# Ratios, Proportions, and Percents

- by Lauren Bertoglio, Megan Fairley, Christine Ofrecio, and Michelle Sanchez (U.C. Irvine, August 2009)

## Contents |

## Cognition and Learning Background

### Methods of Teaching

#### Ratios

Students introduced to ratios are typically moving from a concrete conceptual understanding of fractions, to a more abstract representation of fractions. It is not uncommon for students to mistakenly use the terms ratio and proportion interchangeably. Ratios are comparisons of two numbers using division, unlike proportions which are comparisons of two equal ratios (Drum & Petty, 2001).

In understanding ratios, students must understand four key ideas: being able to differentiate between multiplicative associations and additive associations; recognizing that ratios compare two quantities using division; being able to recognize that ratios can be expressed in three ways, 2:3, 2 to 3, or 2/3; and the significance of the units in shaping the ratios (Lo, et al, 2004).

#### Proportions

Students traditionally develop and sharpen proportional reasoning skills during late elementary school through middle school. Proportional reasoning adeptness provides later support for algebraic and science concepts at the secondary school level(Thompson & Bush, 2003).

The traditional method for teaching proportions generally requires students to find the cross products. The process of forming cross products directs students to cross multiply and divide. This has been seen as ineffective since math professionals view proportional reasoning as a way of conceptualizing, not another algorithm for students to commit to memory (Thompson & Bush, 2003).

#### Percents

Teachers have used a number of representations to assist students in comprehending percentages. These representations have included pie charts, decimals, fractions, and a variety of geometric shapes with portions shaded. The difficulty with teaching percents using these various representations is that the true meaning of percent can be lost in the surface features of the graphic or numeric depictions (Parker, 2004). It is important to promote learning that is meaningful to students.

There are two common ways to numerically solve percent problems: the equation/mathematical sentence (4n = 12) or using a ratio/proportion set-up (3/5 = ?/100). NCTM Principles and Standards for School Mathematics recommends making mathematics meaningful by providing a visual representation to demonstrate part-whole relationships of percentages.

### California State Standards

#### Ratios

*Grade 6*

Number Sense 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).

Number Sense 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring.

*Grade 7*

Number Sense 3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.

#### Proportions

*Grade 6*

Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

Number Sense 2.0 Students analyze and use tables, graphs, and rules to solve problems involving rates and proportions.

Number Sense 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring.

#### Percents

*Grade 5*

Number Sense 1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.

Number Sense 1.3 Use fractions and percentages to compare data sets of different sizes.

*Grade 6*

Number Sense 1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.

Number Sense 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring.

*Grade 7*

Number Sense 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.

Number Sense 1.6 Calculate the percentage of increases and decreases of a quantity.

*Algebra 1*

Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems

## Cognitive Obstacles and Common Misconceptions

### Cognitive Obstacles

- Teaching students to use the unit ratio strategy (find the rate for one and multiplying to get the rate for many) as a standard approach to proportional problems may not help students develop multiplicative reasoning (Singh, 2000).
- Very few secondary students use typical textbook generalizations to solve proportion problems. Rather, they use less formal strategies that vary with the task and with the student's mental development and experience (Fisher, 1988).
- Students often use additive reasoning in solving problems where multiplicative reasoning is required (Hart,1981, 1988; Karplus,Pulos and Stage, 1983b; Noelting, 1980b; Vergnaud, 1988; Lamon 1993a; Resnick and Singer, 1993 as cited by Singh, 2000).
- The language that is used in comparative statements involving percent is additive in form, compressed, requires attention to unstated relationships, and is often in direct conflict with common everyday language usage (Parker and Leinhardt, 1995).
- Natural language does not support percent (Parker and Leinhardt, 1995):
- The language is concise/not explicit
- “Of” has a multiplicity of meanings
- Additive Clues for a Multiplicative Relationship

- Because percent as it has evolved is not a single thing but rather many different, overlapping things at the same time, the task for the learner and teacher is to determine which sense of percent is most meaningful in a given context (Parker and Leinhardt, 1995).

- Percent as a Number
- Percent as Intensive/Relational Quantity (miles per hour, dollars per dollar)
- Percent as Fraction or Ratio
- Percent as a Fraction: Subsets of Sets
- Percent as a Ratio: Separate Sets
- Percent as a Statistic or a Function

(Parker, 1994)

### Common Misconceptions

#### Ratios and Proportions

- “Constant difference” or “Additive” Strategy: The relationship within the ratios is computed by subtracting one term from another, and then the difference is applied to the second ratio (Tourniaire and Pulos, 1985 as cited by Misailidou and Williams, 2002).
- “Constant Sum” Strategy: The sum of the two numbers in one ratio should be equal to the sum of the two numbers in the equivalent ratio (Mellar, 1987 as cited by Misailidou and Williams, 2002).
- “Magical Doubling” Method: Doubling (when doubling is inappropriate) one of the data of a problem in order to find an answer (Mellar, 1987 as cited by Misailidou and Williams, 2002).
- “Incomplete” Strategy: The number asked should be the same as the one given from the same measure space (Karplus, Pulos & Stage, 1983 as cited by Misailidou and Williams, 2002).
- Incorrect application of build up method: Inappropriately combining multiplication and addition to reach the answer (Misailidou and Williams, 2002).

#### Percents

- “Numerator Rule”: The percent sign to the right of the numeral can be replaced by a decimal point to the left of the numeral; correctly applied 50% to 0.55 and incorrectly applied to 110% to 0.110 (Payne and Allinger, 1984 as cited by Parker and Leinhart, 1995).
- "Random Algorithm": Without knowing which operation to perform, divide if the quotient is an integer or otherwise multiply (Payne and Allinger, 1984 as cited by Parker and Leinhart, 1995).
- Students' strong part-whole notion of percent can lead to a serious misconception, making percents greater than 100 counterintuitive, since a part cannot exceed the whole (Parker and Leinhart, 1995).
- Students translate percent symbol as hundredths, a quantity out of a hundred, per hundred, etc. This translation does not encourage students to think about relationships (Parker and Leinhart, 1995).
- Students tend to ignore the percent sign completely, as if it had no significance whatsoever. Many of the students drop the percent sign and then reinsert it at will anywhere within the problem, making no distinction between 1/2 and 1/2% or 1/100 and 1/100% (Kircher, 1926, Edwards, 1930, and Brueckner, 1930 as cited by Parker and Leinhardt, 1995).

## Pedagogical Tools and Strategies

### Background

- In order for students to have conceptual understanding of ratios, proportions, and percents they must have developed proportional reasoning. Students who are introduced to a variety of methods to solve problems in various subjects have a better chance of making connections to ratios, fractions, proportions and part-whole relationships (Cai & Sun 2002). Solving problems using an investigative approach is fundamental to mathematics learning and understanding of the ways proportions can be used to extend knowledge and to solve problems in other fields (Stemn 2008). Research has found that it is important to first teach students proportional and non-proportional situations, and what these concepts mean, before teaching students algorithms that involve cross multiplication. Once rules are given it dramatically decreases the chances of students using reasoning to solve problems (Lamon 2005).

### What is Proportional Reasoning?

- Proportional reasoning involves mathematical relationships which are multiplicative in nature (Ben-Chaim, et al 1998).

### Types of Proportional Reasoning Problems

- Proportional reasoning problems can be described in three broad categories:

- Comparing two parts of a single whole, as in the ratio of girls to boys in a class is 15 to 10, or a segment is divided in the golden ratio.
- Comparing magnitudes of different quantities with an interesting connection, as in miles per gallon, or people per square kilometer, or unit price. These comparisons are not generally called ratios, but rates, or densities.
- Comparing magnitudes of two quantities that are conceptually related, but not naturally thought of as parts of a common whole, as in the ratio of sides of two triangles is 2 to 1. These comparisons are sometimes referred to as scaling and they include questions of stretching and shrinking in similarity transformations.

(Ben-Chaim et al 1998)

### Tasks Involving Proportional Reasoning

Three types of tasks for assessing proportional reasoning:

- Missing value problems, where three pieces of information are given and the task is to find the fourth or missing piece of information.
- Numerical comparison problems, where two complete rates/ratios are given and a numerical answer is not required, but the rates or ratios are to be compared.
- Qualitative prediction and comparison problems which require comparisons not dependent on specific numerical values.

(Ben-Chaim et al 1998)

### Method for Developing Proportional Reasoning

Stemn (2008) discusses a real life problem posed to students who are developing proportional reasoning.

- Tooth Brush Problem:
- “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?”

In order to solve the problem students were asked to collect their own data at home and then bring it to class. The data was then collected and students worked together to make conversions from cups to gallons.

Further development can be made by building on this problem. The missing value method (Stemn 2008) can be introduced to students by asking, “If two non-standard cups of water is equivalent to one quarter of a gallon, then 18 cups will be equivalent to how many gallons?”

### Strategies for Teaching Percents

- Referent reps refer to a set of three rectangles that give a visual display of relative sizes of the three quantities in percent problems. Students are usually confused by the decimal notation for percents, but this strategy displays each element quantified using amounts based on 100, which agrees with the “per 100” translation of the word “percent.” (Parker 2004)

Here is a sample of using referent reps to solve for a percent problem:

- “During a sale, prices were marked down by 20%. The sale price of an item was $84. What was the original price of the item before the discount?"

Looking at the problem we can place the 84 under the final amount. As numbers in a ratio can be scaled down by dividing by a constant, they can also be scaled up by multiplication. Finding the constant multiplier in this problem corresponds to finding the size of each piece in the figure. The 84 must be distributed into four equal sections, so each section has the value of 21. (Parker 2004)

### Strategies for Setting up Ratios

Setting up manipulatives students can relate to can help students make connections to creating ratios. Manipulatives can be strung together in groups for students to create ratios.

For students at higher grade level more age appropriate materials can be used.
(Petty & Drum 2001)

## Curricula and Technological Resources

Curricula & Technological Resources

Curricula

### Options to traditional algorithms

#### Ratio

Ratios are presented to elementary school students in 6th grade. Ratio tables can assist students in achieving conceptual understanding (Abrahamson & Cigan, 2003).

Every day, Robin puts $3 in her bank and Tim puts $5 in his bank.

Students can also use visual representations to assist them with ratios. In a problem which asks students to compare the ratio of chicken legs to cow legs, students sketch the problem in order to better understand the relationships. Students are asked to compare 3 cows to 8 chickens.

#### Proportion

Proportional reasoning begins to develop in mathematics starting in the 6th grade. A more recent, innovative way to teach proportional reasoning involves something called a “proportion-quartet.” A proportion-quartet requires students to create a 2-by-2 table and label according to the word problem (Abrahamson & Cigan, 2003).

Example: Joan used exactly 15 cans of paint to paint 18 chairs. How many chairs can she paint with 20 cans?

Students label the table and factor each term. Columns have the same factors, and rows have the same factors. The product of the factors is inside the 2-by-2 table. Only one number can satisfy the unknown product.

Along with ratio tables, students can also use pictorial representations to assist them in developing their proportional reasoning skills (Sharp & Adams, 2003).

The cookies shown in this picture were enough to give everyone at our 14-person part a nice halftime snack. How many cookies should we make for our 35-person party?

#### Percent

Percents are first introduced in 5th grade mathematics. As a means for enhancing the acquisition of percents, teachers translate the symbolic language of percent. They provide clues for students to use by examining the language of the problem. In a percent problem, teachers would instruct students to translate as follows: of = x (multiply)

so 15% of 80 would translate into .15 x 80

### Technological Resources

#### Interactive Math Software

NumberNut.com

Academy of Math (percent)

Odyssey Math (percent)

The Math World of Ratios and Proportions

Seeing Math (proportions)

National Library of Virtual Manipulatives (percent)
(http://nlvm.usu.edu/)

#### Websites for Skill Development

Cool Math http://www.coolmath.com

Purple Math http://www.purplemath.com

A Plus Math http://aplusmath.com

Math in Daily Life http://www.learner.org/exhibits/dailymath

## Annotated References

- Abrahamson, D. and Cigan, C. (2003) A Design for Ratio and Proportion Instruction. National Council of Teachers of Mathematics, 8(9), 493-501

- This article described an alternate method for teaching ratio and proportion to 5th grade students. The research supported exploring patterns of multiplication tables and moves it to the concept of rate as a single column within the multiplication tables. Teachers believed it supported meaningful transition to ratio and proportion problems.

- Ben-Chaim, D., Fey, J., et al (1998) Proportional Reasoning Among 7th Grade Students with Different Curricular Experiences. Educational Studies in Math. 36(3) 247-273

- This article looked at the differences among seventh grade students who were taught under the Connected Mathematics Project (CMP) curriculum and those who were taught under standard curriculum. The results of the study concluded that students taught under CMP curriculum developed better proportional reasoning skills.

- Billings, E. (2001)Problems That Encourage Proportion. National Council of Teachers of Mathematics 7(1)10-14.

- Esther Billings discusses the problems with simply teaching students how to solve proportions by using the cross multiply and divide method. While this method, when used correctly, does work, it does not let the student understand what it is they are truly doing. Billings argues that qualitative reasoning is what should be addressed in schools and more problems where students should apply this method should show up in classrooms.

- Cai, J., and W. Sun. (2002) Developing students’ proportional reasoning: A Chinese perspective. National Council of Teachers of Mathematics

- This article provides an alternative approach to proportional reasoning. The article focuses on teaching concepts from multiple perspectives so that students can develop conceptual understanding across concepts.

- Drum, R. and Petty, G. (2001) Miniature Toys Introduce Ratio and Proportion with a Real-World Connection. National Council of Teachers of Mathematics. 7(1), 50-54.

- This article identifies novel ways in modeling of ratios and proportions by emphasizing real-world connections through the use of toys. This study provided a concrete strategy as a bridge to the more abstract concepts of ratio and proportion.

- Fisher, Linda. (1988). Strategies Used By Secondary Mathematics Teachers to Solve Proportion Problems. Journal for Research in Mathematics Education, 19(2), 157-168.

- The study investigates the strategies described by secondary mathematics teachers as they solved proportion problems. Collected data indicated that teachers were not well prepared in teaching multiple methods and strategies to their students as research suggests. The conclusion encourages that teachers be well equipped with the multiple techniques for approaching proportion problem solving.

- Lamon, S. (2005) Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New Jersey: Lawrence Erlbaum.

- Lamon discusses instructional strategies for teachers teaching fractions and ratios. It also discusses misconceptions teachers have about teaching specific content areas.

- Lo, 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.

- This article follows a fifth grader, Bruce, on his discovery of ratios and proportions. They give him different problems and ask him to explain his thinking. After multiple exercises Bruce differentiates how he solves the problems depending on the numbers given. Some techniques were scalar reasoning and functional reasoning.

- Lo, J., Watanabe, T. and Cai, J. (2004) Developing Ratio Concepts: An Asian Perspective. National Council of Teachers of Mathematics, 9(7), 362-367

- The authors discuss the unique approaches that Asian cultures utilize in introducing the concepts of ratio and proportion. The Asian text books emphasize the connection to multiplication, rather than introducing it as another method to write a fraction. Pictorial representations are used to reinforce conceptual understanding.

- Mihaila, I. (2004) Farey Sums and Understanding Ratios. National Council of Teachers of Mathematics, 98(3), 158-162.

- The author stresses the importance of visual representations in assisting students in understanding the ratio relationships. Students are encouraged to draw pictures when problem solving ratios.

- Miller, J. and Fey, J. (2000). Proportional Reasoning. National Council of Teachers of Mathematics, 5(5) 310-313.

- This was a study that compared students studying a new standards based curriculum against students from a controlled group. The study was to examine how students think about solving proportion problems and common misconceptions they ran in to. The study concluded that students are able to use their background knowledge to help them with proportional thinking which helps develop flexible strategies when solving proportion problems.

- Misailidou, C. and Williams, J. (2002). Ratio: Raising teachers’ awareness of children’s thinking. Proceedings of the 2nd International Conference on the teaching of Mathematics.

- This study aims to contribute to teachers’ awareness of student strategies and misconceptions in the field of “ratio.” Data is collected from strategies of students as well as those predicted by pre-service teachers. Their analysis indicates that there exists a gap between students’ strategies and errors and their future teacher’s perception of those.

- Parker, M. (2004) Reasoning and Working Proportionally with Percents. Mathematics in the Middle School, 9(6), 326-330.

- This article talked about a specific strategy for teaching percents. The author shows how referent reps can be used to visually show the proportions equated with percent problems. This method can be taught to students in elementary school and then be used as a tool throughout middle school.

- Parker, M., Leinhart, G. (1995). Percent: A Privileged Proportion. American Educational Research Association, 65(4), 421-481.

- The article investigates existing literature to gain a better understanding of percents as a difficult math topic. Some explanations involve the fact that the notion of a percent can take multiple forms. It also considers language as a possible hindrance to fully understanding percents. After exploring different explanations, the authors discuss possible strategies for avoiding these difficulties.

- Petty, W.G. & Drum, R.L. (2001) Miniature Toys Introduce Ratio and Proportion with Real-World Connection. Mathematics Teaching in Middle School. 7(1) 50-54

- This study outlines how using manipulatives can help teachers introduce the concept of a ratio to students. Using objects that students can relate to can create deeper connections for students to create better transfer.

- Sharp, J.M. & Adams, B. (2004) Using a Pattern Table to Solve Contextualized Proportion Problems. Mathematics Teaching in Middle School, 8(8), 432-439.

- These authors focused on instructional strategies involving pictorial representations in order to solve contextualized proportion problems. Students used tables, drawings, and other visual representations to enhance learning.

Develops a week-long, problem-based mathematics curriculum focusing on proportions and explains instructional and assessment strategies.

- Singh, Parmjit. (2000). Understanding the Concepts of Proportion and Ratio Constructed by Two Grade Six Students. Educational Studies in Mathematics, 43(3), 271-292.

- Parmjit studies the strategies used by two students in completing proportion and ratio problems. The study focuses on how students transition from solving ratio problem by iterating composite ratio units to using multiplication and division. Findings suggest that the unit method should not be taught to students until they are well educated in unit coordination schemes.

- Stemn, Blidi S.(2008) Building middle school students' understanding of proportional reasoning through mathematical investigation. Education 36(4) 383-392

- This article discusses a method for building proportional reasoning among seventh grade students. It outlines the procedures taken to teach these methods and the outcomes of student achievements.