PRE-ALGEBRA
LESSON/TITLE PAGE 1-1 WORDS AND EXPRESSIONS 1-2 VARIABLES AND EXPRESSIONS 1-3 PROPERTIES 1-4 ORDERED PAIRS AND RELATIONS 1-5 WORDS, EQUATIONS, TABLES, AND GRAPHS 1-6 SCATTER PLOTS 2-1 INTEGERS AND ABSOLUTE VALUE 2-2 ADDING INTEGERS 2-3 SUBTRACTING INTEGERS 2-4 MULTIPLYING INTEGERS 2-5 DIVIDING INTEGERS 2-6 GRAPHING IN FOUR QUADRANTS 2-7 TRANSLATIONS AND REFL ECTIONS ON THE COORDINATE PLANE 3-1 FRACTIONS AND DECIMALS 3-2 RATIONAL NUMBERS 3-3 MULTIPLYING RATIONAL NUMBERS 3-4 DIVIDING RATIONAL NUMBERS 3-5 ADDING AND SUBTRACTING LIKE FRACTIONS 3-6 ADDING AND SUBTRACTING UNLIKE FRACTIONS 4-1 THE DISTRIBUTIVE PROPERTY 4-2 SIMPLIFYING ALGEBRAIC EXPRESSIONS 4-3 SOLVING EQUATIONS BY ADDING OR SUBTRACTING 4-4 SOLVING EQUATIONS BY MULTIPLYING OR DIVIDING 4-5 SOLVING TWO-STEP EQUATIONS 4-6 WRITING EQUATIONS 5-1 PERIMETER AND AREA 5-2 SOLVING EQUATIONS WITH VARIABLES ON EACH SIDE 5-3 INEQUALITIES 5-4 SOLVING INEQUALITIES 5-5 SOLVING MULTI-STEP EQUATIONS AND INEQUALITIES 6-1 RATIOS 6-2 UNIT RATES 6-3 CONVERTING RATES AND MEASUREMENTS 6-4 PROPORTIONAL AND NONPROPORTIONAL RELATIONSHIPS 6-5 SOLVING PROPORTIONS 6-6 SCALE DRAWINGS AND MODELS LESSON/TITLE PAGE 6-7 SIMILAR FIGURES 6-8 DILATIONS 6-9 INDIRECT MEASUREMENT 7-1 FRACTIONS AND PERCENTS 7-2 FRACTIONS, DECIMALS, AND PERCENTS 7-3 USING THE PERCENT PROPORTION 7-4 FIND PERCENT OF A NUMBER MENTALLY 7-5 USING PERCENT EQUATIONS 7-6 PERCENT OF CHANGE 7-7 SIMPLE AND COMPOUND INTEREST 7-8 CIRCLE GRAPHS 8-1 FUNCTIONS 8-2 SEQUENCES AND EQUATIONS 8-3 REPRESENTING LINEAR FUNCTIONS 8-4 RATE OF CHANGE 8-5 CONSTANT RATE OF CHANGE AND DIRECT VARIATION 8-6 SLOPE 8-7 SLOPE-INTERCEPT FORM 8-8 WRITING LINEAR EQUATIONS 8-9 PREDICTION EQUATIONS 8-10 SYSTEMS OF EQUATIONS 9-1 POWERS AND EXPONENTS 9-2 PRIME FACTORIZATION 9-3 MULTIPLYING AND DIVIDING MONOMIALS 9-4 NEGATIVE EXPONENTS 9-5 SCIENTIFI C NOTATION 9-6 POWERS OF MONOMIALS 9-7 LINEAR AND NONLINEAR FUNCTIONS 9-8 QUADRATIC FUNCTIONS 9-9 CUBIC AND EXPONENTIAL FUNCTIONS 10-1 SQUARES AND SQUARE ROOTS 10-2 THE REAL NUMBER SYSTEM 10-3 TRIANGLES 10-4 THE PYTHAGOREAN THEOREM 10-5 THE DISTANCE FORMULA 10-6 SPECIAL RIGHT TRIANGLES 11-1 ANGLE AND LINE RELATIONSHIPS 11-2 CONGRUENT TRIANGLES 11-3 ROTATIONS 11-4 QUADRILATERALS 11-5 POLYGONS 11-6 AREA OF PARALLELOGRAMS, TRIANGLES, AND TRAPEZOIDS 11-7 CIRCLES AND CIRCUMFERENCE CONTENTS LESSON/TITLE PAGE 11-8 AREA OF CIRCLES 11-9 AREA OF COMPOSITE FIGURES 12-1 THREE-DIMENSIONAL FIGURES 12-2 VOLUME OF PRISMS 12-3 VOLUME OF CYLINDERS 12-4 VOLUME OF PYRAMIDS, CONES AND SPHERES 12-5 SURFACE AREA OF PRISMS 12-6 SURFACE AREA OF CYLINDERS 12-7 SURFACE AREA OF PYRAMIDS AND CONES 12-8 SIMILAR SOLIDS LESSON/TITLE PAGE 13-1 MEASURES OF CENTRAL TENDENCY 13-2 STEM-AND-LEAF PLOTS 13-3 MEASURES OF VARIATION 13-4 BOX-AND-WHISKER PLOTS 13-5 HISTOGRAMS 13-6 THEORETICAL AND EXPERIMENTAL PROBABILITY 13-7 USING SAMPLING TO PREDICT 13-8 COUNTING OUTCOMES 13-9 PERMUTATIONS AND COMBINATIONS 13-10 PROBABILITY OF COMPOUND EVENTS TRANSLATE VERBAL PHRASES INTO EXPRESSIONS A NUMERICAL EXPRESSION CONTAINS A COMBINATION OF NUMBERS AND OPERATIONS SUCH AS ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION VERBAL PHRASES CAN BE TRANSLATED INTO NUMERICAL EXPRESSIONS BY REPLACING WORDS WITH OPERATIONS AND NUMBERS EXAMPLE PLUS THE SUM OF INCREASED BY MORE THAN MINUS THE DIFFERENCE OF DECREASED BY LESS THAN TIMES THE PRODUCT OF OF DIVIDE THE QUOTIENT OF DIVIDED BY AMONG + - × ÷ PHRASE THE PRODUCT OF SEVENTEEN AND THREE EXPRESSION 17 × WRITE A NUMERICAL EXPRESSION FOR EACH VERBAL PHRASE A THE PRODUCT OF SEVENTEEN AND THREE EXERCISES WRITE A NUMERICAL EXPRESSION FOR EACH VERBAL PHRASE 1 ELEVEN LESS THAN TWENTY 2 TWENTY-FIVE INCREASED BY SIX 3 SIXTY-FOUR DIVIDED BY EIGHT 4 THE PRODUCT OF SEVEN AND TWELVE 5 THE QUOTIENT OF FORTY AND EIGHT 6 SIXTEEN MORE THAN FIFTY–FOUR 7 SIX GROUPS OF TWELVE 8 EIGHTY–ONE DECREASED BY NINE 9 THE SUM OF THIRTEEN AND EIGHTEEN 10 THREE TIMES SEVENTEEN B THE TOTAL NUMBER OF PENCILS GIVEN TO EACH STUDENT IF 18 PENCILS ARE SHARED AMONG 6 STUDENTS PHRASE 18 SHARED AMONG EXPRESSION 18 ÷ PRE ALGEBRA STUDY GUIDE AND INTERVENTION (CONTINUED) WORDS AND EXPRESSIONS EXAMPLE EVALUATE EACH EXPRESSION B 4(3 + 6) + 2 4(3 + 6) + 2 11 = 4(9) + 2 = 36 + EVALUATE (3 + 6) MULTIPLY 4 AND 9, AND 2 AND 11 ADD 36 AND 22 MULTIPLY 6 AND 5 DIVIDE 10 BY 2 SUBTRACT 5 FROM 30 A 6 5 - 10 ÷ 6 5 - 10 ÷ 2 = 30 - 10 ÷ = 30 EXERCISES EVALUATE EACH EXPRESSION + 3 2 7 + 7 3 14 - 6 + 4 + 5 10 ÷ 5 6 22 ÷ 11 + 2) - 4 8 5(6 + 1) - 3 9 2[(13 - 4) + 2(2)] 4[(10 - 6) + 6(2)] 11 (67 + 13) ? (34 - 29) 12 6(4 - 2) + 3[(2 + 7) ÷ 9] 14 (8 7) ÷ 14 15 4 (18) ? 2 (9) 16 (9 8) - (100 ÷ 5) ORDER OF OPERATIONS EVALUATE, OR FIND THE NUMERICAL VALUE OF, EXPRESSIONS WITH MORE THAN ONE OPERATION BY FOLLOWING THE ORDER OF OPERATIONS STEP 1 EVALUATE THE EXPRESSIONS INSIDE GROUPING SYMBOLS STEP 2 MULTIPLY AND/OR DIVIDE FROM LEFT TO RIGHT STEP 3 ADD AND/OR SUBTRACT FROM LEFT TO RIGHT 3[(7 + 5) ÷ 4 - 1] 3[(7 + 5) ÷ 4 - 1] = 3[12 ÷ 4 - 1] = 3(3 - 1) = 3(2) = EVALUATE (7 + 5) FIRST DIVIDE 12 BY 4 SUBTRACT 1 FROM 3 MULTIPLY 3 AND 2 PRE ALGEBRA TRANSLATE VERBAL PHRASES AN ALGEBRAIC EXPRESSION IS A COMBINATION OF VARIABLES, NUMBERS, AND AT LEAST ONE OPERATION A VARIABLE IS A LETTER OR SYMBOL USED TO REPRESENT AN UNKNOWN VALUE TO TRANSLATE VERBAL PHRASES WITH AN UNKNOWN QUANTITY INTO ALGEBRAIC EXPRESSIONS, FIRST DEFINE THE VARIABLE EXAMPLE TRANSLATE EACH PHRASE INTO AN ALGEBRAIC EXPRESSION A FIVE INCHES LONGER THAN THE LENGTH OF A BOOK WORDS FIVE INCHES LONGER THAN THE LENGTH OF A BOOK VARIABLE LET B REPRESENT THE LENGTH OF THE BOOK EXPRESSION B + B TWO LESS THAN THE PRODUCT OF A NUMBER AND EIGHT WORDS TWO LESS THAN THE PRODUCT OF A NUMBER AND EIGHT VARIABLE LET N REPRESENT THE UNKNOWN NUMBER EXPRESSION 8N 7D MEANS 7 × D MN MEANS M × N B ? 5 MEANS B ÷ 5 ALGEBRAIC EXPRESSIONS MN THE LETTER X IS MOST OFTEN USED AS A VARIABLE X + B ? A 8S - 2R 8S - 2R = 8(2) - 2(6) = 16 - 12 OR 15 10T ? 4S 16 (T + R) - (R + S) B 3(R + S) 3(R + S) = 3(2 + 6) = 3 8 OR C 5 RS ? 5RS ?= 5RS ÷ = 5(6) (2) ÷ = 60 ÷ 4 OR REWRITE AS A DIVISION EXPRESSION REPLACE R WITH 6 AND S WITH 2 MULTIPLY THEN DIVIDE PRE ALGEBRA EXERCISES 1 STATE WHETHER THE FOLLOWING CONJECTURE IS TRUE OR FALSE: THE MULTIPLICATIVE IDENTITY APPLIES TO DIVISION ALSO IF FALSE, GIVE A COUNTEREXAMPLE NAME THE PROPERTY SHOWN BY EACH STATEMENT 2 75 + 25 = 25 + 3 2 (3 4) = (2 3) 4 14 1 = 5 P 0 = PROPERTY EXPLANATIONS EXAMPLE COMMUTATIVE PROPERTY OF ADDITION A + B = B + A 6 + 3 = 3 + 9 = COMMUTATIVE PROPERTY OF MULTIPLICATION A B = B A 4 5 = 5 20 = ASSOCIATIVE PROPERTY OF ADDITION (A + B) + C = A + (B + C) (3 + 4) + 7 = 3 + (4 + 7) 14 = ASSOCIATIVE PROPERTY OF MULTIPLICATION (A B) C = A (B C) (2 5) 8 = 2 (5 8) 80 = ADDITIVE IDENTITY A + 0 = 0 + A = A 10 + 0 = 0 + 10 = MULTIPLICATIVE IDENTITY A 1 = 1 A = A 5 1 = 1 5 = MULTIPLICATIVE PROPERTY OF ZERO A 0 = 0 A = 15 0 = 0 15 = PROPERTIES OF ADDITION AND MULTIPLICATION IN ALGEBRA, THERE ARE CERTAIN STATEMENTS CALLED PROPERTIES THAT ARE TRUE FOR ANY NUMBERS IS SUBTRACTION OF WHOLE NUMBERS ASSOCIATIVE? IF NOT, GIVE A COUNTEREXAMPLE (9 - 4) - 2 = 9 - (4 - 2) STATE THE CONJECTURE 5 - 2 = 9 SIMPLIFY 3 = SIMPLIFY THIS IS A COUNTEREXAMPLE SO, SUBTRACTION OF WHOLE NUMBERS IS NOT ASSOCIATIVE EXAMPLE NAME THE PROPERTY SHOWN BY THE STATEMENT 15 × B = B × THE ORDER OF THE NUMBERS AND VARIABLES CHANGED THIS IS THE COMMUTATIVE PROPERTY OF MULTIPLICATION EXAMPLE SIMPLIFY ALGEBRAIC EXPRESSIONS TO SIMPLIFY AN ALGEBRAIC EXPRESSION, PERFORM ALL POSSIBLE OPERATIONS PROPERTIES CAN BE USED TO HELP SIMPLIFY AN EXPRESSION THAT CONTAINS VARIABLES SIMPLIFY EACH EXPRESSION A (9 + R) + (9 + R) + 7 = (R + 9) + COMMUTATIVE PROPERTY OF ADDITION = R + (9 + 7) ASSOCIATIVE PROPERTY OF ADDITION = R + ADD 9 AND 7 B 3 (X 5) 3 (X 5) = 3 (5 X) COMMUTATIVE PROPERTY OF MULTIPLICATION = (3 5) X ASSOCIATIVE PROPERTY OF MULTIPLICATION = 15 X MULTIPLY 3 AND 5 EXERCISES SIMPLIFY EACH EXPRESSION 1 24 + (X + 6) 2 3 (4A) + (12 + C) 4 13D + F ) + 6 11 + (M + 5) + 0) + 8 15(A 1) (6) 10 (N + 7) + X Y O A X Y O 2 3 4 5 GRAPH A(4, 3) ON THE COORDINATE PLANE EXAMPLE EXAMPLE " % # $ Y X EXAMPLE RELATIONS A RELATION IS A SET OF ORDERED PAIRS, SUCH AS {(0, 3), (1, 2), (3, 6), (7, 4)} A RELATION CAN ALSO BE SHOWN IN A TABLE OR A GRAPH THE SET OF X-COORDINATES IS THE DOMAIN OF THE RELATION, WHILE THE SET OF Y-COORDINATES IS THE RANGE OF THE RELATION EXPRESS THE RELATION {(0, 0), (2, 1), (4, 2), (3, 5)} AS A TABLE AND AS A GRAPH THEN DETERMINE THE DOMAIN AND RANGE THE DOMAIN IS {0, 2, 4, 3}, AND THE RANGE IS {0, 1, 2, 5} X Y O 2 3 4 5 6 7 X Y EXERCISES EXPRESS EACH RELATION AS A TABLE AND AS A GRAPH THEN DETERMINE THE DOMAIN AND RANGE {(4, 6), (0, 3), (1, 4)} {(2, 5), (5, 3), (2, 2)} {(1, 2), (3, 4), (5, 6)} X Y O X Y X Y O EXERCISES FOR EACH TICKET COST AND SURCHARGE GIVEN BELOW, MAKE A FUNCTION TABLE FOR EACH AND THERE IS A $6 SURCHARGE PER ORDER” TRANSLATES TO 22X + 6 USE THE RULE TO COMPLETE THE TABLE STEP 3 THE DOMAIN IS {1, 2, 3, 4} THE RANGE IS {28, 50, 72, 94} INPUT (X) RULE: OUTPUT (Y) INPUT (X) RULE: OUTPUT (Y) INPUT (X) RULE: 22X + OUTPUT (Y) 22(1) + 22(2) + 22(3) + 22(4) + FILE PROTECTION TORI’S COMPUTER BACKS UP THE FILE SHE IS WORKING ON EVERY 5 MINUTES MAKE A FUNCTION TABLE TO FIND THE TIME FOR 3, 6, 9, AND BACKUPS THEN GRAPH THE ORDERED PAIRS LET M REPRESENT THE NUMBER OF MINUTES AND B REPRESENT THE NUMBER OF BACKUPS SO, THE RULE IS M = 5B INPUT (X) 5B OUTPUT (Y) 5(3) 5(6) 5(9) 5(12) MINUTES NUMBER OF BACKUPS MULTIPLE REPRESENTATIONS FUNCTIONS CAN BE DESCRIBED AS WORDS, EQUATIONS, TABLES AND GRAPHS WORDS THE DISTANCE BIKED IS EQUAL TO 12 MILES PER HOUR TIMES THE NUMBER OF HOURS EQUATION D = 12T GRAPH TABLE TIME (H) DISTANCE (MI) DISTANCE (MI) TIME (H) EXAMPLE EXERCISE VIKTOR’S HEART BEATS 72 TIMES A MINUTE ALGEBRAIC WRITE AN EQUATION TO FIND THE NUMBER OF TIMES VIKTOR’S HEART BEATS FOR ANY NUMBER OF MINUTES TABULAR MAKE A FUNCTION TABLE TO FIND THE NUMBER OF TIMES VIKTOR’S HEART BEATS IN 5, 10, 15, AND 20 MINUTES GRAPHICAL GRAPH THE ORDERED PAIRS FOR THE FUNCTION INPUT (X) OUTPUT (Y) HEARTBEATS 10 15 20 EXAMPLE EXERCISE FOOD THE TABLE BELOW SHOWS THE FAT GRAMS AND CALORIES FOR SEVERAL SNACK FOODS SCHOOL THE TABLE SHOWS MIRANDA’S MATH QUIZ SCORES FOR THE LAST FIVE WEEKS MAKE A SCATTER PLOT OF THE DATA MIRANDA'S QUIZ SCORES WEEK SCORE SINCE THE POINTS ARE SHOWING AN UPWARD TREND FROM LEFT TO RIGHT, THE DATA SUGGEST A POSITIVE RELATIONSHIP 1 MAKE A SCATTER PLOT OF THE DATA IN THE TABLE X Y SNACK FOOD DATA FAT (GRAMS) CALORIES 12 15 18 WEEK SCORE FOOD FAT GRAMS PER SERVING CALORIES PER SERVING DOUGHNUT CORN CHIPS PUDDING CAKE SNACK CRACKERS ICE CREAM (LIGHT) YOGURT CHEESE PIZZA ANALYZE SCATTER PLOTS A SCATTER PLOT MAY SHOW A PATTERN OR RELATIONSHIP OF THE DATA POSITIVE RELATIONSHIP NEGATIVE RELATIONSHIP NO RELATIONSHIP Y X Y X EXAMPLE SHOE SIZE AND HEIGHT DETERMINE WHETHER A SCATTER PLOT OF SHOE SIZE AND HEIGHT OF PEOPLE AT A GYM MIGHT SHOW A POSITIVE, NEGATIVE, OR NO RELATION- SHIP EXPLAIN YOUR ANSWER Y X SHOE SIZE AND HEIGHT HEIGHT (IN INCHES) SHOE SIZE REPLACE EACH ? WITH < OR > TO MAKE A TRUE SENTENCE A - 6 ? B 3 ? COMPARE AND ORDER INTEGERS THE SET OF INTEGERS CAN BE WRITTEN {…, -3, -2, -1, 0, 1, 2, 3, …} WHERE … MEANS CONTINUES INDEFINITELY TWO INTEGERS CAN BE COMPARED USING AN INEQUALITY, WHICH IS A MATHEMATICAL SENTENCE CONTAINING < OR > REPLACE EACH ? WITH <, >, OR = TO MAKE A TRUE SENTENCE 5 4 ? 6 8 ? 8 2 ? 9 -1 ? 11 6 ? 12 -2 ? 16 -10 ? USE THE INTEGERS GRAPHED ON THE NUMBER LINE BELOW EXAMPLE 4 A LOSS OF 8 YARDS PRE ALGEBRA ?-3? = 3, ?6? = 2 UNITS 2 UNITS EVALUATE EACH EXPRESSION THE SYMBOL FOR ABSOLUTE VALUE IS TWO VERTICAL BARS ON EITHER SIDE OF THE NUMBER |2| = AND |-2| = ABSOLUTE VALUE NUMBERS ON OPPOSITE SIDES OF ZERO AND THE SAME DISTANCE FROM ZERO HAVE THE SAME ABSOLUTE VALUE A |-4| B |-3| + |6| |-4| = ON THE NUMBER LINE, -4 IS 4 UNITS FROM 0 EVALUATE EACH EXPRESSION 6| 2 |15| 3 |-12| 4 |21| - |2| 6 |-8| + |-3| 7 |-10| - |-6| 8 |12| + |-4| ALGEBRA EVALUATE EACH EXPRESSION IF X = 8 AND Y = – 3 + |Y| 10 X - |Y| 11 2|X| + 3|Y| + |Y| 13 6|Y| 14 3X - 4|Y| |- 3| + |6| = 3 + 4 UNITS EXERCISES EXAMPLE REPLACE X WITH - 8 = 8 THE ABSOLUTE VALUE OF - 8 IS 8 = SIMPLIFY EXAMPLE A -5 + 5 + 4 = |-5| - |4| = 5 - 4 OR B 6 + (-2) 6 + (-2) = |6| - |-2| = 6 - 2 OR ADD ?-3? AND ?-4? THE SUM IS NEGATIVE FIND EACH SUM 1 6 + (-3) 2 -3 + (-5) 3 7 + (-3) 4 -4 + (-4) 5 -8 + 6 -12 + (-10) 7 6 + (-13) 8 -14 + 9 6 + (-6) 10 -15 + (-5) 11 -9 + 12 20 + (-8) 13 -19 + (-11) 14 17 + (-9) 15 -16 + (-5) 16 -12 + 17 9 + (-25) 18 -36 + 19 7 + (-18) 20 -12 + (-15) 21 10 + (-14) 22 -33 + 23 -20 + (-5) 24 -12 + (-10) 25 -15 + 26 -34 + 27 46 + (-32) SUBTRACT ?4? FROM ?-5? SIMPLIFY THE SUM IS NEGATIVE BECAUSE ?-5? > ?4? SUBTRACT ?-2? FROM ?6? SIMPLIFY THE SUM IS POSITIVE BECAUSE ?6? > ?-2? FIND THE SUM -3 + (-4) FIND EACH SUM ADDING INTEGERS WITH THE SAME SIGN ADD THEIR ABSOLUTE VALUES THE SUM IS: POSITIVE IF BOTH INTEGERS ARE POSITIVE NEGATIVE IF BOTH INTEGERS ARE NEGATIVE ADDING INTEGERS WITH DIFFERENT SIGNS SUBTRACT THEIR ABSOLUTE VALUES THE SUM IS: POSITIVE IF THE POSITIVE INTEGER’S ABSOLUTE VALUE IS GREATER NEGATIVE IF THE NEGATIVE INTEGER’S ABSOLUTE VALUE IS GREATER EXAMPLE EXERCISES PRE ALGEBRA STUDY GUIDE AND INTERVENTION (CONTINUED) ADDING INTEGERS ADD MORE THAN TWO INTEGERS TWO NUMBERS WITH THE SAME ABSOLUTE VALUE BUT DIFFERENT SIGNS ARE OPPOSITES AN INTEGER AND ITS OPPOSITE ARE ALSO CALLED ADDITIVE INVERSES THIS PROPERTY IS USEFUL WHEN ADDING 2 OR MORE INTEGERS ADDITIVE INVERSE PROPERTY WORDS THE SUM OF ANY NUMBER AND ITS ADDITIVE INVERSE IS ZERO EXAMPLE 5 + (–5) = SYMBOLS A + (–A) = FIND EACH SUM A –7 + (-16) + IDENTITY PROPERTY OF ADDITION B 12 + (– 4) + 9 + (–7) 12 + (– 4) + 9 + (–7) = 12 + 9 + (– 4) + (–7) COMMUTATIVE PROPERTY = (12 + 9) + [– 4 + (–7)] ASSOCIATIVE PROPERTY = 21 + (–11) OR SIMPLIFY EXERCISES FIND EACH SUM + 14 + (-2) 2 -8 + (-7) + 13 + 11 + (-4) 4 7 + (-5) + (-6) + 14 + (-12) 6 -9 + 17 + (-3) + (-5) + 8 54 + 39 + (-54) 42 + 20 + (-8) 10 -11 + (-6) + + (-43) + (-4) 12 -100 + 50 + (-25) + (-14) + (-5) + (-6) 14 -18 + 9 + (-7) + + 13 + (-11) + 16 -20 + 15 + (-10) + 17 -33 + (-7) + 20 + 18 16 + (-12) + 21 + (-25) EXAMPLE PRE ALGEBRA STUDY GUIDE AND INTERVENTION SUBTRACTING INTEGERS FIND EACH DIFFERENCE 1 9 2 7 3 12 4 -5 5 -8 6 -13 7 7 - (-4) 8 9 - (-9) 9 -11 - (-2) 10 -6 - (-9) 11 -6 12 -16 - (-20) 13 -14 14 8 - (-6) 15 -10 - (-6) 16 13 - (-17) 17 24 - (-16) 18 17 - (-9) 19 -24 20 18 - (-9) 21 26 22 -45 - (-26) 23 -15 - (-25) 24 29 - (-6) A 4 - (-5) 4 - (-5) = 4 + B -6 - (-2) -6 - (-2) = -6 + A 9 B -7 TO SUBTRACT 17, ADD -17 SIMPLIFY TO SUBTRACT 3, ADD -3 SIMPLIFY TO SUBTRACT -2, ADD +2 SIMPLIFY TO SUBTRACT -5, ADD +5 SIMPLIFY TO SUBTRACT AN INTEGER, ADD ITS ADDITIVE INVERSE FIND EACH DIFFERENCE FIND EACH DIFFERENCE SUBTRACTING INTEGERS EXAMPLE STUDY GUIDE AND INTERVENTION (CONTINUED) SUBTRACTING INTEGERS EVALUATE EXPRESSIONS USE THE RULE FOR SUBTRACTING INTEGERS TO EVALUATE EXPRESSIONS EVALUATE EACH EXPRESSION A X - 16 IF X = 6 X - 16 = 6 ADD 6 AND -16 B A - B - C IF A = 7, B = 2, AND C = -3 A - B - C = 7 - 2 - (-3) REPLACE A WITH 7, B WITH 2, AND C WITH -3 = 5 - (-3) USE ORDER OF OPERATIONS = 5 + TO SUBTRACT -3, ADD ITS ADDITIVE INVERSE, 3 = ADD 5 AND 3 EXERCISES ALGEBRA EVALUATE EACH EXPRESSION IF A = 11, B = -1, AND C = -8 – 2 B 3 12 – C - A 5 C 6 -19 - B - C 8 3 - A 9 B - (-1) - (-7) 11 6 - B 12 C - (-12) - B 14 A - C 15 C - B - C 17 C - A 18 B - A - B - C 20 A + B - C 21 B - C - A 22 C - A + B 23 B - (-A) - C 24 C + B - A EXAMPLE PRE ALGEBRA STUDY GUIDE AND INTERVENTION MULTIPLYING INTEGERS FIND EACH PRODUCT 1 -5(7) 2 6(-9) 3 -10 5 5(-11) 6 -15(-4) 7 -14(2) 8 6(14) 9 -18 10 -9(10) 11 12(-6) 12 -11(-11) 13 -4(-4)(5) 14 6(-7)(2) 15 -10(-4)(-6) 16 -7(-3)(2) 17 -9(4)(2) 18 6(-4)(-12) 19 11(3)(-2) 20 -5(-6)(7) 21 -3(-4)(-8) 22 22(3)(-3) 23 -8(10)(-2) 24 A 6(6) B -7(-4) 6(6) = 7(-4) = 6(-3)(-2) = [6(-3)](-2) = -18(-2) = 18(-2) = THE PRODUCT OF TWO INTEGERS WITH DIFFERENT SIGNS IS NEGATIVE THE PRODUCT OF TWO INTEGERS WITH THE SAME SIGN IS POSITIVE FIND EACH PRODUCT FIND EACH PRODUCT FIND 6(-3)(-2) MULTIPLYING INTEGERS WITH DIFFERENT SIGNS MULTIPLYING INTEGERS WITH THE SAME SIGN EXAMPLE THE PRODUCT OF 12 AND -5 IS NEGATIVE EXERCISES ALGEBRA SIMPLIFY EACH EXPRESSION -3W) 2 2E 9F 3 -8 7M S(-7) 5 10P(-5Q) 6 N 6 A(15B) 8 -9X (-4Y) 9 -C 11D ALGEBRA EVALUATE EACH EXPRESSION IF X = -4 AND Y = 8 11 3Y 12 -12X Y 14 XY 15 -XY XY 17 5XY 18 -3X(-Y) EXAMPLE A 14 ÷ 14 ÷ 2 = ? -5 = -25 ÷ (-5) = FIND EACH QUOTIENT 1 32 ÷ (-4) 2 -18 ÷ (-2) 3 -24 ÷ 4 -36 ÷ (-2) 5 50 ÷ (-5) 6 -81 ÷ (-9) 7 -72 ÷ (-2) 8 -45 ÷ 9 -60 ÷ (-12) 10 99 ÷ (-11) 11 -200 ÷ (-4) 12 38 ÷ (-2) 13 -144 ÷ ? THE DIVIDEND AND THE DIVISOR HAVE THE SAME SIGN THE QUOTIENT IS POSITIVE THE DIVIDEND AND DIVISOR HAVE THE SAME SIGN THE QUOTIENT IS POSITIVE THE SIGNS ARE DIFFERENT THE QUOTIENT IS NEGATIVE THE SIGNS ARE DIFFERENT THE QUOTIENT IS NEGATIVE FIND EACH QUOTIENT FIND EACH QUOTIENT DIVIDING INTEGERS WITH THE SAME SIGN THE QUOTIENT OF TWO INTEGERS WITH THE SAME SIGN IS POSITIVE DIVIDING INTEGERS WITH DIFFERENT SIGNS THE QUOTIENT OF TWO INTEGERS WITH DIFFERENT SIGNS IS NEGATIVE EXAMPLE STUDY GUIDE AND INTERVENTION (CONTINUED) DIVIDING INTEGERS MEAN (AVERAGE) TO FIND THE MEAN, OR AVERAGE, OF A SET OF NUMBERS, FIND THE SUM OF THE NUMBERS AND THEN DIVIDE BY THE NUMBER OF ITEMS IN THE SET USE THE RULES FOR DIVIDING INTEGERS TO FIND THE MEAN OCEANOGRAPHY THE DIVING DEPTHS IN FEET OF 7 SCUBA DIVERS STUDYING SCHOOLS OF FISH WERE -12, -9, -15, -8, -20, -17, AND -10 FIND THE MEAN DIVING DEPTH -12 + (-9) + (-15) + (-8) + (-20) + (-17) + (-10) ???? SIMPLIFY THE MEAN DIVING DEPTH IS -13 FEET, OR 13 FEET BELOW SEA LEVEL EXERCISES 1 WEATHER THE LOW TEMPERATURES IN DEGREES FAHRENHEIT FOR A WEEK WERE -3, 5, -9, 2, 6, -11, AND -4 FIND THE MEAN TEMPERATURE MONEY THE LAST 6 ENTRIES IN MS CAUDLE’S CHECKBOOK LEDGER SHOW BOTH DEPOSITS AND WITHDRAWALS MS CAUDLE WROTE DOWN $100, -$20, -$35, $250, -$150, AND -$85 WHAT IS THE MEAN DOLLAR AMOUNT FOR THESE ENTRIES? GOLF DURING 5 ROUNDS OF GOLF, JAMES HAD SCORES OF 2, -1, 0, -2, AND -4 FIND THE MEAN OF HIS GOLF SCORES TRAINING TO TRAIN HIMSELF FOR A MOTIVATION, JOSH RUNS EVERY DAY LAST WEEK HE RAN 3 MILES, 7 MILES, 3 MILES, 4 MILES, 7 MILES, 10 MILES AND 5 MILES WHAT IS THE MEAN NUMBER OF MILES HE RAN LAST WEEK? ROCK CLIMBING A ROCK CLIMBER MAKES SEVERAL CHANGES IN POSITION WHILE ATTEMPTING TO SCALE A CLIFF FACE SHE ASCENDS 15 FEET, DESCENDS 7 FEET, ASCENDS 22 FEET, DESCENDS 13 FEET, AND THEN ASCENDS ANOTHER 28 FEET WHAT IS HER MEAN CHANGE IN POSITION? EXAMPLE PRE ALGEBRA THE COORDINATES ARE (NEGATIVE, POSITIVE) THE COORDINATES ARE (NEGATIVE, NEGATIVE) THE COORDINATES ARE (POSITIVE, POSITIVE) THE COORDINATES ARE (POSITIVE, NEGATIVE) Y O 2 3 4 X (+, +) (-, +) (-, -) (+, -) M N X Y O Y O 3 4 5 6 7 STUDY GUIDE AND INTERVENTION GRAPHING IN FOUR QUADRANTS PRE ALGEBRA STUDY GUIDE AND INTERVENTION (CONTINUED) GRAPHING IN FOUR QUADRANTS EXAMPLE Y X GRAPH ALGEBRAIC RELATIONSHIPS A COORDINATE GRAPH CAN BE USED TO SHOW RELATIONSHIPS BETWEEN TWO NUMBERS MONEY THE DIFFERENCE BETWEEN ZORA’S AND CHARLIE’S BANK ACCOUNTS IS $1 IF X REPRESENTS ZORA’S BANK ACCOUNT AND Y REPRESENTS CHARLIE’S BANK ACCOUNT, MAKE A FUNCTION TABLE OF POSSIBLE VALUES FOR X AND Y GRAPH THE ORDERED PAIRS AND DESCRIBE THE GRAPH STEP MAKE A TABLE CHOOSE VALUES FOR X AND Y THAT HAVE A DIFFERENCE OF 1 STEP GRAPH THE ORDERED PAIRS THE POINTS ARE ALONG A DIAGONAL LINE THAT CROSSES THE X-AXIS AT X = 1 EXERCISES TEMPERATURE THE SUM OF TWO TEMPERATURES IS 3°F IF X REPRESENTS THE FIRST TEMPERATURE AND Y REPRESENTS THE SECOND TEMPERATURE, MAKE A FUNCTION TABLE OF POSSIBLE VALUES FOR X AND Y GRAPH THE ORDERED PAIRS AND DESCRIBE THE GRAPH X + Y = X Y (X, Y) X - Y = X Y (X, Y) (2, 1) (1, 0) (-2, -3) PRE ALGEBRA STUDY GUIDE AND INTERVENTION TRANSLATIONS AND REFLECTIONS ON THE COORDINATE PLANE Y X Y X 0 / TRANSFORMATIONS A TRANSFORMATION IS AN OPERATION THAT MAPS AN ORIGINAL GEOMETRIC FIGURE ONTO A NEW FIGURE CALLED THE IMAGE A TRANSLATION AND A REFLECTION ARE TWO TYPES OF TRANSFORMATIONS ON THE COORDINATE PLANE AN ORDERED PAIR (A, B) CAN BE USED TO DESCRIBE A TRANSLATION, WHERE EVERY POINT P(X, Y) IS MOVED TO AN IMAGE P'(X + A, Y + B) RECTANGLE MNOP IS SHOWN AT THE RIGHT IF IT IS TRANSLATED 4 UNITS TO THE LEFT AND 5 UNITS UP, FIND THE COORDINATES OF THE VERTICES OF THE IMAGE THIS TRANSLATION CAN BE WRITTEN AS (-4, 5) TO FIND THE COORDINATES OF THE TRANSLATED IMAGE, ADD -4 TO EACH X-COORDINATE AND ADD 5 TO EACH Y-COORDINATE EXERCISES 1 TRIANGLE RST IS SHOWN ON THE COORDINATE PLANE FIND THE COORDINATES OF THE VERTICES OF THE IMAGE IF TRIANGLE RST IS TRANSLATED 6 UNITS TO THE LEFT AND 3 UNITS DOWN VERTEX TRANSLATION IMAGE M(1, -1) + (-4, 5) ? M'(-3, 4) N(4, -1) + (-4, 5) ? N'(0, 4) O(4, -3) + (-4, 5) ? O'(0, 2) P(1, -3) + (-4, 5) ? P'(-3, 2) TRANSLATION CALLED A “SLIDE” IMAGE IS THE SAME SHAPE AND THE SAME SIZE AS ORIGINAL FIGURE ORIENTATION IS THE SAME AS THE ORIGINAL FIGURE REFLECTION CALLED A “FLIP” FIGURES ARE MIRROR IMAGES OF EACH OTHER IMAGE IS THE SAME SHAPE AND SAME SIZE AS ORIGINAL FIGURE ORIENTATION IS DIFFERENT FROM THE ORIGINAL FIGURE EXAMPLE PRE ALGEBRA GRAPH TRANSFORMATIONS WHEN REFLECTING A FIGURE, EVERY POINT OF THE ORIGINAL FIGURE HAS A CORRESPONDING POINT ON THE OTHER SIDE OF THE LINE OF SYMMETRY CORRESPONDING POINTS ARE THE SAME DISTANCE FROM THE LINE OF SYMMETRY REFLECTION OVER THE X-AXIS REFLECTION OVER THE Y-AXIS THE VERTICES OF FIGURE WXYZ ARE W(1, –1), X(3, –2), Y(4, –4), AND Z(2, –3) GRAPH THE FIGURE AND ITS IMAGE AFTER A REFLECTION OVER THE X-AXIS TO FIND THE COORDINATES OF THE VERTICES OF THE IMAGE AFTER A REFLECTION OVER THE -AXIS, USE THE SAME X-COORDINATE REPLACE -COORDINATE WITH ITS OPPOSITE EXAMPLE (1, –1) W '(1, 1) (3, –2) X '(3, 2) (4, –4) Y '(4, 4) (2, –3) Z '(2, 3) OPPOSITES SAME EXERCISES THE VERTICES OF FIGURE JKLM ARE J (–4, –2), (–2, –2), L (–1, –4), AND M (–5, –4) GRAPH THE FIGURE AND ITS IMAGE AFTER A REFLECTION OVER THE Y-AXIS Y X THE X-COORDINATES ARE THE SAME, BUT THE Y-COORDINATES ARE OPPOSITES (?4, 4) (?1, 4) (?4, 2) (?1, 2) (?4, ?2)(?1, ?2) (?4, ?4)(?1, ?4) Y X 2 3 4 STUDY GUIDE AND INTERVENTION (CONTINUED) TRANSLATIONS AND REFLECTIONS ON THE COORDINATE PLANE STUDY GUIDE AND INTERVENTION FRACTIONS AND DECIMALS WRITE FRACTIONS AS DECIMALS SOME FRACTIONS, SUCH AS ? 4 AND ? 5 , CAN EASILY BE WRITTEN AS DECIMALS BY MAKING EQUIVALENT FRACTIONS WITH DENOMINATORS OF 10, 100, OR 1,000 ALL FRACTIONS CAN BE WRITTEN AS DECIMALS BY DIVIDING THE NUMERATOR BY THE DENOMINATOR IF THE DIVISION ENDS OR TERMINATES WITH A REMAINDER OF 0, IT IS A TERMINATING DECIMAL IF THE DECIMAL NUMBER REPEATS WITHOUT END IT IS A REPEATING DECIMAL EXERCISES WRITE EACH FRACTION AS A DECIMAL USE A BAR TO SHOW A REPEATING DECIMAL 1 WRITE ? 8 AS A DECIMAL 0 875 IS A TERMINATING DECIMAL WRITE ? 9 AS A DECIMAL 0 444 IS A REPEATING DECIMAL YOU CAN INDICATE THAT A DECIMAL REPEATS BY WRITING A BAR OR LINE OVER THE REPEATING DIGIT(S) STUDY GUIDE AND INTERVENTION (CONTINUED) FRACTIONS AND DECIMALS COMPARE FRACTIONS AND DECIMALS IT MAY BE EASIER TO COMPARE NUMBERS WHEN THEY ARE WRITTEN AS DECIMALS EXERCISES REPLACE EACH ? WITH < , > , OR = TO MAKE A TRUE SENTENCE ? ? 8 ? 0 09 ? 5 0 3 ? ? 12 ? ? 10 ? 10 0 03 ? ? 30 ? 12 -0 55 ? 15 REPLACE ? WITH < , > , OR = TO MAKE 0 28 ? ? 8 A TRUE SENTENCE 0 28 ? REPLACE ? WITH < , > , OR = TO MAKE -0 37 ? ? 11 A TRUE SENTENCE -0 37 ? 0 37 ? -0 ?? 0 37 < -0 ?? ? 8 AS A DECIMAL COMPARE THE TENTHS PLACE: 2 < 3 WRITE WRITE RATIONAL NUMBERS AS FRACTIONS A NUMBER THAT CAN BE WRITTEN AS A FRACTION IS CALLED A RATIONAL NUMBER MIXED NUMBERS, INTEGERS, TERMINATING DECIMALS, AND REPEATING DECIMALS CAN ALL BE WRITTEN AS FRACTIONS ANY NUMBER THAT CAN BE EXPRESSED AS A ? B , WHERE A AND B ARE INTEGERS AND B ? 0 IS A RATIONAL NUMBER WRITE EACH NUMBER AS A FRACTION A 3 ? 5 = 7 = D 0 ? 0 14 IS 14 HUNDREDTHS 0 ? 5 = 0 555 0 14 = ? 100 OR N = 0 555 10N = 5 555 10N = 5 555 –(N = 0 555 ) 9N = 9N ? 9 = N = WRITE THE MIXED NUMBER AS AN IMPROPER FRACTION LET N REPRESENT THE NUMBER MULTIPLY EACH SIDE BY 10 BECAUSE ONE DIGIT REPEATS SUBTRACT N FROM 10N DIVIDE EACH SIDE BY 9 SIMPLIFY THE DENOMINATOR IS 1 SIMPLIFY EXERCISES WRITE EACH NUMBER AS A FRACTION 1 1 5 – 0 ? 6 0 ?? 8 -7 9 0 ?? 11 -0 0 ? 12 6 EXERCISES IDENTIFY ALL SETS TO WHICH EACH NUMBER BELONGS 12 4 ?? 13 9 09090909 14 -13 INTEGERS WHOLE NUMBERS THIS IS NEITHER A WHOLE NUMBER NOR AN INTEGER SINCE –0 08 CAN BE WRITTEN AS – FIND EACH PRODUCT WRITE IN SIMPLEST FORM A ? 15 SIMPLIFY BEFORE MULTIPLYING BY DIVIDING 5 AND 15 BY THEIR GCF, 5 = -8 ? 3 ? 21 = SIMPLIFY B 7 ? 2 2 ? 3 = ? 2 RENAME MIXED NUMBERS AS IMPROPER FRACTIONS = DIVIDE 15 AND 3 BY 3, AND 8 AND 2 BY 2 = 5 ? 1 MULTIPLY = ? 1 OR SIMPLIFY MULTIPLY FRACTIONS TO MULTIPLY FRACTIONS, MULTIPLY THE NUMERATORS AND MULTIPLY THE DENOMINATORS: A ? B C ? D = A C ? B D , WHERE B, D ? 0 FRACTIONS MAY BE SIMPLIFIED EITHER BEFORE OR AFTER MULTIPLYING WHEN MULTIPLYING NEGATIVE FRACTIONS, ASSIGN THE NEGATIVE SIGN TO THE NUMERATOR EXERCISES FIND EACH PRODUCT WRITE IN SIMPLEST FORM 1 ? 9 ? 5 ? 10 ( ? 4 ) 5 ? 5 (-5) 7 -4 ? 5 1 8 1 ? 7 10 9 – 2 ? 8 (-4 ? 7 ) 10 2 ? 9 (-3 ? 11 ) EVALUATE ? 3 AB IF A = 3 ? 7 AND B = ? 12 WRITE THE PRODUCT IN SIMPLEST FORM ? 3 AB = ? 3 (3 ? 7 ) ( ? 12 ) REPLACE A WITH 3 ? 7 AND B WITH ? 12 = ? 3 ( ? 12 ) RENAME 3 ? 7 AS ? 7 = ? 12 ) THE GCF OF 24 AND 12 IS 12 = 2 2 (-5) ? 3 SIMPLIFY EXERCISES EVALUATE EACH EXPRESSION IF X = ? 10 , Y = -4 ? 5 , AND Z = ? 7 WRITE THE PRODUCT IN SIMPLEST FORM 2 YZ 3 XYZ 5 -5XY 6 ? 2 Y ? 10 Z 8 ? 3 X 9 X X Z 11 -Y 12 Y Y 13 5 ? 6 XZ 14 ? 5 (-X) 15 ? 10 Y EVALUATE EXPRESSIONS WITH FRACTIONS ALGEBRAIC EXPRESSIONS ARE EXPRESSIONS WHICH CONTAIN ONE OR MORE VARIABLES VARIABLES CAN REPRESENT FRACTIONS IN ALGEBRAIC EXPRESSIONS FIND EACH QUOTIENT WRITE IN SIMPLEST FORM A ? 4 ÷ ? 8 = ? 4 MULTIPLY BY THE MULTIPLICATIVE INVERSE OF ? 8 , DIVIDE 4 AND 8 BY THEIR GCF, 4 = ? 5 OR 1 SIMPLIFY B -6 ? 5 ÷ 2 ? 5 ÷ ? 5 , ? 11 OR -2 SIMPLIFY DIVIDE FRACTIONS TWO NUMBERS WHOSE PRODUCT IS 1 ARE CALLED MULTIPLICATIVE INVERSES OR RECIPROCALS FOR ANY FRACTION A ? B , WHERE A, B ? 0, B ? A IS THE MULTIPLICATIVE INVERSE AND A ? B B ? A = 1 THIS MEANS THAT ? 3 AND ? 2 ARE MULTIPLICATIVE INVERSES BECAUSE ? 2 = 1 TO DIVIDE BY A FRACTION, MULTIPLY BY ITS MULTIPLICATIVE INVERSE: A ? B ÷ C ? D = A ? B D ? C = AD ? BC , WHERE B, C, D ? 0 EXERCISES FIND EACH QUOTIENT WRITE IN SIMPLEST FORM 1 ? 16 ÷ ? 9 ÷ ? 21 ÷ ( ? 7 ) 4 5 1 ? 4 ÷ 2 6 -8 ? 7 ÷ 2 ? 21 ÷ 8 -4 ? 8 ÷ (-3 ? 3 ) FIND ? QRS ÷ ? QS WRITE THE QUOTIENT IN SIMPLEST FORM ? QS = ? QRS QS ? MULTIPLY BY THE RECIPROCAL OF ? QS , QS ? 10 = DIVIDE OUT COMMON FACTORS = ? 5R SIMPLIFY DIVIDE ALGEBRAIC FRACTIONS ALGEBRAIC FRACTIONS ARE FRACTIONS WHICH CONTAIN ONE OR MORE VARIABLES YOU CAN DIVIDE ALGEBRAIC FRACTIONS JUST AS YOU WOULD DIVIDE NUMERICAL FRACTIONS EXERCISES FIND EACH QUOTIENT WRITE IN SIMPLEST FORM ÷ ? Y 2 C ? 4D ÷ ? 8D 3 4A ? B ÷ 2AC ? B ÷ M N 5 AB ? 9 ÷ BC ? 6 2ST ? Q ÷ 4T ? Q ? ÷ ? 5XYZ 8 8G ? 3HI ÷ 4G ? 15I 9 7P ? 9QR ÷ 3P ? 18Q ÷ 4X ? 11Z 11 2D ? 3EF ÷ ? 6EF 12 3X ? 5WY ÷ 6X ? 20YZ AB ? ÷ 6B ? 4C 14 14 J K ? 3L ÷ 4 J ? 9L 15 6A ? 11BC ÷ A ? 44B YZ ? ÷ 10Z ? 3X 17 DE ? 20F ÷ E ? 2F 18 6I ? 5GH ÷ 8I ? 3H ? 8 + ( ? 8 ) WRITE IN SIMPLEST FORM ? 8 ) = 3 + (-7) ? ? 8 OR SIMPLIFY ADD LIKE FRACTIONS TO ADD FRACTIONS WITH THE SAME DENOMINATORS, CALLED LIKE DENOMINATORS, ADD THE NUMERATORS AND WRITE THE SUM OVER THE DENOMINATOR SO, A ? C + B ? C = A + B ? C , WHERE C ? 0 FIND ? 12 + ? 12 WRITE IN SIMPLEST FORM ? 12 = 5 + ? 12 OR 1 THE DENOMINATORS ARE THE SAME ADD THE NUMERATORS SIMPLIFY AND RENAME TO A MIXED NUMBER EXERCISES FIND EACH SUM WRITE IN SIMPLEST FORM 1 ? 15 + ? 9 + ? 20 + ( ? 20 ) 5 ? 6 + ? 10 + ? 20 8 9 + 4 9 7 ? 4 + 3 10 -6 ? 12 + (-8 ? 12 ) 11 -4 ? 14 + 3 12 2 ( 1) FIND 1 ? 9 + 3 ? 9 WRITE IN SIMPLEST FORM 1 ? 9 = (1 + 3) + ( ? 9 ) = 4 + 2 + = 4 ? 9 OR 4 ADD THE WHOLE NUMBERS AND FRACTIONS SEPARATELY OR WRITE AS IMPROPER FRACTIONS ADD THE NUMERATORS SIMPLIFY EXAMPLE EVALUATE X - Y WHEN X = 7 ? 3 AND Y = 5 ? 3 WRITE IN SIMPLEST FORM X - Y = 7 ? 3 - 5 REPLACE X WITH 7 ? 3 AND Y WITH 5 ? 3 = 7 ? 3 = 6 SINCE ? 3 < ? 3 , THINK OF 7 ? 3 AS 6 ? 3 + ? 3 , OR 6 ? 3 = 1 SUBTRACT THE WHOLE NUMBERS THEN SUBTRACT THE FRACTIONS SU