Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 56082
Decimal to Fraction Conversion
Students are given several terminating and repeating decimals and asked to convert them to fractions.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, decimal, repeating decimal, terminating decimal, fraction, rational, irrational
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_DecimalToFractionConversion_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Decimal to Fraction Conversion worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not convert a terminating decimal into a fraction.
Examples of Student Work at this Level
The student does not use place value for the denominator, but instead writes a one over the given digits to make the fractions.
page 1 of 5 The student uses the wrong place value when writing the denominators of each fraction.
The student attempts to change the numbers to a percent.
Questions Eliciting Thinking
How did you choose that numerator and that denominator?
How do you read the decimal number? How do you read your fraction? Does the place value you said for each one match?
Why did you change to a percent? What does percent mean? What do the instructions ask you to do?
Instructional Implications
Provide instruction to the student regarding converting terminating decimals to fractions and mixed numbers.
Have the student use mathematically precise vocabulary when saying decimal and fraction names. Emphasizing the place value words helps the student see the relationship
between the two.
Moving Forward
Misconception/Error
The student does not convert a decimal with all repeating digits into a fraction.
Examples of Student Work at this Level
The student does not convert the decimals 0.777…, 0.272727… or 0.27777… to fractions explaining, “They cannot be written as fractions because they are repeating
decimals.”
The student writes the fraction as if it were a terminating decimal:
Over a place value of ten, based on the number of digits shown.
But adds a repeating bar over the numbers in the numerator to show the repeating digits continue.
page 2 of 5 After adding a zero onto the end of the decimal to “finish” the number, before writing it over its new place value denominator.
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Questions Eliciting Thinking
How is a repeating decimal different than a terminating decimal? What do the “dots” mean? How can you rewrite this decimal number using repeating bar notation?
Why did you add a zero onto the repeating digits of the decimal? Does that change the meaning of the number? If this were a terminating decimal, would adding a zero
give you an equivalent decimal?
Instructional Implications
Provide the student with a calculator and a wide variety of fractions (repeating and non-repeating). Have the student change each fraction to a decimal while looking for
patterns. From this they can develop general rules and procedures regarding writing decimals as fractions. Show the student how the patterns they discovered come from
algebraic solutions.
Review with the student why adding zeros onto the end of terminating decimals produces an equivalent decimal, but why this is not possible with a repeating decimal.
Making Progress
Misconception/Error
The student does not convert a decimal that has non-repeating digits before the repeating digits into a fraction.
Examples of Student Work at this Level
The student converts 0.27777… to i.e., as if it were 0.272727… (or with a repeating bar over the repeating seven as ).
Questions Eliciting Thinking
Why did you write the fraction for 0.27777… the same as you did for 0.272727… ? What is the difference between these two decimals? Should there be a difference
between their fraction equivalents?
Instructional Implications
Provide direct instruction with the algebraic method of changing decimals that have non-repeating digits before the repeating digits into fractions.
Almost There
Misconception/Error
The student cannot explain that there are some decimals that cannot be written as fractions.
Examples of Student Work at this Level
There are no decimals that cannot be written as fractions because:
You can always round.
They all have a place value.
A decimal is just a different form of a fraction.
Everything is terminating or repeating.
Decimals are rational numbers.
There are decimals that cannot be written as fractions because:
There are decimals that go on and on.
Some decimals are not tenths, hundredths, thousandths, etc.
Questions Eliciting Thinking
Is it possible to have a decimal with digits that don’t repeat but keep on going? Can you think of an example? How would you read that number?
page 3 of 5 Instructional Implications
If necessary, review the counting numbers, whole numbers, and integers. Then define the rational numbers as numbers of the form
where a and b are integers but b
cannot be zero. Use this as the basis to discuss examples of irrational numbers.
Provide the student a wide range of differing decimal numbers of all types: terminating, non-terminating, repeating and non-repeating. Have the student categorize these
into rational and irrational numbers based on their characteristics, then write fraction equivalents for all rational numbers.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
(1)
(2)
(3)
(4)
or
(5)
or
(6) Yes, irrational numbers (or decimals that are non-terminating and non-repeating) cannot be written as fractions.
Questions Eliciting Thinking
If any of these decimals was greater than one, how would that change your fraction answer?
How do you write an integer as a decimal? How are integers and rational numbers related? Are integers also rational? Are all rational numbers integers?
What are some examples of irrational numbers?
Instructional Implications
Have the student convert decimals greater than one (with and without repeating decimals) into mixed numbers and improper fractions.
Ask the student to draw a diagram that shows the relationships among the following number systems: counting numbers, whole numbers, integers, rational numbers,
irrational numbers, and real numbers. Provide feedback as needed.
Have the student practice converting fractions into repeating decimals and identifying rational numbers. Consider implementing MFAS tasks Fraction to Decimal Conversion
and Rational Numbers (8.NS.1.1).
Make the student aware that there are numbers that are neither rational nor irrational that will be studied in later mathematics courses.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Decimal to Fraction Conversion worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
page 4 of 5 Name
MAFS.8.NS.1.1:
Description
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal
expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion
which repeats eventually into a rational number.
page 5 of 5