Proportions

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Proportions (Pedagogical Content Knowledge Project)

  • by Amy Baril, Samantha Hrabe, Kristen Jackson, Christina Kwon, Kesarin Sripinyo, and Kathy Truong. (UC Irvine, July 2009)


Contents

Cognition and Learning Background

Proportional reasoning is the ability to compare ratios. It is one of the most important goals of the 5-8 grade curriculum. It has been referred to as the capstone topic of the elementary mathematics curriculum and the basis of algebra and beyond (Lesh, Post, & Behr, 1987).

Proportionality is one of the higher order understandings as proportional reasoning is one of the most advanced elementary skills in students' mathematical development (Lesh, Port, & Behr, 1988)

Lamon (2005) states that proportional reasoning is one of the best indicators that a student has attained understanding of rational numbers and related multiplicative concepts. With the understanding of proportional reasoning, students will have a better foundation for future complex concepts in higher grade levels (Lamon, 2005).

Why do we need to teach proportional reasoning?

  • In the past, proportional reasoning referred to working with rational number concepts and contexts. Without proper instruction, conceptual understanding about the topic was very difficult as it continued to be a by-product of a fractions lesson (Lamon, 2005).
  • Lamon makes an interesting point when he states that, "The fact that a large portion of the adult population does not reason proportionally suggests that certain kinds of thinking do not occur spontaneously and that instructions needs to take an active role in facilitating thinking that will lead to proportional reasoning (Lamon, 2005).
  • In order for students to develop a conceptual understanding for proportional reasoning, teachers should understand that it requires more than a brief introduction to part-whole fractions and the computational procedures because students need more time to construct their way of thinking (Lamon, 2005).

Background Knowledge

Multiplicative reasoning needs to be mastered in order to advance to proportional reasoning. (Singh, 2000) One of the most difficult tasks for children is understanding the multiplicative nature of the change in proportional situations. Students who are unable to tell the difference tend to utilize additive transformations because their reasoning may be cued by the wrong key words in the problem (Lamon, 2005). In order to reason proportionally, there needs to be a foundation of associated mathematical topics, particularly multiplication and division, fractions, and fractional concepts of order and equivalence (Vergnaud, 1983; English & Halford, 1995; Behr, et al., 1992, as cited in Dole et al., 2008)

  1. In second grade, students learn how to multiply and divide and learn to recognize when to apply the appropriate mathematical operations in simple, one step word problems. For an example, students would be able to solve "If 1 pound of apples costs 2 dollars. How much do 3 pounds of apples cost?”
  2. Then in the third and fourth grade and so on, proportional reasoning is introduced as they are exposed to more "proportionality problems" with a missing-value structure, which means that in the word problems, three numbers are known, while the fourth must be solved for its value. An example for this type of question is “10 eggs weigh 600 grams. What is the weight of 30 eggs?” Often times, students learn to solve these types of problems by using a “rule of three” such as “1 egg weighs 60 grams, so 30 eggs weigh 1,800 grams” (Van, De Bock, Hessels, Janssens, & Verschaffel; 2005).

Developmental Trajectories and Frameworks

  • NCTM Curriculum Standards (1989) noted that proportional reasoning "was of such great importance that it merits whatever time and effort must be expected to assure its careful development."
  • Piaget's Theory of development
    • Proportional reasoning is the key characteristic of the formal operations stage of development because it involves understanding the “relation between relations” (Inhelder & Piaget, 1958)
    • Piaget et al. (1968) describes this early stage of proportional understanding as “pre-proportionality,” because of “the true understanding of proportion as a formal operation task” (Hart and CSM team, 1981, p. 90).
  • How to help children develop proportional thought processes (Van De Walle, 2007)
    • Provide ratio and proportion tasks in a wide range of contexts
      • Many scholars have argued for the use of so-called model eliciting activities in the classroom wherein students can invent, extend, revise, and refine many of the important mathematical ideas throughout the mathematics curriculum (Lesh & Doerr, 2003; Verschaffel et al., 2000).
      • model-eliciting activities require students to develop, in a more powerful manner, the intended conceptual tools and mathematical ideas by going through a series of modeling cycles in which the givens, goals, and relevant solutions of a certain problem situation are continuously reinterpreted, re thought, and renegotiated (Van, De Bock, Hessels, Janssens, & Verschaffel; 2005).
      • Provide authentic mathematical tasks that is plausible and within reach (Ben-Chaim, Keret, & Ilany, 2007)
    • Encourage discussion and experimentation in predicting and comparing ratios. Help students distinguish between proportional and nonproportional comparisons by providing examples and discussing the differences.
    • Help students relate proportional reasoning to existing processes (e.g. using a unit rate for comparing ratios and solving proportions)
      • The example they gave relates to mixing 1 oz. of orange concentrate and 2 oz. of water, compared to mixing 2 oz of orange concentrate and 4 oz. of water. If the question is which mixture will taste stronger, the ratios should indeed be compared, but if the question is which mixture will make more, a ratio comparison is of course inappropriate. Cramer, Post, and Currier (1993 p. 160) argued that “we cannot define a proportional reasoner simply as one who knows how to set up and solve a proportion” and diagnosed that textbooks do not sufficiently emphasize the ability to discriminate linear and non-linear situations.
    • Recognize that symbolic and mechanical methods for solving proportions do not help proportional reasoning and should not be introduced until students have had many experiences with intuitive and conceptual methods.


Characteristics of Proportional Thinkers (Lamon, 1999)

  • Proportional thinkers have a sense of covariation. They understand relationships in which two quantities vary together and are able to see how the variation in one coincides with the variation of another
  • Proportional thinkers recognize proportional relationships as distinct from nonproportional relationships in real-world contexts
  • Proportional thinkers develop a wide variety of strategies for solving proportions or comparing rations, based on information strategies rather than algorithms.
  • Proportional thinkers understand ratios as distinct entities representing a relationship different from the quantities they compare.
  • Proportional thinkers are able to differentiate between additive and multiplicative situations and apply whichever transformation is appropriate (Lamon, 2005).
  • Proportional thinkers are able to construct and solve algebraic proportions (Lamon, 1993)

Cognitive Obstacles and Common Misconceptions

Cognitive Obstacles

  • Current mathematics instruction practices in elementary school cause students to have a “routine expertise” instead of “adaptive expertise” (Hatano, 1988)
    • Students start to over-generalize the validity and relevance of some well trained procedures and begin to transfer them to settings to which they are neither relevant nor valid (Van, De Bock, Hessels, Janssens, & Verschaffel; 2005). This
    • The problem is partly attributed to teachers and textbooks prematurely introducing students to memorization of the cross-multiplication algorithm (Vanhille & Baroody, 2002)
    • Pupils tend to overgeneralize proportional methods and learn to apply them on the basis of superficial problem characteristics (Dooren, Bock, et. al, 2006)
  • Children’s difficulty with proportional reasoning in the context of conventional fractions leads to the belief that proportional reasoning is a late development (Boyer, Levin, & Huttenlocher, 2008)

Common Misconceptions

  • Students often use additive reasoning in solving problems where multiplicative reasoning is required (Hart, 1981, 1988; Karplus, Pulos & Stage, 1983b, Noelting, 1980b; Vergnaud, 1988; Lamon 1993a; Resnick & Singer, 1993)
  • Students over-use proportional methods (Dooren, Bock, et. al, 2006)
    • The tendency of students to apply properties of linear relations ‘anywhere’-thus also in situations where this is inadequate- has also been called the ‘illusion of linearity’, the ‘linearity trap’, the linear obstacle’, etc. (Bock, Van Dooren, Janssens, & Verschaffel; 2007).
    • In the upper grades of elementary school, students seem to learn to use rather superficial cues (e.g., the missing-value structure of a word problem) to decide on the solution scheme that has to be applied to solve a problem (Van De Bock, Hessels, Janssens, & Verschaffek; 2005).
    • Even when proportionality is manifestly not applicable for that problem, students sometimes tend to apply proportional methods whenever the missing-value character of the problem statement makes this possible (Van, De Bock, Hessels, Janssens, & Verschaffel; 2005).
    • tendency among 12- to 16-year-old students to believe that if a figure enlarges k times, the area and volume of that figure are enlarged k times too (De Bock, Van Dooren, Janssens, & Verschaffel, 2002; De Bock, Verschaffel, & Janssens, 1998, 2002; De Bock, Verschaffel, Janssens, Van Dooren, & Claes, 2003; Van Dooren, De Bock, De Bolle, Janssens, & Verschaffel, 2003; Van Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2004
  • Many people who have not developed their proportional reasoning ability have been able to compensate by using rules in algebra, geometry, and trigonometry courses, but, in the end, the rules are poor substitute for understanding (Lamon, 2005).
  • Children systematically misinterpret traditionally notated fractions (e.g., 3⁄4) and estimate that fractions with larger denominators are quantitatively greater than fractions with smaller denominators (e.g., 4/6 < 4/8). (Boyer, Levin, & Huttenlocher, 2008)
  • Proportional reasoning may be hindered by counting and the overextension of numerical counts to proportional problems (Boyer, Levin, Huttenlocher, 2008)
  • Children’s difficulties stem at least partly from their propensity to compare quantities on the basis of the number of elements in the target quantity rather than on the basis of proportional relations. (Boyer, Levin, & Huttenlocher, 2008)
  • Children have difficulty solving proportional reasoning problems when both the target and the choice proportions are represented with discrete units, but they perform significantly better on problems for which the target, the choice alternatives, or both are represented with continuous amounts. (Boyer, Levin, & Huttenlocher, 2008)

Pedagogical Tools and Strategies

Developing Children's Conceptual Understanding

  • The concepts of proportions and ratios do not develop in isolation. (Lo and Watanabe, 1997)
    • Ratios and proportions are a part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers
    • Development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field
    • Proportional reasoning is a second-order relationship that involves an equivalent relationship between two ratios (Piaget and Inhelder, 1975)
    • Proportions and ratios are an important topic in school mathematics, and have long been a focus of mathematics education research and much has been learned about students' errors and difficulties in solving ratio and proportion tasks
  • Ratio and proportion: Connecting content and children’s thinking (Lamon, 1993)
    • Prior knowledge and a repertoire of sense-making tools helped students to produce creative solutions to problems involving ratio and proportion
    • Exploring ratios in many different settings and contexts will help create the foundation for the students to be able to apply to other concepts
    • Instructionally based research and development based on the cognitive dimensions of proportional reasoning and their connections to semantic problem types may be the most useful route for discovering the best ways to facilitate more sophisticated student thinking in this domain of proportions and ratios, while allowing the students to build on and extend their informal knowledge.
  • Proportional reasoning plays such a critical role in a student's mathematical development that it has been called a watershed concept, a cornerstone of higher mathematics, and the capstone of elementary concepts (Lesh, Post, & Behr, 1988)

Instructional Approaches

  • Teach proportions and ratios using Mulitplication Tables (Fuson, K. C. & Abrahamson, D., 2004)
    • Introduce ratios to students using multiplication stories involving repeatedly adding some amount
      • Sample Problem: "Every day Robin puts $3 in her kitty bank and Tim puts $7 in his doggy bank."
    • The Multiplication Table (MT) is a widely used to display the products of numbers 1 through 10. A multiplication table can be used to make a ratio table by using two columns from the multiplication table and putting them together.
      • Sample: Multiplication table and Ratio Table Image.jpg
    • Ratio tables are vertical or horizontal formats that record the result of the repeated coordinated additions for a ratio pair by using two columns.
    • A proportion is made from any two rows of a ratio table because they are each multiples of a basic ratio
    • To help students solve an unknown proportion, write the proportion as a mini-MT with one empty cell with only three values in it
      • Sample: MT Puzzle
        • Picture11.jpg
    • Given any proportion problem students can set up an MT puzzle to solve for the unknown proportion problem.
    • Use middle-difficulty problem numbers to show ratio pairs that are not multiples of each other but multiples of the smallest basic ratio
      • (i.e. 3:7 such as 6 to 14 and 15 to 35)


  • Teach Proportions Using Multiple Strategies (Stemn, B.S., 2008)
    • Students need to learn proportional and nonproportional situations and what this concept is about before they are taught the cross multiplication algorithm
    • Begin the unit with an investigational approach in order to extend the students' knowledge and help them solve problems in other fields. It also provides students the opportunity to apply mathematics to everyday situations and the application of mathematics to their lives
      • Sample Investigational Task for Proportions: If you brush your teeth twice a day and you leave the tap running each time, how many gallons of water will be wasted? If this trend continues how many gallons of water will be wasted in 5 days, 9 days, 31 days and any number of days? If a gallon of water cost $0.89, how much is the cost of the water wasted in 31 days?
    • Have the students work in small groups to discuss and write how they should go about finding out the actual amount of water wasted and identify the factors that could influence their results. After 10-12 minutes of group discussion, bring the class back together and have each group present their suggestions.
    • Students find their data at home prior to the next class.
      • Build relationships and patterns
        • Have the students use their personal derived data to create a table as a class. Based on the data of the class make a ratio statement. (i.e. 2 nonstandard cups of water is equal to 1/4 gallon of water) The students use this relationship to compute the quantity of water wasted in gallons.
        • Students work in small groups to discover how many gallons were wasted if 3, 4, or 5 cups of water were wasted.
        • Teacher creates a chart on the board once the groups have discovered the pattern.
          • Picture2.jpg
      • Proportional and nonproportional relationships
        • Challenge students to write the symbolic representation of the relationship between two quantities.
        • Guide the students through the process by asking questions
          • Sample Questions: What does it mean to times the number of cups of water by one-eighth? what if the number of gallons of water wasted is represented by the letter G, and the number of cups of water wasted was N, where Nrepresents any number of cups of water wasted, write an equation to represent this relationships?
          • G=Picture1.jpg
        • To relate the proportional concept to prior knowledge link the content to linear equations such as
      • Missing Value Method
        • After students have been taught/introduced to the concepts related to proportions it is time to teach the algorithms.
        • This method shows proportions between relationship and within relationship meaning the relationship between the numerators of the two ratios and between the two denominators is compared, but the within relationship is comparing the numerator and denominator of the same ratio
          • Sample Problem to present: If two nonstandard cups of water is equivalent to one-quarter of a gallon, then 18 cups will be equivalent to how many gallons?
            • Picture3.jpg
        • Students solve for the missing "x" value.
        • Give students time to invent their own strategies and draw on their past experiences in order to solve.
        • Draw students attention to solving the "between relationship"-- balance the equation to find a number that you times by one-fourth and the same number times 2 will give you 18
        • Draw students attention to solving the "within relationship"-- what must be multiplied by 18 to balance the equation
        • Practice other missing value method proportion problems that allow students to solve the different problem sets.
      • Cross Multiplication
        • Final method to introduce to students
          • Show students:Picture4.jpg
        • Show students that to solve for 'x' you multiply both sides of the equation by 1/4. Equation is reduced to:
          • Picture6.jpg
        • Multiply both sides by 4. Equation is: 8x=18.
        • To solve for 'x' multiply both sides by 8. x= 9/4
        • After practicing this method with the students, introduce the cross-multiplication method.
        • Multiply the 18 by 1/4 and the 2 by x.
          • Picture8.jpg
        • The next step would be to divide both sides by 2 which is the same as multiplying by 1/2.
          • Picture9.jpg
        • Simplify the equation:
          • Picture5.jpg
        • So 18 cups is equivalent to 9/4 gallons or 2 1/4.
      • Students should first learn about proportional and nonproportional situations and what this concept is before students learn the cross-multiplication algorithm in order to assist the students to use reasoning to solve the problems not a procedure (Lamon, 1999)


  • Teach Proportions with Equivalent Fractions
    • In order for students to understand proportional situations, they must be able to recognize when different quantities represent the same ratio (Van de Walle).
      • To solve this type of proportional situation, the students must apply the same ratio seen in one quantity to each of the other proportional situations.
      • Students can use prior knowledge of finding equivalent fractions to help solve proportions in this type of proportional situation.
        • Example from Van de Walle (p 355)
          • 3 : 9 = 4 : 12 or 3/9 = 4/12

“These might read “3 is to 9 as 4 is to 12” or “3 and 9 are in the same ratio as 4 and 12”


  • Teach Proportions Using Hands-on Experience
    • Using miniature toys was shown helpful in creating a meaningful understanding of the concept (Drum & Petty, 2001).
    • Students are introduced to proportions in a fun and interactive manner
    • Teacher can model on the overhead using toys with easily recognizable shapes
    • Students are able to experience real world scenarios
    • Toys can be purchased for a fairly small amount of money
    • Toys are available in a wide variety of sizes and styles allowing for real life examples

Academic Language

    • Math Expressions, 2006. and Holt California Mathematics Course 1: Numbers to Algebra Online Edition ©2008
  • Denominator: The bottom number in a fraction that shows the total number of equal parts in the whole.
  • Equivalent ratios: ratios that name the same comparison
  • Numerator: The number above the line in a fraction. The numerator represents how many

pieces of the whole that are discussed.

  • Product: the result of a multiplication expression; factor x factor = product
       Example: 3 x 4 = 12, 12 is the product
  • Proportion: if two ratios are equivalent, they are said to be proportional to each other or in proportion
  • Ratio: a comparison of two numbers


  • Fraction: Picture 1.jpg


Sample Problems and Problem-Solving

  • 1. Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.

Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to 15". Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

        *odds notation:  15 : 20
        *fractional notation:  15/20

You should be able to recognize all three notations; you will probably be expected to know them for your test. Given a pair of numbers, you should be able to write down the ratios. For example:

  • There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.
   16:9 ; 16/9 ; 16 to 9

Consider the above park. Express the ratio of geese to ducks in all three formats.

   9:16 ; 9/16 ; 9 to 16

The numbers were the same in each of the above exercises, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.


  • 2. Example of Cross- Multiplication

Proportions wouldn't be of much use if you only used them for reducing fractions. A more typical use would be something like the following:


Picture 2.jpg


This is not standard notation, but it can be very useful for setting up your proportion. Clearly labelling what values are represented by the numerators and denominators will help you keep track of what each number stands for. In other words, it will help you set up your proportion correctly. If you do not set up the ratios consistently (if, in the above example, you mix up where the "ducks" and the "geese" go in the various fractions), you will get an incorrect answer. Clarity can be very important.

  • 3. Shadow Casting

A building casts a 103-foot shadow at the same time that a 32-foot flagpole casts as 34.5-foot shadow. How tall is the building? (Round your answer to the nearest tenth.)

Picture 14.jpg

Curricula and Technological Resources

Lessons

Literature

  • If You Hopped Like a Frog by David Schwartz (1999)
  • Counting on Frank by Rod Clement (1991)
    • There are three parts in the book that talk about fantasies of proportions. "If I had grown at the same speed as the tree - 6 feet per year - I'd now be almost 50 feet tall!"
  • The Borrowers by Mary Norton (1953)
    • Little people are living in the walls of a house and all their furnishings have been created from full-sized things in the human world.
  • "One Inch Tall" by Shel Silverstein (1974)
    • Readers are asked to imagine being 1 inch tall.
  • Cut Down to Size at High Noon by Scott Sundby
    • Western themed - teaches about proportions and scale drawings
  • If the World Were a Village: A Book about the World’s People by David J. Smith
    • Statistics about the different languages spoken in the world, age distribution, religion, air and water quality, etc

Technology Resources


Annotated References

Ben-Chaim, D., Keret, Y., Ilany, B. (2007). Designing and Implementing Authentic Investigative Proportional Reasoning Tasks: The Impact on Pre-Service Mathematics Teachers' Content and Pedagogical Knowledge and Attitudes. Journal of Mathematics Teacher Education, 10, 333-340. - Proportional reasoning authentic investigative tasks were created, implemented and evaluated on the math content and pedagogical knowledge of pre-service elementary and middle school mathematics teachers. Results showed that the application of a special teaching model lead to a significant positive change in pre-service teachers' mathematical pedagogical content knowledge.

Boyer, T.W., Levine, S.C., & Huttenlocher, J. (2008). Development of proportional reasoning: Where young children go wrong. Developmental Psychology, 44(4), 1478-1490. - This article examines misconceptions and cognitive obstacles that students face when solving problems with proportional reasoning. It also examines where children go wrong in processing proportions that involve discreet quantities. Students are overextending their understanding of numerical and additive equivalence to proportional equivalence problems.

Cramer, K. and Post, T. (1993) Connecting research to teaching proportional reasoning. Mathematics Teacher 86-5 , pp. 404-407. This article is a study about how to teach proportional reasoning and strategies that describe and show how to assess students' proportional reasoning.

Dole S., & Shield, M. (2008). Proportion in middle-school mathematics: It’s everywhere. Australian Mathematics Teacher, 64(3), 10-15.- Proportional thinking type problems and their structure are compared to other problems.

Dole, S., Wright, T., Clarke, D., Hilton, G., Roche, A. (2008). Eliciting growth in teachers’ proportional reasoning: Measuring the impact of a professional development program. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia. - The authors of this paper examine the connections between mathematics and science and try to integrate proportional reasoning across the curriculum through authentic investigative task. The study focus of this paper was on the teacher's knowledge of proportional reasoning as well as measuring the growth of teacher knowledge as a result of participating in a professional development program.

Dooren, W.V., Bock, D.D., Evers, M., & Verschaffel, L. (2006). Pupils’ over-use of proportionality on missing-value problems: How numbers may change solutions. – The article discusses the over-use of proportional methods amongst primary school students when solving non-proportional missing-value word problems.

Fujimura, N. (2001). Facilitating Children's Proportional Reasoning: A Model of Reasoning Processes and Effects of Intervention on Strategy Change. Journal of Educational Psychology, 93(3), pp. 589-603. A study which examines a model for solving proportional situations with the use of two intervention strategies based on student prior knowledge. Depending upon the type of proportional situation, different instructional approaches are necessary for student understand to occur.

Fuson, K.C., Abrahamson, D. (2004). Understanding ratio and proportion as an example of the apprehending zone. In J. Campbell, Handbook of mathematical cognition (213-234). Psychology Press. This chapter focuses on teaching students proportions by using methods according to the Apprehending Zone Model. A fifth grade class receives instruction on proportions and ratios through instruction that encourage critical thinking.

Inhelder, B. and J. Piaget (1958). The Growth of Logical Thinking from Childhood to Adolescence. New York: Basic Books. - This book tries to describe the changes in logical operations between childhood and adolescence as well as describe the formal structures that mark the completion of the operational stage of development. The authors describe proportional reasoning as a key characteristic of the formal operation stage.

Lamon, S. (1993). Ratio and proportion: Connecting content and children’s thinking. National Council of Teachers of Mathematics, 24(1), 41-61.- A case study of twenty-four 6th grade students in which student thinking was mainly analyzed using four semantic type problems critical to proportional reasoning: Well-Chunked Measures, Part-Part Whole, Associated Sets, and Stretchers and Shrinkers.

Lamon, S.J. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum. - This book discusses essential content knowledge and instructional strategies that teachers need to teach fractions and ratios for understanding. These strategies and ideas are the foundations for proportional reasoning. This book should be used as a resource for strategies to assist teachers in developing conceptual understanding.

Lamon, S.J. (2005) Teaching Fractions and More 2nd ed. Taylor and Francis, Inc. This book addresses the topic of proportional reasoning as it provides instructional strategies for teachers to use in their lessons. It also asks the teacher to reflect on their own understanding of proportional reasoning as they solve problems throughout the book. It also provides teachers with problems to ask their students in order to assess their understanding and ways of solving proportional problems.

Lawton, C. A. (1993). Contextual factors affecting errors in proportional reasoning. Journal for Research in Mathematics Education, 25(5), 460-466. A study of students solving proportional satiation problems found that solving problems where the same ratio is used on both quantities was easier for students to answer then nonproportional situations where the ratio relates to both quantities, but is not the same.

Lesh, R.A., Post, T.R., & Behr, M.J. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Erlbaum - This chapter examines several roles that representations and translations play in mathematical learning and problem solving. The authors look at five different types of representations in mathematics and problem solving.

Lesh, R., Post, T., & Behr, M. (1988). Proportional Reasoning. In J. Hiebert and M. Behr (Ed.), Number concepts and operations in the middle grades (pp. 93-118). Reston, Virginia: The National Council of Teachers of Mathematics. The main focus of the book is to clarify and describe the goals of instruction and how it impacts the individual student and mathematical learning.

Lo, J.J., Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28(2), 216-236. Case study of a fifth grader who describes his conceptual understanding and methods used to solve ratios and proportions. Methods are analyzed to study the developmental process of how the concepts of ratios and proportions are developed.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. The NCTM published this book to examine and evaluate the curriculum and standards for school mathematics.

Petty, W. G., Drum, R. L. (2001). Miniature toys introduce ration and proportion with a real-world connection. Mathematics Teaching in the Middle School 7-1. 50-54. Researchers discuss how using miniature toys during proportion lessons help to give students a deeper and more meaningful understanding of proportions based on real life situations.

Piaget, J. & Inhelder, B. (1975) The Origin of the Idea of Chance in Children. Routledge and Kegan Paul PLC. The book presents the research of Piaget and Inhelder based on the cognitive development theory to describe the research found for teaching probability in the elementary school.

Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43, 271-292.- The article covers a case study of two 6th grade students and their ability to reason proportionally.

Stapel, Elizabeth. "Solving Proportions: More Examples." Purplemath. Available from http://www.purplemath.com/modules/ratio7.htm. Accessed 11 August 2009 This website provided the sample problems for proportions used in the Wiki. The website modeled example problems such as cross-multiplication, geometric shapes, and finding the unknown.

Stemn, B.S. (2008).Building middle school students' understanding of proportional reasoning through mathematical investigation. Education 3-13, 36, 383-392. This article focuses on how teach seventh grade students proportion by describing specific strategies and methods based on new pedagogical instruction.

Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally 6th ed. United States of America: Pearson Education, Inc. This is a resource text for elementary and middle school mathematics teachers. It provides cognitive background information, activity ideas, as well as resources for the classroom teacher for mathematical topics in the elementary and middle school mathematics curriculum.

Van D.W., De Bock, D., Hessels, A., Janssens, D; Verschaffel, L. (2005). Not Everything is Proportional: Effects of Age and Problem Type of Propensities for Overgeneralization. Cognition and Instruction 23-1, 57-86. This article states that students tend to overrely on proportional methods when working in other domains of mathematics where it is not applicable. The authors further investigate how students develop their misapplication for proportional reasoning.