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.

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. 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 ¹ 0

5.1.1.3.1. |x| = x, when x > 0

5.1.1.3.2. |-x| = x, when x > 0

5.1.1.3.3. -|x| = -x, when x > 0

5.1.1.3.4. |x| = -x, when x < 0

5.1.1.3.5. |-x| = -x, when x < 0

5.1.1.3.6. -|x| = x, when x < 0

5.1.1.3.7. |x| = |-x| = -|x| = x, when x = 0

5.1.1.3.8. Your turn p. 233: Do the practice and the reflection in your group

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 5.2 and 5.3 p. 234

5.1.2.1.4. Your turn p. 235: Do the practice and the reflection in your group

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 5.4 p. 235

5.1.2.2.4. Your turn p. 236: Do the practice and the reflection in your group

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)

5.1.2.3.3. Your turn p. 237: Do the practice and the reflection in your group

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

· A.V. means absolute value

· Adding two positive integers: Add the A.V. s and keep the sign

· Adding two negative integers: Add the A.V.s and keep the sign

· Adding a positive and a negative integer: Subtract the smaller A.V. from the larger A.V. (disregarding the signs) and keep the sign of the larger A. V. (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 240 Example 5.5

5.1.3.2.2. Your turn p. 240: Do the practice and the reflection in your group

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

5.1.4.1.3. Your turn p. 242: Do the practice and the reflection in your group

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 –

5.1.4.2.3. Your turn p. 243: Do the practice and the reflection in your group

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)

5.1.4.3.3. Your turn p. 244: Do the practice and the reflection in your group

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.

5.1.4.4.4. Your turn p. 246: Do the practice and the reflection in your group

5.1.4.5. Procedures for Subtracting Integers p. 247

· 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. 247-248

5.1.5.2. Your turn p. 249: Do the practice and the reflection in your group

5.1.5.3. see example 5.12 p. 249

5.1.5.4. Your turn p. 249: Do the practice and the reflection in your group

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

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. J