Combined File 013009 Gaylord

Overview of Session 10 Designing Professional Development in Mathematics

• Welcome • Consider “What is a Proportion?” • Work on a mathematical task • Analyze OGAP tasks and student thinking

Michigan Mathematics and Science Teacher Leadership Collaborative

• 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 the article “What is a Proportion? What Does it Mean to be Proportional? What is Proportional Reasoning?”

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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 task

• A, A’, and A” (Paul’s Dog) • B, B’, and B” (Racing Track) • C, C’, and C” (Paper Towel)

Organizing for Facilitation • Thinking Through a Session Protocol:

Whole group • Planning facilitation • A, A’, A’’, B, B’, B’’, C, C’, C’’ • Next Session: A

• Analyzing student thinking

• A, A’, A”, B, B’, B”, C, C’, C” • Sharing analysis on student thinking

B

A’ C

B’

A’’ C’

B’’

C’’

• A, A’, and A” • B, B’, and B” • C, C’, and C”

Wrap up • Expectations for February 20 • Bring five samples of student work on a high level task • Read Chapter 2 from Peg Smith’s 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 is

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

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 ratios

2 girls 12 girls = 3 boys x boys

could use part-to-whole ratios

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

2 girls 12 girls = 5 students x students

to determine how many total students

are in the classroom. Notice that in both of the previous examples three of the four values in each proportion are known and the fourth value is unknown. These types of problems are typically called missing value problems.

When computing equivalent ratios, the backbone of proportions, we use multiplication or division rather than addition or subtraction. For example, to compute ratios that are equivalent to

12 we can either multiply both the 12 and 15 by the same number, or divide 15

both the 12 and 15 by the same number. For example,

12 ! 4 48 12 ÷ 3 4 = , and = are both 15 ! 4 60 15 ÷ 3 5

Copyright 2006 Vermont Mathematics Partnership. Please do not copy or distribute materials without written permission from the Vermont Mathematics Partnership: www.vermontmathematics.org The Vermont Mathematics Partnership is funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

equivalent to

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

number from both the 12 and 15 the resulting ratios are not equivalent to example, neither

12 . For 15

12 + 45 57 12 ! 10 2 12 = = are equivalent to nor . Therefore, we say that 15 + 45 60 15 ! 10 5 15

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)

In every proportion there are two multiplicative relationships: the multiplicative relationship within each ratio and the multiplicative relationship between the two ratios. Using the example above, the multiplicative relationship within the ratio

2 girls is 3 boys

1.5 or

3 boys , since the number of boys is 1.5 times the number of girls.* Since equivalent 2 girl

ratios have the same multiplicative relationship within each ratio, we can use this relationship to determine the missing value in the proportion

2 girls 12 girls = 3 boys x boys

. To

determine the number of boys in the classroom, we simply need to compute

2

12 girls ! 1.5

12 girls x boys

boys = 18 boys . girl

The multiplicative relationship between the ratios

2 girls 3 boys

and

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 3 boys ! 6 = 18 boys in the classroom. Notice that in this example the multiplicative relationship within the ratios is non-integral (1.5

boys girl

) whereas the multiplicative relationship between the ratios is

integral (6).

*

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

0.6666 or

**

2 girls 2 , since the number of girls is times the number of boys. 3 boy 3

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

0.1666 or

1 1 because the number of girls in the sample is of the number of total girls in 6 6

the classroom.

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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/or between 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 1 3

(3 times as many balloons) ratios. Often students that successfully use proportional reasoning to solve the first problem will inappropriately revert to additive reasoning on the second problem because of the increased difficulty in the numerical relationships. For example, in Problem 2 they might reason that since the difference between the number of balloons and the cost in the first ratio is 2, that the difference in the second ratio must also be 2 for a total cost of $12 in the second ratio…or they might reason that because there are 7 more balloons in the second ratio than in the first ratio, the cost must also increase by $7 resulting again in a total cost of $12 in the second ratio.

Therefore,

it is important to remember that evidence of proportional reasoning on one problem does

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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 quantities x and y are said to be proportional or in proportion with each other if all ratios of the form

x y

(where x and y are nonzero* and form the ordered pair ( x, y ) ) are

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

x y

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 form

total cost # of gallons of gas

are

# $0.60 $8.40 $36.00 = = = K!! , and therefore we say % 0.5 gallons 7 gallons 30 gallons " &

equivalent to one another $$

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 y =

$1.20 x, gallon

where x

represents the number of gallons of gasoline purchased and y represents the total cost of the gasoline. In fact, another equivalent way of defining proportional relationships is that the quantities x and y are proportional if y = kx for some nonzero constant k. The constant k is known as the constant of proportionality. Notice that the constant of proportionality is the slope of the line y = kx , and it is the unit rate of y per unit of x. Notice also that from this alternative definition it follows that in a proportional relationship the quotient of the quantities remains constant, that is

y = k (where x and y x

are nonzero and form the ordered pair ( x, y ) ).

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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 the time (t) traveled [or d = r ! t ]. o If a scuba diver starts at sea level and descends 10 meters every 30 seconds, the diver’s height in meters above sea level (h) is proportional to his/her time in 1 3

seconds under water (t) [or h = ! t ]. 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), the length is proportional to the width [or l = 1.5w ]. o When 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 amount of orange juice concentrate (j) used [or w = 2.5 j ]. o If the exchange rate between Euro and US $ is 1 Euro = 1.3 US $, the number of Euro (r) received in an exchange is proportional to the amount of US $ (d) converted [or r =

1 Euro d 1.3 US $

].

o When rolling a fair die the number of fours predicted (f) is proportional to the 1 6

number of rolls (r) [or f = r ].

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*Why do we require that x and y be nonzero members of the same ordered pair? Notice that, in any proportional relationship, that is any relationship in which the two quantities are related by the equation y = kx , the ordered pair (0, 0) satisfies the equation since 0 multiplied by any constant, k, is 0. Therefore in any proportional relationship the ratio

0 units of x 0 units of y

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

equivalent to one another, since the fraction

are

0 is undefined.** 0

**Notice that this provides one of many good explanations of why Suppose that

x y

0 is undefined. 0

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

are an infinite number of possibilities for the constant k in the equation y = kx , and the ordered pair (0, 0) satisfies all of these, there are an infinite number of possible equivalence classes to which

0 could belong. For example, if k=3, since both (0, 0) and 0 0 0

(1, 3) satisfy the equation y=3x , could belong to the equivalence class containing while if k=2, since both (0, 0) and (1, 2) satisfy the equation y=2x, equivalence class containing and

1 ; 3

0 could belong to the 0

1 1 . By transitivity, this would, however, then imply that 3 2

1 belong to the same equivalence class, which we know is untrue. As in many other 2

cases, attempting to define

0 0 results in a contradiction therefore is undefined. 0 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 not reason 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 reasoners. We believe that this is a limited perspective, a necessary but not a sufficient condition, especially since these problems lend themselves to purely algorithmic solutions.” (Post, Behr, & Lesh, 1988) Proportional reasoning certainly requires that reasoning about proportional relationships occur. Therefore, proportional reasoning requires more than a simple application of a rule or procedure to solve proportions; it requires flexibly solving problems involving proportional situations, with meaning and understanding. Proportional reasoners have several methods, including the standard algorithm (cross-products/cross-multiplication), for solving problems involving

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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 missingvalue problem. (For more information on missing-value problems, see p. 1 of this essay.) Problem 3:

Paul’s dog eats 20 pounds of food in 30 days. How long will it take Paul’s dog to eat a 45 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 A’s incorrect use of the equality symbol in the “run-on equation” 20 + 20 = 40 + 5 = 45 , 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 0.666 to 0.66 , and then rounds the result of 45 ÷ 0.66 to 68 with an overall result of 68 days instead of the more accurate 67.5 days. Student C:

Student C reasons proportionally by building up both the amount of dog food and the number of days to find that 40 lbs. of dog food will last 60 days. This student then builds down the rate of 20 lbs. for 30 days to 5 lbs. in 7.5 days. Student C then combines these results to arrive at the correct answer of 67.5 days.

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Student D:

Student D reasons proportionally by recognizing the factor-of-change of 1

1 within the 2

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:

Student E reasons proportionally by applying the cross-products/cross-multiplication algorithm to find the missing value, the number of days 45 pounds of food will last.

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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 dayspounds 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 60 square miles 480 raccoons

Town B 40 square miles 380 raccoons

Karl says that Town A has more raccoons per square mile. Josh says that Town B has more raccoons per square mile. Who is right? Justify your answer.

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

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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 F’s 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. (“Town B has 95 raccoons per square mile. 9.5 is more than 8.0, so Josh is right.”) Student G:

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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 G’s 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, however, that Student H’s work lacks units throughout, so there is some question about whether Student H fully understands the problem and the results of his/her calculations, or arrived at the correct solution by fortuitously selecting the bigger quotient.

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Student I:

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

2 between the 3

sizes of Towns A and B. This student then applies the factor-of-change to the number of raccoons in Town A by finding

1 of the raccoons in Town A and subtracting them from 3

480, the total number of raccoons in Town A, to determine that

2 of the raccoons in 3

Town A is 320 raccoons. This allows Student I to see that, “40 square miles of town A only has 320 raccoons” and to directly compare the number of raccoons in 40 square miles of each town.

For more information on the use of models, unit-rates, factors-of-change, and/or building up/down strategies see the essay “Multiple Ways to Solve Proportional Reasoning Problems.”

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It is very possible for a student, or an adult, to solve proportions correctly by following a procedure without reasoning proportionally. In general, application of the traditional algorithm (cross-products/cross-multiplication) without supporting evidence (e.g., use of appropriate units throughout or justification of the procedure) provides little evidence of any type of reasoning, much less proportional reasoning. In fact, R. Lesh, T. Post, and M. Behr state that the traditional algorithm is, “often used by students to avoid proportional reasoning rather than to facilitate it.” (Lesh, Post, & Behr, 1988)

All too often the traditional algorithm is introduced procedurally without first developing a conceptual understanding of proportional relationships. T. Post, M. Behr, and R. Lesh add, “Unfortunately we sometimes confuse efficiency and meaning, and by default, even with the best intentions, we introduce a concept in the most efficient but least meaningful manner.” (Post, Behr, & Lesh, 1988) K. Cramer, T. Post, and S. Currier state that, “Teachers need to step outside the textbook and provide hands-on experiences with ratio and proportional situations. Initial activities should focus on the development of meaning, postponing efficient procedures until such understandings are internalized by students.” (Cramer, Post, & Currier, 1993) Premature introduction of the traditional algorithm frequently leads both adults and students to apply it both in appropriate and in inappropriate situations. Before proceeding, try to solve the following problems.

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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 in Problem 5, while all 33 correctly applied the cross-products/cross-multiplication algorithm in Problem 6. None of the 33 preservice teachers could explain why Problem 6 represented a proportional situation but Problem 5 did not. (Cramer, Post, & Currier, 1993) Typically the traditional algorithm is introduced as a means of solving missing value problems in a very efficient manner. Therefore, it is not surprising that it is common for adults and students to apply the traditional algorithm to any problem in which three values are given and the question asks for the fourth value to be found, even if the situation is non-proportional. (For a complete discussion of the first problem, see the section “What are Some Examples of Non-Proportional Relationships?”) 19

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. Lamon’s 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 of missing 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, more dense, less dense, etc. A wide variety of missing value problems and ratio comparison problems can be found in most middle level mathematics programs. Typical contexts include concentrations, density, consumption, production, packing, similarity, scale, percents, probability, conversion, etc. Proportional reasoning problems are also found in a third, commonly overlooked category known as qualitative reasoning problems. These problems do not involve any numbers, but instead require reasoning about a situation and the relationship between the quantities involved to answer a question. An example of a qualitative reasoning problem for you to solve is:

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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, 1988)

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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.

1 3

Problem 2: If 3 balloons cost $5, 10 balloons will cost 3 times as much for a cost of $16.67.

Problem 3: Paul’s dog eats 20 pounds of food in 30 days or equivalently Paul’s dog eats 1 pound in 1.5 days. Therefore, a 45 pound bag of dog food will last Paul’s dog 45 pounds ! 1.5

days = 67.5 days . pound

Problem 4: Town A has 480 ÷ 60 = 8

mile, it would have 40 square miles !

raccoons square mile

. If Town B had 8 raccoons per square

8 raccoons = 320 raccoons . square mile

However, Town B has 380

raccoons. Therefore Town B has more raccoons per square mile than Town A, making Josh correct.

Problem 5: For a complete discussion of the first problem see the section “What are Some Examples of Non-proportional Relationships?”

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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, Bryan’s 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 proportional reasoning. Mathematics Teacher. 86(5), 404-407. Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.), Research Ideas for the Classroom: Middle Grades Mathematics. (pp. 159-178) Reston, VA: National Council of Teachers of Mathematics & Macmillan. Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content 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 for School Mathematics. Reston, VA.:NCTM, 2000. Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of prealgebra 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. 78– 90). Reston, VA: NCTM.

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THINKING TH R OU GH A SESSION PROT OCOL (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?

What misconceptions might students have? What errors might they make?

© 2008, University of Michigan

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?

What might you do to encourage participants to share their thinking or to analyze the thinking of others?

© 2008, University of Michigan

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 different strategies that are presented?

What will you see or hear that lets you know that participants in the session understand the mathematical and pedagogical ideas that you intended for them to learn?

What records of practice will you have participants bring to the next session?

© 2008, University of Michigan

PLAN FOR MMSTLC SESSION 10: PRACTICE FACILITATION Materials

Handouts

To do before session

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

Session Goals:

Time

Activity/Task Welcome/Agenda w/ goals:

Detail

Notes