Fractions - Multiplication and Division

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Topic: Fractions-Multiplication and Division

  • by: Ashley Gormin, Teresa Hu, Avindri Jayasekara, Eric Joaquin, Heather McCook (UCIrvine, August 2010)


Contents

Cognition and Learning Background: Multiplication and Division of Fractions

Multiplying and Dividing Fractions

Students are formally introduced to the concept of fractions in the second grade. Students begin to add and subtract fractions in the third grade. Based on California state standards for mathematics, students should learn simple multiplication and division of fractions in the fifth grade. Students should be able to understand the concept of multiplication and division of fractions as well as be able to compute and perform simple multiplication and division of fractions then apply these procedures to solving problems. As students move on to sixth grade, the standard asks them to explain the meaning of multiplication and division of positive fractions and perform the calculations.

Multiplication and division by fractions are among the most difficult concepts in the elementary mathematics curriculum. Many students learn these concepts through procedure-oriented, memory-based instruction that attribute little meaning to operations. Students need to develop number sense and operation sense before understanding the procedure and computation of multiplying and dividing fractions (Tirosh 2000). A common type of error is to have students begin computations before they have an adequate background of such operations. Students must understand the meanings of fractions before performing operations with them (Aksu 1997). To solve fraction multiplication, students are traditionally taught the cancellation algorithm (cancel and multiply). For division, students are asked to follow the computational procedure of inverting the divisor and changing the operation to multiplication (invert and multiply) (Castro 2008). However, student errors arise due to a lack of understanding of the underlying rationale in multiplication and division of fractions. Research suggests that teachers should provide students with meaningful concrete activities that assist students in connecting their informal knowledge of fractions with more formal instruction to build a foundation of understanding (Castro 2008).

Multiplication of Fractions

Multiplying fractions is a relatively simple procedure to teach, but focusing on fraction rules can be dangerous. Rules do not help students think about the meaning behind multiplying fractions (Aksu 1997). Many students know and use the procedural rules for multiplying fractions, but many cannot explain what 3/4 x 1/2 means (Aksu 1997). In a study conducted by Mack (1998), equal sharing situations in which parts of a part can be used was developed as a basis for understanding multiplication of fractions. For example, sharing half a pizza equally among three children results in a child getting one third of a half. Students should understand the concept of fractions before focusing on the operation of multiplication (de Castro 2008). Several researchers suggest that knowledge of partitioning (i.e., the process of dividing a whole or unit into equal sized parts) may provide a foundation for student understanding of multiplication of fractions because knowledge of portioning may lend itself to understanding the concept of fraction (Mack 2001).

Division of Fractions

Division of fractions is often considered the “most mechanical and least understood topic in elementary school” (Tirosh 2000). Children’s success rates on various tasks related to such divisions are usually very low. During the late 1950s and early 1960s, researchers suggested two basic algorithms, invert-and-multiply and common denominator, for division of fractions (Sharp & Adams, 2002). The invert-and-multiply algorithm is more efficient and more closely related to algebraic thinking, but students struggle with this algorithm because they memorize, rather than understand, the procedure (Sharp & Adams, 2002). The problem with the common denominator algorithm arises when the division results in a remainder or when the divisor is greater than the dividend; students find it difficult to compute such division accurately (Sharps & Adams, 2000).

California State Standard Curriculum for Fractions

Prior Knowledge

  • 2nd Grade: 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.
  • 3rd Grade: Number Sense
    • 3.2 Add and subtract simple fractions (e.g., determine that 1⁄8 + 3⁄8 is the same as 1⁄2).
    • 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).
    • 3.3 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).
  • 4th Grade: Number Sense
    • 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.
    • 1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places.

Objective

  • 5th Grade: 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.
    • 2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals
    • 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.
  • 6th Grade: 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
    • 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 calculation.


Cognitive Obstacles and Common Misconceptions

Like the addition and the subtraction of fractions, multiplication and division of fractions can be confusing for students from elementary school to high school. The underlying concepts behind multiplying and dividing fractions are as much to blame for students' confusion as much as the procedures that students are taught. Without conceptual understanding of multiplying and dividing fractions, many students fail to understand why one multiplies by the inverse to divide fractions or when cross multiplication is used. Fractions are very much interrelated to percentages and decimals. However, the teaching of fractions along with decimals and percentages without creating a strong foundation of knowledge for students leaves many students either not understanding how to, when to, or why to multiply and divide fractions. This and other instances of students receiving information without having the proper understanding of the part-whole relation and/or whole numbers.

Multiplication of Fractions

There are several common misconceptions regarding the multiplication of fractions. The first misconception is that a student must find the common denominator before solving a problem. See the problem below:

  • 1/2 multiplied by 3/4

A student will find the common denominator of both fractions (8) before solving the problem, thus rendering the new problem:

  • 4/8 multiplied by 6/8

Another misconception students make is that a student will not know whether to add/multiply numerators/denominators (Brown & Quinn, 2006).

  • A student will multiply the numerators and add the denominators.
  • E.g. 4/6 multiplied by 1/2 equals 4/8

In addition to these misconceptions, students do not understand the formula for multiplying fractions (Brown & Quinn, 2006).

  • Students will not properly apply G/H multiplied by E/J equals GxE/HxJ

Division of Fractions

One of the main problems students will face with the division of fractions is understanding the formula. This is taught commonly as the rule "multiplying by the inverse" of the fraction. See the example below:

  • 4/5 divided by 1/3

A student is taught that before solving the problem, the student must multiply by the inverse of the problem, thus solving this problem:

  • 4/5 multiplied by 3/1

Often students misuse the formula (and many times without understanding the mistake or why to multiply by the inverse). A mistake may be:

  • 4/5 divided by 1/3 turns into 1/3 multiplied by 4/5
  • 4/5 divided by 1/3 turns into 3/1 multiplied by 4/5
  • 4/5 divided by 1/3 turns into 3/1 multiplied by 5/4
  • 4/5 divided by 1/3 turns into 3/1 multiplied by 4/5

Misconceptions and their Effects

Misconceptions related to multiplying and dividing fractions do not only affect a students' ability with fractions, percentages, and decimals. These misunderstandings that are created during a students' elementary education go on to affect a student later during his/her time studying more complex subjects such as algebra (Brown & Quinn, 2007). Brown & Quinn find that early struggles with fractions corresponds with students later not understanding formulas and concepts presented in algebra. As shown, multiplication and division of fractions present different formulas for solving problems than addition and subtraction do. However, all the functions are related because the fraction represents the division of two numbers, often with the difficult concept of numbers between 0 and 1. As higher math classes like algebra and calculus call on a proficient understanding of fraction concepts and proficient execution of solving problems with fractions, multiplying and dividing fractions must be understood for proficiency later in a students' mathematics career.

Pedagogical Tools and Strategies

General Concepts to Abide By

When teaching students to manipulate fractions, whether it be multiplying, dividing, adding, or subtracting, it is important for students to connect the idea of fraction manipulation with whole number manipulation. Also important, students should be encouraged to use their own strategies and methods of modeling to solve problems, as long as they are able to explain their solutions.

According to Van de Walle and colleagues (2010), before finding fractions of fractions, students should find fractions of whole numbers. It would be best for students to be confronted with problems to solve before they know the type of problem they are solving. For example, when given the problem “There were 60 cars on the freeway. ¼ of them were white. How many were white?” students should be asked to solve this problem before being told that this is a multiplication problem. This way, a classroom discussion may evolve over what multiplying fractions means, and multiplication of fractions may be taught with a strong conceptual basis.

After introducing and practicing the multiplication of fractions and whole numbers, teachers should then allow students to practice multiplying fractions where there is no need to subdivide after the initial fractions. For example, a sample problem would be, “We ate ¼ of the cake last night, so there was ¾ left this morning. At lunch today, my brother ate 1/3 of what was left. Then how much is left?” The two fractions begin multiplied are ¾ and 1/3; therefore, the fourths do not need to be subdivided further because the fact that there are three pieces allows it to be divided into thirds easily. (Van de Walle and colleagues 2010).

Fraction Strips and Area Models

Using fraction strips can be a helpful way for students to model the multiplication of fractions. They should have the opportunity to use fraction strips to discover the relationships between quarters, thirds, halves and sixths, by dividing them into their different parts and labeling them. By asking them to model 1/3 of ¼ instead of asking them to compute the algorithm, students will develop a strong conceptual understanding of the meaning of multiplication of fractions. In addition to the fraction strips, area models can be helpful ways to diagram fractions—they are at times easier to draw and to connect to the algorithm (Van de Walle and collegues 2010).


Vertical/Horizontal Cognitive Models

Another successful method of teaching fraction multiplication are the vertical/horizontal cognitive model strips, where students may superimpose the multiplier, drawn horizontally, over the multiplicand, drawn vertically. The overlapping section is the product (de Castro, 2008).

Multiplication Table.JPG

Similar cognitive models can be applied to division. One can draw the divisor and the dividend, both horizontally. The overlapping section becomes the numerator and the total number of shaded regions as the denominator of the quotient (de Castro, 2008).

Division Table.JPG

In addition, students may choose to use area models, length models, or counters to represent the partative and measurement division of fractions (Van de Walle and Colleagues 2010).


Remember!

In every scenario, it is important for students to develop the algorithms themselves, based on their conceptual understanding. Guiding students toward these conclusions is best done through the study of patterns using various cognitive models and figures.

Curricula and Technological Resources

Technological Resources

Online Games

Home School Math

An example of a game provided on the website is Multiply Fractions Jeopardy: Multiply Fractions Jeopardy.jpg


AAA Math

  • features short lessons on many math concepts, including division and multiplication of fractions. This website also includes games, practice exercises, and quizzes to help students focus their learning of the subject.

An example of a practice exercise available on the website is Division of Fractions Practice: Division of Fractions Practice.jpg

Visual Fractions

  • Great site for studying all aspects of fractions: identifying, renaming, comparing, addition, subtraction, multiplication, division. Each topic is illustrated by either a number line or a circle with a Java applet. Also includes a couple of games. For example: make cookies for Grampy.

Who Wants Pizza?

Fraction Model

Clara Fraction's Ice Cream Shop

Equivalent Fractions from National Library of Virtual Manipulatives (NLVM)

Teacher Resources and Literary Resources

Teacher Resources

Brain Pop: Multiplying and Dividing Fractions

Mathematics for Elementary School Teachers

  • This concepts book was written by Tom Bassarear specifically to assist teachers in teaching many different mathematical subjects. Pages 262-272 of the book focus primarily on the multiplication and division of fractions, and provide clear and helpful ways of relaying sometimes intimidating concepts to students.
  • Most effective when utilizing with "Mathematics for Elementary School Teachers: Explorations" (2nd Edition). This supplementary resource contains handouts and worksheets that create a more well-rounded learning environment for the mathematics teacher. Fraction tiles and pie pieces are examples of manipulatives that can be found in this resource.

Teach-Nology

Literary Resources

Marilyn Burns has a great series of books that introduces, emphasizes, and extends fractions with children of all ages. The series is organized among grade levels. Her website is provided below:

Annotated References

Ahia, F., Fredua-Kwarteng, E. (2006). Understanding Division of Fractions: An Alternate View. 1-12

  • The researcher discusses problems students face with the division of fractions.

Aksu, M. (1997). Student performance in dealing with fractions. The Journal of Educational Research, 90(6), 375-380.

  • This research studied the differences between student procedural and conceptual understanding of fractions and its effect on student performance.

Bassarear, T. (2000). Mathematics For Elementary Teachers Second Edition (2 ed.). Boston: Houghton Mifflin Company.

  • This concepts book contains strategies and tips for teaching all math subjects for elementary school teachers. Recommended to be used with "Mathematics for Elementary Teachers: Explorations", a guide that contains supplementary worksheets and blackline masters to use for lessons.

Brown, G., Quinn, R.J. (2006). Algebra Students' Difficulty with Fractions: An Error Analysis. Australian Mathematics Teacher, 62(4), 28-40.

  • The researchers searchers for and discusses the link between understanding fractions and later errors while learning algebra.

Brown, G., Quinn, R.J. (2007). Investigating the Relationship between Fraction Proficiency and Success in Algebra. Australian Mathematics Teacher, 63(4), 8-15.

  • The researchers continue finding links to knowledge of fractions and how it correlates with proficiency with algebra.

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 informs teachers of the mathematical concepts they are required to teach at each grade level.

Callingham, R., Watson, J. (2004). A Developmental Scale of Mental Computation with Part-Whole Numbers. Mathematics Education Research Journal, 16(2), 69-86.

  • The researcher tests students from 3rd to 10th grade on their ability to mentally represent fractions, decimals, and percentages.

de Castro, B.V. (2008). Cognitive models: The missing link to learning fraction multiplication and division. Asia Pacific Education Review. 9(2), 101-112.

  • This research aimed to establish the effectiveness of building students’ understanding of fractions using cognitive models as compared to the traditional algorithmic way of teaching.

Karp, K. S., Walle, J. A., & Williams, J. M. (2009). Texas Edition of Elementary and Middle School Mathematics (with MyEducationLab) (7th Edition) (7 ed.). Boston, MA: Allyn & Bacon.

  • This text was designed to aid teachers in developing research-based methods of teaching elementary and middle school mathematics. The referenced section focused on methods developing students' concept of fraction computation.

Mack, N.K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295.

  • The purpose of this research was to examine the development of students’ understanding of multiplication of fractions. This research provided insight to how students can solve and understand problems with multiplication of fractions by using skills of partitioning.

Sharp, J. & Adams, B. (2002). Children’s construction of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.

  • This research article examined the thinking of children who had the opportunity to construct personal knowledge about division of fractions. The study provided alternative views to the traditional view of teaching division of fractions-invert and multiply.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.

  • This research discussed how elementary educators can promote student understanding of division of fractions by building conceptual understanding of fractions, rather than the procedural computations of multiplying and dividing factions.