AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Unit Overview
Unit Title
Unit Designer
7.NS.1
7.NS.3
The Number System – Adding and
Subtracting Rational Numbers
Will Roble
Duration
12 Lessons – 1 Unit Assessment
IA Period 1
Identify Desired Results: Identify the Standards
Standard
Previous Grade Level Standards / Previously Taught
& Related Standards
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.
Solve real-world and mathematical problems involving the four operations
with rational numbers
6.NS.B Compute fluently with multi-digit numbers and find common
factors and multiples
6.NS.C Apply and extend previous understandings of numbers to the
system of rational numbers
6.EE.A Apply and extend previous understandings of arithmetic to
algebraic expressions
6.EE.B Reason about and solve one-variable equations and inequalities
6.EE.C Represent and analyze quantitative relationships between
dependent and independent variables
5.NF.A Use equivalent fractions as a strategy to add and subtract
fractions
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Enduring Understandings: What do you want students to know in 10 years about this topic? What does it look like, in
this unit, for students to understand this?
Grade Level Enduring Understandings
What it looks like – in this unit
Properties of 2D and 3D figures can be used to
qualitatively and quantitatively describe and
measure attributes of figures as well as solve
geometric problems.
Understanding relationships between operations
and properties of operations can be used to
understand, develop rules for, and perform
operations with rational numbers and when
working with algebraic representations.
-
Mathematical and real-world situations and
structures can be translated and represented
using different forms to make meaning, illustrate
relationships between quantities (known and
unknown), and solve problems.
-
-
-
The way that data are collected, organized and
displayed influences interpretations and analyses.
-
Apply addition and subtraction of rational numbers to measure perimeters of figures or
find the combined area, surface area or volume of a figure composed of triangles,
quadrilaterals, cubes and right prisms (*should appropriately review what was learned
in 6th grade and areas or volumes should be provided to focus on addition).
Understand relationships between addition and subtraction in the context of rational
numbers.
Understand that opposite quantities combine to make 0.
Apply absolute value to understanding addition and subtraction of rational numbers.
Understand subtraction of rational numbers as adding the additive inverse.
Apply properties of operations to add and subtract rational numbers
Essential Questions:
o Do all mathematical operations relate to each other?
o Will addition ever yield a sum smaller than one or both of its addends?
o Will subtraction ever yield a difference greater than the minuend and/or
subtrahend?
Use number lines and colored chips to represent problems and understand operations
with rational numbers.
Translate and solve mathematical and real world situations using expressions and
equations.
Represent situations using addition or subtraction depending on the context and utility
of the operation.
Describe situations in which opposite quantities combine to make 0.
Essential Question: What is the best representation to use for a situation?
As an application of rational number operations, students may be asked to interpret
and analyze data displays (see 6th grade standards to determine appropriate displays
and analyses).
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Identify The Narrative1
During the first unit of grade 7, students learn to conceptually understand, fluently operate with, and apply addition and subtraction of rational
numbers. In 6th grade, students first developed a conceptual understanding of rational numbers “through the use of a number line, absolute
value, and opposites and extended their understanding to include ordering and comparing rational numbers (6NS5, 6NS6, 6NS7).” They further
extended their understanding of working with rational numbers on a number line to the coordinate plane (6NS8). 6th grade also marked the year
when students were expected to fluently work with whole number, fraction and decimal operations (6NS1, 6NS2, 6NS3). Work with number
properties and relationships between operations in earlier grades to understand whole number, decimal and fraction operations serves as a
significant foundation as well, given that early lessons to develop an understanding of addition and subtraction of rational numbers draws on this
understanding.
To develop a conceptual understanding of addition and subtraction with integers, students return to the number line and they use colored chips
to model the operations.
With a number line:
Addition:
1
Content, quotations, and images of unit summary heavily taken from Engage New York, Grade 7 Module 2 Overview, pg. 3 and Lesson 6, pg. 70. Additionally, images are
captured from the CC NS Progression and Math Matter.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Subtraction:
When subtracting a negative from a positive or negative (below), first move to the point that represents the minuend. Then, think, if you were
adding (-7), you would move to the left. Since you are subtracting (-7), move in the opposite direction, to the right.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
With colored chips (+ is one color and – is one color)
Addition:
Zero Pair
Subtraction:
When the subtrahend is represented by the chips used to represent the minuend. Ex.) -5 – (-1)
When the subtrahend is not represented by the chips used to represent the minuend, you add zero pairs as needed. Ex.) -2 – 1
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
They use the models to demonstrate and understand that adding an integer to its opposite equals 0 (7NS1a, 7NS1b), subtracting a number is the
same as adding its opposite (7NS1c), and finding the distance between two integers on a number line is the absolute value of their difference
(7NS1c).
7NS1a, 7NS1b
7NS1c
Find the distance between -3 and 2.
7NS1c
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Students’ “findings are formalized as students develop rules for adding and subtracting integers. Students extend integer rules to include the
rational numbers and use properties of operations to perform rational number calculations (7NS1D).” They apply their understanding of addition
and subtraction of rational numbers to model and solve mathematical and real-world problems (7NS3).
Looking ahead to the remainder of 7th grade, students will continue to apply their understanding of addition and subtraction with rational
numbers to develop an understanding of how to multiply and divide rational numbers (7NS2). They will then apply the four operations with
rational numbers when working with expressions, equations and inequalities (7EEA, 7EEB) as well as when solving problems involving proportional
thinking (7RPA), scale drawings (7G1), area, surface area and volume (7G4, 7G6), angle measure (7G5), and statistics and probability (7SPA,
7SPB, 7SPC). Erring on the side of pointing out the obvious, rational number operations are integrated throughout the entire 7 th grade curriculum.
Later, in 8th grade, students continue to understand rational numbers as they learn about numbers that are not rational, called irrational numbers
(8NSA). They also apply their understanding when working with integer exponents (8EEA), graphing and solving (pairs of) linear equations (8EEC),
performing translations and dilations (8GA), and using functions to model and compare relationships between quantities (8FA, 8FB)
For High School, fluency with rational numbers sets students up to focus on learning new algebraic material in high school that incorporates the
use of these numbers and assumes knowledge of them. An understanding of rational number operations also facilitates the understanding of
rational functions and how to work with them appropriately.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Standards for Mathematical Practices: When and how will you intentionally provide students with opportunities to
practice in this unit…
12 Make sense of
problems and persevere
in solving them
2 Reason abstractly and
quantitatively
3 Construct viable
arguments and critique
the reasoning of others
4 Model with
mathematics
5 Use appropriate tools
strategically
6 Attend to precision
2
Students set out to understand a problem and look for entry points to a solution. They analyze conditions and goals
as well as translate the conditions into mathematical representations to further their understanding and start working
towards a solution path. In this unit, translating could look like writing an expression or equation, using a number line
or drawing a picture to represent a verbal description.
Students explain how different representations of problem conditions relate to each other. For example, students
can navigate among number lines, pictures, and expressions/equations.
Students use inverse operations to check correctness of answers and ask themselves if their answer makes sense
given the context of the problem.
Students use properties of operations to generate equivalent expressions and use the number line to understand
addition and subtraction of rational numbers.
Students consider units involved to both consider an appropriate representation and to understand the desired units
for the solution.
Students represent and solve problems with expressions and equations, and define the variable when appropriate.
Students use examples and counter examples to understand common errors in manipulation of numbers (i.e. thinking
that 3-5 is equivalent to 5-3).
Students make arguments about generalizations involving rules for addition and subtraction of rational numbers.
Students apply mathematics to solve real world problems. Students will apply their understanding of addition and
subtraction of rational numbers to solve problems involving bank accounts, temperature, altitude, etc.
Students write expressions and equations to represent real world problems.
Students interpret the mathematical results in the context of the situation and gauge whether or not they make
sense. For example, a student should recognize that a calculation resulting in a higher temperature when the
problem states that the temperature got colder is incorrect based on the problem’s context.
Students consider the available tools when solving problems. In this unit, students mostly utilize colored chips, and
paper and pencil. While calculators may be used later in the year, fluency with operations is paramount in this unit.
Students communicate precisely to others both verbally and in written form by using appropriate mathematical
language and explaining connections between concepts.
Students take care with identifying appropriate units.
Students calculate accurately and efficiently.
Students express their answer to with a degree of precision appropriate for the context of the problem, i.e. if the
problem asks students to estimate, they do so.
Bolded SMP are the Focal SMP for the unit identified in the Scope and Sequence.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
7 Look for and make use
of structures
8 Look for and express
regularity in repeated
reasoning
Students use the structure of the number line to demonstrate that the distance between two rational numbers is the
absolute value of their difference
Students see subtraction as addition of the opposite and use this as a tool subtracting rational numbers (later they
will use this as a tool for collecting like terms in an expression).
Skills and Procedural Knowledge: What do you want students to be able to do comfortably, accurately, and with
flexibility?
 Represent addition and subtraction of rational numbers on vertical and horizontal number lines
 Model addition and subtraction of integers with colored chips
 Represent real world and mathematical problems using simple expressions and equations
 Fluently add and subtract rational numbers
 Interpret, represent and describe real-world contexts involving addition and subtraction with rational numbers
 Interpret sums and differences by describing real world contexts
 Solve multi-step mathematical and real world problems involving rational numbers
 Reason abstractly using the rules of addition and subtraction of rational numbers
Major Misconceptions & Clarifications
Misconception
Subtraction is commutative.
To find the difference between two numbers,
add their absolute values.
Misconceptions that arise from
misunderstanding of language, i.e. “Anne
assumed $30 more in debt.” Students may think
this means addition.
Clarification
Students can model subtraction on a number line interchanging the same values
for the minuend and subtrahend. You can also use counters. Have students also
rewrite subtraction problems with addition and do the same activity to show that
addition is commutative. This may also clear up the confusion as kids might think
that subtraction is commutative because an equivalent addition expression can
apply the commutative property and maintain its value.
Students may draw this incorrect conclusion after working on understanding how
to find the difference between two numbers with opposite signs. Have them try
this strategy when subtracting two numbers with the same sign using a number
line and the algorithm.
Students draw models when working with real world problems to represent the
values used accurately as well as the situation. This could entail using rational
numbers, number lines or a picture to help clarify the language used to describe
the context.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Unit Vocabulary









Rational numbers: All numbers, both positive and negative, that can be expressed as the ratio of two integers (with
a non-zero denominator); includes integers, fractions, decimals, mixed numbers, etc.
Integer: All positive whole numbers, their opposites, and zero
Additive inverse: The opposite of a number; two numbers are additive inverses when they have the same absolute
value but opposite signs; synonym: zero pairs
Absolute value: A number’s absolute value is its distance from zero on a number line; always a positive value
Distance: The number of units between 2 points on a number line; found by taking the absolute value of the
difference of the two values
Sum: The total amount resulting from the addition or two or more numbers
Difference: The result of subtracting one number from another
Commutative Property: States that changing the order of addends or factors does not change the sum or product
in an addition or multiplication problem, respectively
Associative Property: States that changing the grouping of three or more addends or factors does not change the
sum or product in an addition or multiplication problem, respectively
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Aims Sequence:
LP #
Aim
Standard
Notes
Unit #1: The Number System – Adding and Subtracting Rational Numbers: 13 days
1
SWBAT represent and begin to develop an understanding of addition of integers by using a
horizontal or vertical number line
a.) SWBAT understand p + q as the number located a distance |q| from p in a positive or negative
direction depending on the sign of q.
b.) SWBAT show and understand that an integer and its opposite have a sum of 0 (are additive
inverses)
2
SWBAT represent and continue to develop an understanding of addition of integers using twocolored chips
a.) SWBAT show that an integer and its opposite have a sum of 0 (are additive inverses);
understand that these pairs of numbers represent 0 pairs
3
SWBAT derive, explain and apply a generalized rule for adding integers (from 0-100)
a.) SWBAT explain that the sum takes the sign of the addend with the larger absolute value.
Assessment: Unit 1 Assessment
7.NS.1
Note: p is the starting value and q is the distance moved to the left or right from p.
Note: Students should understand that when q = 0, p + q = p
Note: Introduce Additive Inverse but do not focus on it.
Note: Include real world and mathematical problems.
4
SWBAT calculate and interpret sums of integers in real-world contexts (task)
7.NS.1
5
SWBAT represent and begin to develop an understanding of subtraction of integers (p – q) using a
horizontal or vertical number line when q >0 and when q<0
SWBAT understand and explain why p – q = p + (-q) by creating a model on a number line
SWBAT represent and continue to develop an understanding of subtraction of integers using
colored chips
SWBAT show and explain 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
SWBAT derive, explain and apply a generalized rule for subtracting integers (from 0-100)
7.NS.1
SWBAT apply the properties of operations to add and subtract integers to solve mathematical and
real-world problems (task)
SWBAT apply properties of operations to add and subtract rational numbers
7.NS.1
SWBAT apply properties of operations to add and subtract rational numbers to solve
mathematical and real world problems
Given constraints for the values of p and q, SWBAT determine whether or not a mathematical
inequality is sometimes, always or never true by applying the rules for addition and subtraction of
rational numbers
Unit Assessment 1
7.NS.1
6
7
8
9
10
11
12
13
7.NS.1
Note: Include real world and mathematical problems.
7.NS.1
Note: Students should be making connections to previous lessons in their
explanations.
Note: Include real world and mathematical problems when applicable.
Sub-aim: SWBAT represent real world problems using pictorial representations and
number lines.
Sub-aim: SWBAT describe situations in which opposite quantities combine to make 0
Note: After LP5, students should have a firm grasp on the idea that p – q = p + (-q)
7.NS.1
Show with addition of zero sums – i.e. 2 – (-4) is the same as 2 – (+1/-1, +1/-1, +1/-1,
+1/-1) and then taking away -4 so that you are left with 2 + (1 + 1 + 1 + 1)
7.NS.1
7.NS.1
Note: Students should be making connections to strategies applied in other lessons
(i.e. rewriting the problem as an addition problem)
Sub-aim: SWBAT model real world problems using pictorial representations and
number lines.
7.NS.1
7.NS.1
7.NS.3
Possible grade 6 topics for review at the start of the unit: Defining and representing absolute value, representing positive and negative rational numbers on the number line, comparing and ordering positive and negative rational
numbers, and adding and subtracting positive rational numbers. S should also be able to use bar models and equations to represent and solve word problems involving positive rational numbers, including multi-step problems.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Assessments:
UNIT 1 Addition and Subtraction of Rational Numbers – Unit Assessment
Standards
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 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.
d) Apply properties of operations as strategies to add and subtract 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.
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.3 Solve real-world and mathematical problems involving the four
operations with rational numbers
TOTAL
Questions
1-6
7-10
11-18
18
Percent
Mastery
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Directions: Complete each problem and show your complete work. Please use complete
sentences.
1) What is the sum of -19 and 35? Use a model to represent your thinking (2 pts)
Answer: __________________
3
5
2) Simplify the expression: −2 4 + (− 8). Show your work (2 pts)
Answer: __________________
3) What value of a will make the equation a true statement? Explain how you arrived at
your solution. (2 pts)
(-13.4 + 9) + a = 0
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
4) Diamond used a number line to add. She started counting at 10, and then she
counted until she was on the number -4 on the number line. (3 pts)
a. If Diamond is modeling addition, what number did she add to 10? Use the
number line below to model your answer.
b. Write an equation to represent the addition Diamond did in Part A.
Answer: ____________________________________
c. Write a real-world story problem that would fit this situation.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
5) What is the additive inverse of -7.5? Use a model to show your thinking. (2 pts)
6) Write a real-world story problem for the equation below (1 pt)
-5 + 5 = 0
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
7) Simplify the expression to the right: 12 – (-9) (1 pt)
Answer: ____________________________
8) Jeremiah wrote an expression on the board: -6 – 18. (2 pts)
a. Which expression is equivalent to Jeremiah’s expression?
i. -6 – (-18)
ii. 6 + 18
iii. -6 + 18
iv. -6 + (-18)
b. Use a number line to prove that the two expressions are equivalent.
9) Determine the value of the ? in the equation below. Show your work. (2 pts)
1
3
−6 2 − 3 4 = ?
Answer: _____________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
10)A bird is flying 43
5
9
feet above the sea. A fish is swimming 23
2
3
feet below the surface
of the sea, directly below the bird. What is the vertical distance between the bird and
the fish? (3 pts)
a.) Draw a number line to model this situation.
b.) Write an expression that represents this situation.
Expression: _____________________________
c.) Determine the vertical distance between the bird and the fish.
Answer: _____________________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
11)According to Ray’s bank statement, he is in debt $32.50 right now. He had owed
more, but he has been paying off his debt by depositing $10 per month into his bank
account for the last two months. (3 pts)
a.) Write a rational number to represent Ray’s account balance 2 months ago. Show
your work.
Answer: ______________
b.) After paying off the first two months of his debt, Ray went on another spending
spree. He bought $43.38 worth of books. After these purchases, what is Ray’s new
balance? Show your work.
Answer: ______________
12)A number line is shown below (drawn to scale). The numbers 0 and 1 are marked on
the line, as are two other numbers a and b. (2 pts)
b
0
1
a
Which of the following expressions results in a negative number? Circle all that apply.

a−1

a−2

−b

a+b

a−b
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
13)A number is located a distance of 4.5 units from -3.5. What could the number be?
(1 pt)
a.
b.
c.
d.
only 1
-8 or -1
-8 or 1
only -8
14)The table contains data for the temperature in Portland, Maine, during the month of
January. Complete the last 4 rows of the table. (2 pts)
Temperature at 8:00 AM
Temperature at 8:00 PM
Change in Temperature from
8:00 AM to 8:00 PM
8°
3°
-5°
-2°
-13°
-11°
-13°
11°
-1°
15°
-2°
-8°
-5°
4°
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
15)Which of the following contextual situations accurately models the expression
-125 – 25? (1 pt)
a. Tony owes $125 to his bank. He then deposits $25 dollars. What’s his bank
balance now?
b. The temperature of the water was -125 degrees. Then it rose 25 degrees. What’s
the new temperature of the water?
c. Tony owes 125 dollars on his credit card and then spends 25 dollars on a new
textbook for an SHSAT prep course. What does his bank statement say now?
d. Tony was $125 in debt at the beginning of the week. By the end of the week, he
raised his bank account to be in the positives, at $25. What does his bank
statement say now?
For questions 16-17, assume the following:
 𝑎 > 0
 𝑏 < 0
Determine whether the given statement is always true, sometimes true, never true, or if you
need more information to answer.
16) 𝑎 + 𝑏 < 0 (1 pt)
a. always true
b. sometimes true
c. never true
d. need more information to answer
17) 𝑏 − 𝑎 < 0 (1 pt)
a. always true
b. sometimes true
c. never true
d. need more information to answer
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
18)Jesse and Miya are playing the integer card game. The cards in Jesse’s hand are
shown below: (4 pts)
a. What is the total score in Jesse’s hand? Support your answer by showing your
work.
Answer: _____________________________
b. Jesse picks up two more cards but they do not affect his overall point total.
Neither card is a 0. What could the value of each of the two cards be?
Answer: _____________________________
c. Complete Jesse’s new hand to make this total score equal zero. What must be
the value of the “?” card?
Answer: _____________________________
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Appendix: Teacher Background Knowledge
Rational Number Operations (7.NS.1, 7.NS.2, 7.NS.3, 7.EE.3)
From CC progression on Number Systems:
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Adapted from Math Matters:
By this time in middle school under the New York state standards, 7th graders have learned to add, subtract,
multiply, and divide whole numbers, decimals, and fractions. Students have also learned to add, subtract, multiply,
and divide positive and negative integers (whole numbers and their opposites). The Common Core standards advance
this work in two important ways:
 Students add, subtract, multiply, and divide rational numbers (whole numbers, decimals, and fractions both
positive and negative). Adding, subtracting, multiplying, and dividing rational numbers is the culmination
of numerical work with the four basic operations. Fluency with rational number arithmetic should be the
goal in Grade 7.
 Students must also understand and be able to explain, at a conceptual level, how the operations with
rational numbers are performed. Students justify the operations with rational numbers using visual models,
number line diagrams, the concept of the additive inverse, the relationship between addition and
multiplication, the relationship between multiplication and division, and the properties of operations
(commutative, associative, and distributive).
Note: Students must understand operations with rational numbers at a deeper level than the nonmathematical tricks
they may already know (KCF, keep the chicken in the oven, caveman says “same good; different bad”). While these
tricks may have seemed sufficient to satisfy the NY state standards, the Common Core standards require students to
have a more sophisticated understanding rooted in mathematics. In general, tricks that are not based on
mathematical relationships are easily forgotten or misapplied, since students cannot rely on their understanding of
situations to help them remember.
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
Addition and Subtraction of Signed Numbers
From CC progression on Number Systems:
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
More on addition and subtraction; Adapted from Math Matters:
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
AF Math-Grade 7-Common Core Unit 1 – 2014-2015
From enVision:
Essential Understandings:
 Addition and subtraction of integers can be modeled as moves on the number line. Patterns show why the
rules for adding and subtracting integers make sense.
 Absolute value is used to define the distance from a number to zero, regardless of whether the number is
positive or negative.
Adding Integers
The algorithms for adding integers can be generalized by using the idea of absolute value. The absolute value of a
number is the distance the number is from 0. Thus, the absolute value of a number is always positive since a
distance cannot be negative. We say, for example, -8 is 8 units away from 0 or the absolute value of -8 is 8. We
write this as |-8| = 8.
Adding two negative or two positive integers
-6 + (-4) = -10
+7 + (+5) = +12
Generalization: Add the absolute values of both integers. The sum has the sign of the original addends. In the first
example, you would think, 6 + 4 = 10. Since both integers are negative, the sum is -10.
Adding one negative and one positive integer
-6 + (+4) = -2
+7 + (-5) = +2
Generalization: Find the absolute value of both integers. Subtract the smaller absolute value from the larger absolute
value. The sum has the same sign of the integer with the greater absolute value. In the first example, you would
think, 6 – 4 = 2. Since 6 is greater than 4, and the 6 is negative, the sum is -2.
Subtracting integers
To subtract integers, we change the subtraction calculation to an addition calculation: a – b = a + (-b). For
example, to find 6 – 4, we can write this as 6 + (-4) = 2. Here’s how to show that this makes sense.




If a – b = c, then c + b = a.
c + b + (-b) = a + (-b)
c = a + (-b)
Since a – b = c and a + (-b) = c, then a – b = a + (-b)
(Definition of subtraction)
(Addition Property of Equality)
(Transitive Property of Equality)