Module on Relating and Comparing Fractions and Decimals for Grades 5 and 6

By

Mary Jo Ehrlich, Alyce Montgomery, Linda Schmitz

Table of Contents Math Module Outline Grade 5-6 Pretest Form A Lesson 1: Converting Decimals to Fractions Lesson 2: Converting Fractions to Decimals Lesson 3: Comparing Fractions and Decimals Grade 5-6 Posttest Form B Rubrics For Test Items Module development was partially funded by the Missouri Coordinating Board for Higher Education through the Eisenhower Professional Development Program.

Translations were partially funded by NSF ESIE SGER Project 0086580

Math Module Outline

Strand: Number Sense Grade Level: 5-6 Concepts Included in Module: Relating and Comparing Fractions and Decimals Authors: Mary Jo Ehrlich, Alyce Montgomery, Linda Schmitz District: Lewis County C-1 Statement of Basis for Selection of Strand/Concepts: The module will explore and expand the interrelationship of fractions and decimals. Teacher observations and test scores show a need for this module. Module Resources: Korean Mathematics, Grade 5-1 (2001) Edited by Janice Grow-Maienza, translated by Sue Chung Nugent, Kirksville, MO: Truman State University, From Ministry of Education. Arithmetic Grades 1-6 Seoul, Korea: National Textbooks Inc., 1993. Investigations Grade 5: Fractions, Percents, and Decimal Joan Akers, et.al. Dale Seymour Publications. White Plains, New York (1998) SuccessLink-Great Ideas. Fractions with Pizza-zz. http://www.successlink.org/great/g450.html Retrieved August 7, 2002. Calculators, magazines, newspapers, games, number cubes NCTM Content Standards for Fractions and Decimals Module Number and Operations

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• Understand meanings of operations and how they relate to one another • Compute fluently and make reasonable estimates

Problem Solving

• Build new mathematical knowledge through problem solving • Solve problems that arise in mathematics and in other contexts • Apply and adapt a variety of appropriate strategies to solve problems • Monitor and reflect on the process of mathematical problem solving

Connections

• Recognize and use connections among mathematical ideas • Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole • Recognize and apply mathematics in contexts outside of mathematics

Total Score___________

Grade 5-6 Pretest Form A

Name ____________________________________________ Date_______________ Be sure to follow the directions for each problem. Solve.

1. Express the shaded areas as both a fraction and a decimal.

a) Fraction _________

Decimal _________

b) Fraction ________

Decimal ________

2. Draw a picture that represents 3/5. 3. Draw a picture that represents 0.4.

4. Write a fraction of your choice and its equivalent decimal. Fraction Decimal _______ = _______

5. a) How much money is ¼ of a dollar? Write it with a decimal point. _____________ b) Explain how you got your answer.

6. Juan drank a 0.5 liter bottle of Mountain Dew with his lunch. What fraction of a liter did he drink?

_______________

7. Order the following from least to greatest: 0.75 , 1/3 , 4/5 , 0.2 ___________________________________ 8. a) Susan and Tina were the first to finish in a swimming race. Susan swam

the race in 0.3 hour. Tina’s time was 1/5 hour. Who won the race? _______________

b) Explain how you arrived at your answer.

Lesson 1: Converting Decimals to Fractions Objective: Students will be able to convert decimals to fractions in their lowest form. Standards Addressed: Show-Me Goals:

• 1.2 – Students will demonstrate within and integrate across all content areas the ability to conduct research to answer questions and evaluate information and ideas.

• 2.3 – Students will demonstrate within and integrate across all content areas the ability to exchange information, questions and ideas while recognizing the perspectives of others.

• 4.6 – Students will demonstrate within and integrate across all content areas the ability to identify tasks that require a coordinated effort and work with others to complete those tasks.

Missouri Frameworks for Curriculum Development: • Mathematics #1 – In Mathematics, students in Missouri public schools will

acquire a solid foundation which includes knowledge of addition, subtraction, multiplication and division; other number sense, including numeration and estimation; and the application of these operations and concepts in the workplace and other situations.

• Mathematics #5 – In Mathematics, students in Missouri public schools will acquire a solid foundation which includes knowledge of mathematical systems (including real numbers, whole numbers, integers, fractions), geometry, and number theory (including primes, factors, multiples).

Introduction and application to everyday life: Homework assignment the night before lesson 1: Ask students to bring at least two items from home that use decimals or fractions.. Examples: rain gauge, stop watch, newspaper articles, soda bottle, pizza price list, stock market reports, recipes, etc. Have a large bulletin board and table prepared upon which to display items. Label the bulletin board with DECIMALS AND FRACTIONS. Lesson: Ask students to share the items they brought to class. Have them post each on the bulletin board or the table. Talk briefly how often we see decimals and fractions in our everyday life. Let’s study the relation between fractions and decimals. It’s very easy to change decimals to fractions. Just say the decimal and then write it as a fraction that would be read the same way.

Examples: 0.3 = three tenths = 3/10 0.57 = fifty-seven hundredths = 57/100 0.131 = one hundred thirty-one thousandths = 131/1000 Sometimes the fractions will need to be reduced to their lowest terms. Examples: 0.12 = 12 ÷ 4 = 3

100 ÷ 4 = 25

0.5 = 5 ÷ 5 = 1 10 ÷ 5 = 2 0.214 = 214 ÷ 2 = 107 1000 ÷ 2 = 500

Guided Practice: Have students form groups of three. In each group one student will write a decimal on a piece of paper, the second student will read the decimal, and the third student will write the decimal as an equivalent fraction. Students should rotate roles until each student has performed each role twice. The students will check each other’s answers. Independent Practice: Students complete Worksheet P1

Worksheet P1

Complete the following chart. DECIMAL WRITTEN NAME FRACTION IN

REDUCED FORM

1. 0.12 ___________________ __________ 2. ___________ one tenth __________ 3. 0.607 ______________________________ __________ 4. ___________ three hundred seventy-seven thousandths __________ 5. ___________ forty-two hundredths __________ 6. 0.9 ______________________________ __________ 7. 0.124 ______________________________ __________ 8. ___________ ______________________________ 1/2

Lesson 2: Converting Fractions to Decimals Objective: Students will be able to convert fractions to decimals. Standards Addressed: Show-Me Goals:

• Goal 1.4 – Students will demonstrate within and integrate across all content areas the ability to use technological tools and other resources to locate, select and organize information.

• Goal 1.5 – Students will demonstrate within and integrate across all content areas the ability to comprehend and evaluate written, visual and oral presentations and works.

• Goal 1.6 – Students will demonstrate within and integrate across all content areas the ability to discover and evaluate patterns and relationships in information, ideas and structures.

Missouri Frameworks:

• Mathematics #1 – In Mathematics, students in Missouri public schools will acquire a solid foundation which includes knowledge of addition, subtraction, multiplication and division; other number sense, including numeration and estimation; and the application of these operations and concepts in the workplace and other situations.

• Mathematics #5 – In Mathematics, students in Missouri public schools will acquire a solid foundation which includes knowledge of mathematical systems (including real numbers, whole numbers, integers, fractions), geometry, and number theory (including primes, factors, multiples).

Application to everyday life: The local weatherman recorded 3/4” rain last night. Your science teacher asks you to graph the rainfall amount on your class’ rainfall chart. However, in order to do so, you must convert that fractional amount to a decimal amount. What decimal amount equals 3/5” rain? Lesson: As you recall from our last lesson, decimals can be converted to fractions. It only makes sense that fractions can be converted back to decimals. Method 1: Sometimes you can create an equivalent fraction with a denominator of 10 or 100 (the same name as the first two places on the right side of the decimal point). Examples:

1 x 25 = 25 = 0.25 4 x 25 = 100 2 x 2 = 4 = 0.4 5 x 2 = 10

Practice: (Fill in the missing numbers.) 1 x 2 = 2 = 0.2

# x 2 10

# x 25 = 75 = 0.75 4 x 25 = 100

3 x # = # = 0.12

25 x # = 100 Method 2: A fraction is nothing more than a division problem. A fraction, read from the top downward, is the numerator divided by the denominator. Example: 3 = 3 divided by 8 OR 8 3 8 This problem can be solved either :

a) manually by using long division

0.375

8 3.000 24 60 56 40 40 0

OR

(answer)

b) by calculator by pressing: 8 ÷ 3 = 0.375

NOTE: If the quotient does not come out even, you may have to round to the nearest tenth, hundredth, or thousandth, depending on the accuracy level

you are seeking.

Guided Practice: (Practice using all three methods, when possible.) 4/5 = ___________ 3/4 = ________________ 7/10 = ___________ 6/8 = ________________ 2/3 = ____________ 4/7 = ________________ 6/25 = ____________ 5/12 = ________________ Point of Discussion: Two different fractions above were equivalent to the same decimal. Why did that occur? Write your explanation below. Pair Activity: Have students form pairs. Give each pair two number cubes that have numbers from 1 to 6. Both students roll their number cube. Have students record the smaller number as the numerator and the larger number as the denominator. Both students use calculators to find the equivalent decimal. Students should keep a record of their results on Worksheet P2. Class Activity: Show Transparency 1 on the screen. Supply students with calculators. Have all students use their calculators to help the teacher fill in the chart. Have students take turns vocalizing the answers.

Worksheet P2

Numerator Denominator Fraction Equivalent Decimal _________ ___________ _______ ___________ _________ ___________ _______ _________________ __________ ___________ _______ _________________ ___________ ___________ _______ _________________ ___________ ___________ _______ _________________ ___________ ___________ _______ _________________ ____________ ___________ _______ _________________ ____________ ___________ _______ _________________

Transparency 1

Fraction to Decimal Division Table

N/D 1 2 3 4 5 6

1 1

2 3

3 1.333

4

5

6 0.333

Lesson 3: Comparing Fractions and Decimals Objective: Students will be able to compare fractions to decimals and decimals to fractions. Standards Addressed: Show-Me-Goals:

• Goal 1.5 – Students will demonstrate within and integrate across all content areas the ability to comprehend and evaluate written, visual and oral presentations and works.

• Goal 1.6 – Students will demonstrate within and integrate across all content areas the ability to discover and evaluate patterns and relationships in information, ideas and structures.

• Goal 3.4 – Students will demonstrate within and integrate across all content areas the ability to evaluate the processes used in recognizing and solving problems.

Missouri Frameworks for Curriculum Development:

• Mathematics #1 – In Mathematics, students in Missouri public schools will acquire a solid foundation which includes knowledge of addition, subtraction, multiplication and division; other number sense, including numeration and estimation; and the application of these operations and concepts in the workplace and other situations.

• Mathematics #5 - In Mathematics, students in Missouri public schools will acquire a solid foundation which includes knowledge of mathematical systems (including real numbers, whole numbers, integers, fractions), geometry, and number theory (including primes, factors, multiples).

Application to everyday life: Tom finished his homework in 3/5 of an hour. Frank completed his homework in .65 of an hour. Who took longer doing his homework?

Lesson: • From Korean Mathematics Grade 5-1

Korean Mathematics p. 48

Korean Mathematics p. 49

Korean Mathematics p. 50

Korean Mathematics p. 51

Total Score _________ Grade 5-6 Posttest Form B

Name __________________________________________ Date _________________ Be sure to follow the directions for each problem. Solve.

1. Write a decimal of your choice and its equivalent fraction. Decimal Fraction

________ = __________

2. Draw a picture that represents 0.2

3. Express the shaded areas as both a fraction and a decimal.

a)

Fraction ________ Decimal __________ b)

Fraction _______________ Decimal ______________

4. Peter ate 3/5 of a pizza last Saturday night. Andy ate 0.7 of a pizza. Who ate the most pizza?

__________________ Explain how you arrived at your answer.

5. Hans ran the 50 yard dash in 0.9 of a minute. How would you show that amount as a fraction of a minute?

__________________

6. a) How much money is 4/5 a dollar? Write it with a decimal point.

_____________________

b) Explain how you got your answer.

7. Draw a picture that represents 7/10. 8. Order the following from greatest to least: 0.3 , 6/7 , 2/3 , 0.45

_____________________________________

9. Ralph’s dog, Pogo, is 0.4 ft. tall. Express Pogo’s height as a fraction of a foot. __________________________

10.

Rubric For

Multi-Point Questions On

Grade 5 – 6 Pretest Form A #2:

2 pts. – Student draws appropriate picture and indicates the correct number of parts

1 pt. - Student draws appropriate picture or shades the correct number of parts 0 pts. - Student indicates no mathematical understanding of the task #3: (use question #2 rubric) #5b:

2 pts. - Student demonstrates complete knowledge of mathematical concept and principles needed to complete the task

1 pt. - Student demonstrates partial knowledge of mathematical concept and

principles needed to complete the task

0 pts. - Student demonstrates no knowledge of mathematical concept and principles needed to complete the task

#8b: 3 pts. - Response includes all three of the following expectations: 1) clear and understandable explanation of solution processes

2) correctly converts the fraction to a decimal or the decimal to a fraction 3) correctly compares decimal to decimal or like fraction to like fraction

2 pts. - Response includes two of the three expectations above 1 pt. - Response includes one of the three expectations above 0 pts. - Response, or lack thereof, indicates no mathematical understanding of the task

Rubric For

Multi-Point Questions on

Grade 5 – 6 Posttest Form B #2 : 2 pts. – Student draws appropriate picture and indicates the correct number of parts 1 pt. - Student draws appropriate picture or shades the correct number of parts 0 pts. - Student indicates no mathematical understanding of the task #4b: 3 pts. - Response includes all three of the following expectations:

1) clear and understandable explanation of solution processes 2) correctly converts the fraction to a decimal or the

decimal to a fraction 3) correctly compares decimal to decimal or like

fraction to like fraction

2 pts. - Response includes two of the three expectations above 1 pt. - Response includes one of the three expectations above 0 pts. - Response, or lack thereof, indicates no mathematical understanding

of the task #6b :

2 pts. - Student demonstrates complete knowledge of mathematical concept and principles needed to complete the task

1 pt. - Student demonstrates partial knowledge of mathematical concept and

principles needed to complete the task 0 pts. - Student demonstrates no knowledge of mathematical concept and principles needed to complete the task #7 : (use question #2 rubric)

RUBRIC FOR

PERFORMANCE EVENT 4 pts. - The student’s response fully addresses the performance event.

The response: • demonstrates knowledge of the mathematical concepts and

principles needed to complete the event • communicates all process components that lead to an appropriate

and systematic solution • may have only minor flaws with no effect on the reasonableness of

the solution 3 pts. - The student’s response substantially addresses the performance event.

The response: • demonstrates knowledge of the mathematical concepts and

principles needed to complete the event • communicates most process components that lead to an

appropriate systematic solution • may have only minor flaws with minimal effect on the

reasonableness of the solution 2 pts. - The student’s response partially addresses the performance event. The response:

• demonstrates a limited knowledge of the mathematical concepts and principles needed to complete the event

• communicates some process components that lead to an appropriate and systematic solution

• may have flaws or extraneous information that indicates some lack of understanding or confusion

1 pt. - The student’s response minimally addresses the performance event. The response:

• demonstrates a limited knowledge of the mathematical concepts and principles needed to complete the event

• communicates few or no process components that lead to an appropriate and systematic solution

• may have flaws or extraneous information that indicates lack of understanding or confusion

0 pts. - The student’s response is non-existent or indicates no mathematical

understanding of the task.