ratio and proportion pd template

RATIO, PROPORTION, AND PROPORTIONAL REASONINGEssential understandings for teachers and students.

Professional Development Guide

E SS E N T I A L U N D E R S TA N D I N G S F O R T E A C H E R S A N D S T U D E N T S :

Ratio, Proportion, and Proportional Reasoning

Northbrook Academy Mathematics Department

Table of ContentsIntroduction: 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

middle school 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 opportunity to 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 InformationThe 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).

ContextAs 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

Interpreting Ratios

Overview

n this session, participants will develop meaning of the term ratio. In particular, participants will examine both additive and multiplicative thinking. Participants will also consider ratio as a multiplicative comparison of two quantities or the joining of

two quantities in a composed unit. Finally, participants will complete a task which will have them apply some of what they have learned and to think about how the task might reveal certain student misconceptions of ratio.

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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?

Session 1

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- 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. 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 ThinkingGoals:

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:

Overhead Projector 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? 5. Record responses onto a transparency or chart paper. 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.

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8. Present a pizza problem (adapted from some ratio lessons developed by Cynthia Lanius). Participants will consider visual and pictorial representations for developing multiplicative thinking in creating ratios.

9. The second essential understanding will be formalized: Ratios are often meaningfully interpreted as quotients.

Part 3: Composing the UnitGoals:

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: Overhead Projector 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. Participants will 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

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discussed in this session (e.g. multiplicative thinking, composing a unit, ratio as quotient).

Part 4: ClosingGoals:

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.

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.

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T I T L E

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 used to solve missing

values in proportions. There will be some discussion and analysis of student work and thinking as well.

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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:

Overhead Projector Copy of Handout: Proportions: What is Equal?


1. Brief recap of the three key understandings from the previous session.

Session 2

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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.

4. Differences between ratios and fractions will be discussed.5. Record responses onto a transparency or chart paper.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 multiplicative relationships between and within the ratios, which are constant in a proportion. Participants will be asked to think about why knowledge of these multiplicative relationships 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 multiplicative comparisons – are related (Lobato et al, 2010).

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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 ReasoningGoals:

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.


Overhead Projector 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)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.)

ClosingA. Summarize essential understandings covered in this session.B. Teacher reflection

- Will you use this information?- If so, how will it impact your teaching?

REFERENCES:

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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.


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 multiplicative reasoning in the learning of mathematics (pp. 235 – 287). New York: State University of New York Press.


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Attributes Needed for Ratio Formation. Rates. Evaluating Teaching. Superficial Cues.

Overview

his section of the program is devoted to learning what is involved in ratio formation by examining the steepness of a ramp. Participants will examine ratio from a graphical perspective in

order to further develop a deep conceptual understanding of ratio from a symbolic perspective. This in turn, will lead to discussion and an examination of rates. Finally, participants will examine some superficial cues in problems that appear proportional but are really not.

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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.


Overhead Projector Copy of Handout: The Steepness of a Ramp? Copy of Short Article: Rates and Ratios. Rulers Paper Index Cards with different definitions of rates.


Session 3

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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.


Overhead Projector 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.

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multiplicative thinking, 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. 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).

ClosingA. 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:


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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.

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ReflectionCreating 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

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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.

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 AgreeSD 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

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Please feel free to write down any other ideas you may have thought of through our discussions.

Please use bullets to answer the following questions:

What changes or improvements would you suggest to the presenter?

What follow-up support/assistance is needed?

Comments/Questions:

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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.


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

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affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning 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). Rational Numbers 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.


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.

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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: State University of New York Press.

Which Tastes More Juicy?

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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 taste and 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.

Lindsay is making a mixture but she does not particularly care for a strong juicier taste. Her reasoning is shown below.

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“ 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?

22

T I T L E

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.

23

T I T L E

Examining a special property of certain ratio pairs.

Consider the following sets of ratios.

1.) 300meters7minutes

; 750meters

17.5minutes 2.)

8boys20 students

;12boys

30 students

3.) 200miles3hours

;900miles14.5hours

4.) 21correct

30questions;

49 correct70questions

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?

24

T I T L E

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?

25

T I T L E

Rates: Varying Definitions

Definition Source"a fixed ratio between two things" http://mathforum.org/

library/drmath/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/math/prealg/mathnet/

pr01/pdf/0901a.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/

studyaids/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/


"a" per "b" = a/b http://www.algebralab.org/


"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 anoth

er thing and used as a standard or

http://dictionary.reference.com/

browse/rate

26

http://dictionary.reference.com/browse/rate



http://www.mathpropress.com/glossary/glossary.html#R

http://www.mathpropress.com/glossary/glossary.html#R

http://en.wikipedia.org/wiki/Rate_(mathematics)#cite_ref-0

http://en.wikipedia.org/wiki/Measurement

http://en.wikipedia.org/wiki/Ratio

http://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra1_2-8.xml









http://www.glencoe.com/sec/math/prealg/mathnet/pr01/pdf/0901a.pdf



T I T L E

measure" "a fixed charge per unit of quantity" http://

dictionary.reference.com/browse/rate

27




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