Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 57151
Quotients of Integers
Students are given an integer division problem and asked to identify fractions which are equivalent to the division problem.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, quotient, integers, rational numbers, division, divide, fraction, expressions, equivalent
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_QuotientsOfIntegers_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 Quotients of Integers worksheet.
2. The teacher asks follow-up questions, as needed.
Note: Do not allow students to use calculators on this task.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand the meaning of the term equivalent.
Examples of Student Work at this Level
The student makes decisions based on whether or not the expressions “look like” the original expression in terms of placement of signs and numbers, e.g., where the five is
located or where the negative is located in the fraction.
Student explanations may include:
a. “it doesn’t match” or “it’s dividing one and four, not five and 20”
b. “neither the 20 nor the five is negative” or “the numbers are in the wrong place”
page 1 of 4 c. “the number four is not even in the problem” or “it is not a fraction”
d. “no parenthesis” or “there’s too many negative signs”
e. “it cannot be a fraction” or “it doesn’t match up”
f. “the 20 shouldn’t be negative” or “there should only be one negative” or “it’s kind of the same”
Questions Eliciting Thinking
What do you mean that it “matches?”
What does it mean for two fractions to be equivalent? Can you give me an example of two fractions that are equivalent?
Instructional Implications
Provide instruction on the meaning of equivalent as it applies to expressions that are equal in value. Provide opportunities for the student to evaluate expressions containing
quotients of rational numbers and to determine its equivalency.
Provide instruction on the meaning of equivalent as it applies to fractions. Be sure the student understands that p/q means p ÷ q and –(p/q) = (–p)/q = p/(–q). Expose
the student to effective strategies of his or her classmates to determine equivalency of expressions containing rational numbers.
Moving Forward
Misconception/Error
The student makes errors with the order of division.
Examples of Student Work at this Level
The student interprets ­5 ÷ 20 as 20 ÷ (­5) or the student interprets as ­4 ÷ 1.
Questions Eliciting Thinking
What is the difference between 5 ÷ 20 and 20 ÷ 5?
What is the difference between ­5 ÷ 20 and 20 ÷ (­5)?
How would you rewrite
as a division problem?
How would you rewrite 5 ÷ 20 as a fraction?
Instructional Implications
Guide the student to understand the order of division implied in simple division statements and in fractions. Give the student a general statement to follow such as “p ÷ q =
p/q and that means how many times q divides into p.” Also address writing each form using a standard long division symbol, since that is the method most students will use
to evaluate fractions or convert them to decimals. Ask the student to redo the problems on the Quotients of Integers worksheet. Provide feedback as needed.
Note: The Quotients of Integers worksheet is editable and can be rewritten with new integers and the problems rearranged to give the student further practice.
Almost There
Misconception/Error
The student uses a correct strategy but makes some sign errors.
Examples of Student Work at this Level
The student explains the equivalency of most expressions correctly, but has trouble with the use of three negatives within one fraction (d).
The student interprets fractions of the form
as –a ÷ (–b).
page 2 of 4 The student writes ­5 ÷ 20 as a fraction and uses a cross­multiplication strategy to check for equivalency, but makes some errors when applying the strategy to fractions
with multiple negatives.
Questions Eliciting Thinking
What did you find difficult about (d)? How did you determine whether the sign of the answer is positive or negative?
What does it mean for a fraction to be negative?
When using cross-multiplication to check for equivalency, what did you do with the negatives in each fraction?
Instructional Implications
Provide guidance on how to evaluate a fraction with three negatives, such as (d). Many students learn rules for multiplication and division of pairs of negative integers but do
not realize that these rules must be generalized when there are more than two negatives. Guide the student to systematically consider what happens when two, three,
four, five, or more negative numbers are multiplied or divided, and help the student to develop a general rule for the sign of the result.
Be sure the student understands that
. Ask the student to reconsider his or her responses to (b) and (e), if the meaning of the sign was
misinterpreted in these problems.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
a. It is equivalent because it also equals
b. It is not equivalent because ­5 ÷ 20 = c. It is not equivalent because
or ­0.25; ­5 ÷ 20 simplifies to but
, which can also be written as
.
.
does not equal -4.
d. It is equivalent because the two negatives in the parentheses make the fraction positive, so the negative on the outside of the parentheses makes the whole expression
negative.
e. It is equivalent because the negative of a fraction can be in the numerator or the denominator or out in front of the whole fraction (those are all equal).
f. It is not equivalent because ­5 ÷ 20 is negative but is positive.
Questions Eliciting Thinking
What would happen if the expression in (d) had only one negative sign in the parentheses? Would the answer change depending on whether the negative were in the
numerator or the denominator?
What are some other fractions that are equivalent to ­5 ÷ 20? Can you write one that uses the integer ­100 somewhere in the fraction?
Instructional Implications
Ask the student to evaluate more complex expressions involving operations on rational numbers. Guide the student to apply properties of operations as strategies to add,
subtract, multiply, and divide rational numbers.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Quotients of Integers 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.7.NS.1.2:
Description
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational
numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue
to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) =
1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world
contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with
non­zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients
of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number
terminates in 0s or eventually repeats.
Remarks/Examples:
Fluency Expectations or Examples of Culminating Standards
Adding, subtracting, multiplying, and dividing rational numbers is the culmination of numerical work with the four
basic operations. The number system will continue to develop in grade 8, expanding to become the real numbers
by the introduction of irrational numbers, and will develop further in high school, expanding to become the complex
numbers with the introduction of imaginary numbers. Because there are no specific standards for rational number
arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic,
fluency with rational number arithmetic should be the goal in grade 7.
page 4 of 4