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Transcript
1-1 Properties of Real Numbers M11.A.1.3.1: Locate/identify irrational numbers at the approximate location on the number line. M11.A.1.3.2: Compare and/or order any real numbers Objectives Graphing and Ordering Real Numbers Properties of Real Numbers Number Classification • Natural numbers are the counting numbers. • Whole numbers are natural numbers and zero. • Integers are whole numbers and their opposites. • Rational numbers can be written as a fraction. • Irrational numbers cannot be written as a fraction. • All of these numbers are real numbers. Number Classifications Subsets of the Real Numbers Q - Rational I - Irrational Z - Integers W - Whole N - Natural Classify each number -1 6 real, rational, integer real, rational, integer, whole, natural real, irrational 1 2 real, rational 0 real, rational, integer, whole -2.222 real, rational Properties of Real Numbers Graph the numbers – 3 , 4 number line. 7 , and 3.6 on a – 34 is between –1 and 0. Use a calculator to find that 7 2.65. Work on Quick Check #2 on Page 6 Properties of Real Numbers Compare –9 and – 9 = 3, so – 9. Use the symbols < and >. 9 = –3. Since –9 < –3, it follows that –9 < – 9. Work on Quick Check #3 on Page 6 Properties of Real Numbers Inverses The Additive Inverse of any number a is -a. The sum of opposites is 0. The Multiplicative Inverse of any nonzero number a is 1/a. The product of reciprocals is 1. Properties of Real Numbers Find the opposite and the reciprocal of each number. 1 a. –3 7 b. 4 1 Opposite: –(–3 7 Reciprocal: 1 1 –3 7 Opposite: –4 ) = 3 17 = 1 – 22 =– 7 22 Reciprocal: 1 7 Work on Quick Check #4 on Page 7 4 Properties of Real Numbers Commutative Property • Think… commuting to work. • Deals with ORDER. It doesn’t matter what order you ADD or MULTIPLY. • a+b = b+a •4 • 6 = 6 • 4 Properties of Real Numbers Associative Property • Think…the people you associate with, your group. • Deals with grouping when you Add or Multiply. • Order does not change. Properties of Real Numbers Associative Property • • (a + b) + c = a + ( b + c) (nm)p = n(mp) Properties of Real Numbers Additive Identity Property • s + 0 = s Multiplicative Identity Property • 1(b) = b Properties of Real Numbers Distributive Property • a(b + c) = ab + ac • (r + s)9 = 9r + 9s Name the Property •5=5+0 • • • • Additive Identity 5(2x + 7) = 10x + 35 Distributive 8•7=7•8 Commutative 24(2) = 2(24) Commutative (7 + 8) + 2 = 2 + (7 + 8) Commutative Name the Property • 7 + (8 + 2) = (7 + 8) + 2 • Associative • 1 • v + -4 = v + -4 • (6 - 3a)b = 6b - 3ab • 4(a + b) = 4a + 4b • Multiplicative Identity • Distributive • Distributive Properties of Real Numbers The absolute value of a real number is the distance from zero on the number line. Simplify | 4 1 |, |–9.2|, and |3 – 8|. 3 1 1 1 1 43 is 4 3 units from 0, so | 4 3 | = 4 3 . –9.2 is 9.2 units from 0, so |–9.2| = 9.2. |3 – 8| = |–5| and –5 is 5 units from 0. So, |–5| = 5, and hence |3 – 8| = 5. Work on Quick Check #6 on Page 8 1-2 Algebraic Expressions M11.A.3.1.1 – Simplify/Evaluate expressions using the order of operations to solve problems Objectives Evaluating Algebraic Expressions Simplifying Algebraic Expressions Vocabulary A variable is a symbol, usually a letter, that represents one or more numbers. An algebraic expression is an expression that contains at least one variable. You can evaluate an algebraic expression by replacing each variable with a value and then applying the Order of Operations. Order of Operations Parenthesis Exponents Multiply & Divide from Left to Right Add & Subtract from Left to Right Example: Evaluate a(5a + 2b) if a=3 and b=-2 Substitute the values into the expression. 3[5(3) + 2(-2)] Now apply the Order of Operations: Inside the brackets, perform multiplication and division before addition and subtraction 5(3) = 15 and 2(-2)= -4 3[15 + -4] then 15 + -4 = 11 3[11] = 33 Evaluating an Algebraic Expression Evaluate 7x – 3xy for x = –2 and y = 5. 7x – 3xy = 7(–2) – 3(–2) (5) Substitute –2 for x and 5 for y. = –14 – (–30) Multiply first. = –14 + 30 To subtract, add the opposite. = 16 Add. Work on Quick Check #1 on page 12 Evaluating an Algebraic Expression with Exponents Evaluate (k – 18)2 – 4k for k = 6. (k – 18)2 – 4k = (6 – 18)2 – 4(6) Substitute 6 for k. = (–12)2 – 4(6) Subtract within parentheses. = 144 – 4(6) Simplify the power. = 144 – 24 Multiply. = 120 Subtract. Work on Quick Check #2 on page 12 Vocabulary Simplifying Algebraic Expressions A term is a number, a variable, or product of a number and one or more variables. The numerical factor in a term is the coefficient. Like terms have the same variables raised to the same powers. Like terms: 3r 2 and r 2 2 xy3and 3xy3 Combining Like Terms Simplify by combining like terms. 2h – 3k + 7(2h – 3k) 2h – 3k + 7(2h – 3k) = 2h – 3k + 14h – 21k Distributive Property = 2h + 14h – 3k – 21k Commutative Property = (2 + 14)h – (3 + 21)k Distributive Property = 16h – 24k Work on Quick Check #4 on page 14 Finding Perimeter Find the perimeter of this figure. Simplify the answer. c c P = c + 2 + d + (d – c) + d + 2 + c + d c c = c + 2+ d + d – c + d + 2 + c + d c c = 2 + 2 + c + 4d = 2c + c + 4d 2 = c + c + 4d = 2c + 4d Work on Quick Check #5 on page 14