Decimal Operations No Calculators!!! Directions: Perform the indicated operation. Show all work. Use extra paper if necessary.

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2 Decimal Operations No Calculators!!! Directions: Perform the indicated operation. Show all work. Use etra paper if necessar. Find = = = = Find. 0 Estimate 0 = Place the decimal point. 0 ).0 0 Place a zero in the quotient. 0 0 Anne a zero and continue dividing = 0.0 Compared to the estimate, the quotient is reasonable ) )

3 Fraction Operations No Calculators!!! Write 6 as an improper fraction. R ) 6 number of fifths left Find + 6 Write the quotient as a whole number Find the least common denominator of and : 6 multiples of :, 8,, 6, multiples of 6: 6,, 8,, The LCD is. Write the Rename using Add the problem. the LCD,. fractions

4 Multipl and Divide Mied Numbers - No Calculators!!! Rewrite as improper fractions.. 6 =. = Find. Write in simplest form. Estimate. = 8 Rename as an improper fraction,. = 9 9 Rename as an improper fraction, Divide and 8 b their GCF,. 9 = Multipl. = 8 or Simplif. Check for Reasonableness The answer matches the estimate. Simplif =. =. = 6. --

5 Adding and Subtracting Mied Numbers No Calculators!!! Simplif. Write our answer in lowest terms.. + = =. 9 = 8 9. = --

6 Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifing epressions that contain multiple operations. One eas wa to remember the order of operations process is to remember the acronm PEMDAS or the old saing Please Ecuse M Dear Aunt Sall. P E Parentheses first Eponents (e: Powers and Square Roots, etc.) MD Multiplication and Division (left-to-right) Divide & Multipl rank equall. AS Addition and Subtraction (left-to-right) Add & Subtract rank equall. Evaluate + ( 8 ( )). [ 8 ( )] = [8 8 ( )] Evaluate power. = [8 ( )] Divide 8 b. = 0 Perform the operation in the parentheses. = 0 Add and. = Subtract 0 from. Evaluate each epression. Remember our order of operations process (PEMDAS) =. 6 8 =. =. 9 (0 ) =. 0 = 6. [0 ( 8)] =. 0 ( 8) = 8. 8 ( ) = --

7 Multipling and Dividing Signed Numbers If the signs are the same, the answer is positive. Like Signs If the signs are different, the answer is negative. Different Signs Simplif. Do NOT use a calculator for this section.. ( )( ) =. =. 6 = 8. ()( ) =. ()() = 9. 8 =. = 0. ( )() =. ( )( ) =. 0 = =. ()( ) = -6-

8 Operations with Signed Numbers Adding Numbers with the Same Sign: Add the numbers and keep the sign. Add and. Keep the answer positive. So, ( ) or. Add and. Keep the answer negative. So, + ( ) = 6 Adding Numbers with Different Signs: Subtract the numbers (without their signs) and take the sign of the larger number. Subtract and. There are more positives () than negatives (); Your answer is positive. So, ( ) Subtract and. There are more negatives () than positives (); Your answer is negative. So, Subtracting Integers: To subtract an integer, add its opposite (Change the problem to addition.). When subtracting, change the sign of the second number. 0 ( ) = (+) = 6 ( ) = + (+) = Simplif. Do not use a calculator for this section ( ) =. 8 + =. 6 =. 0 + ( 9) =. 0 = 6. + =. 0 ( 6) = 8. 0 = 9. 6 ( ) = 0. 9 =. 8 =. ( ) = --

9 Rounding Numbers Step : Underline the place value (the digit) to be rounded. Step : Look at the number to the right of the place value (digit) to be rounded. Step : If the number is less than ( or less), keep the digit the same and drop all digits after the underlined digit. If the number is or more (more than ), add to the underlined digit and drop all digits after the underlined digit. Eample: Round the following numbers to the tenths place. Tenths is less than, so keep the digit the same...6 Answer:. 6 is or more, so add to the digit Answer: 6. is or more, so add to the digit Answer: 8.0 Round the following numbers to the tenths place

10 Evaluating Epressions Eamples a. d. b. c. e. Evaluate each epression given that =, =, z = ( + z) z 6. + z 8. + z -9-

11 Combining Like Terms What is a term? What are like terms? The parts of an algebraic epression separated b an addition or subtraction sign are called terms. Eample: The epression 6 + has terms:, 6,, a. b. Simplif 6. 6 ( 6) c. d. Simplif = + + Commutative Prop. ( ) + ( ) Distributive Prop. Substitution e. Simplif b c b 6c Identit Prop. of = b b + c 6c Commutative Propert ( )b ( 6)c Distributive Propert b ( c) Substitution b c Definition of Subtraction, Identit Propert of Practice: Simplif each epression.. 6n n. b b. z z.. n n f f t 9 t 8. k 8k 9. r r g 9h g h. m n m n. a b a 9b -0-

12 Graphing The Coordinate Plane Divided into Sections (Quadrants) ordered pair used to name points in the coordinate plane -coordinate first number (left/right) negative -coordinate: left of origin positive -coordinate: right of origin -coordinate second number (up/down) negative -coordinate: down from origin positive -coordinate: up from origin (, ) (, +) (, ) (, ) (, ) (+, +) (, ) (+, ) ordered pair: (-coordinate, -coordinate) Eamples: Plot each point on the graph provided. (6, ) go 6 units to right of origin, then go units up (6, ) (, ) go units to left of origin, then go unit up (, 0) (, ) go units to right of origin, then go units down (, 0) go units to the right of origin, do not go up/down (, ) (0, 6) do not go left/right, go 6 units down from origin (0, 6) Plot each point on the graph below. Remember, ordered pairs are labeled (, ). Label each point on the graph with the letter given. Eample: F( 6, ) +. A(, ). B(, 0) ( 6, ). C(, ) +. D(, ). E(0, ) --

13 Naming Ordered Pairs Determine the coordinates of each point below. Eample: (, ) 6. (, ). (, ) O O O (, ) 9. (, ) 0. (, ) O O O (, ). (, ). (, ) O O O

14 Graphing Linear Equations b Using a Table of Values How to make a table of values: Use each -value as the input. Substitute the -value for and evaluate (where it sas Work ) to find the -value. Your ordered pair to plot is (-value, -value). Graph = Work: (, ) ( ) = 6 = (, ) ( ) = = (, ) ( ) = = (, ) 0 (0) = 0 = (0, ). = + Work: (, ) 0. = Work: (, ) 0 6. = Work: (, ) --

15 Algebraic Translations Translations from English to Mathematics Ke words for Translations Order changes with: less than, fewer than, smaller than, and shorter than Add Subtract Multipl Divide Inequalities Variable = plus minus times divided b is less than a number equals sum increased b more than greater than total in all and together difference decreased b less than fewer than smaller than diminished b reduced b product of per for ever for each twice double quotient split equall each part one-half one-third is greater than is less than or equal to is greater than or equal to some number a quantit same as is total was result outcome answer longer than shorter than triple differ. Translate to an algebraic epression: less than times some number Same as: times some number decreased b Answer: n n. Translate into a mathematical statement: increased b one-third of a number is n Answer: n. Translate into a mathematical statement: the quotient of a number and, less 8, is. Same as: a number divided b, less 8, is. n ( ) 8 Answer: n 8 =. Translate into a mathematical statement: four plus three times a number is less than or equal to 8. + n 8 Answer: + n 8 --

16 Algebraic Translations Practice Practice: Translate each phrase into a mathematical statement.. Seven plus five times a number is greater than or equal to 9.. Eight times a number increased b 6 is 6.. One-half of a number is equal to.. 6 less than 8 times some number. a number divided b 9 6. p decreased b. Twice a number decreased b is equal to less than times some number is The sum of a number and eight is less than. 0. Eleven increased b a number is. Matching Place the letter of the algebraic epression that best matches the phrase.. two more than a number a.. two less than a number b.. half of a number c.. twice a number d.. two decreased b a number e. Be Careful! In division and subtraction, the order makes a difference. Translate to an algebraic epression, then reread to check. --

17 Solving Equations I The ke in equation solving: isolating the variable (to get the letter b itself). In one-step equations, we undo the operation (do the inverse operation) in the equation.. Eamples: Addition is the inverse of Subtraction. Multiplication is the inverse of Division.. 6 Check: 6. t 6 Check: t t. 0 Check: 0. Check: () ( ) Check: ( ) 6. Check: ( ) Eercises: Solve each of the following equations. No Calculators!!! SHOW ALL WORK. Use a separate sheet of paper if necessar and staple to this page. Golden Rule of Equation Solving: If we do something to one side of the equation, we must do the same thing to the other side of the equation.. 8. t ( 9). t. r. 6. h 8. p 6 8. k p 8-6-

18 Solving Equations II The ke in equation solving: isolating the variable (to get the letter b itself). In two-step equations, we undo operations (do the inverse operations) in the equation. In most cases: Undo the Addition/Subtraction first. Then, undo the Multiplication/Division. Addition is the inverse of Subtraction. Multiplication is the inverse of Division. Golden Rule of Equation Solving: If we do something to one side of the equation, we must do the same thing to the other side of the equation. Eamples:. 6 Check: 6. ( ) 8 Check: 6 6 ( ) ( ) ( ) ( ) Check: 9 Eercises: Solve each of the following equations. ( 6) + 9 No Calculators!!! + 9 SHOW ALL WORK. Use a separate sheet of paper if necessar and staple to this page t 6. m 6. r. ( ) 6. g ( ) 6. ( ) 8 --

19 Inequalities inequalit statement containing one of the following smbols:,,, open circle, open circle, closed circle, closed circle, shade to the left shade to the right shade to the left shade to the right no line underneath smbol (no sign) open circle (unshaded circle) line underneath smbol (has sign) closed circle (shaded circle) Eamples: Practice: Write an inequalit to represent the solution set that is shown in the graph

20 Solving Inequalities inequalit statement containing one of the following smbols:,,, solving inequalities goal: to isolate the variable Steps to solving inequalities the same as solving equations with one eception: If both sides of an inequalit are multiplied/divided b a negative number, the inequalit sign is reversed (changes direction). The solution of an inequalit can be represented b a graph on the number line. 6 Multipl both sides of inequalit b a negative #. The inequalit smbol reverses E: Integers between and All numbers between and The positive numbers ( 0) Solve: 8 8 Divide both sides b a negative number. The inequalit smbol reverses (changes direction) from dividing both sides b. Graph: Eercises: Solve the following inequalities. Then, graph the solution set. No Calculators!!!. 9. t.. -9-