Combined File 013009 Gaylord

8/14/2019 Combined File 013009 Gaylord

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1

Designing Professional

Development in Mathematics

Michigan Mathematics and Science

Teacher Leadership Collaborative

Overview of Session 10

Welcome

Consider What is a Proportion?

Work on a mathematical task

Analyze OGAP tasks and student thinking

Prepare for practice facilitation

Plan for facilitating a session

Wrap-up

Goals of the Session

To build connections among different solution

strategies for proportional problemsTo

develop the knowledge and skills for

analyzing student thinkingTo identify

difficulties that students might have when

working on ratio comparison problemsTo

develop skills for facilitating professional

development around examining student work

Considering Proportionality

How is proportionality defined in your

textbooks?

How does it compare to the key ideas in thearticle What is a Proportion? What Does it

Mean to be Proportional? What is

Proportional Reasoning?


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2

Organizing Our Work

Work individually and discuss the math in

small groups:A, A, A; B, B, B; C, C, C

Share out on common taskA, A, and A (Pauls Dog)

B, B, and B (Racing Track)

C, C, and C (Paper Towel)

Analyzing student thinking

A, A, A, B, B, B, C, C, C

Sharing analysis on student thinking

A, A, and A

B, B, and B

Organizing for Facilitation

Thinking Through a Session Protocol:

Whole group

Planning facilitationA, A, A, B, B, B, C, C, C

Next Session:

A

B C

A

CB

A

CB

Wrap up

Expectations for February 20Bring five samples of student work on a high

level task

Read Chapter 2 from Peg Smiths book,

Practiced-Based Professional Development for

Teachers of Mathematics


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What is a Proportion? What Does it Mean to be Proportional? What is

Proportional Reasoning?

What is a Proportion?

A proportion is an equation composed of two equivalent ratios

. (For a more detailed

discussion of ratios see the Rates and Ratios essay.) A common, generic way of writing a

proportion isd

c

b

a= ; another way of writing a proportion is a:b=c:d. The ratios in a

proportion can be part-to-part ratios or part-to-whole ratios. For example, if there are 3

boys for every 2 girls in a classroom and a total of 12 girls in the classroom, we could use

part-to-part ratiosboys

girls12

boys3

girls2

x

= to determine how many boys are in the classroom, or we

could use part-to-whole ratiosstudents

girls12

students5

girls2

x

= to determine how many total students

are in the classroom. Notice that in both of the previous examples three of the four


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equivalent to15

12. If we add the same number to both the 12 and 15, or subtract the same

number from both the 12 and 15 the resulting ratios are not equivalent to15

12. For

example, neither60

57

4515

4512=

+

+

nor5

2

1015

1012=

!

!

are equivalent to15

12. Therefore, we say that

ratios are multiplicative as opposed to additive in nature. Similarly, since proportions are

composed of two equivalent ratios, we say that proportions are multiplicative structures

(as opposed to additive structures). Elementary students typically spend the first several

years of their mathematical careers focusing solely on additive situations, and frequently

have difficulty transitioning to upper-elementary and middle grades mathematics which

requires them to discriminate between additive and multiplicative situations and apply the

appropriate type of reasoning for a given situation. See S. Lamon for more information

on additive versus multiplicative reasoning and on ways to encourage the development of

multiplicative reasoning skills. (Lamon, 2005)


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boys18girl

boys5.1girls12 =! . The multiplicative relationship between the ratios

boys3

girls2and

boys

girls12

x

is 6 because the total number of girls in the classroom is 6 times the number of

girls in the sample.** Using the multiplicative relationship between the two ratios

provides us with another way of determining that there are boys186boys3 =! in the

classroom. Notice that in this example the multiplicative relationship within the ratios is

non-integral (1.5girl

boys) whereas the multiplicative relationship between the ratios is

integral (6).

*Alternatively we could also say that the multiplicative relationship within the ratio is

6666.0 or3

2

boy

girls, since the number of girls is

3

2times the number of boys.

**Alternatively we could also say that the multiplicative relationship between the ratios is


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Research Implications for Teaching:

In general, research shows that it is easier for students to solve problems in which the

multiplicative relationships within and between ratios are integral, and that it is more

difficult for students to solve problems in which the multiplicative relationships within

and/orbetween ratios are non-integral. (Cramer & Post, 1993) Before proceeding,

consider the following two problems and the questions that follow:

Problem 1: If 3 balloons cost $6, how much will 12 balloons cost?

Problem 2: If 3 balloons cost $5, how much will 10 balloons cost?

Which of these problems do you think would be more difficult for students?

Why?

The first problem has integral relationships within ($2/balloon) and between (4 times as

many balloons) ratios and is therefore much easier for students to solve than the second

problem, which has non-integral relationships within ($1.67/balloon) and between

(31

times as many balloons) ratios Often students that successfully use proportional


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not necessarily indicate a solid understanding of the concept that can be extended and

transferred to other problems. For more information on additional strategies for solving

proportions see the essay Multiple Ways to Solve Proportional Reasoning Problems.


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What Does it Mean for Two Quantities to be Proportional?

Two quantitiesx andy are said to be proportional or in proportion with each other if all

ratios of the formy

x(wherex andy are nonzero* and form the ordered pair ),( yx ) are

equivalent to one another or, in other words, if all ratios y

x

create an equivalence class.

For example, when purchasing gasoline at a price of $1.20 per gallon, we see that

regardless of how much gasoline we buy, all ratios of the formgasofgallonsof#

costtotalare

equivalent to one another !!"

#$$%

&=== K

gallonsgallonsgallons 30

00.36$

7

40.8$

5.0

60.0$, and therefore we say

that the total cost of the gasoline is proportional to the amount of gasoline purchased.

Notice that the multiplicative relationship within each ratio is $1.20 per gallon, and that

we can use variables to succinctly portray this relationship as xgallon

y 20.1$= , wherex

represents the number of gallons of gasoline purchased and y represents the total cost of


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A few examples of quantities that are related proportionally include:

o If traveling at a constant rate (r), the distance traveled (d) is proportional to thetime (t) traveled [or trd != ].

o If a scuba diver starts at sea level and descends 10 meters every 30 seconds, thedivers height in meters above sea level (h) is proportional to his/her time in

seconds under water (t) [or th3

1!= ]. Note that this situation cannot continue

indefinitely. Typically recreational divers do not descend below -120m or -130m.

Therefore these two quantities are proportional from the start of the descent until

the maximum depth of the dive is reached around 6 minutes after beginning the

descent.

o In the set of all rectangles for which the length (l) is 1.5 times the width (w), thelength is proportional to the width [or wl 5.1= ].

oWhen making orange juice from concentrate, one can of concentrate calls for 2.5

cans of water. Therefore, the amount of water (w) needed is proportional to the


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*Why do we require thatx andybe nonzero members of the same ordered pair?

Notice that, in anyproportional relationship, that is any relationship in which the two

quantities are related by the equation kxy = , the ordered pair (0, 0) satisfies the equation

since 0 multiplied by any constant, k, is 0. Therefore in any proportional relationship the

ratioyofunits

xofunits

0

0will be the exception to the rule that all ratios of the form

y

xare

equivalent to one another, since the fraction0

0is undefined.**

**Notice that this provides one of many good explanations of why0

0is undefined.

Suppose that0

0were defined, to which equivalence class should it belong? Since there

are an infinite number of possibilities for the constant kin the equation kxy = , and the

ordered pair (0, 0) satisfies all of these, there are an infinite number of possible

0


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What is Proportional Reasoning?

Now that we know what a proportion is, and what it means for two quantities to be

proportional, it would seem as though defining the concept of proportional reasoning

would simply be a formality. Unfortunately, proportional reasoning has long been, as S.

Lamon states, an umbrella term, a catch-all phrase that refers to a certain facility with

rational number concepts and contexts. The term is ill-defined and researchers have been

better at defining when a student or an adult does notreason proportionally than at

defining characteristics of one who does. (Lamon, 2005) T. Post, M. Behr, and R. Lesh

add, The majority of past attempts to define proportional reasoning (e.g., Karplus, Pulas,

and Stage 1983; Noelting 1980) have been primarily concerned with individual responses

to missing value problems where three of the four values in two rate pairs were given and

the fourth was to be found. Those students who were able to answer successfully the

numerically awkward situations containing non-integer multiples within and between

rate pairs were thought to be at the highest level and were considered proportional


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proportional situations and they employ appropriate and efficient methods depending on

the complexity of the situation. The following examples of student work provide an

overview of several commonly used proportional reasoning strategies. We will look at

student work from two different problems. The first problem is an example of a missing-

value problem. (For more information on missing-value problems, see p. 1 of this essay.)

Problem 3:

Pauls dog eats 20 pounds of food in 30 days. How long will it take Pauls dog to eat a45 pound bag of dog food?Explain your thinking.

Solve this problem yourself before examining the student work that follows.

Student A:


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Student A uses proportional reasoning to build down both the number of pounds and the

number of days to determine how long 5 lbs. of food will last. Using this information

and the given rate of 20 lbs. eaten in 30 days, Student A then builds both the number of

pounds and the number of days back up to correctly determine that 45 lbs. of food are

consumed in 67.5 days. Notice, however, Student As incorrect use of the equality

symbol in the run-on equation 455402020 =+=+ , where the leftmost and rightmost

expressions are not equal. For more information on the building up/down strategy see the

essay Multiple Ways to Solve Proportional Reasoning Problems.

Student B:


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Student B reasons proportionally by computing a unit-rate of 0.66 pounds per day, and

then divides 45 pounds of food by this unit-rate to find the number of days 45 pounds of

food will last at this rate. Notice, however, that Student B rounds 666.0 to 66.0 , and then

rounds the result of 66.045 to 68 with an overall result of 68 days instead of the more

accurate 67.5 days.

Student C:


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Student D:

Student D reasons proportionally by recognizing the factor-of-change of2

11 within the

given rate of 20 pounds per 30 days, and applies this factor-of-change to the known

amount of 45 pounds of dog food to find the unknown number of days that the dog food

will last.

Student E:


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Notice that after setting up the proportion and performing the cross-multiplication,

Student E omits the units in the product of 30 days and 45 pounds. This is extremely

common in the use of cross products, presumably because the appropriate units days-

pounds are incomprehensible.

Next we examine examples of student work from another type of problem frequently

used to elicit proportional reasoning strategies. The following problem is an example of

a ratio comparison problem. Try to solve it before continuing. For more discussion on

ratio comparison problems, see p. 21 of this essay.

Problem 4:

The chart below shows the population of raccoons in two towns.

Town A Town B

60 square miles 40 square miles

480 raccoons 380 raccoons

Karl says that Town A has more raccoons per square mile. Josh says that Town B hasil Wh i i ht?


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Student F:

Student F reasons proportionally using a model to effectively partition the raccoons in

each town into 10 square mile blocks. Notice, however, that Student Fs explanation

refers to raccoons per square mile while his/her model is in terms of raccoons per 10

square mile block and that his/her use of decimal points in the explanation is inconsistent


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Student G reasons proportionally using a building down strategy resulting in a common

number of square miles, 20, thereby allowing for a direct comparison of the number of

raccoons in each town (Town A has 160 raccoons in 20 square miles, while Town B has

190 raccoons in 20 square miles). Notice, however, that Student Gs work completely

lacks units throughout and may be a cause of concern.

Student H:

Student H reasons proportionally by dividing the number of raccoons by the number of

square miles to find the unit-rate of raccoons per square mile in each town. Notice,


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Student I:

Student I reasons proportionally by recognizing the factor-of-change of3

2between the

sizes of Towns A and B. This student then applies the factor-of-change to the number of

raccoons in Town A by finding3

1of the raccoons in Town A and subtracting them from


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Problem 5: Sue and Julie were running equally fast around a track. Sue started

first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15

laps, how many laps had Sue run?

Problem 6: 3 U.S. Dollars can be exchanged for 2 British pounds. How many

pounds for 21 U.S. Dollars?

Can the traditional cross-products/cross-multiplication algorithm be applied to

both of these problems? Why or why not?

Problems from: K. Cramer, T. Post, S. Currier, Learning and Teaching Ratio and Proportion: Research Implications, p159

K. Cramer, T. Post, and S. Currier gave Problems 5 and 6 to 33 preservice elementary

education teachers enrolled in a mathematics methods course. Of the 33 preservice

teachers, 32 of them incorrectly applied the cross-products/cross-multiplication algorithm


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Hence another characteristic of proportional reasoning is that it should be reserved only

for those situations in which it is appropriate. In other words, proportional reasoning

requires discrimination between proportional and non-proportional situations.

In summary, S. Lamons definition of proportional reasoning, the ability to scale up and

down in appropriate situations and to supply justification for assertions made about

relationships involving simple direct proportions and inverse proportions (Lamon, 2005)

provides us with a succinct and useable definition that supports the thoughts and ideas

developed above.


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Categories of Problems Used to Encourage Proportional Reasoning

There are two broad categories of problems that are typically used to encourage

proportional reasoning. The first is the category ofmissing value problems illustrated

above with the example of the number girls to boys in a classroom. The second category

is known as ratio (or rate) comparison problems. The raccoon problem above is an

example of a ratio comparison problem. Another example of a ratio comparison problem

for you to solve is:

Problem 7:

Amy and Bryan are mixing paint. Amy mixes 2 quarts of blue paint with 5 quarts

of white paint. Bryan mixes 4 quarts of blue paint with 7 quarts of white paint.

Whose mix is more blue? Explain your reasoning.

In general, in a ratio comparison problem two ratios are given and the task is to determine

which is darker, lighter, faster, slower, more expensive, less expensive, stronger, weaker,


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Problem 8:

Alice ran more laps in more time today than she did yesterday. Did she run faster,

slower, or the same speed today as she did yesterday? Or is there not enough

information to compare her speed today with her speed yesterday?

Qualitative reasoning problems require thinking about questions such as, Is this answer

reasonable? As one quantity increases, what happens to the other quantity? One

advantage to using qualitative problems is that they require more than procedural

knowledge of an algorithm; they require reasoning. According to T. Post, M. Behr, and

R. Lesh, It is well known that experts in a wide variety of areas use qualitative

approaches to problems as a means to better understand the situation before proceeding to

actual calculations and the generation of an answer. Novices, however, tend to proceed

directly to a calculation or a formula without the benefit of prior qualitative analyses. It

should also be pointed out that novices often answer problems incorrectly, suggesting

that they could benefit from the use of qualitative procedures. (Post, Behr, & Lesh,


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Solutions to Problems:

Problem 1: If 3 balloons cost $6, 12 balloons will cost four times as much for a cost of

$24.

Problem 2: If 3 balloons cost $5, 10 balloons will cost3

13 times as much for a cost of

$16.67.

Problem 3: Pauls dog eats 20 pounds of food in 30 days or equivalently Pauls dog eats

1 pound in 1.5 days. Therefore, a 45 pound bag of dog food will last Pauls dog

days5.67pound

days5.1pounds45 =! .

Problem 4: Town A has

milesquare

raccoons860480 = . If Town B had 8 raccoons per square

mile, it would have raccoons320milesquare

raccoons8milessquare40 =! . However, Town B has 380


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Problem 6: 3 U.S. Dollars can be exchanged for 2 British pounds. Since 21 U.S. Dollars

is seven times 3 U.S. Dollars, seven times 2 British pounds will be received, or 14 British

pounds.

Problem 7: If Amy doubled her mix, she would mix 4 quarts of blue paint with 10 quarts

of white paint. Bryan mixes 4 quarts of blue paint with only 7 quarts of white paint.

Therefore, Bryans mix will be more blue (because there are fewer quarts of white paint

to dilute the same amount of blue paint).

Problem 8: If Alice ran more laps in more time today than she did yesterday, there is no

way to tell whether her running speed was faster, slower, or the same as it was yesterday.


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Bibliography

Cramer, K., & Post, T. (1993). Connecting research to teaching proportionalreasoning. Mathematics Teacher. 86(5), 404-407.

Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio andProportion: Research Implications. In D. Owens (Ed.),Research Ideas for theClassroom: Middle Grades Mathematics. (pp. 159-178) Reston, VA: NationalCouncil of Teachers of Mathematics & Macmillan.

Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essentialcontent knowledge and instructional strategies for teachers. Mahwah, New Jersey:Lawrence Erlbaum Associates.

Lesh,R., Post, T., & Behr,M. (1988). Proportional Reasoning. In J. Heibert & M.Behr (Eds.)Number concepts and operation in the middle grades. (pp. 93-118)Reston,VA: Lawrence Erlbaum & National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (NCTM).Principles and Standards forSchool Mathematics. Reston, VA.:NCTM, 2000.

Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of pre-algebra understanding. In A.F. Coxford & A.P. Schulte (Eds.), The ideas of algebra,K-12 (1988 Yearbook of the National Council of Teachers of Mathematics, pp. 7890). Reston, VA: NCTM.


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THINKING THROUGH A SESSIONPROTOCOL (TTSP) Part 1: Setting up the Task/Session

What are your mathematical and pedagogical goals for the task/session?

In what ways does the task build on participants previous knowledge and experiences?

How will you help participants make these connections?

What are all the ways the task can be solved?


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Part 2: Supporting Participants Exploration of the Task/Activity

As participants are working independently or in small groups:What might you do (questions, suggestions, directions, etc.) to focus their participants

thinking on the key mathematical ideas/concepts of the task?

What will you see or hear that lets you know how participants are thinking about the

mathematical ideas or aspects of practice?

What assistance will you give or what questions will you ask participants who become

frustrated or finish the task almost immediately?


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Part 3: Sharing and Discussing the Task/Activity

Which solution paths do you want to have shared during the discussion?

Which common misconceptions do you want to discuss publicly?

What specific questions will you ask so that participants will make sense of the

mathematical ideas that you want them to learn?

What specific questions will you ask so that participants will make connections among the


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PLAN FOR MMSTLC SESSION 10: PRACTICEFACILITATIONMaterials Handouts

To do before session Basic sketch of session activities

Welcome participants, agenda of the session. (X min)

Session Goals:


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Time Activity/Task Detail Notes

Welcome/Agenda w/ goals:

Documents

Combined File 013009 Gaylord