1- 1 VARIABLES AND EXPRESSIONS Variable- a letter that stands for a number Variable expression- a mathematical phrase that uses variables, numerals, and operation symbols. It is an efficient way to express a mathematical expression. Ex) Identify each expression as a numerical expression or a variable expression. If it is a variable expression, name the variable. a) 5 – 5 numerical expression b) c – 5 variable expression the variable is c Page 5 Quick Check # 1 a-c a. b. c. Translating word phrases into variable expressions. Word Phrase Variable Expression Nine more than a number y y + 9 4 less than a number n n – 4 3 times z z * 3, 3z, 3(z) A number a divided by 12 a – 12, a/12 5 times the quantity 4 plus 5(4 + c) a number c page 6 1-2 ORDER OF OPERATIONS
Order of Operations
EX) 1.) 4 + 15 – 3 2.) 2 + 5 x 3 3.) 12 – 3 – 1 4.) 10-1*7 Ex) 1) 3 * 5 * 8 – 4 + 6 2) 4 – 1 * 2 + 6 – 3 3) 5 + 6 * 4 – 3 – 1 EX) 24 – [ 6 – (2 * 2) ] EX) 1 + 10 – 2 4 EX) 20 – 3 [ ( 5 + 2) – 1 ] Use >, <, or = to complete the statements. (22 + 8 ) __________ 22 + 8 – 2 Insert grouping symbols to make each number sentences true. 3 + 6 * 2 = 18 1-3 WRITING AND EVALUATING EXPRESSIONS Evaluate (solve) – to evaluate a variable expression, you first replace each variable with a number. Then you use the Order of Operations to simplify. EX) Evaluate 1) 4y – 15 for y = 9 4(9) – 15 36 – 15 21 EX) 2) 63 – 5x for x = 7 63 – 5(7) 63 – 35 28 EX) 3) 4(t + 3) + 1 for t = 8 4(8 + 3) + 1 4(11) + 1 44 + 1 45 EX) 4) 3ab + c/2 for a = 2, b = 5, c = 10 3(2)(5) + 10/2 30 + 5 35 1-4 INTEGERS AND ABSOLUTE VALUE ______________________________ Negative Positive Graph Order from least to greatest -1, 4, -5 -5, -1, 4 Graph order: 0, 2, -6 Order from least to greatest -6, 0, 2 Opposites - numbers that are the same distance from zero on a number line but in opposite direction. Integers- whole numbers and their opposites. Absolute Value - a number’s distance from zero. 1-5 ADDING INTEGERS Addition of Opposites The sum of an integer and its opposite is zero. Arithmetic Algebra 1 + (-1) = 0 x + (-x) = 0 Using a number line On two plays, a football team loses 8 yards and then gains 3 yards. Find –8 + 3 to find the result of the two plays. Loses 5 yards -5 Page 25 Quick Check 2 USING RULES TO ADD INTEGERS Same Sign – The sum of two positive integers is positive. The sum of two negative integers is negative. Different Signs - To add two integers with different signs, find the difference of their absolute values. The sum has the sign of the integer with the greatest absolute value. -12 + -31 = -43 7 + -18 = -11 -22 + (-16) = -38 60 + (-13) = 47 -125 + 35 = -90 -12 + -6 + 15 + -2 -20 + 15 = -5 1 + (-3) + 2 + -10 3 + -13 = -10 RULES TO ADDING INTEGERS Positive + Positive means positive Positive + Negative means positive or negative Negative + Positive means positive or negative Negative + Negative means negative Same signs add then give them the same sign. Different signs subtract then give the sign of the larger ___number_______________. (What you have more of) 1-6 SUBTRACTING INTEGERS -6 – (-2) -4 -7 – (-2) -4 – (-3) -8 – (-5) 3 – 5 4 – 8 -1 – 5 -2 – ( -7) Subtracting Integers To subtract an integer, add its OPPOSITE. 2 – 5 = 2 + -5 = -3 a – b = a + -b 2 – (-5) = 2 + 5 = 7 a- -b = a + b Subtracting Rules A subtraction sign creates a negative integer EXCEPTSubtracting a negative creates a positive integer. 1-7 INDUCTIVE REASONING Inductive reasoning – making conclusions based on patterns you observe. Conjecture – a conclusion you reach by inductive reasoning. Page 35 Counter example – an example that proves a statement false. You need only one counterexample to prove that a conjecture is incorrect. EX) 30, 25, 20, 15 Start with 30 and subtract 5 repeatedly 2, -2, 2, -2 Alternate 2 and its opposite 1, 3, 4, 12, 13 Start with 1, alternate multiplying by 3 and adding 1 Page 35 OC # 2 a-c 1-9 MULTIPLYING AND DIVIDNG INTEGERS The product of two integers with the same sign is positive.(multiplying) The product of two integers with different signs is negative. (multiplying) The product of zero and any integer is zero. (multiplying) EX) 3(4) = 12 3(-4) = -12 -3(-4) = 12 -3(4) = 12 3(0) = 0 -4(0) = 0 Page 44 EX) -3 * 5 (-4) = 60 -2 (-3) (-2) = 4 (2) (-3) = Quick Check -4 * 8 (-2) = 64 6(-3)(5) = -90 -7*(-14) * 0 = 0 The quotient of two integers with the same sign is positive. (dividing) The quotient of two integers with different signs is negative. (dividing) Division by zero is undefined. EX) 12 – 3 = 4 12 – (-3) = -4 -12 – (-3) = 4 -12 – 3 = -4 EX) -32 – 8 = -4 -48 – (-6) = 8 -56 – (-4) = 14 Find the average of: -33 + (-33) + (-35) + (-36) + (-29) = -166 – 5 = -33.2 4, -3, -5, 2, -8 = -2 Page 76,77
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AuthorMr. Porcaro Links to NotesArchives
March 2020
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