Relating Decimals, Fractions, and Percents

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Relating Decimals, Fractions and Percents (Pedagogical Content Knowledge Project)

  • revised by Nirali Bhatt, Emily Vandever, and Pauline Yee (UC Irvine, August 2008)
  • started by Andrea Lubcke, Megan LeBel, Christine Williamson, & Jacqueline Stutz (UC Irvine, 2006)


Contents

Mathematical Background

Students need to know that one number can be represented in several different formats. For example, 1/4 is the same as 0.25 which is the same as 25%. Not only do they need to know what this looks like numerically, but pictorially as well. The emphasis shouldn’t be on memorizing the conversions of benchmark numbers (i.e. 1/4, 1/2, 3/4) but to understand what those numbers are actually representing.


Conversion Procedures

Students need to be able to quickly manipulate numbers, for example, knowing which way to move the decimal point when changing between decimals and percents. The emphasis has traditionally been on the rules and tricks, with less emphasis on the understanding of why the numbers are able to convert in the manner that they are. Having a better understanding of what the fractions, decimals, and percents represent will result in students being able to convert numbers more quickly will allow the student to become more competent in comparing and ordering fractions, decimals, and percents.


Comparison and Ordering

Students understand comparison and ordering when it comes to whole numbers, but confusion arises when fractions, decimals, and percents are used, especially when students are asked to compare numbers in a variety of formats. For example, it is difficult for students to compare a fraction and a decimal, such as comparing .3 to 1/4. However, students should be able to compare and order numbers regardless of their format.

Place Value

Because early instruction focuses on the numbers to the left of the decimal point, students are familiar with the place value of whole numbers. However, the numbers to the right of the decimal point aren’t introduced until later, and aren’t as familiar to the students as a result.


Related Standards by Grade

Grade 2, Number Sense

4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole:

  • 4.1 Recognize, name, and compare unit fractions from 1/12 to 1/2.
  • 4.2 Recognize fractions of a whole and parts of a group (e.g., one-fourth of a pie, two-thirds of 15 balls).
  • 4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one.


Grade 3, Number Sense

3.0 Students understand the relationship between whole numbers, simple fractions, and decimals:

  • 3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4).
  • 3.2 Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2).
  • 3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors.
  • 3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is 1/2 of a dollar, 75 cents is 3/4 of a dollar).


Grade 4, Number Sense

1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers:

  • 1.1 Read and write whole numbers in the millions.
  • 1.2 Order and compare whole numbers and decimals to two decimal places.
  • 1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand.
  • 1.4 Decide when a rounded solution is called for and explain why such a solution may be appropriate.
  • 1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0).
  • 1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 = 1.75).
  • 1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
  • 1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing").
  • 1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places.

2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals:

  • 2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places.
  • 2.2 Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer.


Grade 5, Number Sense

1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers:

  • 1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.
  • 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.
  • 1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication.
  • 1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).
  • 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals:

  • 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.
  • 2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.
  • 2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.
  • 2.4 Understand the concept of multiplication and division of fractions.
  • 2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.


Grade 6, Number Sense

1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages:

  • 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
  • 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 ).
  • 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.
  • 1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.

2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division:

  • 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
  • 2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations (e.g., 5/8 ÷ 15/16 = 5/8 x 16/15 = 2/3).
  • 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
  • 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).

Cognition and Learning Background

Current Curriculum

Fractions → Decimals → Percents

Within the current curriculum, fractions are taught first, which is in second grade. From that point forward, each topic is generally addressed separately. It isn’t until after the students have learned about fractions, decimals, and percents as separate entities that they are taught that they are actually different representations of each other. Students aren’t asked to interchange fractions, decimals, and percents within a problem until fifth or sixth grade, and even then such a task doesn’t occur too often.

Operational and abstract

Current instruction gives a brief introduction to part-whole fractions and then proceeds to introduce computation procedures. This method doesn’t allow children the time they need to construct important ideas and ways of thinking. For example, when a student sees 3/7 he knows that it means “three out of seven,” but then translating that into a division problem they simply begin performing a series of actions until they satisfy some criterion for stopping, such as a remainder of 0. The student might say that they understand 3/7, but then show him 7/3 and that doesn’t make any sense to him because he doesn’t understand how there can be “seven out of three things.” (Thompson & Saldanha, 2003) Click to see more information on the Conceptual Understanding of Fractions

Furthermore, it is not sufficient to use only part-whole thinking skills as the foundation for understanding rational numbers. Teachers should ask students more questions that involve pictures rather than numerical symbols to encourage flexible thinking (Lamon, 1999).

Providing students with mathematical experiences that incorporate components of proportional reasoning before engaging students in more abstract and formal presentations will improve chances of developing proportional reasoning (Lamon, 1999). The 6 areas that contribute to proportional reasoning are: relative thinking, unitizing, partitioning, ratio sense, rational numbers, and quantities & change. Understanding develops over time through gradual exposure and practice with ratios.

Case and Moss (1999): Teaching Percents First

To prevent misconceptions and increase the understanding of decimals, fractions, and percents prominent math scholars Robbie Case and Joan Moss (1999) developed a new model for teaching rational numbers. In this new trajectory percents were introduced first, two-place decimals next, and fractional notation last. Case and Moss (1999) rationalized that by age 10 or 11 students have well-developed qualitative intuitions about both proportions and whole numbers from 1 to 100. This new model for understanding rational numbers adds to this prior proportional and whole number knowledge rather than interfering with it (Case & Moss, 1999). This previous understanding, in conjunction with the prior exposure to percents and the relation of percents to fractions and decimals, led Case and Moss (1999) to develop students’ understanding of rational numbers through percents first.

Psychophysical research suggested that humans easily see objects in proportional terms, such as full, half full, nearly empty, etc (Case & Moss, 1999). Utilizing this innate ability Case and Moss (1999) prompted students to estimate the “fullness” of beakers filled with water by assigning a numerical value from 1 to 100. This initial introduction of rational numbers allowed students to apply their proportional and whole number knowledge. Throughout the new trajectory the proportional nature of a quantity was emphasized instead of the traditional view of units or “shares” (Case & Moss, 1999).

Another reason percents were introduced first was because they are a familiar part of life. When transferring data on a computer the “number ribbon” indicates the percent completed. This visual representation provides a foundation for understanding percents and can be applied toward further understanding of rational numbers (Case & Moss). Case and Moss (1999) used the “number ribbon” throughout their new model to establish a uniform representation of rational numbers. Students also encounter percents when shopping or watching television. A clothing sale might boast of 50% off or an advertisement could speak of the product being 99% accurate. Exposure to percents can be frequent and can strengthen comprehension, whereas students rarely come across fractions and decimals in daily life (Case & Moss, 1999).

The traditional trajectory requires students to formulate an understanding of completely new concepts. The traditional trajectory also asks students to prematurely compare or change ratios with different denominators (Moss & Case, 1999). The process of comparison can be challenging and can create confusion and misunderstanding early on in the students’ learning of rational numbers. Introducing percents first postpones (hopefully avoids) this problem and enables students to develop a stronger conceptual understanding. Another added benefit of beginning with percents is that every percentage has a corresponding fractional or decimal equivalent that is simple to determine. For example, 77% can easily be written as 77/100 or 0.77. This conversion process is effortless and allows students to develop their own procedures rather than memorizing algorithms (Case & Moss, 1999). Unfortunately, the same is untrue of fractions and decimals. When converting fractions or decimals the corresponding decimal, fraction, or percent might not be easily found. The fraction 1/7 has no quickly calculated equivalent. Thus, students are forced to change the denominator and implement an algorithm they might not comprehend. Percents avoid this problem and enable students to make their initial conversions between percents, decimals, and fractions in a direct, intuitive manner (Case & Moss, 1999).

Next, Case & Moss (1999) transitioned from focusing on percents to two-place decimals. Two-place decimals were described as an indication of the percent of the way between two adjacent whole numbers (Case & Moss, 1999). For example, 5.25 is 25% of the way between the whole numbers five and six. This integration of percents into the learning of decimals allowed students to utilize prior knowledge and created a foundation for later comparison.

In the last phase of the new trajectory Case & Moss (1999) introduced opportunities where fractions, decimals, and percents were used interchangeably. Fraction terminology was utilized throughout the teaching of percents and decimals. For example, students understood that 50% and one half could be used interchangeably. Case and Moss (1999) also informed students that half of 50% could be expressed as either 25%, 0.25, or ¼.. Though Case and Moss (1999) proposed that the foundations for rational number understanding should be based on percents they also suggest that greater comprehension can be established by creating exercises where percents, decimals, and fractions are used interchangeably.

NZ Maths and Van de Walle

Although the new trajectory mentioned above finds that students will learn fractions, decimals, and percents more effectively in a different order, the research to support this is limited and still in the preliminary stages. Even so, there are still those who support the traditional order of instruction. The NZ Maths website and John A. Van de Walle (both reputable sources) continue to present these concepts in the traditional order, however, they offer a different approach than traditional curriculum. While the traditional method teaches these concepts separate and disconnected from one another, NZ Maths and Van de Walle place a strong emphasis on the connections and representations between fraction, decimal, and percent concepts. This key difference in instruction provides students with a much deeper understanding.


NZ Maths

All of the NZ Maths material takes a cognitive approach and is based upon the seven development of strategy stages:

NZ Maths Stages.jpg


The Teaching Fractions, Decimals, and Percentages book outlines instructional activities that will help students to understand these three concepts and progress through the strategy stages. While fraction concepts are introduced first, ratios, decimals, and percentages shortly follow. Connections and relationships between fractions, decimals, and percents are highlighted throughout the activities.

More information on Decimals, Fractions, and Percents is available from their website at http://www.nzmaths.co.nz/Numeracy/Other%20material/Tutorials/DecimalsFractionsPercentages.aspx. They emphasize the importance connecting fractions to decimals, and connecting percents to hundredths. They also recommend that students "develop some fraction-decimal relationships as benchmarks."


Van de Walle

Van de Walle, author of Elementary and Middle School Mathematics, finds that ultimately we are trying to help our students make connections between representational systems. He uses the standard order of instruction (fraction>decimals>percents) to demonstrate how these concepts can be taught effectively. Van de Walle states that the “fractions- first, decimals-later sequence is arguably the best approach.” He finds that students should be first given a strong fraction foundation and then introduced to decimals. While students are developing an understanding of decimals, fraction/ decimal relationships should be outlined. Van de Walle states that “A significant goal of instruction in decimal and fraction numeration should be to help students see that both systems represent the same concepts” (p.280) Once these concepts are understood, Van de Walle looks at introducing percents as the third operating system. Since percent is another name for hundredth, it can be easily integrated into instruction. Making the connection between percents and hundredths (both in decimal and fraction forms) greatly assists students in converting from one representation to another.

Although students eventually need to learn the material with non-contextual difficult numbers, Van de Walle believes that students should frequently be presented with nice numbers in realistic situations. These types of problems help students understand the underlying principles and ideas behind the concept. In addition, students will be more interested if the problem has some meaning to it.

Cognitive Obstacles and Common Misconceptions

In the case of understanding the relationship between mathematical representations, children’s efforts to make sense of these new concepts can sometimes be hindered by the knowledge that they already possess. Although it is important to consider students’ background knowledge in classroom instruction, so as to provide context for new material, teachers also need to ensure that students are not over generalizing concepts from a familiar domain of mathematics in order to interpret a new one. When learning to relate decimals, fractions, and percentages to one another, students harbor many misconceptions based on the current learning trajectory, which treats each mathematical representation as a separate entity and presents them in an illogical sequence.

Decimals and whole numbers share many common characteristics: they are embedded in a structure based on features of place value, where column values decrease when moving from left to right, and zero serves as a place holder. In their efforts to help students make sense of the decimal system, teachers encourage students to focus on the commonalities it shares with the whole number system. Students wrongly assume that because the systems are similar that they are identical, and therefore ignore all of the key differences which are at the core of each. This same principle applies to the relationship between fractions and decimals. Since the current curriculum places fraction instruction before decimals, students attempt to apply their conceptual knowledge about the relationship between fractional size of parts and number of parts in making sense of decimal value.

To further explain the above mentioned conflict between fraction and decimal conceptual understandings, Sackur-Grisvard and Leonard (1985) conducted a study to examine the how the earlier learned information (of fractions) serves to skew the understanding of the newly acquired (of decimals). The researchers found that there are three basic erroneous rules students apply when trying to make sense of the decimal system:

Rule 1: The number with more decimal places is the larger one. For example, 3.214 is greater than 3.8 because it has more digits. Students who rely on creating a relationship between the decimal and whole number systems often will apply this rule.

Rule 2: The number with fewer decimal places is the larger one. Referred to as the “Fraction Rule”: since smaller parts lead to larger fractions, students can wrongly apply this reasoning and assume that longer decimals are smaller, since they have a greater number of parts.

Rule 3: When one or more zeros are immediately to the right of decimal in one number, it is the smallest number, but the next number in the sequence will be that with the most digits.


Roche (2005) interviewed students doing decimal comparison tasks. She noted two strategies that that students use to compare decimals:

Strategy 1: Using fractional language or benchmarking Example: “0.567 is greater than 0.3 because five hundredths is greater than three tenths” Example: “0.567 is more than half and 0.3 is less than half.” This is a more valid because it draws on one’s number sense and decimal understanding.

Strategy 2: Adding zeros to the shorter decimal to make it the same length as the larger decimal and then comparing the two. Example: “0.37 is greater than 0.217 because 370 is greater than 217.” This strategy is procedural, uses whole number thinking, and students can apply it without understanding decimals.


Because the common misconceptions are essentially based on conflict between previously taught content and newly acquired information, a new trajectory for a cognitively sound curriculum is required in an effort to eliminate the possibility for occurrence of these misunderstandings.

Pedagogical Tools and Strategies

Although the new trajectory might be more cognitively sound than the curriculum that schools are using, it is not practical to think that major overhauls can be done very easily. As a result, we would like to use ideas from NZ Maths, Van de Walle, and others to outline some of the more realistic steps that teachers can take in their own classrooms while keeping with the traditional order of instruction. Models are a great way for students to visualize and understand difficult concepts. They allow teachers to explicitly show students how fractions, decimals, and percents relate to each other and to other types of numbers. In order for teachers to help their students understand fraction, decimal, and percentage relationships, they need to use multiple model types so that the students understand the broad range of these concepts.

Hiebert, Wearne, and Taber (1991) studied a fourth grade class learning about decimal representations and decimal functions. Especially for struggling learners, students need practice with a lot of different physical representations in order to learn decimal concepts. In the study, the researchers used base 10 blocks and varied which block represented the units. They tried to help students understand the decimal system by breaking the unit into 10 pieces and further subdividing these pieces into 10 more pieces. They also had students use the physical representation of straws that were bundled together and a clock divided into tenths and then hundredths of a second.

The other major finding in Hiebert and colleagues' 1991 study was that understanding happens gradually, not all at once. They found that students learned "in a halting, back-and-forth sort of way" (p. 324). The process of learning is not just cumulative, but students are "disconnecting, connecting, and reorganizing" the knowledge they have acquired. This is important for teachers to realize so that they give students enough time to construct and process the knowledge in their minds.


Here are various models that can be used for teaching fraction, decimal, and percent relationships:

10 x 10 Grids

10 x 10 grids are useful for students as they are learning decimals, fractions, and percents. Students can shade in a certain number of squares and then discuss the corresponding fraction, decimal, and percent.

10by10grid.jpg


The Blackline Master can be found here

Base Ten Blocks

Base Ten blocks can be used in conjunction with a mat and a paper decimal point. The decimal can be placed between any of the columns, for example, placing the decimal between the hundreds and the tens transforms the hundreds place into ones, the tens place into tenths, and the ones place into hundredths. Although this provides students with a great base-ten model for decimals, this process can be confusing if they don’t already have a good conceptual foundation.


Basetenblocks.jpg


Fraction Circles

Fraction Circles are commonly used for fraction instruction; however, these can also be used for decimal and percent concepts.


Fractioncircles.jpg


Fraction Strips

Fraction Strips can be used to represent fractions, decimals, and percents simultaneously-


Fractionstrips.jpg


Rotating Hundredths Disk

Rotating hundredths disks show two parts of a whole. It is particularly helpful in showing the relationships between decimals, fractions, and percents because it is already divided into tenths and hundredths. The Blackline Master can be found at: [1]


Hundredths disk.jpg

Counters

Counters allow students greater challenge. Students each get a set of two-sided counters. They dump them out on their desktop and then they must decide what fraction/ decimal/ percent are red or yellow-

Counters.jpg

Number Lines

Students should also be exposed to these concepts (primarily fractions and decimals) on a number line. This enables them to see the relationship to whole numbers.

A great cooperative classroom activity with number lines would be to have students create a Mega-Number-Line (Gibbons & Tracy, 1999). Students first measure out ten meters of tape. Then they construct a number line from zero to one. Within the zero and one mark, students mark the tenths, hundredths, and thousandths. Through this conceptual activity, students discover that one tenth of ten meters is one meter, one one-hundredth of ten meters is one decimeter, and so on.

Pagni (2004) suggests using number lines to show fraction and decimal representations of different quantities. He says that when fractions and decimals are taught separately, students may see them as two different sets of numbers, rather than different representations of the same number. Putting decimal/fraction equivalents next to each other on a number line will help students see the connection between the two.

Metric Measuring Tools

Metric measurement allows students to experience the base ten system in a concrete learning way.


Decimal Squares

Decimal Squares is a program that helps develop conceptual understanding of decimal fractions with visuals (Gibbons & Tracy, 1999). There are two sets of colored cards. The first set shows a pictorial form of decimal fractions and the second set shows the corresponding numerical symbol and word form of the decimal fraction. The program allows students to connect the concrete and numerical representation of decimal fractions. http://www.decimalsquares.com/dsGames/

Playdough

A Sequence for Developing Decimal Sense (Caswell, 2004):

  • Give students a lump of playdough and have them divide it into two.
  • Have the students divide one-half of the dough into ten equal parts.
  • Talk about decimals as being parts of wholes, just like fractions.
    • Ask, “Do these tenths still equal the same amount as the whole piece of playdough?” This is to help students see that the tenths are part of a whole.
  • Introduce the written notation on cards. Students use the playdough to represent the decimal on the card.
    • Example: The card has 1.7, the student gets one big ball of playdough and 7 little pieces of playdough.
  • Extend to hundredths and thousandths. Challenge the students to represent hundredths using the playdough.

Full-playdough-small.jpg

Classroom Advice

  • Use ragged decimals (Roche, 2005). Ragged decimals are decimals of different lengths. It is important to have students compare ragged decimals because it can bring to light a whole-number thinking misconception. For example, when ordering 0.9 and 0.10, some students might think that “point 9” goes before “point 10.” In contrast, comparing decimals of equal length can mask this misconception because students can use whole-number thinking and still get the right answer.
  • Use representations with perceptual distracters (Roche 2005). A perceptual distracter is a visual that does not match the task. It challenges the whole-number thinking and tests fraction/decimal understanding


Non-example of perceptual distractor: Shade in four tenths (0.4)

Fourtenths.jpg

To do this task, students just need to count four segments. This example affirms counting and whole-number thinking.


Example of perceptual distractor: Shade in 0.3 of this shape.

Threetenths.jpg

To do this task, students cannot just count out three parts. They are challenged to use their fraction/decimal knowledge. One way to do this problem would be to shade one quadrant, which would be 0.25. Then to divide another quadrant into fifths, and to shade one of those fifths, which would be 0.05, to make a total of 0.3.


  • Use fractional language when describing decimals (Roche, 2005). This requires a more conceptual understanding of decimals. For example 2.75 would be “2 and 75 hundredths” rather than “2 point 75."

5 categories of representations (Clement, 2004)

Clement (2004) suggests five different kinds of representations for teaching students the concepts of fractions, decimals and percents. The five types of representations she suggests are pictures, manipulatives, spoken language, written symbols, and relevant situations. She states that students should learn the concepts in each of the representations and be able to make connections among different representations. For example, students should be able to represent 3/4 as a written symbol, in words, in pictures, in manipulatives, and in a relevant situation.

Pictures are shapes or figures that are student drawn. Manipulatives are physical representations of a number that students can stack together, take apart, etc. Many forms of manipulatives have been suggested above. Spoken language is a number stated in words, and can be oral or written. It is important that students know how to read a written number to be able to communicate about mathematics. Written symbols are numerals. Relevant situations are contexts that are meaningful for students. Clement (2004) uses the example of asking students to subtract 4 - 1/8. Some students mistakenly said 3/8. However, when she stated the problem as four brownies minus 1/8 of a brownie, the students knew that there were more than three brownies remaining. Putting the problem in a relevant context helped the students to understand it and conceptualize it. These different representations are useful in teaching students decimals, fractions, and percents.

Curricula and Technological Resources

In addition to the information provided, there are many legitimate internet-based resources, which teachers can use for teaching the relationships between fractions, decimals, and percents.


NCTM Illuminations Website

This tool was found on the NCTM Illuminations website. It is a model that offers a fraction circle representation and gives the percent and decimal equivalents for the shaded portion. http://illuminations.nctm.org/ActivityDetail.aspx?ID=11

Arcytech Fraction Bars Applet

This is a similar tool that was found on the Arcytech website. It is a Fraction Bars applet, which can represent fractions, decimals, and percents. http://www.arcytech.org/java/fractions/fractions.html

Math Munchers Deluxe

The computer game, Math Munchers Deluxe, provides students with fraction- decimal equivalence drills. This type of game is really only useful once students have developed their conceptual knowledge.

Concentration

Sweeney and Quinn (2000) developed and implemented a series of lessons using the game Concentration to help students learn that decimals, fractions, and percents are different representations of the same amount. They used a series of three lessons to teach the material and then play concentration to give students the chance to practice.

In the first lesson, they used a pre-assessment quiz with seven questions: 1. Explain what a fraction means to you. 2. Explain what a decimal means to you. 3. Explain the relationship between fractions and decimals. 4. Explain what a percent means to you. 5. Write the words that you would use to say 3/4. 6. Write the words that you would use to say 0.7. 7. Write the numeric or mathematical expression for three-tenths.

Through the pre-assessment, Sweeney and Quinn (2000) found that many students don't know that fractions, decimals and percents are related. The next step was to provide instruction on the relationship between fractions, decimals, and percents. They used circles with different fractions shaded in and created a chart on the board showing the shaded area as a fraction of the circle, as a decimal, and as a percent. They used several circles and continued making the chart.

After finishing the whole class instruction, they put students in groups of four to make cards with a shaded circle and the fraction, decimal, or percent that it represents. After the groups made four cards, the teacher put all of the cards together in a class set and played the game concentration with the whole class. To play concentration, each player on his or her turn flips two cards and tries to get a pair. If the cards are a pair, the player takes the cards and gets a bonus turn. If the two cards are not a pair, the player returns the cards to the face down position and then it becomes the next player's turn. The game continues until all of the cards have been matched, and the player with the most pairs is the winner.

After teaching the students to play Concentration as a whole class, the teacher then split the students back into groups of 4. In groups, students made a whole set of concentration cards and played in their groups.

During the next lesson, the teacher gave the students the same assessment and after playing concentration found that more students understood the relationship between decimals, fractions and percents. More students answered the questions correctly on the assessment and they wrote more than one representation of three-tenths.

Conceptual Bingo

Another fun game is Conceptual Bingo (a board game), which provides students opportunities to practice their understanding of fraction, decimal, and percent equivalence. This game should be used once students understand the concepts involved. Find this game at: http://www.conceptualmathmedia.com/bingo/convert_fraction_decimal_percent.asp

Mind Research Institute Applets

The Mind Research Institute provides innovative visual approaches to teaching math concepts. Activities do not require students to have a strong acquisition of the English language, therefore there is no language barrier when engaging in mathematical activities. http://www.mindinst.org/video/demo.html

Van de Walle text

Van de Walle’s 2003 (revised, 2007) book, Elementary and Middle School Mathematics, is another wonderful source. Included in the book are cognitively based instructional strategies and activities, making it a great resource for teachers. The companion website for the book can be found at: http://wps.ablongman.com/ab_vandewalle_math_5/0,7959,796751-,00.html

Rational Number Project

The Rational Number Project is a research development project funded by the National Science Foundation (NSF). The project is a culmination of years of research done on how to organize and teach deep conceptual learning of fractions for students grades 4-8. The compilation of lessons allows students to learn fractions using manipulatives such as fraction circles, chips, and paper folding. The lessons aim to develop number sense for fractions by focusing on the development of concepts, order, and equivalence ideas (Cramer, et.al, 1997). The lessons do not introduce operations with fractions until students have developed meaning for the fraction symbols first. A link to the lessons are available at: http://cehd.umn.edu/rationalnumberproject/rnp1.html


Literature

Piece = Part = Portion: Fractions = Decimals = Percents by Scott Gifford (2003)

Piecepartportion.jpg

Available for purchase at Amazon.com

This is a great book to introduce the idea that fractions, decimals, and percents are all saying the same thing. The pictures are bright and real context is provided.


The Phantom Tollbooth by Norton Juster (1988)

Available for purchase at Amazon.com

Several chapters of this book are devoted to silly numerical concepts. Fractions, decimals, and percents are all visited. This is a classic story that is a great read for upper elementary students.


Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers by Susan Lamon (1999)

Available for purchase at Amazon.com

The activities in Susan Lamon’s book are based on using reasoning skills and abandoning the traditional fraction rules and procedures that are commonly emphasized. Attempting problems using reasoning skills allows for deeper understanding and conversation to emerge.

Annotated References

California Board of Education (1997). Mathematics content standards for California public schools: Kindergarten through grade twelve. Sacramento, CA: California Department of Education. Available online at http://www.cde.ca.gov/be/st/ss/mthmain.asp -- The California Math Standards tell teachers the standards that they need to teach and that students need to learn at each grade level.


Case, J. & Moss, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30 (2), 122 - 147. -- Case and Moss recommend that students be introduced to percentages first rather than fractions or decimals. They recommend teaching percentages, then decimals, and finally fractions, based on the idea that students are more familiar with percentages because they are exposed to percentages in daily life.


Caswell, R. (2006). Developing decimal sense. Australian Primary Mathematics Classroom, 11(4), 25-28. --Caswell describes a sequence of tasks to help students acquire a number sense for decimals.


Clement, L. L. (2004). A model for understanding, using, and connecting representations. Teaching Children Mathematics, 11 (2), 97 - 102. -- Clement details representations that can be sued to help students understand fractions and decimals: relevant situations, pictures, manipulatives, spoken language, and written symbols.


Cramer, K., Behr, M., Post T., Lesh, R. (1997). Rational number project: Fraction lessons for the middle grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa. -- The Rational Number Project is the product of years of research development on teaching fractions, ratios, decimals, and proportionality. It offers strategically planned lesson plans that use manipulatives and promote conceptual based learning.


Gibbons, M., Tracy, D., (1999). Deci-mania! Teaching teachers and students conceptual understanding of our decimal system. (Eric Document Reproduction Service No. ED433237). --This article suggests activities such as using metric tools, number lines, and Decimal-Squares to help students understand decimal concepts in a more meaningful way. The authors believe their collection of curricular activities and materials is necessary and cognitively appropriate for students.


Hiebert, J., Wearne, D., & Tabor, S. (1991). Fourth graders' gradual construction of decimal fractions during instruction using different physical representations. The Elementary School Journal, 91 (4), 321-341. -- Emphasizes the importance of using physical representations in teaching decimal fractions and the gradual way that students learn. Cannot expect students to understand the concept all at once, after being exposed to something new only once.


Lamon, Susan (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New Jersey: Lawrence Erlbaum Associates. --This is a resource book for teachers to help teach fraction concepts. The book suggests ways to build proportional reasoning in students’ understanding of ratios.


Ministry of Education (2008). New Zealand Maths Project. Wellington, NZ. Available at http://www.nzmaths.co.nz/Numeracy/Other%20material/Tutorials/DecimalsFractionsPercentages.aspx -- This website outlines major ideas in mathematics education and provides summaries of important math topics, including one on the importance of relating decimals, fractions, and percents.


Pagni, D. (2004). Fractions and decimals. Australian Mathematics Teacher, 60(4), 28-30. --Pagni argues that fractions and decimals should not be taught separately. Since they are two representations for the same numbers, teachers should highlight the connection between the two by placing the equivalent fractions and decimals on a number line.


Roche, A. (2005). Longer is larger—or is it? Australian Primary Mathematics Classroom, 10(3), 11-16. --Roche says that a common misconception is for students to treat decimals as whole numbers. She gives suggestions on tasks that counter this whole number thinking.


Thompson, P. & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research Companion to the Principles and Standards for School Mathematics (pp. 95-114). Reston, VA: National Council of Teachers of Mathematics.


Sweeney, E. S. & Quinn, R. J. (2000). Concentration: Connecting fractions, decimals, and percents. Mathematics Teaching in the Middle School, 5 (5), 324 - 328. -- Sweeney and Quinn described a series of lessons using the game Concentration to help students see that fractions, decimals, and percents are different ways of representing a part of a whole.


Van De Walle, J. A. (2007). Elementary and middle school mathematics: teaching developmentally. Boston: Pearson Education, Inc. --Van De Walle provides insight into how students learn decimals, fractions, and percents and offers instructional activities.