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Integer Operations Example 1 Let yellow = positive This piece has a value of 1. 1 53 8 Write the value of these pieces. I will add more pieces. Write a numerical expression and simplify. Example 2 Let red = negative This piece has a value of –1. 1 6 5 11 Write the value of these pieces. I will add more pieces. Write a numerical expression and simplify. Rules of Addition To add two numbers with the same sign: Step 1 ADD Step 2 ATTACH the common sign –8 + (–5) = – 13 8 + 5 = 13 If the common sign is positive, the + is not written! Example 3 Let yellow = positive Let red = negative 1 1 0 Write a numerical expression and simplify. This expression has a special name. Do you know what it is? zero pair Adding or subtracting zero pairs will not affect the value of an expression. Let yellow = positive Let red = negative 4 7 3 When you add two numbers with opposite signs, you will need to subtract (take away) the zero pairs. Example 4 Let yellow = positive Let red = negative 7 3 4 Write the value of these pieces. I will add more pieces. Write a numerical expression. To simplify remove all the zero pairs. The pieces that remain model the simplified answer. Example 5 Let yellow = positive Let red = negative 83 5 Write the value of these pieces. I will add more pieces. Write a numerical expression. To simplify remove all the zero pairs. The pieces that remain model the simplified answer. Rules of Addition To add two numbers with the opposite signs: Step 1 SUBTRACT Step 2 ATTACH the sign of the number with the larger absolute value. 4 + (–9) –5 Example 6 Write a numerical expression and simplify. 94 5 or 4 9 5 Can you tell at a glance what the sign is for the answer ? Example 7 Write a numerical expression and simplify. 5 11 6 or 11 5 6 Can you tell at a glance what the sign is for the answer ? Example 8 Find the sum. 1. Write problem. 2. Signs alike – add write the common sign. Example 9 Find the sum. 1. Write problem. 2. Signs unlike – subtract write sign of the largest absolute value. 12 30 –42 7 15 –8 12 30 Properties of Addition Commutative Property: The order in which numbers are added does not change the sum. x +y=y +x 9+4=4+9 Associative Property: The way numbers are grouped when added does not change the sum. (2 + 5) + 10 = 2 + (5 + 10) (a + b) + c = a + (b + c) Closure Property: The sum of any two real numbers is a unique real number. 5+3=8 Properties of Addition Identity Property: The sum of a number and 0 is the number. –16 + 0 = –16 x+0=x Inverse Property of addition: number and its opposite is 0. 45 45 0 The sum of a a a 0 45 45 0 a a 0 When two signs are needed use parentheses or a superscript! Example 10 Name the property shown by the statement. 1. Write problem. 12 + –36 = –36 + 12 2. Name the property. commutative Example 11 Name the property shown by the statement. 1. Write problem. 0 + –8 = –8 2. Name the property. identity Example 12 Name the property shown by the statement. 1. Write problem. (4 + 5) + 6 = (5 + 4) + 6 2. Name the property. commutative Example 13 Name the property shown by the statement. 1. Write problem. 2. Name the property. –7 +7=0 inverse Example 14 Name the property shown by the statement. 1. Write problem. 2. Name the property. (9 + 6) + 5 = 9 + (6 + 5) associative Example 15 Find the sum. 1. Write problem. 37 29 2 2. Follow the rules for 8 2 the order of operations 10 to simplify. If you override the left to right rule, you must show the step which gives support to your thinking! 37 29 2 37 2 29 39 29 10 Example 16 Find the sum. 1. Write problem. 12 6 15 2 12 6 15 2 2. Follow the rules for 12 9 2 the order of operations to simplify. 3 2 5 If you override the left to right rule, you must show the step which gives support to your thinking! 12 6 15 2 18 15 2 18 2 15 20 15 5 Example 17 Find the sum. 58 100 78 1. Write problem. 2. Use associative 58 100 78 property to regroup. 58 178 Then simplify. 120 Inductive Reasoning is making conclusions on patterns you observe. What are the missing numbers in the following: Does the pattern suggest a general rule for multiplying a positive number by a negative number? The product of a positive number and a negative number is negative. 33 9 32 6 3 1 3 30 0 3 1 3 3 2 6 3 3 9 What are the missing numbers in the following: Does the pattern suggest a general 3 3 9 rule for multiplying a negative number 3 2 6 by a negative number? 3 1 3 The product of a negative number 30 0 and a negative number is positive. 3 1 3 3 2 6 3 3 9 Rules for Multiplication The product of a positive number and a negative number is negative. 12(–10) = –120 The product of a negative number and a negative number is positive. –12(–10) = 120 Summary: If the signs are the same, the product is positive. If the signs are not the same, the product is negative. Properties of Multiplication Commutative Property: The order in which numbers are multiplied does not change the product. 9(4) = 4(9) x y = y x Associative Property: The way numbers are grouped when multiplied does not change the product. (2 • 5) • 10 = 2 • (5 • 10) (a • b) • c = a • (b • c) Properties of Multiplication Identity Property: The product of a number and 1 is the number. –16 • 1 = –16 x•1=x Property of Zero: The product of a number and 0 is zero. –16 • 0 = 0 x•0=0 Property of Negative One: The product of a number and –1 is the opposite of the number. x • (– 1) = – x –16 • (–1) = 16 Inverse Property of multiplication: The product of a number and it’s reciprocal is 1. 2 3 · =1 3 2 Example 1 Find the product. 1. Write problem. 2. Follow rules for the order of operations to simplify. – 4(–3)(–5) 12(–5) –60 Example 2 Find the product. 1 1. Write problem. 65 5 2. Follow order of operations 1 to simplify. 65 5 ─6 If you alter the left to right rule, you must show the step which gives support to your thinking! 1 30 5 6 What property justifies this step? Example 3 Find the product. 1. Write problem. 14 2. Write in factor form. Optional step. (–1 ) (–1 ) (–1 ) (–1 ) 3. Simplify. Notice the negative one is in parentheses. What is the answer if it is not in parentheses? 14 1 1 Example 4 Find the product. 15 1. Write problem. 2. Write in factor form. (–1) (–1) (–1) (–1) (–1) Optional step. –1 3. Simplify. Can you formulate a rule that works with the product of negatives? A product is negative if it has an odd number of negative factors. A product is positive if it has an even number of negative factors. Example 5 Simplify the expression. 1. Write problem. – 6(y)(–y) 2. Multiply left to right. – 6y(–y) 3. Multiply. Write variables in power form. Notice the simplified answer does NOT have parentheses! 6y2 Example 6 Simplify the expression. 1. Write problem. 2. Write factored form. 3. Simplify. Writing the factored form helps to prevent errors! 4(–b)3 4 b b b – 4b3 Notice the simplified answer does NOT have parentheses! Example 7 Simplify the expression. 1. Write problem. 2(–x)(–x)(–x)(–x) 2. Multiply. Write the answer in power form. 2x4 Notice the simplified answer does NOT have parentheses! Example 8 Evaluate the expression when x = –7. 1. Write problem. 2. Substitute. 2 5 x 7 2 5 7 3. Follow rules for order 7 of operations to simplify. 5 2 35 7 –10 Example 9 Evaluate the expression when x = –2. 1. Write problem. 2. Substitute. 3 x 3 3 23 3. Follow order of operations: 323 simplify within parentheses. 4. Evaluate the power. 5. Simplify. 3(8) 24 Example 10 A leaf floats down from a tree at a velocity of –12 cm/sec. Find the displacement, which is the change in position, of the leaf after 4.2 seconds. Note: An object’s change in position when it drops can be found by multiplying its velocity by the time it drops. Let d = displacement 1. Write let statement. 2. Write verbal model. Displacement = Velocity • Time 3. Write algebraic model. d = –12(4.2) 4. Solve. 5. Sentence. = –50.4 The leaf’s displacement is –50.4 cm. Subtracting Real Numbers Rules of Subtraction Adding the opposite of a number is equivalent to subtracting the number. Addition Problem 6 + (– 4) = 2 6 4 2 Equivalent Subtraction Problem 6– 4 = 2 6 4 2 Example 1 Find the difference. 1. Write problem. – 9 – (–4) 2. Rewrite as addition. 9 4 3. Use rules of addition. –5 For tonight’s homework, you must rewrite subtraction as addition! Example 2 Find the difference. 1. Write problem. –9–4 2. Rewrite as addition. 9 4 3. Use rules of addition. –13 Example 3 Find the difference. 5–7 5 + –7 –2 Example 4 Simplify the expression. 1. Write problem. 10 – 5 + 14 – (–18) 10 + (−5) + 14 + 18 5 + 14 + 18 3. Follow rules for order Leave of operations to simplify. 19 + 18 addition 37 alone! 2. Rewrite as addition. Example 5 Simplify the expression. 1. Write problem. 2 5 7 3 3 2 5 7 3 3 2. Rewrite as addition. 3. Follow rules for order of operations 21 2 5 to simplify. 3 3 3 What property 19 5 allows you to 3 3 move the 7 to the far right? 14 3 2 5 7 3 3 2 5 7 3 3 2 5 7 3 3 7 7 3 21 7 3 3 14 3 Terms of an Expression When an expression is written as a sum, the parts that are added are the terms of the expression. Find the terms of the expression. The problem. List the terms. 4x + y + 6 4x, y and 6 The problem. Rewrite as addition. List the terms. –5 – x –5 + – x –5 and – x Example 6 Find the terms of the expression. 1. Write problem. 3x – 14y – –36 2. Rewrite as addition. 3x + (–14y) + 36 3. List the terms. 3x, –14y and 36 Example 7 In February 1956 the temperature in Bismarck, North Dakota fell 95 degrees overnight. The initial temperature was 42 degrees Fahrenheit. What was the final temperature? 1. Write an expression. 42 – 95 2. Rewrite as addition. 42 + (–95) 3. Use rules of addition. –53 4. Write a sentence. The final temperature was –53 Fahrenheit. Combining Like Terms An algebraic expression is easier to evaluate when it is simplified. The distributive property allows you to combine like terms by adding their coefficients. What is a term? A term is a number or the product of a number and variable/s. one term 7 x one term Terms are separated by 7x one term addition. 7 one term 1 7 x x two terms 7x two terms –– x 7+ An algebraic expression is easier to evaluate when it is simplified. The distributive property allows you to combine like terms by adding their coefficients. What is a coefficient? In a term that is the product of a number and a variable, the number is called the coefficient of the variable. – 1 is the coefficient of x –1x + 3x2 3 is the coefficient of x2 Like terms are terms in an expression that have the same variable raised to the same power. 8x and 3x 4x2 and 4x 7m and –2m 25 and 10 Like Terms Not Like Terms Like Terms Like Terms Constants are considered like terms. Example 1 Identify the like terms in the expression. 1. Write problem. x2+–– 3x + 2x2 +–– 5 + 4x 2. Change subtraction to addition. 3. Identify the like terms. x2, 2x2 and –3x, 4x Example 2 Identify the like terms in the expression. 1. Write problem. 3m2 +– – 6m + 4m2 +–– 7 + m + 15 2. Change subtraction to addition. 3. Identify the like terms. 3m2, 4m2 and – 6m, m and –7, 15 An expression is simplified if it has: no grouping symbols, no like terms, and no double signs. The problem. 4c – c Use the Identity Property 4c– – 1c– to name the coefficient. (4 – 1 )c Distribute. 3c Simplify. If you have 4 cookies and you eat one cookie, how many cookies are left? This is the mathematical proof for combining like terms! Example 3 Simplify the expression. 1. Write problem. 5x2 +–– 7 + 3x2 2. Change subtraction to addition. 8x2 + – 7 3. Combine like terms. 8x2 – 7 4. Undo the double sign. It may be helpful to circle like terms! Write the variable term before the constant term! Example 4 Simplify the expression. 1. Write problem. 2. Change subtraction to addition. 3. Combine like terms. 4. Undo the double sign. I bet she wants me to write that in my notes so I‘ll remember it! 3x2 +–– 5x + 4x2 –+– 7x 7x2 + – 12x 7x2 – 12x Good form is alphabetical descending order!