Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70510
Product of Rational Numbers
Students are asked to define a rational number and then explain why the product of two rational numbers is rational.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, rational number, product
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProductOfRationalNumbers_Worksheet.docx
MFAS_ProductOfRationalNumbers_Worksheet.pdf
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 Product of Rational Numbers worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student cannot correctly define a rational number.
Examples of Student Work at this Level
The student may be able to provide an example of a rational number. However, the student is unable to write a complete and correct definition. The student says:
A rational number cannot be negative.
A rational number is a number that can be written as a fraction (or a ratio).
page 1 of 3 A rational number is a positive whole number.
A rational number is any number that does not have a repeating decimal.
Questions Eliciting Thinking
You said that a rational number can be written as a fraction. Can you be more specific about the numerator and denominator? What type of numbers must they be? Are
there any restrictions on the denominator?
Is
a rational number? How do you write
as a decimal number? So repeating decimal numbers must be what?
Can a rational number be negative?
Can you give me an example of a number that is not rational?
Instructional Implications
Remind the student that the integers consist of the set {…­3, ­2, ­1, 0, 1, 2, 3, …}. Then, review the definition of a rational number as a number that can be written in
the form
where a and b are integers and
. Use the definition as a way to “build” rational numbers by substituting integers for a and b to form a variety of rational
numbers. Then use the definition as a way to show a number (e.g., 0, -8, 12,
,
) is rational by rewriting it as a fraction of integers. Finally, ask the student to
convert a variety of rational numbers written in fraction form to decimals and to observe that the decimal representation of a rational number will either terminate or repeat.
Challenge the student to review his or her initial explanation of a rational number. Ask the student to correct any errors and revise the explanation. Also, ask the student to
provide additional examples of rational numbers.
Making Progress
Misconception/Error
The student is unable to explain why the product of two rational numbers must be rational.
Examples of Student Work at this Level
The student can define a rational number and provide examples. However, the student is unable to explain why the product of two rational numbers must be rational. The
student:
Indicates that he or she does not understand why the product of two rational numbers must be rational.
Offers an incorrect explanation such as:
The product of two rational numbers is always rational because you are multiplying a rational number a rational amount of times.
The product always has an ending and is not continuous.
The product of two rational numbers is rational because nothing about their “state” changes.
Questions Eliciting Thinking
What do you mean by the “state” of a number?
What happens when you multiply two integers? Will the product be an integer?
Can you use the fact that the product of two integers is always an integer to explain why the product of two rational numbers is always rational?
Instructional Implications
Review the fact that the integers are closed for addition, subtraction, and multiplication. Guide the student to understand that the rational numbers are closed for addition,
subtraction, multiplication, and division. Show that the product of two rational numbers must be rational by reasoning that if
and
are rational then their product,
, must be rational since both ac and bd are integers. Ask the student to reason in a similar fashion to show that the rational numbers are closed for subtraction, addition,
and division.
Challenge the student to find an example that shows the irrational numbers are not closed for multiplication.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student explains that rational numbers can be written as a ratio of two integers with a nonzero denominator. Rational numbers written as decimals are repeating or
terminating. The student is able to provide correct examples.
Suppose p and q are rational numbers. Since p and q are rational, they can be represented as fractions of integers with nonzero denominators, for example, as
where a, b, c, d, are integers such that
integers. Additionally, since
and
then
and
. Then
. So,
and
. Since integers are closed under multiplication, both ac and bd are
is a fraction of integers with
which means it is a rational number. Therefore, a rational
number times a rational number will always be rational.
page 2 of 3 Questions Eliciting Thinking
To what number system do both the rational numbers and irrational numbers belong?
Do you think the real numbers are closed under multiplication?
Instructional Implications
Challenge the student to determine whether each of the following statements is always true, sometimes true, or never true:
The sum of two rational numbers is rational.
The difference between two integers is an integer.
The quotient of two integers is an integer.
For statements that are sometimes or never true, ask the student to provide a counterexample.
Consider using the MFAS tasks Sum of Rational Numbers (N-RN.2.3), Sum of Rational and Irrational Numbers (N-RN.2.3), and Product of Non-Zero Rational and Irrational
Numbers (N-RN.2.3) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Product of Rational Numbers 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
Name
MAFS.912.N-RN.2.3:
Description
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational
number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
page 3 of 3