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Name _________________________________________ Date ___________________ 6th – Everyday Math Page 1 Study Guide for Unit 6 Assessment 1) Multiplying Mixed Numbers and Fractions Example: 3 1/2 * 2/3 First, changed the mixed number 3 1/2 to a fraction. + 3 1/2 = 7/2 * Second, multiply the numerators and multiply the denominators. 7 2 = 14 2 * 3 = 6 Third, Change back to a mixed number and reduce. 14/6 = 14 divided by 6 = 2 2/6 = 2 1/3 Solve: a) 5 1/2 * 3/4 = _____ b) 3 2/5 * 1/2 = _____ 2) Dividing Fractions by Fractions - Multiply the first fraction by the reciprocal of the second fraction. *Reciprocal fractions are upside down fractions of the original. The reciprocal fraction of 2/3 is 3/2. Example: 7/2 divided by 2/3. So, 7 2 2 3 a) 4/5 7 3 = 21 2 * 2 = 4 Solve: 2/3 = _____ b) 2/3 1/2 = _____ 3) Adding and Subtracting Negative Numbers When adding two numbers, think of the first addend as a starting point on a number line, the second addend as a move away from that starting point to the sum or difference. (Start at 1, Add –5) Example: -6 –5 –4 –3 –2 –1 0 1 2 3 1 + (-5) = -4 4 5 6 6th Unit 6 Study Guide page 2 Solve: a) -3 + 5 = _____ c) -8 – (-3) = _____ b) 5 + (-3) = _____ d) (OPP) (OPP) -9 + (-3) = _____ (the two negatives make a plus) 4) Multiplying and Dividing Negative Numbers Rule 1: If the numbers are both positive or both negative, the result is a positive number. Rule 2: If one of the numbers is negative and the other is positive, the result is a negative number. (Neg) (Neg) (Pos) (Pos) (Neg) (Neg) Example 1: -6 * (-2) = 12 (Neg) (Pos) (Neg) Example 2: 6 * (-2) = -12 Example 3: -12 / 2 = -6 Solve: a) -3 * 8 = _____ b) 64 / (-8) = _____ c) -5 * (-5) = _____ Working with Expressions Involving Positive and Negative Numbers Compare the Add/Sub. Chart with the Mult./Div. Chart below. Remember when Adding or Subtracting to think of it as putting together positives and negatives Remember when Multiplying & Dividing some rules are different from add/sub. Pos. & Pos. / Add / answer is Pos. Neg. & Neg. / Add / answer is Neg. Pos. & Neg. / Sub. / answer is the sign of larger # Pos. & Pos. Neg. & Neg. Pos. & Neg answer is Pos. answer is Pos. answer is Neg When adding two negative numbers your answer is always negative, but... When multiplying or dividing two negative numbers the answer is always positive. When adding a positive and negative number, the sign depends on which number is larger (pos or neg.) but... When multiplying or dividing a positive and negative number the sign is always negative. 5) The Order of Operations – Please Excuse My Dear Aunt Sally. or PEMDAS Parenthesis, Exponents, ( Multiply “or” Example: 52 + (3 * 4 – 2) / 5 = Parenthesis first. (multiply) (subtract) Exponents next. Divide then add. Answer: Divide), (Add “or” Subtract). (Work left to right.) 52 + (3 * 4 – 2) / 5 = ? 52 + ( 12 – 2) / 5 = 52 + 10 /5 = 25 + 10 /5 = 25 + 2 = 27 52 + (3 * 4 – 2) / 5 = 27 6th Unit 6 Study Guide page 3 Solve: Follow the rules for the order of operations. (Show each step.) (You would multiply here) a) 6 + 3(14 – 6) = _____ b) (6 + 14) / 5 * 8 = _____ c) 4 * 7 + 16 / 4 = _____ d) 2 * 62 + (27 / 3 – 4) = _____ e) (42 / 2) + (14 / 2 – 5)3 = ____ f) 0.4 * (-6) = _____ g) 2 + 24 / 4 = ____ h) 102 * (21 / 3) i) (-3 + 5) 9 / 3 = _____ j) (12/4 + 7)3 + (24/6 –2)4 = _____ = _____ 6) Insert parenthesis to make each number sentence true. a) 7 * 3 - 3 = 18 b) 9.2 * 2 - 2 > 10 c) 3 * 4 - 2 = 3 + 0 * 2 d) 6 / 2 + 3 > 6 e) 12 - 3 * 4 f) 6 * 9 = 46 + 4 * 2 = 36 6th Unit 6 Study Guide page 4 7) Equations – Number sentences that contain the relation symbol “=” are called equations. Example: 7 * 5 = 35 8) Variables A variable is a letter or symbol that represents a number. Example 1: 7*5=x Example 2: 7 * b = 35 Find the solution to each equation. Show your steps. (*Hint: When all else fails, try plugging in numbers to see if it makes the number sentence true – Guess & Test) a) c) (1/4)t – 4 = 6 j + 6 = 23 b) Operation ____ ____________ Operation ____ ____________ ____ ____________ Solution: Solution: ____________ ____________ (e – 7) + 6 = 24 d) 3x + 4 = 25 – 4x Operation ____ ____________ ____ ____________ Operation ____ ____________ ____ ____________ ____ ____________ Solution: Solution: ____________ ____________ Evaluate each expression for x = 5. a) 9) 6 + x = ______ b) 4 x – 7 = ______ Joe is giving each student in his class 2/5 of a bag of jellybeans for his birthday. He has 20 students in his room. a) How many bags will he need to buy? ___________________ b) Write down the equation you used to solve this. ___________________ 6th Unit 6 Study Guide page 5 10) Pan Balance Problems – If several objects are placed in the pans and if the pans are in balance, then you can find the weight of one kind of object in terms of the other kind of object. Example: 1) 2) 3) Here you see 2 blocks, 6 stars on one side and 1 block , 8 stars on the other side. You have at least one block on each side that balance so they cancel each other out. You have at least 6 stars on each side that balance so they cancel each other out. That leaves one block which equals 2 stars. Solve the pan balance problems. a) b) 1 circle = _____ cylinders 1 heart = _____ triangles c) 1 star = ______ lightning bolts 11) Circle each number sentence that is true. a) x–5>x b) 42 > c) (42 * 65) + 12 = 12 + (65 * 42) d) r*0>0 e) 786.6 = 786.60 f) sixty-two is half of thirty-one g) 12 = 14 h) x/1=x i) 376 < 234 6th Unit 6 Study Guide page 6 12) Inequalities – are any number sentences that contain one of the relation symbols below. = < > > < An inequality that contains one or more variables is an open sentence. Example: The inequality x + 3 > 8 is an open sentence. To graph this we would draw a closed circle beginning at 5 (since 5 + 3 = 8) Then draw an arrow through all numbers greater than 5 on the number line. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Graph the solution for each inequality. a) y = 8 -10 -9 b) -8 -10 -9 13) -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 y > 8 -10 -9 c) -7 -6 -8 -7 -6 y < 8 -8 -7 -6 Write >, <, or = to complete each number sentence. a) -34 + (-18) _____ 12 - 64 b) 5 * (-80) _____ - 800 / 2 c) 5/6 – 2/3 _____ 3/4 – 3/8 d) 1/2 + 5/12 _____ 1