Unit #2: Rational Numbers
Time: 30 Days
In Grade 6, students formed a conceptual understanding of integers through the use of the number line, absolute value, and opposites and
extended their understanding to include the ordering and comparing of rational numbers. Students build on their understanding of rational
numbers to add, subtract, multiply, and divide signed numbers. Previous work in computing the sums, differences, products, and quotients of
fractions serves as a significant foundation.
In this unit students will be able to add, subtract, multiply, and divide rational numbers fluently as well as solve real-world and mathematical
problems. Students will represent their calculations using number line models, equations or expressions, and real world applications.
Major Cluster Standards
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.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
Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world
c) Understand subtraction of rational numbers as adding 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.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/1) = (-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
7. NS.3 Solve real-world and mathematical problems involving the four operations with rational number.
Use properties of operations to generate equivalent expressions.
7.EE.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. For example, a +0.05a = 1.05a means that “increase by 5% is the same as multiply by 1.05.”
*In this unit, this standard is applied to expressions with rational numbers in them.
Solve real-life and mathematical problems using numeral and algebraic expressions and equations.
7.EE.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. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
*In this unit, the equations include negative rational numbers.
Major Cluster Standards Unpacked
7.NS.1 Students add and subtract rational numbers. Visual representations may be helpful as students begin this work; they become less
necessary as students become more fluent with these operations. The expectation of the CCSS is to build on student understanding of number
lines developed in 6th grade.
Use a number line to add -5 + 7.
Students find -5 on the number line and move 7 in a positive direction (to the right). The stopping point of 2 is the sum of this expression.
Students also add negative fractions and decimals and interpret solutions in given contexts.
In 6th grade, students found the distance of horizontal and vertical segments on the coordinate plane. In 7 th grade, students build on this
understanding to recognize subtraction is finding the distance between two numbers on a number line.
In the example, 7 – 5, the difference is the distance between 7 and 5, or 2, in the direction of 5 to 7 (positive). Therefore the answer would be 2.
Use a number line to subtract: -6 – (-4)
This problem is asking for the distance between -6 and -4. The distance between -6 and -4 is 2 and the direction from -4 to -6 is left or negative.
The answer would be -2. Note that this answer is the same as adding the opposite of -4: -6 + 4 = -2
Use a number line to illustrate:
ie. 7 – 4
p + (-q)
ie. 7 + (– 4)
Is this equation true p – q = p + (-q)?
Students explore the above relationship when p is negative and q is positive and when both p and q are negative. Is this relationship always
Morgan has $4 and she needs to pay a friend $3. How much will Morgan have after paying her friend?Solution:
4 + (-3) = 1 or (-3) + 4 = 1
This cluster builds upon the understandings of rational numbers in Grade 6:
• quantities can be shown using + or – as having opposite directions or values,
• points on a number line show distance and direction,
• opposite signs of numbers indicate locations on opposite sides of 0 on the number line,
• the opposite of an opposite is the number itself,
• the absolute value of a rational number is its distance from 0 on the number line,
• the absolute value is the magnitude for a positive or negative quantity, and
• locating and comparing locations on a coordinate grid by using negative and positive numbers.
Learning now moves to exploring and ultimately formalizing rules for operations (addition, subtraction, multiplication and division) with integers.
Using both contextual and numerical problems, students should explore what happens when negatives and positives are combined. Number
lines present a visual image for students to explore and record addition and subtraction results. Two-color counters or colored chips can be
used as a physical and kinesthetic model for adding and subtracting integers. With one color designated to represent positives and a second
color for negatives, addition/subtraction can be represented by placing the appropriate numbers of chips for the addends and their signs on a
board. Using the notion of opposites, the board is simplified by removing pairs of opposite colored chips. The answer is the total of the
remaining chips with the sign representing the appropriate color. Repeated opportunities over time will allow students to compare the results of
adding and subtracting pairs of numbers, leading to the generalization of the rules. Fractional rational numbers and whole numbers should be
used in computations and explorations. Students should be able to give contextual examples of integer operations, write and solve equations
for real world problems and explain how the properties of operations apply. Real-world situations could include: profit/loss, money, weight, sea
level, debit/credit, football yardage, etc.
Using what students already know about positive and negative whole numbers and multiplication with its relationship to division, students
should generalize rules for multiplying and dividing rational numbers. Multiply or divide the same as for positive numbers, then designate the
sign according to the number of negative factors. Students should analyze and solve problems leading to the generalization of the rules for
operations with integers. For example, beginning with known facts, students predict the answers for related facts, keeping in mind that the
properties of operations apply (See Tables 1, 2 and 3 below).
Using the language of “the opposite of” helps some students understand the multiplication of negatively signed numbers ( -4 x -4 = 16, the
opposite of 4 groups of -4). Discussion about the tables should address the patterns in the products, the role of the signs in the products and
commutativity of multiplication. Then students should be asked to answer these questions and prove their responses.
• Is it always true that multiplying a negative factor by a positive factor results in a negative product?
• Does a positive factor times a positive factor always result in a positive product?
• What is the sign of the product of two negative factors?
• When three factors are multiplied, how is the sign of the product determined?
• How is the numerical value of the product of any two numbers found?
Students can use number lines with arrows and hops, groups of colored chips or logic to explain their reasoning. When using number lines,
establishing which factor will represent the length, number and direction of the hops will facilitate understanding. Through discussion,
generalization of the rules for multiplying integers would result.
Division of integers is best understood by relating division to multiplication and applying the rules.
In time, students will transfer the rules to division situations. (Note: In 2b, this algebraic language (–(p/q) = (–p)/q = p/(–q)) is written for the
teacher’s information, not as an expectation for students.)
Ultimately, students should solve other mathematical and real-world problems requiring the application of these rules with fractions and
In Grade 7 the awareness of rational and irrational numbers is initiated by observing the result of changing fractions to decimals. Students
should be provided with families of fractions, such as, sevenths, ninths, thirds, etc. to convert to decimals using long division. The equivalents
can be grouped and named (terminating or repeating). Students should begin to see why these patterns occur. Knowing the formal vocabulary
rational and irrational is not expected.
7.NS.2 Students understand that multiplication and division of integers is an extension of multiplication and division of whole numbers. Students
recognize that when division of rational numbers is represented with a fraction bar, each number can have a negative sign.
Which of the following fractions is equivalent to
? Explain your reasoning.
Example Set 2:
Examine the family of equations in the table below. What patterns are evident? Create a model and context for each of the products. Write and
model the family of equations related to 3 x 4 = 12.
Number Line Model
Selling two packages of apples
at $3.00 per pack
2 • -3 = -6
Spending 3 dollars each on 2
packages of apples
-2 • 3 = -6
Owing 2 dollars to each of your
-2 • -3 = 6
Forgiving 3 debts of $2.00 each
Using long division from elementary school, students understand the difference between terminating and repeating decimals. This
understanding is foundational for the work with rational and irrational numbers in 8 th grade.
Using long division, express the following fractions as decimals. Which of the following fractions will result in terminating decimals; which will
result in repeating decimals?
Identify which fractions will terminate (the denominator of the fraction in reduced form only has factors of 2 and/or 5)
7.NS.3 Students use order of operations from 6th grade to write and solve problem with all rational numbers.
Calculate: [-10(-0.9)] – [(-10) • 0.11]
Jim’s cell phone bill is automatically deducting $32 from his bank account every month. How much will the deductions total for the year?
-32 + (-32) + (-32) + (-32)+ (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) + (-32) = 12 (-32)
It took a submarine 20 seconds to drop to 100 feet below sea level from the surface. What was the rate of the descent?
- 5 feet
= -5 ft/sec
A newspaper reports these changes in the price of a stock over four days:
The sum is
; dividing by 4 will give a daily average of
1 5 3 9
. What is the average daily change?
8 8 8
7.EE.2 Students understand the reason for rewriting an expression in terms of a contextual situation. For example, students understand that a
20% discount is the same as finding 80% of the cost, c (0.80c).
All varieties of a certain brand of cookies are $3.50. A person buys peanut butter cookies and chocolate chip cookies. Write an expression that
represents the total cost, T, of the cookies if p represents the number of peanut butter cookies and c represents the number of chocolate chip
Students could find the cost of each variety of cookies and then add to find the total.
T = 3.50p + 3.50c
Or students could recognize that multiplying 3.50 by the total number of boxes (regardless of variety) will give the same total.
T = 3.50(p +c)
Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an additional $27 dollars in overtime. Write an
expression that represents the weekly wages of both if J = the number of hours that Jamie worked this week and T = the number of hours Ted
worked this week? What is another way to write the expression?
Students may create several different expressions depending upon how they group the quantities in the problem.
Possible student responses are:
Response 1: To find the total wage, first multiply the number of hours Jamie worked by 9. Then, multiply the number of hours Ted worked by 9.
Add these two values with the $27 overtime to find the total wages for the week. The student would write the expression 9J + 9T + 27.
Response 2: To find the total wages, add the number of hours that Ted and Jamie worked. Then, multiply the total number of hours worked by
9. Add the overtime to that value to get the total wages for the week. The student would write the expression 9(J + T) + 27.
Response 3: To find the total wages, find out how much Jamie made and add that to how much Ted made for the week. To figure out Jamie’s
wages, multiply the number of hours she worked by 9. To figure out Ted’s wages, multiply the number of hours he worked by 9 and then add
the $27 he earned in overtime. My final step would be to add Jamie and Ted wages for the week to find their combined total wages. The student
would write the expression (9J) + (9T + 27).
Given a square pool as shown in the picture, write four different expressions to find the total number of tiles in the border. Explain how each of
the expressions relates to the diagram and demonstrate that the expressions are equivalent. Which expression is most useful? Explain.
As students begin to build and work with expressions containing more than two operations, students tend to set aside the order of operations.
For example having a student simplify an expression like 8 + 4(2x - 5) + 3x can bring to light several misconceptions. Do the students
immediately add the 8 and 4 before distributing the 4? Do they only multiply the 4 and the 2x and not distribute the 4 to both terms in the
parenthesis? Do they collect all like terms 8 + 4 – 5, and 2x + 3x? Each of these show gaps in students’ understanding of how to simplify
numerical expressions with multiple operations.
Students solve multi-step equations derived from word problems. Students use the arithmetic from the problem to generalize an algebraic
The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a concert ticket and the cost of 2 passes,
one for the rides and one for the game booths. Each of the passes cost the same price. Write an equation representing the cost of the trip and
determine the price of one pass.
x = cost of one pass
2x + 11 = 52
2x = 41
x = $20.50
x – 4 = -16
x – 4 = -16
x = -12
• x = -12 •
Added 4 to both sides
Multiply both sides by
x = -18
could also reason that if
of some amount is -12 then is -6. Therefore, the whole amount must be 3 times -6 or -18.
Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How much did each pen cost including tax?
x = number of pens
26 = 14.30 + 10x
Solving for x gives $1.17 for each pen.
The sum of three consecutive even numbers is 48. What is the smallest of these numbers?
x = the smallest even number
x + 2 = the second even number
x + 4 = the third even number
x + x + 2 + x + 4 = 48
3x + 6 = 48
3x = 42
x = 14
x + 3 = -5
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.
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.
Skills and Concepts
Prerequisite Skills/Concepts: Students should already be able
Advanced Skills/Concepts: Some students may be ready to…
• Know that there are numbers that are not rational, and approximate
Use equivalent fractions as a strategy to add and subtract
them by rational numbers. (8.NS.1-2)
• Interpret and apply positive and negative slopes of lines and positive
and negative coefficients in equations; develop understanding of square
5.NF.A.1 Add and subtract fractions with unlike denominators
roots and irrational numbers.
(including mixed numbers) by replacing given fractions with
• Understand relationships between positive and negative coefficients or
equivalent fractions in such a way as to produce an equivalent
values for variables; use positive and negative integers to communicate
sum or difference of fractions with like denominators. For
example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d directions in two dimensions.
• Evaluate algebraic expressions involving positive and negative
= (ad + bc)/bd.)
coefficients or values for variables; interpret isometrics in the plane given
Apply and extend previous understandings of multiplication
in symbolic form.
and division to multiply and divide fractions.
5.NF.B.3 Interpret a fraction as division of the numerator by the • Graph equations on coordinate grids; locate square roots on the number
denominator (a/b = a ÷ b). Solve word problems involving
• Use the properties and order of operations to write equivalent
division of whole numbers leading to answers in the form of
expressions and solve equations.
fractions or mixed numbers, e.g., by using visual fraction
models or equations to represent the problem. For example,
interpret 3/4 as the result of dividing 3 by 4, noting that 3/4
Skills: Students will be able to …
multiplied by 4 equals 3, and that when 3 wholes are shared
equally among 4 people each person has a share of size 3/4. If
• Add and subtract rational numbers. (7.NS.1)
9 people want to share a 50-pound sack of rice equally by
• Represent addition and subtraction on a horizontal or vertical number
weight, how many pounds of rice should each person get?
line diagram. (7.NS.1)
Between what two whole numbers does your answer lie?
• Use words, visuals and symbols to describe situations in which opposite
quantities combine to make 0. (7.NS.1)
5.NF.B.4 Apply and extend previous understandings of
• Represent addition of quantities with symbols, visuals and words by
multiplication to multiply a fraction or whole number by a
showing positive or negative direction from one quantity to the other.
Interpret the product (a/b) × q as a parts of a partition of q into b (7.NS.1)
• Show that a number and its opposite have a sum of 0 using visuals,
equal parts; equivalently, as the result of a sequence of
symbols, words and real-world contexts. (7.NS.1)
operations a × q ÷ b. For example, use a visual fraction model
to show (2/3) × 4 = 8/3, and create a story context for this
equation. Do the same with (2/3) × (4/5) = 8/15. (In general,
(a/b) × (c/d) = ac/bd.)
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. For example, create a story context for (2/3) ÷
(3/4) and use a visual fraction model to show the quotient; use
the relationship between multiplication and division to explain
that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general,
(a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get
if 3 people share 1/2 lb of chocolate equally? How many 3/4‐cup
servings are in 2/3 of a cup of yogurt? How wide is a
rectangular strip of land with length 3/4 mi and area 1/2 square
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 (e.g., temperature above/below zero, elevation
above/below sea level, credits/debits, positive/negative electric
charge); use positive and negative numbers to represent
quantities in real‐world contexts, explaining the meaning of 0 in
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
• Use the term “additive inverse” to describe 2 numbers whose sum is
• Use commutative, distributive, associative, identity, and inverse
properties to add and subtract rational numbers. (7.NS.1)
• Use the term “absolute value” to describe the distance from zero on
number line diagram and with symbols. (7.NS.1)
• Multiply and divide rational numbers. (7.NS.2)
• Use the distributive property to multiply positive and negative rational
numbers using symbols, visuals, words and real-life contexts. (7.NS.2)
• Interpret products of rational numbers by describing real-world contexts.
• Identify situations when integers can and cannot be divided. (7.NS.2)
• Use words and real-world contexts to explain why the quotient of two
integers is a rational number. (7.NS.2)
• Identify and apply properties used when multiplying and dividing rational
• Convert a rational number to a decimal using long division. (7.NS.2)
• Identify terminating or repeating decimal representations of rational
• Solve real world and mathematical problems involving the four
operations with rational numbers. (7.NS.3).
in the plane with negative number coordinates.
Recognize opposite signs of numbers as indicating locations on
opposite sides of 0 on the number line; recognize that the
opposite of the opposite of a number is the number itself, e.g., –
(–3) = 3, and that 0 is its own opposite.
6.NS.C.7 Understand ordering and absolute value of rational
c. Understand the absolute value of a rational number as its
distance from 0 on the number line; interpret absolute value as
magnitude for a positive or negative quantity in a real‐world
situation. For example, for an account balance of –30 dollars,
write |–30| = 30 to describe the size of the debt in dollars.
Apply and extend previous understandings of arithmetic to
6.EE.A.2 Write, read, and evaluate expressions in which letters
stand for numbers.
Write expressions that record operations with numbers and with
letters standing for numbers. For example, express the
calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms
(sum, term, product, factor quotient, coefficient); view one or
more parts of an expression as a single entity. For example,
describe the expression 2 (8 + 7) as a product of two factors;
view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables.
Include expressions that arise from formulas used in real‐world
problems. Perform arithmetic operations, including those
involving whole‐number exponents, in the conventional order
when there are no parentheses to specify a particular order
(Order of Operations). For example, use the formulas V = s3
and A = 6 s2 to find the volume and surface area of a cube with
sides of length s = 1/2.
6.EE.A.3 Apply the properties of operations to generate
equivalent expressions. For example, apply the distributive
property to the expression 3 (2 + x) to produce the equivalent
expression 6 + 3x; apply the distributive property to the
expression 24x + 18y to produce the equivalent expression 6
(4x + 3y); apply properties of operations to y + y + y to produce
the equivalent expression 3y.
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). For example, the
expressions y + y + y and 3y are equivalent because they name
the same number regardless of which number y stands for.
Reason about and solve one‐variable equations and
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
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
Additive Identity (The additive identity is 0.)
Additive Inverse (The additive inverse of a real number is the opposite of that number on the real number line. For example, the opposite
of −3 is 3. A number and its additive inverse have a sum of 0.)
Break-Even Point (The point at which there is neither a profit nor loss.)
Distance Formula (If and are rational numbers on a number line, then the distance between and is |−|.)
Loss (A decrease in amount; as when the money earned is less than the money spent.)
Multiplicative Identity (The multiplicative identity is 1.)
Profit (A gain; as in the positive amount represented by the difference between the money earned and spent)
Repeating Decimal (The decimal form of a rational number, For example, 13=0.3�.)
Terminating Decimal (A decimal is called terminating if its repeating digit is 0.)
Associative Property (of Multiplication and Addition)
Commutative Property (of Multiplication and Addition)
Distributive Property (of Multiplication Over Addition)