Chapter 4: Integers and Number Theory
4.1 Integers and the operations of Addition and Subtraction
4.1.1. Vocabulary
4.1.1.1. opposites – negative integers are opposites of positive integers
4.1.1.2. absolute value – The distance between the points corresponding to an integer and zero
4.1.1.3. additive inverse – the opposite of an integer is its additive inverse
4.1.1.4. signed numbers – integers are signed numbers (positive or negative) The only exception is zero which is neither positive nor negative
4.1.2. Representations of integers
4.1.2.1. What does –x mean?
4.1.2.1.1. –x = -3 then x = 3
4.1.2.1.2. –x = 5 then x = -(-5) or -5
4.1.2.1.3. –x = 0 then x =0
4.1.2.1.4. Does –x always represent a negative integer? Explain your answer on your paper. (to be turned in)
4.1.3. Integer addition
4.1.3.1. Integer addition rules
4.1.3.1.1. So sometimes addition is like subtraction with integers
4.1.3.2. Chip model
4.1.3.2.1. use chips that are two different colors: i.e. Othello chips are white on one side and black on the other
4.1.3.2.2. Choose one color to represent positive the other to represent negative numbers
4.1.3.2.3. A white and a black cancel each other out, similar to subtraction
4.1.3.2.4.
4.1.3.3. Charged field model
4.1.3.3.1. Similar to chip model
4.1.3.3.2. uses + and – to represent positive and negative integers
4.1.3.3.3. = +
4.1.3.3.4. Positive 4 plus negative 3 equals positive 1
4.1.3.4. Pattern model
4.1.3.4.1. see page 189
4.1.3.4.2. patterns help students to understand how/why integer rules work
4.1.3.5. Number Line model
4.1.3.5.1.
4.1.3.5.2. Positive 4 plus negative 3 equals positive 1
4.1.3.5.3. +5 + +3 = ?
4.1.3.5.4. -5 + +3 = ?
4.1.3.5.5. +3 – 5 = ?
4.1.3.5.6. -3 – 5 = ?
4.1.4. Absolute value
4.1.4.1. Definition of Absolute Value: |x| = x if x ³ 0 and |x| = -x if x < 0
4.1.4.2. |+5| = 5
4.1.4.3. |-5| = 5
4.1.4.4. -|+5| = -5
4.1.4.5. -|-5| = -5
4.1.5. Properties of integer addition
4.1.5.1. Properties
4.1.5.1.1. Addition of integers is closed: a + b is a unique number
4.1.5.1.2. Commutative property of integers: a + b = b + a
4.1.5.1.3. Associative property of integers: (a + b) + c = a + (b + c)
4.1.5.1.4. Identity element of addition of integers: 0 + a = a + 0 = a
4.1.5.2. Uniqueness Property of Additive Inverse: a + (-a) = (-a) + a = 0
4.1.5.3. Properties of the Additive Inverse: for any integers a and b:
4.1.5.3.1.1. –(-a) = a
4.1.5.3.1.2. (–a) + (-b) = -(a + b)
4.1.5.3.1.3. (– a) – b = -(a + b)
4.1.6. Integer Subtraction
4.1.6.1. Chip model
4.1.6.1.1.
4.1.6.2. Charged-field model
4.1.6.2.1. Similar to chip model
4.1.6.2.2. uses + and – to represent positive and negative integers
4.1.6.3. Patterns model
4.1.6.3.1. see page 173
4.1.6.3.2. patterns help students to understand how/why integer rules work
4.1.6.4. Number Line model
4.1.6.4.1. +3 - +5 = ? think of as +3 + -5 = ? and do as in addition before
4.1.6.4.2. +7 - +6 = ? think of as +7 + -6 = ? and do as in addition before
4.1.6.4.3. -3 - -5 = ? think of as -3 + +5 = ? and do as in addition before
4.1.6.4.4. -5 - -3 = ? think of as -5 + +3 = ? and do as in addition before
4.1.6.4.5. Now Try This 4-1: p. 196
4.1.7. Subtraction as the inverse of addition
4.1.7.1. Definition of Subtraction – for integers a and b, a – b is the unique integer n such that a = b + n
4.1.7.2. Property – For all integers a and b, a – b = a + (-b)
4.1.7.3. Now try this 4-2 p. 197
4.1.8. Order of operations
4.1.8.1. subtraction is neither commutative nor associative
4.1.8.2. Picture eleven million dollars actually spent
4.1.9. Ongoing Assessment p. 199
4.1.9.1. Home work: 1b, 1e, 3a, 3d, 4a, 4c, 6a, 6c, 10a, 11e, 17b, 17e, 20c