Chapter 5: Understanding Integer Operations and Properties
5.1 Addition, Subtraction, and Order Properties of Integers
5.1.1. Integer Uses and Basic Ideas
5.1.1.1. Definition of Integers: The set of integers, I (more often seen as Z), consists of
the positive integers (the Natural numbers), the negative integers (the opposites of
the Natural numbers), and zero. Z = {…, -2, -1, 0, 1, 2, …}
5.1.1.2. The opposite of an integer is the mirror image of the integer around zero on the
number line
5.1.1.3. -5 is read negative 5
5.1.1.4. –(-5) is read the opposite of negative 5
5.1.1.5. 8 – (-5) is read 8 minus negative 5
5.1.1.6. Definition of Absolute Value: The absolute value of an integer is the number of
units the integer is from 0 on the number line. The absolute value of an integer n is
written |n|, and is positive for all n ≠ 0
5.1.1.6.1. |x| = x, when x > 0
5.1.1.6.2. |-x| = x, when x > 0
5.1.1.6.3. -|x| = -x, when x > 0
5.1.1.6.4. |x| = -x, when x < 0
5.1.1.6.5. |-x| = -x, when x < 0
5.1.1.6.6. -|x| = x, when x < 0
5.1.1.6.7. |x| = |-x| = -|x| = x, when x = 0
5.1.2. Modeling Integer Addition
5.1.2.1. Using Counters Model
5.1.2.1.1. A black counter and a red counter cancel each other
5.1.2.1.2. Concrete way of representing the addition of integers
5.1.2.1.3. see figure 5.3 p. 251 and figure 5.4 p. 252
5.1.2.1.4. For tests we will use gray or solid for positive and clear or empty for negative
5.1.2.2. Using a Charge Field Model
5.1.2.2.1. Another model for adding integers
5.1.2.2.2. + cancels out –
5.1.2.2.3. see figure 5.5 p. 253
5.1.2.3. Using the Number Line
5.1.2.3.1. Allows students an opportunity to “act out” the mathematics
5.1.2.3.2. Great for kinesthetic/visual learners (most kids)
5.1.2.4. Using a Calculator
5.1.2.4.1. Great tool for exploring patterns and ideas associated with integers
5.1.2.4.2. See TI-83 Integer Practice Programs
5.1.2.5. Formulating procedures for Adding Integers
5.1.2.5.1. Procedures for Adding Integers
• Adding two positive integers: Add the digits and keep the sign
• Adding two negative integers: Add the digits and keep the sign
• Adding a positive and a negative integer: Subtract the smaller from the
larger digit (disregarding the signs) and keep the sign of the larger digit (if
the sign is disregarded)
5.1.3. Properties of Integer Addition
• The set of integers is closed for addition
• The opposite of any given integer is a unique number
• Zero has the same properties with integers as it had with whole numbers
• The commutative property holds for integers
• The associative property for integers holds
5.1.3.1. Basic Properties of Integer addition
• Additive Inverse Property: For each integer a, there is a unique integer, -a, such
that a + (-a) = 0
• Closure Property: For integers a and b, a + b is a unique integer
• Additive Identity Property: Zero is the unique integer such that for each integer a,
a+0=0+a=a
• Commutative Property: For all integers a and b, a + b = b + a
• Associative Property: For all integers a, b, and c, (a + b) + c = a + (b + c)
5.1.3.2. Using the Basic Ideas of Integer Addition in a Proof
5.1.3.2.1. See page 256-7 for examples
5.1.3.2.2. These are potentially on the unit test ☺
5.1.4. Modeling Integer Subtraction
5.1.4.1. Using Counters Model
5.1.4.1.1. A black counter and a red counter cancel each other
5.1.4.1.2. Concrete way of representing the addition of integers
5.1.4.2. Using a Charge Field Model
5.1.4.2.1. Another model for adding integers
5.1.4.2.2. + cancels out –
5.1.4.3. Using the Number Line
5.1.4.3.1. Allows students an opportunity to “act out” the mathematics
5.1.4.3.2. Great for kinesthetic/visual learners (most kids)
5.1.4.4. Using Mathematical Relationships and Patterns
5.1.4.4.1. Apply “Addend + Missing Addend = Sum” model to integer subtraction (See
example 5.9)
5.1.4.4.2. Definition of Integer Subtraction: For all integers a, b, and c, a – b = c if
and only if c + b = a
5.1.4.4.3. Theorem: Subtracting an Integer by adding the Opposite – For all
integers a and b, a – b = a + (-b). That is, to subtract an integer, add its
opposite.
5.1.4.5. Procedures for Subtracting Integers
• Take Away: To find 5 – (-2), take 2 red counters from a counter model for 5
• Missing Addend: To find 5 – (-2), think, “What number adds to -2 to give 5?”
• Add the Opposite: To find 5 – (-2), find 5 + 2
5.1.5. Applications of Integer Addition and Subtraction
5.1.5.1. see example 5.11 p. 264-265
5.1.5.2. see example 5.12 p. 266
5.1.6. Comparing and Ordering Integers
5.1.6.1. Using the Number Line to Order Integers
5.1.6.1.1. Numbers on the right of a given point on the number line are larger than
numbers to the left of that point
5.1.6.1.2. Graphing guys help us to mark the number line appropriately
5.1.6.2. Using Addition to Order Integers
5.1.6.2.1. Definition of Greater Than (>) and Less Than (<) for Integers: b > a if and
only if there is a positive integer p such that a + p = b. Also, a < b whenever b > a
5.1.6.2.2. See example 5.13 p. 267-8
5.1.6.3. Using Your Calculator to Order Integers
5.1.6.3.1. Try to modify one of the programs given previously to help with this task. ☺
5.1.7. Problems and Exercises p. 268
5.1.7.1. Home work: 1-8, 11, 12, 14, 17, 19, 27, 30, 36abc, 37ab, 38, 39abd, 40a, 44bcfh