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`Integers Addition and Subtraction of IntegersSec. 4.1 The set of integers is the set I = {... -3, -2, -1, 0, 1, 2, 3, ... }. The numbers 1, 2, 3, ... are called the The number -1, -2, -3, ... are called the Zero is&quot;Opposites&quot; Model for IntegersSo far, our number line has looked like: Now with the integers, it looks like:0 1 2 3 4 5Chip Model for IntegersRequires two colors of chips:Black chips represent positive #s (credits) Red chips represent negative #s (debits)When a black chip and a red chip meet, they cancel each other out, making 0.1Chip Model for IntegersFind the value of each set:Addition with the Chip ModelSolve the following using chip model: 5 + ( ¯3) = ? ( ¯2) + ( ¯3) = ?5 There are many ways to represent each number.Charged-Field &amp; Pattern ModelsCharged-Field Model Just like the chip model but use + and +++ _ _ _ _Number-Line ModelPositives move to the right (forward). Negatives move backward. Ex: 3 + 2 = ?; 3 + (¯2) = ?; (¯1) + (¯2)= ?Pattern Model3 3 3 3 + + + + 3 2 1 0 = = = = 6 5 4 3 ¯5¯4¯3¯2¯10123452Absolute ValueAbsolute value is often thought of as the _________ from the number to __. Ex: | 3 | = ___ and | ¯ 4 | = ___ Definition: |x| = if x  0 |x| = if x &lt; 0Properties of Addition for IntegersLet a, b, and c Closure Commutative Associative Identity be integers a + b = c , c is an integer a+b=b+a a + (b + c) = (a + b) + c a+0=a=0+aProperties of Additive InverseFor any integers a and b : 1. ¯ ( ¯a) = a 2. ¯a + ¯b = ¯(a + b)Subtraction for Integers using Chip Model3 ­ 5 = ___3 ­ (¯5) =3Subtraction of IntegersCharged-Field Just like Chip Model 2 ­ ( ¯1) = ? Patterns Model 3­2=1 3­1=2 3­0=3 3 ­ ( ¯1) = 3 ­ ( ¯2) = 3 ­ ( ¯3) =Number-Line ModelEx: 3 ­ 2 = ?; 3 ­ (¯2) = ?; (¯1) ­ (¯2)= ?¯5¯4¯3¯2¯1012345Subtraction and a propertyFor integers a and b, a ­ b is the unique integer n such that ____________Subtraction as Opposite of Addition (Chip Model)3 ­ 5 = ___Subtraction is the inverse of addition:3 ­ 5 = 3 + (-5)4Order of OperationsWhat is the value of: 6 + 4 ÷ 2 + 5 × 3 ­ 1 = ? Is it 11? 22?Multiplication and Division of IntegersSec. 4.2Order or Operations:PEMDAS (Please excuse my dear aunt Sally)Multiplication of Integers Using the Set (Chip) ModelRecall that 3 × 4 means 3 groups of 4 Solve the following problems: 3 × 2 = ___ 3 × ( ¯ 2) = ___Multiplication of Integers Using the Set (Chip) ModelWhat about ¯3 × 2? What does it mean? With 3 × 2, we_____________________ ¯ 3 × 2 = ___5Multiplication of Integers Using the Set (Chip) ModelSolve the following problems: ¯3 × (¯2) = ___ 3 × 0 = __Multiplication of IntegersCharged-Field ModelJust like chip modelNumber-Line Model Ex: 3 × (¯5)= __¯3 × (¯5)= __¯25 ¯20 ¯15 ¯10 ¯ 50510152025Which Properties Hold for Integer MultiplicationClosure Commutative Associative Identity Distributive over Addition Zero Multiplication PropertyMore Properties of Integer MultiplicationFor any integers a and b, ( ¯ 1)a = ¯ a ( ¯ a)( ¯ b) = ab &amp; (¯ a)b = (¯ab) Proof of ( ¯ a)( ¯ b) = ab1. 2. 3. 4.(¯a)(¯b) = [(¯1)a]× [(¯1)b] = [¯1×¯1](ab) = (1)(ab) = ab -6More Properties of Integer MultiplicationFor any integers a and b, Proof of (¯ a)b = ¯(ab) Step Reason1. 2. 3.Distributive Property of Multiplication over SubtractionFor any integers a, b, &amp; c : a (b ­ c) = ab ­ ac.1.2. 3. 4. 5. 6. 7. 8.(¯a)(b) = [(¯1)a]×b = (¯1) (ab) = ¯(ab)  a (b ­ c) = a (b + ¯c) = ab +a( ¯c) = ab +a(¯1c) = ab +(a ×¯1)c = ab +(¯1×a)c = ab +(¯1)ac = ab +¯(ac) = ab ­ ac       A Couple of ExamplesFind the value of ¯3(7 ­ ¯2)= Using the Distributive property over ­ :¯3(7­ ¯2)=Difference-of-SquaresFor any integers a and b: (a + b)(a ­ b) = a 2 ­ b 2.Find the value (mentally) of 36 × 24 = __1. (a+b)(a­b) = (a+b)a ­ (a+b)b  2. = (aa + ba) ­ (ab+bb)  3. = a2 + ba ­ (ab + b2)  4. = a2 + ab ­ (ab + b2)  5. = a2 + ab ­ ab ­ b2  6. = a2 + 0 ­ b2  7. = a2 ­ b2 Proof:7Division of IntegersWhole numbers: a ÷ b = c iff _______ Integers: a ÷ b = c if &amp; only if _______ 15÷(¯3) =__ ¯15÷3 =__ ¯15÷(¯3) =__Ordering Integers (&gt; &amp; &lt;)Number line: Ex: ¯3 &lt; ¯1 because¯5¯4 ¯3 ¯2 ¯1012345Definition of Less than: a is less than b if &amp; only if ___________ _________________________________ Ex: ¯3 &lt; ¯1 becauseSubtraction of IntegersChip ModelAddition with the Chip ModelSolve the following using chip model: 5 + ( ¯3) = ? ( ¯2) + ( ¯3) = ?28Number-Line ModelPositives move to the right (forward). Negatives move backward.¯5¯4¯3¯2¯10123459`

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