Sum of Rational Numbers - | CPALMS.org


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Resource ID#: 70526


Primary Type: Formative Assessment


Sum of Rational Numbers

Students are asked to define a rational number and then explain why the sum of two rational numbers is rational.

General Information

Subject(s): Mathematics
Grade Level(s): 8, 9, 10, 11, 12
Intended Audience: Educators
   
 
Freely Available: Yes
Keywords: MFAS, rational number, irrational number, sum, repeating decimals, terminating decimals, proof, integers, nonrepeating nonterminating decimals
Resource Collection: MFAS Formative Assessments

Attachments

mfas_sumofrationalnumbers_worksheet.docx
mfas_sumofrationalnumbers_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 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 is not a whole number.



  • A rational number is a fraction.



  • A rational number “doesn’t go on forever.”



  • A rational number does not have an infinite number of random numbers following the decimals.



  • A rational number is a real number that can be written as a fraction.

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?

What happens when you try to write a rational number as a decimal?

Can you explain what you mean by “doesn’t go on forever”?

Can you explain what you mean by “can be written as a fraction”? Can you provide examples?

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 begin mathsize 12px style a over b end style where a and b are integers but begin mathsize 12px style b space not equal to space 0 end style. 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, begin mathsize 12px style 2 over 3 end style, begin mathsize 11px style negative 15 over 4 end style) 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 sum of two rational numbers must be rational.

Examples of Student Work at this Level

The student can define and provide examples of rational numbers. However, the student is unable to explain why the sum of two rational numbers must be rational. The student:

  • Indicates that he or she does not understand why the sum of two rational numbers must be rational.
  • Offers an incorrect explanation such as:
    • Since a rational number can be written as a fraction, then combining two fractions requires a common denominator and the result is another fraction. Therefore, the result is another rational number.
    • Since a rational number has a repeating or a terminating decimal, when the decimals are combined, the result is another (larger) repeating or terminating decimal. Therefore, the sum is a rational number.

Questions Eliciting Thinking

How do you know that the common denominator is an integer?

How could you use variables to model rational numbers and then model the sum of the two rational numbers?

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 sum of two rational numbers must be rational by reasoning that if begin mathsize 11px style a over b end style and begin mathsize 11px style c over d end style are rational then their sum, begin mathsize 11px style fraction numerator a d space plus space b c over denominator b d end fraction end style, 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, multiplication, and division.

 

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 fraction of two integers with a nonzero denominator. As a result, rational numbers written as decimals are repeating or terminating. The student is able to provide a variety of examples of rational numbers (e.g., whole numbers, integers, fractions, repeating and terminating decimals, and radicals that reduce to a rational number). To show that the sum of two rational numbers must be rational, the student provides a proof such as: 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 begin mathsize 12px style p space equals space a over b end style and begin mathsize 12px style q space equals space c over d end style where a, b, c, d, are integers such that begin mathsize 12px style b space not equal to space 0 end style, size 12px d begin mathsize 12px style space not equal to space 0 end style. Then begin mathsize 12px style p space plus space q space equals space a over b space plus space c over d space equals space fraction numerator a d space plus space c b over denominator b d end fraction end style. Since integers are closed under multiplication and addition then ad + cb and bd are also integers. Additionally, since begin mathsize 12px style b space not equal to space 0 end style, size 12px d begin mathsize 12px style space not equal to space 0 end style then size 12px b size 12px d begin mathsize 12px style space not equal to space 0 end style. So, begin mathsize 12px style fraction numerator a d space plus space c b over denominator b d end fraction end style is a fraction of integers with size 12px b size 12px d begin mathsize 12px style space not equal to space 0 end style, which means it is a rational number. Therefore, the sum of two rational numbers must be rational.

Questions Eliciting Thinking

To what number system do both the rational numbers and irrational numbers belong?

What is an irrational number? Are there any rational numbers that are also irrational?

Instructional Implications

Challenge the student to determine whether each of the following statements is always true, sometimes true, or never true:

  • The product of two rational numbers is rational.
  • The difference between two rational numbers is rational.
  • The quotient of two rational numbers is rational.


For statements that are sometimes or never true, ask the student to provide a counterexample.

 

Accommodations & Recommendations


  • Special Materials Needed:

    • Sum 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


Aligned Standards


Benchmark(s) of focus

The benchmark(s) of focus is the primary focus for student learning and instruction to be taught or reinforced and provides an intentional opportunity for students to work with that concept or skill.

Name Description
MA.8.NSO.1.1: Extend previous understanding of rational numbers to define irrational numbers within the real number system. Locate an approximate value of a numerical expression involving irrational numbers on a number line.
Clarifications:
Clarification 1: Instruction includes the use of number line and rational number approximations, and recognizing pi (π) as an irrational number.

Clarification 2: Within this benchmark, the expectation is to approximate numerical expressions involving one arithmetic operation and estimating square roots or pi (π).



Printed On:4/25/2024 8:47:26 PM
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