will focus around the multiplicativerelationships between and within
the ratios, which are constant in a proportion. Participants will be asked to think
about why knowledge of these multiplicativerelationships is important and what
potential benefits exist for students who are taught these relationships.
13. Proportional reasoning is complex and involves understanding that a.) equivalent
ratios can be created by iterating and/or partitioning a composed unit, b.) if one
quantity in a ratio is multiplied or divided by a particular factor, then the other
quantity must be multiplied or divided by the same factor to maintain the
proportional relationship, and c.) the two types of ratios – composed units and
multiplicativecomparisons – are related (Lobato et al, 2010).
14. Participants will be presented with a problem requiring proportional reasoning,
and asked to think about some of the ways in which the problem could be solved.
Part 2: The Complexity of Proportional Reasoning
Goals:
 Participants will explore the cross-product technique and why it works.
 Participants will examine and evaluate student work for understanding and what essential
understandings are reflected in their thinking.
Materials:
 Copy of Handout: Proportions: Examining a Special Property of Certain Ratio Pairs.
 Copy of Handout: Exploring the Cross Product Property of Proportions.
 Copy of "Investigating Mathematical Thinking and Discourse with Ratio Triplets" (Canada et
al, 2008)

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Directions:
1. Participants will complete the handout “A Special Property of Proportions.”
2.  Participants will first have the opportunity to see if they can justify for themselves
why the cross product method works.
3. Some formal ideas will be presented as techniques for why the cross product method
works (use of lowest common denominators or factor of change).
4. Participants will be provided with examples of student work (Canada et al, 2008).
5.  Discussion will center around being able to identify the methods students use to
answer the task as well as to think about how/why a student’s idea works (e.g. the
student interpreted a ratio as a quotient, the student used cross multiplication, the
student used between/within ratios, etc.)
Closing
A. Summarize essential understandings covered in this session.
B. Teacher reflection
- Will you use this information?
- If so, how will it impact your teaching?
REFERENCES:
Canada, D., Gilbert, M, and Adolphson, K. (2008). Investigating Mathematical Thinking
and Discourse with Ratio Triplets. Mathematics Teaching in the Middle School. Vol. 14, No. 1,
pp. 12 – 17.
6 – 8, 2nd
Kaput, J. & West, M.M. (1994). Missing value proportional reasoning problems: Factors
affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of
multiplicativereasoning in the learning of mathematics (pp. 235 – 287). New York: State
University of New York Press.

T I T L E
8
AttributesNeeded for Ratio
Formation. Rates. Evaluating
Teaching. SuperficialCues.
Overview
his section of theprogram isdevotedto learningwhat is involved in ratio formation by
examining the steepness of a ramp. Participants will examine ratio from a graphical
perspectivein order to further develop a deep conceptualunderstanding of ratio from a
symbolic perspective. Thisin turn, will lead to discussion and an examination of rates.
Finally, participants willexamine somesuperficialcuesin problems that appear proportional but
are really not.
Part 1: Steepness of a ramp and rates.
Goals:
 To determine the attributes of a ramp needed to maintain its steepness.
 To interpret steepness as a rate of change.
 To examine and evaluate different definitions of rate and determine the extent to which rates
and ratios are one in the same.
Materials:
 Copy of Handout: The Steepness of a Ramp?
 Copy of Short Article: Rates and Ratios.
 Rulers
 Paper
 Index Cards with different definitions of rates.
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Directions:
1. Briefly recap important understandings and key ideas from previous two sessions.
2. Participants will complete the handout The Steepness of a Ramp.
3.  Discuss possible student misconceptions on maintaining a ramp’s steepness.
4. Forming a ratio as a measure of a real-world attribute involves isolating that
attribute from other attributes and understanding the effect of changing each quantity
on the attribute of interest (p. 23) (Lobato et al, 2010).
5. A wrap up discussion of the rise and run of the ramp will serve as a transition into
more discussion on rates and rates of change.
6. Participants will examine small cards containing different definition of rates and come
up with what they believe is a good definition of rate.
7. Participants will explore the dual nature of rates (e.g. miles per gallon versus gallons
per mile) by calculating two different unit rates for some given situations.
8.  Discussion will center around how and why students might struggle with the dual
nature of rates and how we as teachers might help them overcome those difficulties.
Part 2: Evaluating teaching and superficial cues.
Goals:
 To evaluate the existence of essential understandings addressed in a case study of a teacher
teaching ratios.
 To engage in problem solving, focusing primarily on some of the mistakes students might
make when solving problems which appear to be directly proportional.
 To synthesize many of the ideas encountered in the whole program with respect to teaching
and student learning.
Materials:
 Copy of the Case Study: The Case of Marie Hanson
Directions:
1. Participants will read through the case study of Marie Hanson, making notes on
essential understandings present or absent from Ms. Hanson’s teaching (e.g.
multiplicativethinking, composed units, complexity of proportional reasoning, ratios
as quotients).
2.  Discussion will primarily focus on comparing and comparing strategies of
proportional reasoning used by Ms. Hanson and her students and the participants of
the workshop.
3. Ideas for instructional changes proposed by participants, if any, will be recorded.
4. Participants will think and discuss through four problems, two of which are not
proportional.

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10
5.  Discussion will center around what misconceptions students might have, what
words act as “superficial cues,” and how as teachers, we might be able to foster more
problem solving and thought on inverse proportion problems or problems that
present three out of four numbers.
6.  Superficial cues present in the context of a problem do not provide sufficient
evidence of proportional relationships between quantities (p. 46) (Lobato et al, 2010).
Closing
A. Summarize essential understandings covered in all three sessions.
B. Teacher reflection
- Will you use this information?
- If so, how will it impact your teaching?
C. Final survey to be collected to further improve program development in ratio,
proportion, and proportional reasoning.
REFERENCES:
Schwan-Smith, M., Silver, E.A., & Stein, M.K. (2005). Improving Instruction in Rational
Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and
Learning, Vol. 1. New York, Teachers’ College Press.
Thompson, P.W. (1994). The development of the concept of speed and its relationship to
concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative
reasoning in the learning of mathematics (pp. 179 - 234). New York: State University of
New York Press.

T I T L E
11
Reflection
Creating this short professional development program has been a challenging but very
rewarding learning experience. Initially I thought it would not be as time consuming as I expected.
In the past I’ve been to several professional development sessions but I did not know how much
planning was put into those sessions.
In order to truly make a difference in teaching practices, it’s a much more thoughtful and
extensive process. Also, having an impact on pupil learning would require ongoing professional
development that was carefully planned and supported during implementation. Something else I
realized during as I planned was the amount and quality of knowledge required to create a quality
session. Some of the knowledge out there was useful for research but not necessarily for teachers.
Therefore, I needed to learn how to locate information that would be useful to teachers. Some ideas
and definitions proposed by researchers would prove to be more of a hindrance for teachers rather
than a help. As such, some of those ideas would either need to be watered down if appropriate for
the session or omitted entirely.
Finally, I also learned that teaching adults is much different than teaching children. Granted that
adults are older, generally more mature, and more knowledgeable, some tasks that might take
children a longer time would take adults much less time. In an effort to maximize the time and
benefits for teachers, I have come to realize that through interactive discussion, we can all learn from
one another. It is through the flexibility of fruitful discussion that the sessions can become enjoyable
and participants can become engaged with ideas presented in the sessions. Furthermore, in the
selection of activities, I came to realize that the purpose of the activities for teachers is similar, yet
different to the purposes for students. The activities are aimed at developing a deep conceptual
understanding of ratio, proportion, and proportional reasoning for both teachers and students, but
for teachers, the activities are meant to elicit such questions as how might this help my students or
what misconceptions might be revealed through these activities.

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This program is my first experience with the creation of professional development. I am also
thinking about other topics in mathematics in middle school and high school that might be good to
create professional programs for teachers in my school. The creation and implementation of
professional development can serve as a vehicle towards improved teaching, improved student
learning, and improved curriculum development in mathematics. It is my goal to continue to
consider creating professional development sessions for not only my school but also sessions that
might be useful for other schools and other teachers.

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Professional Development Initial Survey:
Ratio, Proportion, and Proportional Reasoning
The results of this survey will be confidential and will only be used to help develop a
professional development program. They will not be shared with any administrators. Thank
you for your time and participation. Please complete and return to Matthew Leach.
1.) How long have you been teaching or tutoring mathematics?
2.) What is your certification or specialty area(s) and discipline(s)?
3.) Have you ever taken any professional development workshops in general? If so,
please describe what and when.
4.) Please rate your overall experience as a prior participant in professional
development?
Very Poor 1 2 3 4 5 6 7 8 9 10 Excellent or N/A
5.) Have you ever taken any mathematics professional development workshops? If so,
please describe when you attended the workshop(s), the grade level(s), and topics
covered.
6.) Please rate your overall experience as a prior participant in mathematics professional
development.
Very Poor 1 2 3 4 5 6 7 8 9 10 Excellent or N/A
7.) What do you hope to gain from this workshop on ratio, proportion, and
proportional reasoning?

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Professional Development Evaluation:
Ratio, Proportion, and Proportional Reasoning
Activity Evaluation
SD= Strongly Disagree, D=Disagree, A=Agree, SA=Strongly Agree
SD D A SA
Content presented was interesting.
Content was relevant to my classroom.
Presenter was knowledgeable.
Session was well organized.
Session was engaging.
My understanding was enhanced.
I would recommend the PD to a colleague.
Content Implementation
Which of the following ideas do you plan on implementing in your classroom?
(check all that apply)
____ Comparing Ratios and Fractions ____ Which Tastes More Juicy?
____ Exploration of Cross-Multiplication/Why It Works ____ Steepness of a Ramp
____ Dual nature of Rates
Please feel free to write down any other ideas you may have thought of through our
discussions.

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Please use bulletsto answer the following questions:
What changes or improvements would you suggest to the presenter?
What follow-up support/assistance is needed?
Comments/Questions:

T I T L E
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References
Canada, D., Gilbert, M, and Adolphson, K. (2008). Investigating Mathematical Thinking
and Discourse with Ratio Triplets. Mathematics Teaching in the Middle School. Vol. 14, No. 1,
pp. 12 – 17.
6 – 8, 2nd
Donovan, M.S., Bransford, J. & Pellegrino, J. (Eds.). (1999) How People Learn: Bridging
Research and Practice. Washington: National Academy Press.
Hiebert, J. & Carpenter, T.P. (1992). “Learning and Teaching With Understanding.”
Handbook of Research on Mathematics Teaching and Learning: 65 – 89.
Herron-Thorpe, F.L., Olson, J.C., and Davis, D. (2010). Shrink Your Class. Mathematics
Teaching in the Middle School, Vol. 15, No. 7, pp. 386 – 391
Kaput, J. & West, M.M. (1994). Missing value proportional reasoning problems: Factors
affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of
multiplicativereasoning in the learning of mathematics (pp. 235 – 287). New York: State
University of New York Press.
Lamon, S. (1999). Teaching Fractions and Ratios for Understanding: Essential Content
Knowledge and Strategies for Teachers. Mahwah, NJ. Lawrence Erlbaum Associates, Inc
Lamon, S. (2007). RationalNumbers and Proportional Reasoning: Toward a Theoretical
Framework for Research. In F.K. Lester, Jr. (Ed.) Second Handbook of Research on
Mathematics Teaching and Learning. (pp. 629 – 667). Charlotte. Information Age Publishing.
Lo, J-J., Watanabe, T., and Cai J. (2004). Developing Ratio Concepts: An Asian Perspective.
Mathematics Teaching in the Middle School, Vol. 9, No. 7, pp. 362 – 367.

T I T L E
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Lanius, C.S. & Williams, S.A. (2003). Proportionality: A Unifying Theme for the Middle
Grades. Mathematics Teaching in the Middle School, 8(8), 392 – 396.
Loucks-Horsley, S., Stiles, K., Mundry, S., Love, N., & P. Hewson. (2010). Designing
professional development for teachers of science and mathematics (3rd
edition). Thousand
Oaks, CA: Corwin Press.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum
Associates.
Principles and Standards for School Mathematics (2000). Reston, VA: National Council
of Teachers of Mathematics.
Roberge, M.C. and Cooper, L.L. (2010). Map Scale, Proportion, and Google Earth.
Mathematics Teaching in the Middle School, Vol. 15, No. 8, pp. 448 – 457
Schwan-Smith, M., Silver, E.A., & Stein, M.K. (2005). Improving Instruction in Rational
Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and
Learning, Vol. 1. New York, Teachers’ College Press.
Thompson, P.W. (1994). The development of the concept of speed and its relationship to
concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative
reasoning in the learning of mathematics (pp. 179 - 234). New York: StateUniversity of
New York Press.

T I T L E
18
Which Tastes More Juicy?
George is making a juice mixture, which requires three cups of water and two cups of juice.
Consider his thinking below:
“Hmmm…three cups of water and two cups of juice is not a lot so I am going to use 9 cups
of water and 6 cups of juice. Not only will I have more to drink but the mix will taste even
juicier than if I made two cups of juice and three cups of water.”
1.) Is George right? Explain your reasoning.
2.) As a teacher, how would you react to George’s reasoning about the “juiciness” of his
drink?
Kincaid is making a juice mixture, which requires three cups of water and two cups of juice.
Like George, Kincaid really likes a slightly juicer tasteand realizes that she does need to
make more juice so she has some more for later. Consider her thinking below:
“Let’s see now. I am going to first mix 2 cups of juice with three cups of water. I want
more juice for later so I will mix 15 cups of water in all. I will use 11 cups of juice to create a
more juicy mixture.”
1.) Will Kincaid’s idea make her drink juicer? Explain.
2.) Using the idea of ratio as a composed unit, show that Kincaid’s drink is indeed juicier.

T I T L E
19
Lindsay is making a mixture but she does not particularly care for a strong juicier taste. Her
reasoning is shown below.
“ First, I will mix 2 cups of juice with 3 cups of water. As I want more for later, I will mix
15 cups of water. I don’t like it too juicy so I will only mix in 9 cups of juice.”
“Using my calculator, 2/3 is .66666 and 9/15 = .6 so my drink is less juicy. Yeah!”
1.) Explain how Lindsay’s reasoning does indeed support her claim that her drink is less
juicy.
2.) Despite Lindsay’s correct claim, she is not fully demonstrating an understanding of
ratio. Thinking of the definition of ratio, how might you help Lindsay center her
explanation around ratios to explain her claim of less juiciness?
3.) How else might Lindsay answer the question to demonstrate a more thorough
understanding of ratios?

T I T L E
20
Proportions: What is equal?
Write two ratios for each situation described. Then determine whether your ratio pairs
are equivalent using any method or technique you'd like (e.g. quotients, composed units).
Explain what in each situation is equal.
1.) Mike can run 800 meters in 6 minutes. Lindsay can run 1000 meters in 7.5 minutes.
2.) Alaisha used 12 eggs to make 30 batches of cookies. Jessica used 9 eggs to make
22.5 batches of cookies.
3.) Amelia, who measures 54 inches, casts a shadow that is 117 inches long. Sam, who
is only 48 inches tall, casts a shadow that is 102 inches long.
4.) Melinda is able to type 70 words in 6 minutes. Alex, on the other hand, can type 175
words in 15 minutes.

T I T L E
21
Examining a special property of
certain ratio pairs.
Consider the following sets of ratios.
1.)
300 𝑚𝑒𝑡𝑒𝑟𝑠
7 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
;
750 𝑚𝑒𝑡𝑒𝑟𝑠
17.5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
2.)
8 𝑏𝑜𝑦𝑠
20 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
;
12 𝑏𝑜𝑦𝑠
30 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
3.)
200 𝑚𝑖𝑙𝑒𝑠
3 ℎ𝑜𝑢𝑟𝑠
;
900 𝑚𝑖𝑙𝑒𝑠
14.5 ℎ𝑜𝑢𝑟𝑠
4.)
21 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
30 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠
;
49 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
70 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛𝑠
A.) In each set of ratios, cross-multiply each numerator with the other denominator.
Ignore the units.
B.) What appears to happen?
C.) Can you offer a convincing argument as to why this property is true?

T I T L E
22
The steepness of a ramp.
On graph paper, draw the ramp shown below. Let each unit represent 1 ft.
height = 6 ft
8 ft
8 ft
base = 8 ft
1.) John decides to add 6 ft to the base to can create a longer ramp.
a.) Draw the ramp described by John. Does it appear to be more steep, less steep, or
equally steep to the original ramp? Explain your reasoning.
2.) Suppose John decides to add 6 ft. to the height. Does this ramp appear to be more steep,
less steep, or as steep as the original ramp? Explain your reasoning.
3.) Suppose John decides to add 6 ft to both the height and the base. Does this ramp appear
to be more steep, less steep, or as steep as the original ramp? Explain your reasoning.
4.) Suppose John wishes to add 6 feet to the base. How many feet must he add to the height
to maintain the steepness of the ramp?
5.) Suppose John wishes to lower the height of the ramp by 1 ft. What will the base of the
ramp have to be to maintain the ramp’s steepness?
6.) How is the steepness of the ramp related to the base and height of the ramp?

T I T L E
23
Rates: Varying Definitions
Definition Source
"a fixed ratio between two things" http://mathforum.org/library/drm
ath/view/58042.html
"a special ratio that compares two measurements with
different units of
measure, such as miles per gallon or cents per pound.
A rate with a denominator
of 1 is called a unit rate."
http://www.glencoe.com/sec/mat
h/prealg/mathnet/pr01/pdf/0901
a.pdf
"When two different types of measure are being
compared, the ratio is usually called a rate."
Chapin & Johnson, 2006, p. 166
"The quotient of two quantities measured in different
units."
http://www.algebralab.org/studyai
ds/studyaid.aspx?file=Algebra1_2-
8.xml
"It’s the speed at which something is happening such
as miles per hour, meters per second, or words per
minute."
http://www.algebralab.org/studyai
8.xml
"a" per "b" = a/b http://www.algebralab.org/studyai
8.xml
"a set of infinitely many equivalent ratios" Thompson, 1994, p. 42
"ratio between two measurements, often with
different units"
^ "On-line Mathematics
Dictionary". MathPro Press.
January 14, 2006.as cited by
wilipedia.org
"a certain quantity or amount of one thing considered
inrelation to a unit of another thing and used as a stan
dard or measure"
http://dictionary.reference.com/br
owse/rate
"a fixed charge per unit of quantity" http://dictionary.reference.com/br
owse/rate

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