Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70488
Sum of Rational and Irrational Numbers
Students are asked to describe the difference between rational and irrational numbers and then explain why the sum of a rational and an irrational
number is irrational.
Subject(s): Mathematics
Grade Level(s): 9, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, rational number, irrational number, sum
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_SumOfRationalAndIrrationalNumbers_Worksheet.docx
MFAS_SumOfRationalAndIrrationalNumbers_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 Sum of Rational and Irrational Numbers worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student cannot distinguish between rational and irrational numbers.
Examples of Student Work at this Level
The student may be able to provide an example of a rational number and an irrational number. However, the student is unable to completely and correctly distinguish
between the two types of numbers. The student says:
A rational number is a decimal while an irrational number is a whole number.
page 1 of 4 A rational number is a fraction while an irrational number is a decimal.
A rational “has an end” while an irrational “goes on forever.”
A rational “has a pattern” while an irrational “goes on forever in a random sequence.”
A rational can be expressed as a fraction while an irrational cannot.
Questions Eliciting Thinking
What happens when you try to write a rational number as a fraction of integers? What happens when you try to write an irrational number as a fraction of integers?
What happens when you try to write a rational number as a decimal? What happens when you try to write an irrational number as a decimal?
Can you explain what you mean by “has an end” and “goes on forever”?
Can you explain what you mean by “has a pattern”?
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 but
. 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.
Introduce irrational numbers by describing them as numbers that are not rational (i.e., numbers that cannot be expressed as a quotient of integers). Provide examples of
irrational numbers other than
. Explain to the student that a way to generate examples of irrational numbers is by taking the square root of a non-perfect square (e.g.,
). Next, compare the decimal representations of rational and irrational numbers by explaining that when an irrational number is written as a decimal, it neither terminates
nor repeats.
Challenge the student to review his or her initial explanation of the difference between rational and irrational numbers. Ask the student to correct any errors and revise the
explanation. Also, ask the student to provide additional examples of rational and irrational numbers.
Making Progress
Misconception/Error
The student is unable to explain why the sum of a rational and irrational must be irrational.
Examples of Student Work at this Level
The student can explain the difference between a rational and an irrational number and provide examples of each. However, the student is unable to explain why the sum
of a rational number and an irrational number is irrational. The student:
Indicates that he or she does not understand why the sum of a rational and an irrational must be irrational.
Offers an incorrect explanation such as:
Since an irrational number cannot be written as a fraction, when a non-fraction (an irrational number) is combined with a fraction (rational number), a common
denominator cannot be determined, so the result cannot be a fraction. Therefore, the result is a non-fraction or an irrational number.
page 2 of 4 Since an irrational number has a nonrepeating, nonterminating decimal, when this type of decimal is combined with a repeating or terminating decimal (rational
number), the decimal becomes another nonrepeating, nonterminating (an irrational number) because the decimal of the irrational number does not reduce.
Therefore, the sum is a larger irrational number.
Questions Eliciting Thinking
Why can’t an irrational number be written as a fraction? What if I rewrite as
? Does this mean that
is rational?
What do you mean by “the decimal does not reduce”? What happens when you add negative five to ? Is the sum larger than five? Is the sum larger than
?
What happens when you add two rational numbers? Will the sum be rational?
Can you use the fact that the sum of two rational numbers is always rational to explain why the sum of a rational and irrational can never be rational?
Instructional Implications
Introduce the student to proof by contradiction. Explain that a proof by contradiction contains the following features: (1) the negation of the conclusion is assumed (e.g.,
assume that the sum of a rational and an irrational is rational); and (2) then one shows how this assumption leads to the contradiction of something known or established.
The contradiction of something known or established indicates the negation of the conclusion cannot be true. Therefore, the conclusion must be true.
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, but the irrationals are not. Show that the sum of two rational numbers must be rational by reasoning that if
then their sum,
and
are rational,
, must be rational since ad + bc and bd are integers. Ask the student to reason in a similar fashion to show that the rational numbers are closed for
subtraction, multiplication, and division.
Next, ask the student to consider if a number such as
cannot be rational. If it is rational, then
is rational or irrational. Guide the student to reason that (given
is equal to some rational number x which means that
if x is rational, x - 5 is rational, which contradicts the fact that
is irrational and 5 is rational)
. But the rational numbers are closed for subtraction, so
is irrational. Ask the student to use similar reasoning to explain why
is irrational. Then, ask the
student to develop a general explanation for why the sum of a rational number and an irrational number must be irrational.
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 but irrational numbers cannot. As a result, rational numbers
written as decimals are repeating or terminating while irrational numbers are nonrepeating, nonterminating decimals. The student is able to provide examples of each type of
number.
To show that the sum of a rational and an irrational must be irrational, the student provides a proof by contradiction such as: Suppose a is a rational number and b is an
irrational number. Then let c = a + b and assume c is rational. Then b = a - c. Since a and c are rational numbers (and the rational numbers are closed for subtraction) then
b must be rational which contradicts the assumption that b is irrational. Therefore, the sum of a rational number and an irrational number must be irrational.
Questions Eliciting Thinking
Are the irrational numbers closed for multiplication?
To what number system do both the rational numbers and irrational numbers belong? Do you think the real numbers are closed for addition?
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 irrational numbers is irrational.
The product of two irrational numbers is irrational.
For statements that are sometimes or never true, ask the student to provide a counterexample.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Sum of Rational and Irrational 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
page 3 of 4 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 4 of 4