Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS


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1 ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right. 4) Do the addition and/or subtraction in order from left to right. Example: (6 + 3) work inside grouping smbols = evaluate the powers = multiplication/division = = addition/subtraction = 18 = Use order of operations to simplif. Show all steps in the space provided below each problem ) (4 1) ) 5 (6 ) ) 4) ) INTEGER OPERATIONS Addition: Subtraction: Multiplication/Division: If the signs are the same, Subtraction is the same as If ou multipl or divide add the absolute values and use adding the opposite. Therefore, numbers with the same sign, the given sign. change the sign of the value the answer is positive. If the signs are different, being subtracted, then follow If ou multipl or divide subtract the absolute values and the rules for addition. numbers with different signs choose the sign of the larger absolute the answer is negative. value. Examples: 1) = 7 4) ) 63 = 18 ) = (15) 6) = 5 3) = 3 = 9 7) 19 = 9 Use the rules of integer operations to add, subtract, multipl and divide. 1) 1 5 ) 9 3) ) ) ) 4 (10) 7) 918 S.Fischbein 1
2 PROPERTIES OF REAL NUMBERS Commutative: In certain situations, order doesn t matter. Commutative propert of addition a + b = b + a Commutative propert of multiplication a b = b a 1) How does commutative work with subtraction? ) Does commutative work for division? Wh or wh not? Associative: In certain situations, grouping can be changed. Associative propert of addition (a + b) + c = a + (b + c) Associative propert of multiplication (a b) c = a (b c) Distributive: A value is multiplied times each member of a sum or difference. a(b + c) = ab + ac or a(b c) = ab ac Identit: Appling an identit, doesn t change the value of the number. Additive Identit a + 0 = a (Zero is the additive identit.) Multiplicative Identit a 1 = a (One is the identit for multiplication.) Inverse: An inverse changes a number into an identit. Additive Inverse a + (a) = 0 (The additive inverse of a number is its opposite.) Multiplicative Inverse 1 a 1 (The multiplicative inverse of a number is its reciprocal.) a Match the sample to the propert it illustrates. Column A Column B Answer Column 1) 6 1 = 6 a) Commutative Propert of Addition 1) ) 5 = 5 b) Associative Propert of Multiplication ) 3) m(3 + n) = 3m + mn c) Inverse for Addition 3) 4) 9 + ( ) = (9 + 41) + 37 d) Identit for Multiplication 4) 5) 4 + (4) = 0 e) Distributive Propert 5) 6) 6(3t) + (63)t f) Identit for Addition 6) 7) a + 0 = 0 + a g) Associative Propert of Addition 7) p q q p 8) = 1 h) Commutative Propert of Multiplication 8) j) Inverse for Multiplication S.Fischbein
3 SIMPLIFYING ALGEBRAIC EXPRESSIONS To simplif algebraic expressions, combine like terms, use order of operations, use rules for integer operations and real number properties to simplif the result. Examples: 4x 7x = 4x + (7x) = 3x 3x 4 x 3 4 x 3x x 1x 3x 1x 1x x x 6 x 3 x 3 x 3 a a 3 a a a(3 a) a a 3 a a a 3a a 5a a Simplif the following algebraic expressions. 1) ax +4ax ) (5 x) 4x 3) 5a 3(7a 6) 4) 6m n 4m 1( 5n m) 5) 8 pq 6q EVALUATING ALGEBRAIC EXPRESSIONS Evaluate means replace the variables with given values and simplif completel. Example: Find the value of Find the value of a b 3 when a = 5 and b = x  (z) when x = 1, =  and z = 3 = (1)() ()(3) 10 7 = = 14 Evaluate the following expressions when a = , b = 5, and c = 1. Show ever step of work. 1) 10abc ) b  a + 3c 3) 6b(a + c) 4) 3a b c 5) 4b ac S.Fischbein 3
4 SOLVING SIMPLE EQUATIONS Solving equations means find the value that makes the sentence true. Make sure to keep the equation balanced b doing exactl the same thing to both sides of the equation. Ask what do ou want the answer to look like? Remove the numbers that are in the wa of the answer one step at a time. Use the opposite operation to remove numbers. Make sure everthing is simplified. Complete addition or subtraction steps before the multiplication or division steps. Check to make sure our answer makes the original sentence true. Examples: 1) 5 + x =  You want x = a number for the answer so ou must remove the x = Subtract 5 from both sides. x = 7 Simplif Check does equal ? ) 4a 7 = 1 You want a = a number for the answer so ou must remove the 7 (1 st ) and the 4. 4a 7 = Add 7 to both sides. 4a = 8 4a a = Divide both sides b 4 Simplif Check does 4 7 equal 1? 3) You want = a number for the answer so ou must remove the 7 and then the Subtract 7 from both sides. 4 5 Simplif Multipl both sides b 5. S.Fischbein 4
5 0 Simplif Check is 0 divided b 5 plus 7 equal to 3? Solve the following equations. Show all steps in the space provided and check our work. c 1) x 5 = 1 ) 8 = 56 3) 4) 3 = 4 + x ) + 3x = 4 6) 8a = 9 7) 3 c = 5 8) 4x 5 3 SOLVING EQUATIONS USING THE DISTRIBUTIVE PROPERTY Solve the equation and check our answer. Example: (x + 3) = 10 Check x +6 = 10 Distribute times x and times 3 (x + 3) = 10 original problem x = 10 6 Subtract 6 from both sides ( + 3) = 10 replace x with the answer x = 4 Simplif (5) = 10 simplif x 4 Divide both sides b 10 = 10 x = Simplif Solve the following equations and check our answers. Show all steps of work. 1) 4( + 3) = 4 ) (x 5) = 1 3) 5 = 5(x + 1) 4) 3 1 S.Fischbein 5
6 TRANSLATING WORDS AND SOLVING WORD PROBLEMS Operation Addition Subtraction Multiplication Division Equals Common Words more than, sum, increased b, total less than, difference, decreased b, fewer than product, times quotient, divided b is, is equal to, equals Example: Translate the words using math smbols. Write an equation and solve. Make sure ou answer the question asked and that our answer is reasonable. 1) Seven more than twice a number is 9. Find the number. 7 more than means +7 twice a number means multipl a number b, or n is 9 means equals n = 9 (the equation!) n = 9 7 begin solving b subtracting 7 from both sides n = simplif, next ou will divide both sides b n = 1 Does this answer make sense? Seven more than twice one is nine? The answer is 1. ) Five less than twice a number is 7. Find the number. 5 less than means 5 is subtracted from something twice a number means multipl a number b, or n is 7 means equals 7 n 5 = 7 (the equation!) n = begin solving b adding 5 to both sides n = 1 simplif, next ou will divide both sides b 6 n = 6 Does the answer make sense? Five less than twice 6 is seven? The answer is 6. Translate into smbols, solve the equation, and state the correct answer. 1) Sixteen is the sum of three times ) The product of 6 and 3) The difference between ten a number and one. Find the number. half a number is 10. and four times a number is Find the number. three. Find the number. (The answer could be a fraction.) S.Fischbein 6
7 PERCENT OF CHANGE The amount of percentage a number increases or decreases from an original number is called the percent of change. The percent of change increases if the original number is smaller than the new number. The percent of change decreases if the original number is larger than the new number. To find the percent of change, use the ratio: amount of increase or decrease original number Ex: 1) Find the percent of change if the rent on our apartment increases from $500 to $550 per month. % increase = amount of increase original number = % Ex: ) Find the percent of change if a shirt originall costs $65 and is on sale for $40. % decrease = amount of decrease original number = or38.46% Find the percent of change requested in each problem. Show the ratio ou create and work to finish the problem. 1) new number is 15 ) new number is 150 3) original number is 80 original number is 75 original number is 0 new number is 45 RATIOS AND PROPORTIONS A ratio is a comparison of two numbers written as a fraction. to 3 means 3 9 out of 18 means 18 9 or 1 6 : **Fractions should alwas be reduced to simplest form. S.Fischbein 7
8 **If a problem uses units for the same tpe of measure, convert one to the other e.g., change weeks into das, pounds into ounces, etc. to match the wa it is written elsewhere in the same problem. A proportion is an equation with two equal ratios. To solve a proportion, cross multipl and simplif. Examples: 1) 8 ) 4 x x 4 Solving problems with proportions: 3.0 Example: It takes 40 a 4x 4.5 lb. chicken 1.5 hours to roast. 1.5 How long will it take a 3 lb. chicken to roast? x 10 1) Set up a proportion lbs. of chicken 4.5 hours x ) Cross multipl 4.5x x 4.5 3) Solve for x. x 1 4) Answer the question It will take one hour to roast a 3 lb. chicken. Complete the following ratio/proportion problems. Show all steps. Write each ratio in simplest form: 1) 13 out of 5 ) 9 : 7 3) 150 to 5 4) das to 3 weeks Solve the following proportions. 5) 6) 7) x a 5 0 8) If 5 apples cost $1.85, how much will apples cost? S.Fischbein 8
9 9) If a car moving at constant speed travels 55 miles in hours, how man miles will it travel in 7 hours? GRAPHING ORDERED PAIRS Ordered pairs, written (x, ) where x represents the horizontal move and represents the vertical move, represent points on the Cartesian coordinate plane. (5, 1) means move from the origin: (, 0) means move from the origin: horizonall verticall horizontall do not move to the right 5 down 1 to the left axis (, 0) xaxis (5, 1) Find the points b writing the correct letter associated with the points. A axis D (1, 3) (0, 4) (3, ) G (, 5) C xaxis (5, 0) B F (3, 1) S.Fischbein 9 E
10 (1, ) GRAPHING LINES BY PLOTTING POINTS You will need to figure out the coordinates of at least 3 points on the given line. Plot those points on an xaxis, connect them, and extend the line in both directions. Ever line except horizontal or vertical lines, eventuall will run through ever value of x and ever value of. So ou should choose an x value and figure out the value that goes with it to complete an ordered pair. Just plug the x of our choice into the equation and solve the equation for. Example: Graph x + = 6 Choose 1 for x. Choose 3 for x. Choose 4 for x. (1) + = 6 = 4 (1, 4) is a point on the line (3) + = 6 = 0 (3, 0) is a point on the line (4) + = 6 =  (4, ) is a point on a line If ou prefer, ou could choose values for. Choose for. x + = solve the equation b subtracting from each side x = 4 x = (, ) is a point on the line. axis It is common to displa our ordered pairs on a Tchart: x xaxis Select at least 3 ordered pairs and graph the following lines. Show work below each axis. 1 1) x + = 5 ) = x 4 3) x axis axis axis S.Fischbein 10
11 axis xaxis Finding slope from a line: SLOPE P x, The slope of a line is the rise of the line the run of the line You can find the rise and the run b counting spaces. = the change in values the change in x values P 1 x 1, 1 run x x 1 rise 1 x The smbol for slope is m which comes from the French verb monter meaning to climb. m rise change in = slope = = = run change in x 1 x x 1 To find the slope of a line without a grid to count spaces: 1) choose an two points on the line ) call one point P 1 and the other point P 3) write down the coordinates of the points 4) in the formula, plug in the numbers for x 1, x, 1, 5) simplif to find the slope m x x P 1 x 1, 1 (, 3) (, 0) P x, x Find the slopes of the following lines. ) axis 4) P x, (4, 8) 1) P 1 x 1, 1 S.Fischbein (3, ) 11 xaxis 3) x
12 axis xaxis VOCABULARY Please write a brief definition of the following math terms. Select the definition that relates to mathematics. 1. square root. sstem of equations 3. function 4. quadratic equation 5. factor (noun) 6. factor (verb) S.Fischbein 1
13 7. rational 8. irrational this line this line this line S.Fischbein 13
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