7th Grade Mathematics
UNIT 1: OPERATIONS WITH RATIONAL NUMBERS
Operations with Rational Numbers (6 weeks)
This five-week unit focuses on the concept of rational numbers. It
encompasses many aspects of working with rational numbers including
performing operations with rational numbers in various forms, representing
addition and subtraction on a number line, expressing rational numbers in
different forms, finding percentages, and applying such skills to multi-step
real-life and mathematical problems. Throughout this unit, students are given
the opportunity to discover relationships between numbers, make
connections to the real world and other mathematical topics, and reinforce
skills through large groups, small groups and independent activities.
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7th Grade Mathematics
Key Ideas:
 Numbers can be represented in a
variety of equivalent forms.
 Subtraction of rational numbers is
adding the additive inverse.
 When adding two rational numbers p
+ q as the number located a distance
|q| from p in the positive or negative
direction, depending on where q is
positive or negative
 Multiplication is extended from
fractions to rational numbers by
requiring that operations continue to
satisfy the properties of operations
such as (-1)(-1) = 1 and the rules for
multiplying signed numbers
 Integers can be divided provided that
the divisor is not zero and every
quotient of integers is a rational
number
 Solve real-world mathematical
problems involving the four
operations with rational numbers
Guiding/Focus Questions:
 What are rational numbers?
 How can we tell if two rational numbers are
equal?
 How do we convert any form of a rational
number to at least two other forms?
 What are some ways of working with
rational numbers that make sense?
 What are some real-life situations that are
better suited to fractions or decimals or
percents?
 How can you find the total value of a
combination of positive and negative
integers?
 How can you use a number line to compare
two numbers?
 How can you write a number sentence to
represent a change on a number line, and
how can you use a number line to
represent a number sentence?
 How can you predict whether the result of
addition of two numbers will be positive,
negative, or zero?
 How are the algorithms for addition and
subtraction of integers related?
 How is multiplication of two integers
represented on a number line and a chip
board?
 What algorithm can you use for multiplying
integers?
 What algorithm can you use for dividing
integers?
 How are multiplication and division of
integers related?
 Does the Order of Operations work for
integers?
 How can you use the Distributive Property
to expand an expression or factor an
expression that involves integers?
 What information in a problem is useful to
help you decide which operation to use to
solve the problem?
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7th Grade Mathematics
NYS Common Core Standards for Mathematics
Focus Standards:
Apply and extend previous understandings of operations with fractions to add, subtract,
multiply, and divide rational numbers.
7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and
subtract rational numbers; represent addition and subtraction on a horizontal or vertical number
line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0
charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction
depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are
additive inverses). Interpret sums of rational numbers by describing real‐world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the
distance between two rational numbers on the number line is the absolute value of their difference, and apply
this principle in real‐world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2 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 non‐zero 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.
7.NS.A.3 Solve real‐world and mathematical problems involving the four operations with rational
numbers.2
Use properties of operations to generate equivalent expressions.
7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed
light on the problem and how the quantities in it are related.
Solve real‐life and mathematical problems using numerical and algebraic expressions and
equations.
7.EE.B.4 Use variables to represent quantities in a real‐world or mathematical problem, and
construct simple equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r, are
specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an
arithmetic solution, identifying the sequence of the operations used in each approach.
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7th Grade Mathematics
Foundational Standards(Prerequisites):
Apply and extend previous understandings of multiplication and division to divide fractions by
fractions.
6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division
of fractions by fractions, e.g., by using visual fraction models and equations to represent the
problem.
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values
6.NS.C.6 Understand a rational number as a point on the number line. Extend number line
diagrams and coordinate axes familiar from previous grades to represent points on the line and in
the plane with negative number coordinates.
6.NS.C.7 Understand ordering and absolute value of rational numbers.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3 Apply the properties of operations to generate equivalent expressions.
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the
same number regardless of which value is substituted into them).
Reason about and solve one‐variable equations and inequalities.
6.EE.B.6 Use variables to represent numbers and write expressions when solving a real‐world or
mathematical problem; understand that a variable can represent an unknown number, or,
depending on the purpose at hand, any number in a specified set.
6.EE.B.7 Solve real‐world and mathematical problems by writing and solving equations of the form
x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Focus Standards for Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them. When problem-solving, students
use a variety of techniques to make sense of a situation involving rational numbers. For example,
they may draw a number line and use arrows to model and make sense of an integer addition or
subtraction problem. Or when converting between forms of rational numbers, students persevere
in carrying out the long division algorithm to determine a decimal’s repeat pattern. A tape diagram
may be constructed as an entry point to make sense of a working-backwards problem. As students
fluently solve word problems using algebraic equations and inverse operations, they consider their
steps and determine whether or not they make sense in relationship to the arithmetic reasoning
that served as their foundation in earlier grades.
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7th Grade Mathematics
MP.2 Reason abstractly and quantitatively. Students make sense of integer addition and
subtraction through the use of an integer card game and diagramming the distances and directions
on the number line. They use different properties of operations to add, subtract, multiply, and
divide rational numbers, applying the properties to generate equivalent expressions or explain a
rule. Students use integer subtraction and absolute value to justify the distance between two
numbers on the number line. Algebraic expressions and equations are created to represent
relationships. Students know how to use the properties of operations to solve equations. They
make “zeros and ones” when solving an algebraic equation, thereby demonstrating an
understanding of how their use of inverse operations ultimately lead to the value of the variable.
MP.4 Model with mathematics. Through the use of number lines, tape diagrams, expressions, and
equations, students model relationships between rational numbers. Students relate operations
involving integers to contextual examples. For instance, an overdraft fee of $25 that is applied to
an account balance of -$73.06, is represented by the expression -73.06 – 25 or -73.06 + (-25) using
the additive inverse. Students compare their answers and thought process in the Integer Game and
use number line diagrams to ensure accurate reasoning. They deconstruct a difficult word problem
by writing an equation, drawing a number line, or drawing tape diagram to represent quantities. To
find a change in elevation, students may draw a picture representing the objects and label their
heights to aid in their understanding of the mathematical operation(s) that must be performed.
MP.6 Attend to precision. In performing operations with rational numbers, students understand
that the decimal representation reflects the specific place value of each digit. When converting
fractions to decimals, they carry out their calculations to specific place values, indicating a
terminating or repeat pattern. In stating answers to problems involving signed numbers, students
use integer rules and properties of operations to verify that the sign of their answer is correct. For
instance, when finding an average temperature for temperatures whose sum is a negative number,
students realize that the quotient must be a negative number since the divisor is positive and the
dividend is negative.
MP.7 Look for and make use of structure. Students formulate rules for operations with signed
numbers by observing patterns. For instance, they notice that adding -7 to a number is the same as
subtracting seven from the number, and thus, they develop a rule for subtraction that relates to
adding the inverse of the subtrahend. Students use the concept of absolute value and subtraction
to represent the distance between two rational numbers on a number
line. They use patterns related to the properties of operations to justify the rules for multiplying
and dividing signed numbers. The order of operations provides the structure by which students
evaluate and generate equivalent expressions.
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7th Grade Mathematics
Content
Skills
 Properties of operations
 Convert between fractions, decimals,
whole numbers and percents mentally
 Opposite quantities and additive inverses
and on paper
 Add and subtract rational numbers on the
 Write equivalency statements involving
number line
rational numbers
 Equivalent operations
 Add, subtract, multiply, and divide
 Multiply a positive and a negative
 Multiply two negatives
rational numbers
 Divide with negative numbers
 Use models to represents fractions,
 Write an addition and a subtraction equation
decimals, and percent operations
in context
 Find a fraction or percent of a number
 Signed numbers and data
in a variety of ways
 Applications in physical science
 Use fractions, decimals and percents in
 Percents and proportions
real world situations including sales tax,
 Rates
discounts, and commission situations.
Vocabulary/ Key Terms (with definitions and Spanish translations)
 Absolute Value
 Additive Identity
 Additive Inverse
 Associative Property (of Multiplication and Addition)
 Break-Even Point
 Commutative Property (of Multiplication and Addition)
 Credit
 Debit
 Deposit
 Distance Formula
 Distributive Property (of Multiplication Over Addition)
 Expression
 Equation
 Integer
 Inverse
 Loss
 Multiplicative Identity
 Multiplicative Inverse
 Negatives
 Opposites
 Overdraft
 Positives
 Profit
 Rational Numbers
 Repeating Decimal
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7th Grade Mathematics


Terminating Decimal
Withdraw
ASSESSMENT EVIDENCE
Initial Assessment: Unit Readiness Test
Formative Assessments:
Quizzes
Exit Slips
Checks for Understanding
Short and Extended-Response questions used throughout the unit.
Mathematical Reflections
Formative Assessments Tasks
 Performance Task 1 (Adding and Subtracting Rational Numbers)
 Performance Task 2 (Multiplying and Dividing Rational Numbers)
Summative Assessment/s: Unit Test
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7th Grade Mathematics
TEACHING PLAN
Teaching and Learning Activities:
I.
Administer the Unit Readiness Test to assess students’ performance against the
Foundational Standards.
II.
Use the guiding questions to focus each lesson.
[Suggested instructional approach: Launch-Explore-Summarize]
Lesson 1: Opposite Quantities Combined to Make Zero
Lesson 2: Using the Number Line to Model the Addition of Integers
Lesson 3: Understanding Addition of Integers
Lesson 4: Adding Integers and Other Rational Numbers
Lesson 5: Understanding Subtraction of Integers and Other Rational Numbers
Lesson 6: The Distance Between Two Rational Numbers
Lesson 7: Applying the Properties of Operations to Add and Subtract Rational Numbers
Lesson 8: Understanding Multiplication of Integers
Lesson 9: Develop Rules for Multiplying Signed Numbers
Lesson 10: Division of Integers
Lesson 11: Converting Between Fractions and Decimals Using Equivalent Fractions
Lesson 12: Converting Rational Numbers to Decimals Using Long Division
Lesson 13: Multiplication and Division of Rational Numbers
Lesson 14: Applying the Properties of Operations to Multiply and Divide Rational Numbers
Lesson 15: Writing, Evaluating, and Finding Equivalent Expressions with Rational Numbers
Lesson 16: Investments—Performing Operations with Rational Numbers
III.
Assess students on the unit.
 Final Performance Assessment
 Unit Test
Resources Needed:
 Connected Math Project 3 (CMP3): Accentuate the Negative
 NYS Common Core Math Module 2: Rational Numbers
https://www.engageny.org/resource/grade-7-mathematics-module-2
 Impact Mathematics Course 2
 Math Handbook
 MathXL (Pearson’s online homework, tutorial, and assessment system)
 Glencoe New York State Review Series
 http://www.livebinders.com/play/play?id=953710#anchor
 http://www.algebrahelp.com/
 https://grade6commoncoremath.wikispaces.hcpss.org
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7th Grade Mathematics
CALENDAR
Time
Spent
3 weeks
Standards
7.NS.1
Topics To Cover




2 weeks
7.NS.2




1 week
7.NS.3
7.EE.3



Definition of rational
numbers
Properties of
operations
Opposite quantities and
additive inverses
Add and subtract
rational numbers on the
number line
Equivalent operations
Multiply a positive and
a negative
Multiply two negatives
Divide with negative
numbers
Resources Needed
 Connected Math Project 3 (CMP3):
Accentuate the Negative
 NYS Common Core Math Module 2:
Rational Numbers
 https://www.engageny.org/resource/grade7-mathematics-module-2
 Impact Mathematics Course 2
 Math Handbook
 MathXL (Pearson’s online homework,
tutorial, and assessment system)
 Glencoe New York State Review Series
 njctl.org
 http://www.algebrahelp.com/
Write an addition and a
subtraction equation in
context
Signed numbers and
data
Applications in physical
science
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