Interpreting Graphs. Interpreting a Bar Graph
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(2) P s CLIFF DIVING
(3) P s TIGER BEETLES
(4) P s PARACHUTING
(5) P . . . . . . . . . .BUI BUDMBTT[POFDPN. s -OUNTAIN #LIMBING 3UBSTITUTION 2ACE
(6) P s 0OWERS AND %XPONENTS
(7) P s 0ERIMETER AND !REA
(8) P . 4LJMM'PDVT 'JOEJOHUIFTVNTPGXIPMFOVNCFST BOEEFDJNBMT. /N A FAMILY VACATION TRIP
(9) THE FIRST STOP IS A MOUNTAIN LOOKOUT TOWER 4HE FOUR LADDERS LEAD TO THE TOP OF THE TOWER 7HICH CHILD GETS TO THE TOP FIRST . s &IND THE SUM OF THE NUMBERS ON THE RUNGS OF EACH LADDER s 4HE LADDER WITH THE LEAST SUM IS WHERE THE FASTEST PERSON CLIMBS . $IBQUFS7BSJBCMFTBOE&RVBUJPOT.
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(18) . . . 4LJMM'PDVT 3PVOEJOHBOEJEFOUJGZJOHQMBDFWBMVF (ELP THE KIDS FIGURE OUT WHAT THE NEXT STOP IS ON THE FAMILY TRIP BY SOLVING THE PUZZLE. s %ACH NUMBER HAS BEEN ROUNDED TO ONE OF ITS DIGITS 3ELECT THE ANSWER THAT SHOWS THE NUMBER ROUNDED CORRECTLY s /N THE RIGHT
(19) WRITE EACH LETTER BELOW THE PLACE VALUE IT WAS ROUNDED TO s 0UT LETTERS IN CORRECT PLACE VALUE ORDER TO FIGURE OUT THE NEXT STOP. 4UPQBOE5IJOL $3*5*$"-5)*/,*/( ,IST THE NAMES IN 6ACATION 6IEWS FROM WHO GETS TO THE TOP FIRST TO WHO GETS THERE LAST THE LEAST SUM TO THE GREATEST 7HOSE NAME IS FIRST ON THIS LIST 83*5*/( )N .EXT 3TOP
(20) A STUDENT THINKS THAT THE RESULT OF ROUNDING IS BECAUSE THE LAST DIGIT IS REMOVED 7HAT IS WRONG WITH THE STUDENTS REASONING . .
(21) . 3FWJFX1SFSFRVJTJUF4LJMMT 70$"#6-"3:$)&$,. 3&7*&8803%4 s WHOLE NUMBER
(22) . #OPY AND COMPLETE USING A REVIEW WORD FROM THE LIST AT THE LEFT. 7HEN YOU ADD TWO NUMBERS
(23) THE RESULT IS CALLED THE . P . 7HEN YOU MULTIPLY TWO NUMBERS
(24) THE RESULT IS CALLED THE . s SUM
(25) P s DIFFERENCE
(26) P . 7HEN YOU SPLIT UP AN AMOUNT INTO EQUAL PARTS
(27) THE RESULT IS CALLED THE . s PRODUCT
(28) P s QUOTIENT
(29) P . 4,*--$)&$, 7RITE THE PLACE VALUE OF THE RED DIGIT Q. . . . . &IND THE SUM
(30) DIFFERENCE
(31) PRODUCT
(32) OR QUOTIENT Q . Q . Q Q . Q. Q. 9OU ARE SHOPPING FOR NEW CLOTHES )F YOU HAVE AND BUY A PAIR OF JEANS FOR
(33) HOW MUCH MONEY DO YOU HAVE LEFT Q. . 1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN. .OTETAKING 3KILLS )N EACH CHAPTER YOU WILL LEARN A NEW NOTETAKING SKILL )N #HAPTER YOU WILL APPLY THE STRATEGY OF KEEPING A NOTEBOOK TO %XAMPLE ON P . 9OUR MATH NOTEBOOK IS AN IMPORTANT TOOL FOR LEARNING AND REVIEWING THE TOPICS OF THIS COURSE (ERE ARE SOME TIPS FOR ORGANIZING YOUR NOTES /RGANIZE YOUR NOTES IN THE SAME WAY FOR EACH LESSON 4UBSUXJUIUIF EBUFBOEUPQJD. . 3EPTEMBER $ECIMALS. . . . $PQZFYBNQMFT TIPXOJODMBTT. 4O ADD DECIMALS
(34) LINE UP THE DECIMAL POINTS %XAMPLE !DDING $ECIMALS ?? . . ,FFQJOHB/PUFCPPL.
(35) 1.1 Before Now. Interpreting Graphs You compared quantities. You’ll use graphs to analyze data.. Why?. So you can make conclusions about data, as in Example 1.. KEY VOCABULARY. Volcanoes The bar graph at the right. • data, p. 3 • frequency table, p. 4 • histogram, p. 4. shows the number of historically active volcanoes in four countries. Which country has the most historically active volcanoes? A bar graph g is a type of graph in which the lengths of bars are used to represent and compare data a in categories. Data a are facts, numbers, or numerical information.. EXAMPLE 1. Volcanoes. Historically active volcanoes. • bar graph, p. 3. 180 150 120 90 60 30 0 Canada Chile Mexico. U.S.. Country. Interpreting a Bar Graph. Use the bar graph above about volcanoes. Answer the question or explain why you can’t answer the question using the graph. a. Which country has the most historically active volcanoes? b. Which country has the most volcanic eruptions in a given year?. SOLUTION. READING MATH Read carefully to make sure you don’t misinterpret the graph. Just because the U.S. has the most active volcanoes of the countries shown does not mean it has the most in the world.. ✓. a. The vertical axis in the bar graph is labeled Historically active volcanoes,. so the tallest bar represents the country with the most historically active volcanoes. Because the United States has the tallest bar, it has the most historically active volcanoes. b. Having more historically active volcanoes doesn’t necessarily mean having. more eruptions, so you can’t answer this question from the bar graph.. GUIDED PRACTICE. for Example 1. Use the bar graph above about historically active volcanoes. 1. About how many more historically active volcanoes does Chile have. than Mexico? 2. Which country has the least number of historically active volcanoes? 3. About how many historically active volcanoes do Canada, Mexico, and. the U.S. have altogether? 1.1 Interpreting Graphs. 3.
(36) )JTUPHSBNT ! LARGE SET OF DATA VALUES CAN BE GROUPED INTO INTERVALS IN A . FREQUENCY TABLE 4HE FREQUENCY OF DATA IN AN INTERVAL IS THE NUMBER OF VALUES IN THE INTERVAL 9OU CAN GRAPH DATA ORGANIZED IN EQUAL INTERVALS IN A HISTOGRAM
(37) WHERE THE HEIGHT OF EACH BAR INDICATES ITS FREQUENCY. & 9 " . 1 - & .BLJOHB'SFRVFODZ5BCMF 3PMMFS$PBTUFST 4HE DATA SHOW THE HEIGHTS
(38) IN METERS
(39) OF SOME OF THE TALLEST ROLLER COASTERS IN THE WORLD -AKE A FREQUENCY TABLE OF THE DATA.
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(59) . 40-65*0/. 34%0 # HOOSE INTERVALS OF EQUAL SIZE . !6/)$ %22/23 .BLFTVSFUIBUUIF JOUFSWBMTEPOPUPWFSMBQ. FOR THE DATA 4HE INTERVALS SHOULD INCLUDE NUMBERS FROM TO . 34%0 4ALLY THE DATA IN EACH INTERVAL 5SE TALLY MARKS TO RECORD EACH OCCURRENCE OF A HEIGHT IN ITS INTERVAL. 34%0 7RITE THE FREQUENCY FOR EACH INTERVAL BY TOTALING THE TALLY MARKS. )FJHIU N. 5BMMZ. 'SFRVFODZ. o. . . o o o. . . . o. . o . . o. . & 9 " . 1 - & .BLJOHB)JTUPHSBN -AKE A HISTOGRAM OF THE DATA IN THE FREQUENCY TABLE ABOVE. 34%0 $RAW AND LABEL THE HORIZONTAL . FREQUENCY OF EACH INTERVAL 4HE BARS OF NEIGHBORING INTERVALS SHOULD TOUCH. 34%0 7RITE A TITLE FOR THE HISTOGRAM. . (6*%&%13"$5*$&. . . . . . . . . . . . . .
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(61) . 34%0 $RAW A BAR TO REPRESENT THE . .
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(63) . . AND VERTICAL AXES 3TART THE VERTICAL SCALE AT AND END AT A POINT GREATER THAN 5SE EQUAL INCREMENTS. . GPS&YBNQMFTBOE. 8IBU*G 5SING THE DATA FROM %XAMPLE
(64) MAKE A NEW FREQUENCY TABLE . USING INTERVALS 4HEN MAKE A HISTOGRAM FROM YOUR FREQUENCY TABLE. . $IBQUFS7BSJBCMFTBOE&RVBUJPOT.
(65) . &9&3$*4&4. ( 45"/%"3%*;&%5&4513"$5*$& &YT BOE. )0.&803, ,&: . . )*/54"/%)0.&803,)&-1 GPS&YT BUDMBTT[POFDPN. 4,*--13"$5*$& 70$"#6-"3: #OPY AND COMPLETE ! HISTOGRAM IS A GRAPH THAT SHOWS DATA THAT ARE DIVIDED INTO EQUAL 70$"#6-"3: #OPY AND COMPLETE 4HE LENGTHS OF BARS ARE USED TO REPRESENT AND COMPARE DATA IN CATEGORIES IN A */5&313&5*/("(3"1) 5SE THE BAR GRAPH )T SHOWS THE NUMBER OF BUSINESSES AT A MALL BY CATEGORY. 3%% %8!-0,% POQ GPS&YTo. . . 7HICH CATEGORY HAS THE GREATEST NUMBER OF BUSINESSES . . . 7HICH CATEGORY HAS THE LEAST NUMBER OF BUSINESSES .
(66) . !BOUT HOW MANY MORE SHOE STORES ARE THERE THAN JEWELRY STORES . . #AN YOU TELL FROM THE GRAPH WHICH CATEGORY OF BUSINESSES USES THE MOST FLOOR SPACE IN THE MALL 7HY OR WHY NOT 3%% %8!-0,% POQ GPS&Y. . . . . .6-5*45&1130#-&. 4HE DATA SHOW THE NUMBERS OF HOURS STUDENTS IN A CLASS SPENT ON THE )NTERNET IN A WEEK . .
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(95) . A .BLFB'SFRVFODZ5BCMF -AKE A FREQUENCY TABLE OF THE DATA . 5SE n AS THE FIRST INTERVAL B .BLFB'SFRVFODZ5BCMF -AKE A FREQUENCY TABLE OF THE DATA . 5SE n AS THE FIRST INTERVAL C $PNQBSF $OES THE FREQUENCY TABLE IN PART A OR PART B GIVE A . CLEARER REPRESENTATION OF THE DATA %XPLAIN YOUR REASONING 3%% %8!-0,% POQ GPS&YTo. &3303"/"-:4*4 *ACK USED THE INTERVALS n
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(99) AND n TO MAKE A HISTOGRAM $ESCRIBE THE ERROR *ACK MADE (3"1)*/(%"5" 4HE FREQUENCY TABLE SHOWS WHEN NEW STATES WERE ADDED TO THE 5NITED 3TATES -AKE A HISTOGRAM OF THE DATA :FBS 4UBUFT. o. o. o. o. . . . . $)"--&/(& %XPLAIN WHY A FREQUENCY TABLE AND A HISTOGRAM ARENT . APPROPRIATE TO SHOW THE DATA
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(107) 130#-&.40-7*/( (6*%&%130#-&.40-7*/( 4HE TABLE SHOWS THE NUMBERS OF MOVIES RELEASED IN THE 5NITED 3TATES FROM THROUGH 5SE A BAR GRAPH TO HELP YOU DESCRIBE HOW THE DATA HAVE CHANGED FROM YEAR TO YEAR. :FBS. .PWJFT. . . . . A #HOOSE INTERVALS FOR THE SCALE FOR A BAR GRAPH. . . B -AKE A BAR GRAPH. . . C #OMPARE THE VALUES FOR DIFFERENT YEARS TO DESCRIBE HOW . . . . . THEYVE CHANGED. .&%*" )N %XERCISES n
(108) USE THE TRIPLE BAR GRAPH )T SHOWS HOW MANY . 53 HIGH SCHOOLS PER SCHOOLS HAVE THE GIVEN MEDIA ACTIVITIES
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(110) . (4)0353&410/4& 7HICH OF THE MEDIA SAW . . . 9 !BOUT TIMES AS MANY HAD NEWSPAPERS. . . . 8 !BOUT TWICE AS MANY HAD NEWSPAPERS. . . . . AS MANY HAD NEWSPAPERS 7 !BOUT ]z . . . 6 !BOUT ]z AS MANY HAD NEWSPAPERS. . .
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(115) HOW DID THE NUMBER OF HIGH SCHOOLS WITH NEWSPAPERS COMPARE TO THE NUMBER OF HIGH SCHOOLS WITH RADIO OR 46 STATIONS .
(116) . . . THE MOST GROWTH FROM TO %XPLAIN HOW YOU CAN TELL THAT FROM THE GRAPH. . POQ GPS&YTo. 7HICH OF THE GIVEN MEDIA DID THE GREATEST NUMBER OF SCHOOLS OFFER IN EACH OF THE YEARS . . 3%% %8!-0,% . )633*$"/&4 )N %XERCISES n THE HISTOGRAM SHOWS THE NUMBERS OF . HURRICANES IN THE !TLANTIC /CEAN FROM THROUGH (OW MANY YEARS OF DATA DOES THIS GRAPH SHOW .
(117) . #AN YOU DETERMINE THE NUMBER OF HURRICANES IN . OF HURRICANES IN n 7HY OR WHY NOT (01&/&/%&%."5) 'IVE AN EXAMPLE OF A SET OF DATA THAT CAN BE GRAPHED WITH A BAR GRAPH BUT NOT WITH A HISTOGRAM 3&"40/*/( 9OU ARE LOOKING FOR DATA ON HIGH TIDES )S IT EASIER TO FIND THE EXACT NUMBER OF HIGH TIDES BETWEEN !- AND !- LAST YEAR FROM A FREQUENCY TABLE OR A HISTOGRAM %XPLAIN. . (45"/%"3%*;&%5&4513"$5*$&. . . #AN YOU USE THE HISTOGRAM TO PREDICT THE NUMBER . . THE !TLANTIC /CEAN IN %XPLAIN. )*/54"/%)0.&803,)&-1BUDMBTT[POFDPN.
(118) . .6-5*1-&3&13&4&/5"5*0/4 4HE DATA SHOW THE AVERAGE NUMBERS OF METEORS THAT FALL PER HOUR DURING ANNUAL METEOR SHOWERS &OR EXAMPLE
(119) IN THE FIRST METEOR SHOWER
(120) AN AVERAGE OF METEORS FALL EACH HOUR. .
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(158) . A 'SFRVFODZ5BCMF -AKE A FREQUENCY TABLE OF THE DATA . 5SE INTERVALS OF STARTING WITH n B )JTUPHSBN -AKE A HISTOGRAM OF THE DATA FROM YOUR . FREQUENCY TABLE C $PNQBSF #OMPARE THE DISPLAYS $ISCUSS ONE THING THAT . IS EASIER TO TELL ABOUT THE DATA FROM EACH DISPLAY. .FUFPSTIPXFS. (01&/&/%&%."5) 9OU WANT TO COLLECT DATA ABOUT THE PETS OWNED . BY STUDENTS IN YOUR CLASS 7HAT DATA WOULD YOU COLLECT TO MAKE A BAR GRAPH #OULD YOU USE THE SAME DATA TO MAKE A HISTOGRAM %XPLAIN YOUR REASONING (&95&/%&%3&410/4& 4HE DATA BELOW ARE THE NUMBER OF MEDALS THE . 5NITED 3TATES WON IN EACH OF THE 7INTER /LYMPIC 'AMES AS OF .
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(176) A #REATE A HISTOGRAM OF THE DATA WITH INTERVALS OF AND . A SECOND HISTOGRAM WITH INTERVALS OF
(177) . B 7HICH HISTOGRAM GIVES A CLEARER PICTURE OF HOW MANY . MEDALS YOU CAN EXPECT THE 53 TO WIN IN FUTURE 7INTER /LYMPICS 7HAT MIGHT ACCOUNT FOR THE MEDALS . . . . . $)"--&/(& 5SE THE GRAPH AT THE RIGHT $ESCRIBE HOW THE DATA CHANGED DURING THE YEARS SHOWN 7HAT MIGHT EXPLAIN THE RAPID GROWTH IN SALES OF $6$ PLAYERS 7HAT MIGHT CAUSE THE GROWTH TO SLOW DOWN IN THE FUTURE %XPLAIN YOUR REASONING. . .*9&%3&7*&8 %VALUATE THE EXPRESSION Q. 1SFQBSFGPS -FTTPO JO&YTo. . . . . &IND THE PRODUCT OR QUOTIENT Q . Q . Q . Q. Q . Q . Q . Q. (.6-5*1-&$)0*$& 2OUND TO THE NEAREST TENTH Q. . 6 . 7 . &953"13"$5*$&GPS-FTTPO Q. 8 . 9 . 0/-*/&26*;BUDMBTT[POFDPN. .
(178) . 0SEFSPG 0QFSBUJPOT. . #FGPSF . . /PX . :PVMMVTFPSEFSPGPQFSBUJPOTUPFWBMVBUFOVNFSJDBMFYQSFTTJPOT. . 8IZ . 4PZPVDBOGJOEUPUBMDPTUT BTJOUIFBRVBSJVNWJTJUJO&YBNQMF. :PVQFSGPSNFEPQFSBUJPOTPOOVNCFST. 7JTJUJOHBO"RVBSJVN 9OU AND FOUR FRIENDS VISIT AN AQUARIUM
(179) BUT ONLY THREE OF YOU GO TO THE MOVIE AT THE AQUARIUM 7HAT IS THE TOTAL COST OF THE VISIT 9OU WILL FIND THE TOTAL COST OF THE VISIT USING THE PRICES IN THE TABLE ON THE RIGHT IN %XAMPLE AND APPLYING THE ORDER OF OPERATIONS. ,&:70$"#6-"3: s NUMERICAL EXPRESSION
(180) P s EVALUATE
(181) P s ORDER OF OPERATIONS
(182) P s VERBAL MODEL
(183) P . "RVBSJVN1SJDFT. "ENJTTJPO. . 4FB-JPOTIPX .PWJF. 'SFF . ! NUMERICAL EXPRESSION CONSISTS OF NUMBERS AND OPERATIONS 4O FIND THE VALUE OF A NUMERICAL EXPRESSION IS TO EVALUATE IT 7HEN A NUMERICAL EXPRESSION HAS MORE THAN ONE OPERATION
(184) USE A SET OF RULES CALLED THE ORDER OF OPERATIONS. &OR 9OUR .OTEBOOK. ,&:$0/$&15 0SEFSPG0QFSBUJPOT. 4O EVALUATE AN EXPRESSION THAT HAS MORE THAN ONE OPERATION %VALUATE EXPRESSIONS INSIDE GROUPING SYMBOLS -ULTIPLY AND DIVIDE FROM LEFT TO RIGHT !DD AND SUBTRACT FROM LEFT TO RIGHT. & 9 " . 1 - & 6TJOHUIF0SEFSPG0QFSBUJPOT %VALUATE THE EXPRESSION . ).4%202%4 39-"/,3 :PVDBOFYQSFTT NVMUJQMJDBUJPOCZVTJOH QBSFOUIFTFTPSUIF TZNCPMT+PS . . . +. . . . . (6*%&%13"$5*$&. %JWJEFCZ. . .VMUJQMZCZ. . "EEBOE. BUDMBTT[POFDPN. GPS&YBNQMF. %VALUATE THE EXPRESSION . . $IBQUFS7BSJBCMFTBOE&RVBUJPOT. . + + .
(185) (SPVQJOH4ZNCPMT 4HE MOST COMMON GROUPING SYMBOLS ARE PARENTHESES
(186) . BRACKETS ; =
(187) AND FRACTION BARS. &9".1-&. 6TJOH(SPVQJOH4ZNCPMT. A + + . . "EEJOTJEFQBSFOUIFTFTGJSTU. . 5IFONVMUJQMZ. . B ]z ]z . !./4(%2 7!9 :PVDBOFYQSFTTEJWJTJPO VTJOHFJUIFSPSB GSBDUJPOCBS5PFWBMVBUF BOFYQSFTTJPOXJUIB GSBDUJPOCBS FWBMVBUF UIFOVNFSBUPSBOEUIF EFOPNJOBUPSCFGPSFZPV EJWJEF. &WBMVBUFOVNFSBUPS. . ]z . &WBMVBUFEFOPNJOBUPS. . . %JWJEF. . C ; = ; = . %JWJEFJOTJEFUIFJOOFSNPTU TFUPGHSPVQJOHTZNCPMT. . . %JWJEFJOTJEFCSBDLFUT. . . %JWJEF. . BUDMBTT[POFDPN. . 7FSCBM.PEFM 7HEN YOU SOLVE A PROBLEM
(188) IT MAY HELP TO WRITE A VERBAL MODEL 5SE SYMBOLS FOR OPERATIONS
(189) AND USE WORDS TO LABEL NECESSARY INFORMATION. & 9 " . 1 - & 6TJOHB7FSCBM.PEFM 4O FIND THE TOTAL COST OF THE VISIT TO THE AQUARIUM DESCRIBED ON THE PREVIOUS PAGE
(190) YOU CAN USE A VERBAL MODEL TO WRITE AND EVALUATE AN EXPRESSION
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(195) . 2%6)%7 $%#)-!,3. z. . 4VCTUJUVUFWBMVFT. /FFEIFMQXJUIEFDJNBM PQFSBUJPOT 4FFQQ BOE. z. . .VMUJQMZGJSTU. z. . 5IFOBEE. C"OTXFS 4HE TOTAL COST OF THE VISIT IS . . (6*%&%13"$5*$&. GPS&YBNQMFTBOE. %VALUATE THE EXPRESSION . + ; = . . ]z . 8IBU*G 3UPPOSE STUDENTS GO TO THE AQUARIUM AND MOVIE IN %XAMPLE )T COSTS TO PARK THEIR SCHOOL BUS 5SE A VERBAL MODEL TO WRITE AND EVALUATE AN EXPRESSION FOR THE TOTAL COST OF THEIR VISIT 0SEFSPG0QFSBUJPOT. .
(196) . ( 45"/%"3%*;&%5&4513"$5*$& &YT BOE. &9&3$*4&4. )0.&803, ,&: . . )*/54"/%)0.&803,)&-1 GPS&YT BUDMBTT[POFDPN. . 4,*--13"$5*$& 70$"#6-"3: 5SE THE ORDER OF OPERATIONS TO LIST
(197) IN ORDER
(198) THE STEPS . NEEDED TO EVALUATE THE FOLLOWING EXPRESSION 70$"#6-"3: #OPY AND COMPLETE !N CONSISTS OF NUMBERS AND OPERATIONS 3%% %8!-0,%3 !.$ POQQo GPS&YTo. (.6-5*1-&$)0*$& )N WHAT ORDER SHOULD THE OPERATIONS BE PERFORMED . TO EVALUATE THE EXPRESSION 6
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(210) . 64*/(03%&30'01&3"5*0/4 %VALUATE THE EXPRESSION. . . . . . . . . ]z . ]z . ; + = . + ]z ]z . + ]z ]z . . . . . . . &|. . + zz] z z. ; =. |. . '. |. . . . . ]]z . ]z . (.6-5*1-&$)0*$& 7HICH OPERATION SHOULD BE PERFORMED FIRST WHEN . FINDING THE SUM OF TWENTY AND THE QUOTIENT OF EIGHTEEN AND SIX 6 . 7 . 8 . &3303"/"-:4*4 $ESCRIBE AND CORRECT THE ERROR MADE IN THE SOLUTION. 9 . ((+(..+(. ,'. -. /6.#&34&/4& )NSERT PARENTHESES TO MAKE THE STATEMENT TRUE. + . . . . . . $)"--&/(& #OPY AND COMPLETE THE STATEMENT USING ONE OF THE OPERATIONS .
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(213) AND
(214) AND ONE SET OF PARENTHESES TO OBTAIN THE GREATEST POSSIBLE VALUE OF THE EXPRESSION . . $IBQUFS7BSJBCMFTBOE&RVBUJPOT. . .
(215) 130#-&.40-7*/( 3%% %8!-0,% POQ GPS&YTo. (.6-5*1-&$)0*$& 9OUR FRIEND PLEDGES TO YOU FOR A FUNDRAISING WALK AND FOR EACH MILE YOU WALK 9OU WALK MILES 7HICH EXPRESSION COULD YOU USE TO FIND YOUR FRIENDS CONTRIBUTION . 6 7 8 9 . (83*5*/( 9OU BUY NOTEBOOKS AT EACH AND PENS AT EACH . %XPLAIN HOW TO FIND THE TOTAL COST (4)0353&410/4& 9OUR SCHOOL SOFTBALL TEAM HAS MEMBERS 4HE SCHOOL CONTRIBUTES TOWARD EACH UNIFORM 7RITE AND EVALUATE A VERBAL MODEL TO FIND HOW MUCH MONEY THE TEAM NEEDS TO RAISE TO BUY UNIFORMS FOR EVERY MEMBER )S MORE THAN ONE MODEL POSSIBLE %XPLAIN (01&/&/%&%."5) 7RITE A VERBAL MODEL AND AN EXPRESSION FOR A REAL WORLD PROBLEM WHOSE ANSWER IS 5SE AT LEAST THREE OPERATIONS .6-5*45&1130#-&. ,IZ AND 4Y ARE BAKING FOR THE SCHOOL BAKE SALE ,IZ MAKES BATCHES OF DOZEN COOKIES
(216) AND 4Y MAKES BATCHES OF DOZEN COOKIES A 7FSCBM.PEFM 7RITE A VERBAL MODEL FOR THE TOTAL . NUMBER OF COOKIES BAKED B &WBMVBUF %VALUATE THE EXPRESSION C &YUFOE ,IZ AND 4Y DECIDE TO MAKE PACKAGES OF . THREE COOKIES 7RITE AND EVALUATE AN EXPRESSION TO FIND THE NUMBER OF PACKAGES THEY CAN MAKE $)"--&/(& 4O MAKE THE STATEMENT BELOW TRUE
(217) COPY AND COMPLETE USING ONE OR MORE OF THE OPERATORS
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(220) AND $)"--&/(& 9OUR SISTER IS YEARS OLD 9OUR BROTHER IS YEARS LESS THAN TWICE YOUR SISTERS AGE 9OUR COUSIN IS HALF OF YOUR BROTHERS AGE 7RITE AND EVALUATE AN EXPRESSION TO FIND YOUR COUSINS AGE. .*9&%3&7*&8 'RAPES COST PER POUND (OW MUCH DOES THE GIVEN AMOUNT COST Q. 1SFQBSFGPS -FTTPO JO&YTo. POUNDS . POUNDS . POUNDS . POUNDS. &IND THE PRODUCT 3IMPLIFY IF POSSIBLE Q. . ]z . ]z . . ]z . . ]z . (.6-5*1-&$)0*$& 9OU BUY GROCERIES COSTING (OW MUCH . CHANGE SHOULD YOU RECEIVE IF YOU PAY WITH A BILL Q. . 6 . 7 . &953"13"$5*$&GPS-FTTPO Q. 8 . 9 . 0/-*/&26*;BUDMBTT[POFDPN. .
(221) (0"-. 6TFBGUFS-FTTPO. 6TFBDBMDVMBUPSUP FWBMVBUFOVNFSJDBM FYQSFTTJPOTXJUI EFDJNBMT. 6TJOH0SEFSPG0QFSBUJPOT & 9 " . 1 - & 9OU AND THREE FRIENDS ARE ORDERING A PIZZA 4HE COST OF THE PIZZA IS
(222) BUT YOU HAVE A COUPON FOR OFF (OW MUCH SHOULD EACH OF YOU PAY IF YOU WANT TO DIVIDE THE TOTAL COST EQUALLY . 40-65*0/ 0RICE OF A PIZZA #OUPON .UMBER OF FRIENDS 9OURSELF. #OST PER PERSON ]]]z z z 8SJUFBWFSCBMNPEFM ]]z . 4VCTUJUVUF. . 4O FIND THE COST PER PERSON
(223) USE THE ORDER OF OPERATIONS 5SE PARENTHESES TO SEPARATE EXPRESSIONS THAT SHOULD BE GROUPED TOGETHER . +EYSTROKES . $ISPLAY . . . . . . . . C"OTXFS %ACH PERSON SHOULD PAY FOR THE PIZZA. 1 3 "$ 5 * $ & 5SE A CALCULATOR TO EVALUATE THE EXPRESSION . . . + . ]z . . ]z . . ]z . 4/"$,4 9OU BUY BAGS OF SNACK MIX FOR EACH
(224) BOXES OF RAISINS FOR EACH
(225) AND LEMONADE FOR &IND THE TOTAL COST USING THE EXPRESSION + + $-05)*/( 9OU BUY 4 SHIRTS AT EACH
(226) A PAIR OF SNEAKERS FOR
(227) A HAT FOR
(228) AND A PAIR OF SOCKS AT A COST OF PAIRS FOR &IND THE TOTAL COST .64*$ 9OU PICK OUT #$S FOR EACH
(229) #$S FOR EACH
(230) AND A #$ FOR !T THE REGISTER
(231) YOU FIND OUT THAT
(232) WHEN YOU BUY #$S
(233) YOU GET THE SIXTH #$ FOR HALF OFF &IND THE TOTAL COST IF YOU GET THE LEAST EXPENSIVE #$ FOR HALF OFF 2OUND TO THE NEAREST CENT. . $IBQUFS7BSJBCMFTBOE&RVBUJPOT. . ]z .
(234) . 7BSJBCMFTBOE &YQSFTTJPOT. . #FGPSF . :PVFWBMVBUFEOVNFSJDBMFYQSFTTJPOT. . /PX . . 8IZ . :PVMMXSJUFBOEFWBMVBUFWBSJBCMFFYQSFTTJPOT 4PZPVDBOGJOEUPUBMEJTUBODFTUSBWFMFE BTJO&YBNQMF. ,&:70$"#6-"3: s VARIABLE
(235) P s VARIABLE EXPRESSION
(236) P . ! VARIABLE IS A SYMBOL
(237) USUALLY A LETTER
(238) THAT REPRESENTS ONE OR MORE NUMBERS ! VARIABLE EXPRESSION CONSISTS OF NUMBERS
(239) VARIABLES
(240) AND OPERATIONS 4O EVALUATE A VARIABLE EXPRESSION
(241) SUBSTITUTE A NUMBER FOR EACH VARIABLE 4HEN FIND THE VALUE OF THE NUMERICAL EXPRESSION 9OU CAN WRITE THE PRODUCT OF A NUMBER AND A VARIABLE BY WRITING THE NUMBER NEXT TO THAT VARIABLE &OR EXAMPLE
(242) YOU CAN WRITE + N AS N. &9".1-&. 6TJOHB7BSJBCMF&YQSFTTJPO. YZ )PU"JS#BMMPPOT 9OU ARE RIDING IN A HOT AIR BALLOON . !FTER TRAVELING MILES
(243) THE BALLOON SPEED CHANGES TO MILES PER HOUR 3O
(244) THE TOTAL DISTANCE TRAVELED IS ORIGINAL DISTANCE SPEED + TIME
(245) WHICH IS T
(246) WHERE T IS THE NUMBER OF HOURS TRAVELED AT MIH . 2%!$).' 5IFTQFFENJIJTSFBE iTJYNJMFTQFSIPVSw. 7HAT IS THE TOTAL DISTANCE YOU TRAVEL IF THE BALLOON MOVES AT THIS SPEED FOR HOUR FOR HOURS . 40-65*0/. 34%0 . 34%0 . 34%0 . 7RITE HOURS TRAVELED T . 3UBSTITUTE FOR T IN THE EXPRESSION T . %VALUATE TO FIND TOTAL DISTANCE. . . . . . . . . C "OTXFS )F THE BALLOON TRAVELS AT MILES PER HOUR FOR HOUR
(247) YOU TRAVEL A TOTAL OF MILES !FTER HOURS YOU TRAVEL A TOTAL OF MILES. . (6*%&%13"$5*$&. GPS&YBNQMF. 8IBU*G 5SE THE INFORMATION ABOVE ABOUT HOT AIR BALLOONS. )F YOU TRAVEL AT MILES PER HOUR FOR HOURS
(248) WHAT IS THE TOTAL DISTANCE )F YOU TRAVEL AT MILES PER HOUR FOR ]z HOUR
(249) WHAT IS THE TOTAL DISTANCE . )F YOU TRAVEL AT MILES PER HOUR FOR HOURS
(250) WHAT IS THE TOTAL DISTANCE . 7BSJBCMFTBOE&YQSFTTJPOT. .
(251) & 9 " . 1 - & &WBMVBUJOH7BSJBCMF&YQSFTTJPOT YZ %VALUATE THE EXPRESSION WHEN X AND Y A X Y . 4VCTUJUVUFGPSYBOEGPSZ. . . .VMUJQMZ. . . 4VCUSBDU. B X Y . 4VCTUJUVUFGPSYBOEGPSZ. . . "EE. . . .VMUJQMZ. BUDMBTT[POFDPN. 8SJUJOH&YQSFTTJPOT -ANY WORDS AND PHRASES SUGGEST MATHEMATICAL . OPERATIONS 4HE FOLLOWING COMMON WORDS AND PHRASES INDICATE ADDITION
(252) SUBTRACTION
(253) MULTIPLICATION
(254) AND DIVISION "EEJUJPO. 4VCUSBDUJPO. .VMUJQMJDBUJPO. %JWJTJPO. QMVT. NJOVT. UJNFT. EJWJEFECZ. UIFTVNPG. UIFEJGGFSFODFPG. UIFQSPEVDUPG. UIFRVPUJFOUPG. JODSFBTFECZ. EFDSFBTFECZ. NVMUJQMJFECZ. UPUBM. GFXFSUIBO. PG. TFQBSBUFJOUP FRVBMQBSUT. NPSFUIBO. MFTTUIBO. UXJDF. BEEFEUP. TVCUSBDUFEGSPN. & 9 " . 1 - & 5SBOTMBUJOH7FSCBM1ISBTFT 2%!$).'. YZ 6ERBAL 0HRASE . 0SEFSJTJNQPSUBOUXIFO USBOTMBUJOHTVCUSBDUJPO BOEEJWJTJPOFYQSFTTJPOT 5IFEJGGFSFODFPGB OVNCFSBOENFBOT O OPUO5IF RVPUJFOUPGBOVNCFS BOENFBOTO OPUO. . 6ARIABLE %XPRESSION. 4HE SUM OF A NUMBER AND . N . 4HE DIFFERENCE OF A NUMBER AND . N . 4HE PRODUCT OF AND A NUMBER . N. 4HE QUOTIENT OF AND A NUMBER . ]z. N N ]z . /NE THIRD OF A NUMBER . (6*%&%13"$5*$&. BUDMBTT[POFDPN. GPS&YBNQMFTBOE. %VALUATE THE EXPRESSION WHEN A AND B A . AB . BA . 7RITE THE PHRASE AS A VARIABLE EXPRESSION USING X A NUMBER INCREASED BY . . $IBQUFS7BSJBCMFTBOE&RVBUJPOT. TIMES A NUMBER. A A B. ]z .
(255) & 9 " . 1 - & 8SJUJOHBOE&WBMVBUJOHBO&YQSFTTJPO YZ )FBSU3BUF 4O MEASURE YOUR HEART RATE
(256) COUNT THE . NUMBER OF HEARTBEATS N IN SECONDS 4HEN MULTIPLY BY TO FIND YOUR HEART RATE IN BEATS PER MINUTE A 5SE N TO WRITE AN EXPRESSION FOR HEART RATE IN . BEATS PER MINUTE B !FTER EXERCISING
(257) YOU COUNT BEATS IN . SECONDS &IND YOUR HEART RATE. 40-65*0/ A 4HE PHRASE MULTIPLY BY SUGGESTS MULTIPLICATION . 3O
(258) MULTIPLY THE NUMBER OF HEART BEATS N BY C "OTXFS ! VARIABLE EXPRESSION FOR HEART RATE IN BEATS PER MINUTE IS N B 3UBSTITUTE FOR N IN THE EXPRESSION N TO FIND YOUR HEART RATE. . N . C "OTXFS 9OUR HEART RATE IS BEATS PER MINUTE. . (6*%&%13"$5*$&. GPS&YBNQMF. 4BWJOHT 9OU HAVE SAVED 9OU SAVE MORE PER WEEK FOR N WEEKS . 7RITE AND EVALUATE AN EXPRESSION FOR YOUR TOTAL SAVINGS AFTER WEEKS. . &9&3$*4&4. ( 45"/%"3%*;&%5&4513"$5*$& &YT BOE. )0.&803, ,&: . . )*/54"/%)0.&803,)&-1 GPS&YT BUDMBTT[POFDPN. 4,*--13"$5*$& 70$"#6-"3: #OPY AND COMPLETE THE STATEMENT. T IS AN EXPRESSION AND IS AN EXPRESSION !N IS A SYMBOL
(259) USUALLY A LETTER
(260) THAT REPRESENTS ONE OR MORE NUMBERS &7"-6"5*/(&913&44*0/4 %VALUATE THE EXPRESSION FOR THE GIVEN VALUES . OF THE VARIABLES 3%% %8!-0,% . X
(261) X . N
(262) N . POQ GPS&YTo . C D
(263) C
(264) D . F G
(265) F
(266) G . P Q
(267) P
(268) Q . A B
(269) A
(270) B . X Y
(271) X
(272) Y D C D. ]z
(273) C
(274) D . R Q
(275) Q
(276) R S P . ]z
(277) P
(278) S 7BSJBCMFTBOE&YQSFTTJPOT. .
(279) 3%% %8!-0,% . &3303"/"-:4*4 $ESCRIBE AND CORRECT . *)+m*)+, .+, +(. THE ERROR MADE IN EVALUATING + X WHEN X . POQ GPS&Y. 53"/4-"5*/(7&3#"-1)3"4&4 7RITE THE PHRASE AS A VARIABLE EXPRESSION ,ET X REPRESENT THE VARIABLE. 3%% %8!-0,% POQ GPS&YTo . TWO FIFTHS OF A NUMBER . A NUMBER SUBTRACTED FROM . INCREASED BY A NUMBER . THE QUOTIENT OF A NUMBER AND . THE SUM OF A NUMBER AND . A NUMBER DECREASED BY . TIMES A NUMBER . THE PRODUCT OF A NUMBER AND . (.6-5*1-&$)0*$& 7HICH VALUES MAKE THE EXPRESSION X + ; Y = . HAVE A VALUE OF 6 X
(280) Y 7 X
(281) Y . 8 X
(282) Y 9 X
(283) Y . 3&"40/*/( %VALUATE THE EXPRESSION WHEN X AND Y 4HEN TELL WHETHER SWAPPING THE VALUES OF X AND Y WILL INCREASE
(284) DECREASE
(285) OR NOT CHANGE THE VALUE OF THE EXPRESSION %XPLAIN YOUR REASONING. X Y . X Y . XY . X Y . Y X . ]z . X Y Y. Y X. X zzz] z . &7"-6"5*/(&913&44*0/4 %VALUATE THE EXPRESSION WHEN X
(286) Y
(287) AND Z 2OUND TO THE NEAREST HUNDREDTH X Z XY Z. ]]z . X Y Z Y Z. ]]z . Y Z X X YZ. ]z . $)"--&/(& 4ELL WHETHER THERE IS A WHOLE NUMBER X THAT WILL MAKE THE STATEMENT TRUE )F SO
(288) FIND THE VALUE OF X. X . X X . X X . X X. 130#-&.40-7*/( 3%% %8!-0,% POQ GPS&YTo . #*$:$-&4 %VALUATE THE EXPRESSION T TO FIND HOW MANY MILES YOU GO IF YOU RIDE YOUR BIKE FOR T HOURS AT MILES PER HOUR. (OW FAR DO YOU GO IF YOU RIDE YOUR BIKE FOR HOURS (OW FAR DO YOU GO IF YOU RIDE YOUR BIKE FOR ]z HOURS . 3%% %8!-0,% POQ GPS&Y. (6*%&%130#-&.40-7*/( 9OU BUY A HAT FOR AND RENT VIDEOS FOR EACH (OW MUCH DO YOU SPEND ALTOGETHER IF YOU RENT VIDEOS A 7RITE A VARIABLE EXPRESSION FOR THE COST TO RENT M VIDEOS B !DD THE COST OF THE HAT TO THIS EXPRESSION C %VALUATE THE EXPRESSION FOR M . . (45"/%"3%*;&%5&4513"$5*$&. )*/54"/%)0.&803,)&-1BUDMBTT[POFDPN.
(289) 3%% %8!-0,% POQ GPS&YTo. 5&-&7*4*0/ 9OU WATCH X THIRTY MINUTE 46 SHOWS AND Y SIXTY MINUTE 46 SHOWS IN ONE WEEK
(290) BUT YOU DONT WATCH Z THREE MINUTE COMMERCIAL BREAKS DURING THE SHOWS 7RITE A VARIABLE EXPRESSION REPRESENTING THE TOTAL NUMBER OF MINUTES YOU SPEND WATCHING TELEVISION THAT WEEK /653*5*0/ 2ICE HAS GRAMS OF PROTEIN PER SERVING
(291) BEANS HAVE GRAMS PER SERVING
(292) AND AN ORANGE HAS GRAMS PER SERVING 7RITE A VARIABLE EXPRESSION FOR THE TOTAL GRAMS OF PROTEIN IN X SERVINGS OF RICE
(293) Y SERVINGS OF BEANS
(294) AND Z ORANGES .07*&4 !T THE MOVIES
(295) POPCORN COSTS AND DRINKS COST 7RITE AN EXPRESSION TO FIND THE CHANGE YOU WILL RECEIVE FROM IF YOU BUY P POPCORNS AND D DRINKS &IND YOUR CHANGE FROM IF YOU TREAT PEOPLE TO POPCORN AND PEOPLE TO DRINKS (4)0353&410/4& ! PERSONAL #$ PLAYER COSTS AND #$S COST EACH 7RITE AN EXPRESSION TO REPRESENT THE TOTAL COST OF THE #$ PLAYER AND N #$S (OW MUCH WILL YOU SPEND FOR A PERSONAL #$ PLAYER AND #$S 7ILL YOU SPEND TWICE AS MUCH FOR #$S %XPLAIN (.6-5*1-&$)0*$& 9OU ARE SAVING MONEY TO BUY A BIKE THAT COSTS 9OU WANT TO BUY THE BIKE IN N WEEKS BY SAVING THE SAME AMOUNT OF MONEY EACH DAY (OW MUCH MONEY SHOULD YOU SAVE EACH DAY . N. 6 ]z . 7 ]z N. N 8 ]z . 9 ]z N. .&"463&.&/5 4HE VINE SHOWN BELOW GROWS THE SAME DISTANCE EVERY WEEK -EASURE THE VINES CURRENT LENGTH IN CENTIMETERS 7RITE A VARIABLE EXPRESSION FOR THE LENGTH OF THE VINE W WEEKS FROM NOW 4HEN ESTIMATE THE VINES LENGTH IN CENTIMETERS WEEKS FROM NOW.
(296) .
(297) . (01&/&/%&%."5) $ESCRIBE A REAL WORLD SITUATION THAT CAN BE MODELED BY THE EXPRESSION D #HOOSE ONE REASONABLE VALUE FOR D AND EVALUATE THE EXPRESSION 4HEN CHOOSE ONE UNREASONABLE VALUE FOR D AND EXPLAIN WHY IT IS UNREASONABLE (83*5*/( !N )NTERNET MUSIC CLUB OFFERS UNLIMITED DOWNLOADS FOR PER MONTH !NOTHER CLUB OFFERS DOWNLOADS FOR PER SONG 7RITE AN EXPRESSION FOR THE DIFFERENCE IN COST BETWEEN A MONTH SUBSCRIPTION AND THE PURCHASE OF N DOWNLOADED SONGS %XPLAIN WHAT IT MEANS WHEN THE EXPRESSION HAS A VALUE OF ZERO. 7BSJBCMFTBOE&YQSFTTJPOT. .
(298) 3 & " % * / ( * / . " 5 ). 3FBEUIFJOGPSNBUJPOCFMPXGPS&YFSDJTFTo. *DF4DVMQUVSFT )CE SCULPTURES CAN BE BEAUTIFUL . ADDITIONS TO SPECIAL PARTIES AND LARGE CELEBRATIONS 4HE SCULPTURES CAN LAST A SURPRISINGLY LONG TIME INDOORS !FTER TO HOURS
(299) THE SCULPTURE WILL BEGIN TO LOSE ITS DETAIL 4O MELT AN ENTIRE CM THICK ICE SCULPTURE WOULD TAKE ABOUT HOURS AT ROOM TEMPERATURE )T MELTS AT A CONSTANT RATE OF ABOUT CM IN ONE HOUR &TUJNBUF (OW MUCH DOES THE SCULPTURE . DESCRIBED ABOVE MELT IN HOURS )N HOURS 8SJUFBOE&WBMVBUF 7RITE AND EVALUATE AN EXPRESSION FOR HOW LONG IT . WOULD TAKE A SCULPTURE Y CENTIMETERS THICK TO MELT IF IT MELTS AT THE SAME RATE 4HEN EVALUATE WHEN Y CM 2OUND TO THE NEAREST HOUR 3FBTPOJOH 7HAT FACTORS MIGHT AFFECT THE MELTING RATE OF THE . ICE SCULPTURE . $)"--&/(& *IMS CAR GETS MILES PER GALLON IN CITY TRAFFIC AND MILES . PER GALLON ON THE HIGHWAY (E HAS GALLONS OF GASOLINE IN HIS TANK AND IS GOING ON A TRIP A 7RITE AN EXPRESSION FOR THE AMOUNT OF GAS *IM USES DRIVING H HIGHWAY . MILES AND C CITY MILES B #AN *IM MAKE A MILE TRIP WITH MILES OF CITY TRAFFIC WITHOUT . BUYING MORE GASOLINE %XPLAIN YOUR REASONING $)"--&/(& 4HE POPULATION OF !MESVILLE CAN BE REPRESENTED BY . X
(300) AND THE POPULATION OF "OOMVILLE CAN BE REPRESENTED BY X
(301) WHERE X IS THE NUMBER OF YEARS AFTER )N HOW MANY YEARS WILL THE POPULATION OF THE TWO TOWNS BE EQUAL 7HAT WILL THE POPULATION OF EACH TOWN BE IN THE YEAR . .*9&%3&7*&8 %VALUATE THE EXPRESSION Q. 1SFQBSFGPS -FTTPO JO&YTo. + . . + . . . + . + . . . (.6-5*1-&$)0*$& 9OU BUY NOTEBOOKS %ACH NOTEBOOK COSTS (OW MUCH DO YOU PAY FOR ALL NOTEBOOKS Q. . . 6 . 7 . &953"13"$5*$&GPS-FTTPO Q. 8 . 9 . 0/-*/&26*;BUDMBTT[POFDPN.
(302) . 1PXFSTBOE &YQPOFOUT. . #FGPSF . . /PX . :PVMMFWBMVBUFFYQSFTTJPOTXJUIQPXFST. . 8IZ . 4PZPVDBODBMDVMBUFTQFFE BTJO&Y. :PVFWBMVBUFEOVNFSJDBMBOEWBSJBCMFFYQSFTTJPOT. 8BUFSGBMM ! STONE FALLS OVER THE EDGE OF A CLIFF NEXT TO A WATERFALL 4HE STONE HITS THE WATER SECONDS LATER (OW TALL IS THE CLIFF . ,&:70$"#6-"3: s POWER
(303) P s EXPONENT
(304) P . 9OU WILL FIND THE HEIGHT OF THE CLIFF IN %XAMPLE USING AN EXPRESSION WITH A POWER ! POWER IS A PRODUCT MADE OF REPEATED FACTORS 4HE EXPONENT TELLS HOW MANY TIMES THE BASE IS USED AS A FACTOR. s BASE
(305) P . #BTF . &YQPOFOU. B B + B + B + B + B + B + B + B 1PXFS . . &9".1-&. C JTBGBDUPSUJNFT. 3FBEJOH1PXFST. 0OWER . 2EPEATED -ULTIPLICATION . $ESCRIPTION IN 7ORDS. !6/)$ %22/23. . + . TO THE SECOND POWER
(306) OR SQUARED. .BLFTVSFUIBUZPVXSJUF BOEOPU+. . + + . TO THE THIRD POWER
(307) OR CUBED. Y . Y + Y + Y + Y + Y . Y TO THE FIFTH POWER. &9".1-&. &WBMVBUJOHB1PXFS. %VALUATE FIVE CUBED . + + . . . (6*%&%13"$5*$&. 8SJUFBTBGBDUPSUJNFT. . .VMUJQMZ. BUDMBTT[POFDPN. GPS&YBNQMFTBOE. 7RITE THE PRODUCT AS A POWER . + + + . W + W. $ESCRIBE THE POWER IN WORDS AND THEN EVALUATE . . . . 1PXFSTBOE&YQPOFOUT. .
(308) 0SEFSPG0QFSBUJPOT 7HEN YOU EVALUATE EXPRESSIONS WITH POWERS
(309) EVALUATE ANY POWERS BEFORE MULTIPLYING OR DIVIDING. &OR 9OUR .OTEBOOK. ,&:$0/$&15 0SEFSPG0QFSBUJPOT. %VALUATE EXPRESSIONS INSIDE GROUPING SYMBOLS %VALUATE POWERS -ULTIPLY AND DIVIDE FROM LEFT TO RIGHT !DD AND SUBTRACT FROM LEFT TO RIGHT. &9".1-&. 6TJOHB1PXFS. $MJGG)FJHIU 4O FIND THE HEIGHT OF THE CLIFF FROM THE PREVIOUS PAGE
(310) USE THE EXPRESSION T
(311) THE DISTANCE IN FEET THAT AN OBJECT FALLS IN T SECONDS AFTER IT BEGINS TO FALL. 4!+% ./4%3 *OZPVSOPUFT ZPVNBZ XBOUUPMBCFMFBDI FYBNQMFUIBUZPVDPQZ XJUIBUJUMFTPUIBUZPV DBOUFMMXIBUJUJTBCPVU. . T . . . &WBMVBUFUIFQPXFS. . . .VMUJQMZ. 4VCTUJUVUFGPSU. C "OTXFS 4HE HEIGHT OF THE CLIFF IS FEET. &9".1-&. 6TJOHUIF0SEFSPG0QFSBUJPOT. %VALUATE THE EXPRESSION A . &WBMVBUFJOTJEFHSPVQJOHTZNCPMT. . . &WBMVBUFQPXFST. . . "EEBOETVCUSBDUGSPNMFGUUPSJHIU. B + + . &WBMVBUFJOTJEFHSPVQJOHTZNCPMT. + . &WBMVBUFQPXFST. . .VMUJQMZBOEEJWJEFGSPNMFGUUPSJHIU. BUDMBTT[POFDPN. . (6*%&%13"$5*$&. GPS&YBNQMFTBOE. %VALUATE THE EXPRESSION . . . 8IBU*G 3UPPOSE THE FALLING STONE IN %XAMPLE TOOK SECONDS TO HIT THE WATER %VALUATE THE EXPRESSION TO FIND THE HEIGHT OF THE CLIFF. . $IBQUFS7BSJBCMFTBOE&RVBUJPOT.
(312) . ( 45"/%"3%*;&%5&4513"$5*$& &YT BOE. &9&3$*4&4. )0.&803, ,&: . . )*/54"/%)0.&803,)&-1 GPS&YT BUDMBTT[POFDPN. 4,*--13"$5*$& 70$"#6-"3: 7RITE A POWER AND LABEL THE BASE AND THE EXPONENT 70$"#6-"3: 5SING THE ORDER OF OPERATIONS
(313) WHAT SHOULD YOU DO FIRST . BEFORE EVALUATING POWERS 3%% %8!-0,% . 3&"%*/(108&34 7RITE THE PRODUCT AS A POWER AND DESCRIBE IT IN WORDS. POQ GPS&YTo. + + + + . + + . N + N + N + N + N + N. &7"-6"5*/("108&3 %VALUATE THE POWER. 3%% %8!-0,% . THREE SQUARED . POQ GPS&YTo. . . . . . . . . . ELEVEN CUBED . ONE TO THE NINTH. &3303"/"-:4*4 $ESCRIBE AND CORRECT THE ERROR MADE IN EVALUATING . , ',' &). YZ "-(" %VALUATE THE EXPRESSION WHEN G . 3%% %8!-0,% . G . G . G . POQ GPS&YTo. G . G . G . 64*/(5)&03%&30'01&3"5*0/4 %VALUATE THE EXPRESSION. 3%% %8!-0,% . . . . POQ GPS&YTo. . . . . . . . ]z . . . BUDMBTT[POFDPN. . (.6-5*1-&$)0*$& 7HICH OPERATION SHOULD BE DONE LAST WHEN EVALUATING . 6 &IND . 7 &IND . 8 3UBTRACT . 9 $IVIDE. $0.1"3*/(7"-6&4 #OPY AND COMPLETE THE STATEMENT USING
(314)
(315) OR . . . |]z |]z . . . | . . | . ]z ]z . . | . . | . ]z ]z . 1PXFSTBOE&YQPOFOUT. .
(316) YZ "-(" %VALUATE THE EXPRESSION WHEN A
(317) B
(318) AND C . A B . C A . + B A. -00,'03"1"55&3/ %VALUATING
(319)
(320)
(321)
(322)
(323)
(324)
(325) AND GIVES THE SEQUENCE
(326)
(327)
(328)
(329)
(330)
(331)
(332) AND &IND THE PATTERN WITH THE ONES DIGITS 0REDICT THE ONES DIGIT IN -00,'03"1"55&3/ %VALUATE
(333)
(334)
(335)
(336)
(337)
(338)
(339) AND 0REDICT THE ONES DIGIT IN *USTIFY YOUR REASONING -00,'03"1"55&3/ %VALUATE
(340)
(341)
(342)
(343)
(344) AND 0REDICT THE ONES DIGIT IN
(345) AND IN *USTIFY YOUR REASONING $)"--&/(& &IND THE VALUE OF X. X . X . X . 130#-&.40-7*/( 3%% %8!-0,% POQ GPS&YTo. (&0.&53: 4HE AREA OF A SQUARE IS S
(346) WHERE S IS THE LENGTH OF ONE SIDE OF . THE SQUARE &IND THE AREA OF A SQUARE THAT HAS A SIDE LENGTH OF INCHES (.6-5*1-&$)0*$& 4HE VOLUME OF A CUBE IS S
(347) WHERE S IS THE LENGTH OF ONE SIDE OF THE CUBE ! CUBE HAS A SIDE LENGTH OF CENTIMETERS 7HAT IS THE VOLUME OF THE CUBE IN CUBIC CENTIMETERS . 6 . 7 . 8 . 9 . .6-5*45&1130#-&. !T +AUNOLO IN (AWAII
(348) DIVERS JUMP FROM A PLATFORM ON TOP OF A CLIFF FEET ABOVE THE WATER !T TIME T SECONDS
(349) A DIVER HAS FALLEN T FEET A $OES A DIVER REACH THE SURFACE OF THE WATER IN . SECONDS %XPLAIN YOUR REASONING B (OW MUCH LOWER WOULD A DIVER BE AFTER SECONDS . THAN AFTER SECONDS %XPLAIN (4)0353&410/4& $ESCRIBE HOW TO FIND THE VALUE OF USING THE FACT THAT %XPLAIN YOUR REASONING. -00,'03"1"55&3/ 7RITE AN EXPRESSION FOR THE NTH NUMBER OF THE SEQUENCE OF NUMBERS
(350)
(351)
(352)
(353) (.6-5*1-&$)0*$& *OE PUTS INTO A RETIREMENT ACCOUNT THAT EARNS INTEREST SO THAT THE ACCOUNT HAS TWO TIMES THE VALUE EVERY YEARS 7HAT EXPRESSION REPRESENTS THE AMOUNT OF MONEY IN THE ACCOUNT AFTER YEARS . 6 . 7 . 8 . 9 . (83*5*/( %XPLAIN WHY X X FOR X p . . (45"/%"3%*;&%5&4513"$5*$&. )*/54"/%)0.&803,)&-1BUDMBTT[POFDPN.
(354) 60. FOOTBALL The season attendance at your school’s football games is. 1000 people one year. For each of the next 3 years, the attendance is double the previous year’s attendance. Write an expression with a power that shows the season attendance at the football games after 3 years. 61. INLINE SKATING An inline skater is at the top of a hill. The hill is 30 feet. long and is angled such that the speed of a skater going down the hill doubles every 6 feet. A skater is going 3 miles per hour after the first 6 feet. What is the speed of the skater when the skater reaches the bottom of the hill? 62.. MULTIPLE REPRESENTATIONS At an archeological dig, the lead scientist decides that the dig should be organized by repeatedly dividing the square area into four smaller squares as shown.. Stage 1. Stage 2. Stage 3. a. Make a Table Make a table that shows the number of squares at each. stage from the 1st stage through the 6th stage. b. Write an Expression Write an expression as a power for the number. of squares in the dig at the 7th stage. c. Write a Description How many squares are along each side of the dig. at the 7th stage? If the whole dig is 64 feet on a side, what is the side length of each individual square at the 7th stage? 63.. ★ EXTENDED RESPONSE. On an ostrich farm a single breeding pair hatches 30 eggs per season. These hatchlings will form breeding pairs the following season. Each new breeding pair will hatch about 30 eggs per season. How many breeding pairs would be on the farm after 1 breeding season, assuming all survive? after 2 seasons? after n seasons? Explain your answer.. 64. EVALUATING POWERS Some powers of 11 read the. same forward and backward. For example 111 5 11, 112 5 121. What is the smallest n such that 11n does nott read the same forward and backward? 65. REASONING Evaluate 74, 73, 72, and 71. Based on this pattern, what. is the value of 7 0? Predictt the value of x 0, where x is a whole number greater than 0. Explain your reasoning. 66. CHALLENGE The personal computers of the early. 1980s had 64 kilobytes of memory. Today they have more than one gigabyte of memory. Use the table. Write, as a power, the number of bytes of memory that personal computers of the early 1980s had.. Name. Bytes. Kilobyte. 210. Megabyte. 220. Gigabyte. 230. 1.4 Powers and Exponents. 23.
(355) .*9&%3&7*&8 $)004&"453"5&(: 5SE A STRATEGY FROM THE LIST TO SOLVE THE FOLLOWING . PROBLEM %XPLAIN YOUR CHOICE OF STRATEGY 9OU HAVE SHIRTS AND PAIRS OF PANTS THAT YOU . 0ROBLEM 3OLVING 3TRATEGIES. ARE PACKING FOR A TRIP 9OU CAN WEAR EACH SHIRT WITH EACH PAIR OF PANTS (OW MANY DIFFERENT OUTFITS ARE POSSIBLE . $RAW A $IAGRAM e#,-, -AKE A ,IST e#,.% N ,OOK FOR A 0ATTERN e#,.& N N . %VALUATE THE EXPRESSION WHEN X AND Y Q. 1SFQBSFGPS -FTTPO JO&YTo. X . X Y . Y X. 1"35:"55&/%"/$& 4HE HISTOGRAM . . 6 . 7 . . 8 . 9 . ]z .
(356) . SHOWS THE TIMES THE PEOPLE ARRIVED AT A PARTY (OW MANY PEOPLE ATTENDED Q. (.6-5*1-&$)0*$& 5SE THE HISTOGRAM (OW MANY PEOPLE ARRIVED BETWEEN 0- AND 0- Q. X. ]z . . . . . . 26*;GPS-FTTPOTo )0.&36/4 )N %XERCISES n
(357) USE THE TABLE AT THE RIGHT )T SHOWS THE RECORD NUMBER OF HOME RUNS HIT IN A SINGLE SEASON BY POSITION AS OF Q. -AKE A BAR GRAPH THAT DISPLAYS THE DATA (OW MANY MORE HOME RUNS IS THE RECORD FOR OUTFIELD THAN FOR PITCHER . %VALUATE THE EXPRESSION Q. 1PTJUJPO. )PNF3VOT. $BUDIFS TUCBTF 1JUDIFS OECBTF 4IPSUTUPQ SECBTF 0VUmFME. . + . . . . . + . 1-"/54 ! PLANT IS INCHES TALL AND GROWS INCHES EACH YEAR !NOTHER PLANT IS INCHES TALL AND GROWS INCHES EACH YEAR 7RITE A VARIABLE EXPRESSION FOR EACH PLANTS HEIGHT IN X YEARS 4HEN EVALUATE THE EXPRESSIONS TO FIND THE HEIGHTS IN YEARS #OMPARE THE RESULTS Q. %VALUATE THE EXPRESSION FOR THE GIVEN VALUE OF THE VARIABLE Q. . A WHEN A . B + WHEN B . + Z WHEN Z . X + WHEN X . &953"13"$5*$&GPS-FTTPO Q. 0/-*/&26*;BUDMBTT[POFDPN.
(358) STATE TEST PRACTICE. classzone.com. Lessons 1.1–1.4 1. MULTI-STEP PROBLEM Muntz metal is an. alloy of copper and zinc. A sample of Muntz metal has a volume of 15 cubic centimeters. The volume of the copper in the sample is x cubic centimeters. The mass of this copper is 80.1 grams. a. Write an expression in terms of x for the. volume of zinc in the sample.. 4. EXTENDED RESPONSE You are participating. in a 20 mile fundraising walk. People can contribute a certain amount per mile that you walk or just make a fixed contribution. The fixed individual contributions that you collected total $150. The per mile contributions total $15 per mile. a. Write an expression for the amount of. b. Use the fact that 1 cubic centimeter of. zinc has a mass of 7.1 grams to write an expression in terms of x for the total mass of the sample. c. What is the sample’s total mass if. money you raise if you walk the entire 20 miles. b. How much will you raise if you complete. the walk? c. How far will you have to walk to raise. the volume of its copper is 9 cubic centimeters? Show your calculations.. $300?. 2. EXTENDED RESPONSE The frequency table. shows the heights of 30 students. Height (inches). Frequency. 54–55.9. 1. 56–57.9. 2. 58–59.9. 4. 60–61.9. 5. 62–63.9. 7. 64–65.9. 6. 66–67.9. 3. 68–69.9. 2. a. Make a histogram of the data shown in. the frequency table. b. Which height interval has the most. students? c. Can you use the histogram to determine. the number of students who are between 60 and 65 inches tall? Explain.. 5. SHORT RESPONSE Your car’s fuel gauge is. broken. The car can go 22 miles on one gallon of gasoline. You start a trip with 13 gallons of gasoline, and you want to always have at least two gallons in the tank. What is the farthest you should drive before stopping for more gasoline? Explain. 6. OPEN-ENDED Consider the expression. 2. 3. GRIDDED ANSWER The expression 16t is. used to find the distance in feet that an object has fallen t seconds after being dropped. When dropping a rock off of a 512 foot cliff, how many more feet does it fall in 4 seconds than in 3 seconds?. 3 1 18 4 3 1 3 3 5 1 3. Find at least six values for the expressions that result from inserting one pair of parentheses. For one of these expressions, write a real-world problem that can be modeled by the expression. Show your calculations. Mixed Review of Problem Solving. 25.
(359) . &RVBUJPOT BOE4PMVUJPOT. . #FGPSF . :PVXSPUFBOEFWBMVBUFEWBSJBCMFFYQSFTTJPOT. . /PX . . 8IZ . ,&:70$"#6-"3: s EQUATION
(360) P . :PVMMXSJUFBOETPMWFFRVBUJPOTVTJOHNFOUBMNBUI 4PZPVDBOGJOEVOLOPXONFBTVSFT BTJO&Y. "$5*7*5:. s SOLUTION
(361) P . 9OU CAN USE ALGEBRA TILES TO SOLVE EQUATIONS. s SOLVING AN EQUATION
(362) P . )N AN EQUATION
(363) THE QUANTITIES ON EACH SIDE OF THE EQUAL SIGN HAVE THE SAME VALUE 4HE ALGEBRA TILE MODEL BELOW REPRESENTS THE EQUATION X . 34%0 7ITH HOW MANY TILES SHOULD YOU REPLACE THE X TILE SO THAT THE QUANTITIES ON BOTH SIDES OF THE EQUAL SIGN HAVE THE SAME VALUE %XPLAIN. 34%0 7HAT VALUE OF X MAKES THE EQUATION X A TRUE STATEMENT -AKE A MODEL TO REPRESENT THE EQUATION 4HEN TELL WHAT VALUE OF X MAKES THE EQUATION A TRUE STATEMENT X . X . X . !N EQUATION IS A MATHEMATICAL SENTENCE FORMED BY PLACING AN EQUAL SIGN BETWEEN TWO EXPRESSIONS ! SOLUTION OF A VARIABLE EQUATION IS A NUMBER THAT YOU CAN SUBSTITUTE FOR A VARIABLE TO MAKE THE EQUATION TRUE &INDING ALL THE SOLUTIONS OF AN EQUATION IS CALLED SOLVING AN EQUATION. & 9 " . 1 - & 6TJOH.FOUBM.BUIUP4PMWF&RVBUJPOT 3OLVE THE EQUATION USING MENTAL MATH A N . B X . C R . 40-65*0/ 4O SOLVE AN EQUATION USING MENTAL MATH
(364) THINK OF THE EQUATION AS A QUESTION A MINUS WHAT NUMBER EQUALS .
(365) SO N . B TIMES WHAT NUMBER EQUALS .
(366) SO X . C 7HAT NUMBER DIVIDED BY EQUALS
(367) SO R . . $IBQUFS7BSJBCMFTBOE&RVBUJPOT.
(368) & 9 " . 1 - & $IFDLJOH4PMVUJPOT 4ELL WHETHER THE VALUE OF THE VARIABLE IS A SOLUTION OF N A N . B N . 40-65*0/ 3UBSTITUTE FOR N AND THEN SIMPLIFY A. 2%!$).'. N . B. . . 5IFTZNCPMNFBOT BSFUIFTFWBMVFTFRVBM 5IFpTZNCPMNFBOTJT OPUFRVBMUP. N . . . p . . . C "OTXFS IS NOT A SOLUTION . C "OTXFS IS A SOLUTION. & 9 " . 1 - & 8SJUJOHBO&RVBUJPO 5JNFT4RVBSF 4HE 4IMES 3QUARE .EW 9EARS %VE "ALL . DROPS A TOTAL OF FEET IN SECONDS !FTER SECONDS IT HAS DROPPED FEET (OW MANY MORE FEET WILL IT DROP . 40-65*0/ 9OU CAN USE A VERBAL MODEL TO WRITE AN EQUATION ,ET D REPRESENT THE DISTANCE LEFT TO DROP
(369) . . .
(370) . .
(371) . 8SJUFBWFSCBMNPEFM. . . . . . . D . 4VCTUJUVUF. . . . . . 6TFNFOUBMNBUI. C "OTXFS "ECAUSE D
(372) THE BALL WILL DROP MORE FEET $IFDL 9OU CAN CHECK YOUR ANSWER BY FINDING THE SUM OF AND . . . . (6*%&%13"$5*$&. GPS&YBNQMFT BOE. 3OLVE THE EQUATION USING MENTAL MATH X . N . T . 4ELL WHETHER THE VALUE OF THE VARIABLE IS A SOLUTION OF THE EQUATION A A . Y Y . N
(373) N . -BVOESZ *ULIE HAS TO WASH HER CLOTHES AT THE LAUNDROMAT %ACH LOAD COSTS TO WASH AND TO DRY (OW MANY LOADS CAN SHE DO &RVBUJPOTBOE4PMVUJPOT. .
(374) (. &9".1-&. 4UBOEBSEJ[FE5FTU1SBDUJDF. )FJHIUT #ESAR
(375) ,UIS
(376) AND -ARCO ARE BROTHERS 4HE DIFFERENCE OF #ESARS AND -ARCOS HEIGHTS IS INCHES ,UIS IS INCHES TALLER THAN #ESAR -ARCO IS INCHES TALL (OW TALL IS ,UIS . %,)-).!4% #(/)#%3. 6 INCHES . -VJTJTUBMMFSUIBO$FTBS BOE$FTBSJTUBMMFS UIBO.BSDP XIPJT JODIFTUBMM4P ZPV DBOFMJNJOBUFDIPJDF". 7 INCHES . 8 INCHES . 9 INCHES. 40-65*0/. 34%0 7RITE AND SOLVE AN EQUATION TO FIND #ESARS HEIGHT #ESARS HEIGHT -ARCOS HEIGHT . 8SJUFBWFSCBMNPEFM. . C . . . . 4VCTUJUVUFGPS.BSDPTIFJHIU. . . . . . 6TFNFOUBMNBUI. #ESAR IS INCHES TALL. 34%0 5SE #ESARS HEIGHT TO FIND ,UISS HEIGHT !DD C "OTXFS ,UIS IS INCHES TALL 4HE CORRECT ANSWER IS $ 6 7 8 . . (6*%&%13"$5*$&. GPS&YBNQMF. 8IBU*G )N %XAMPLE
(377) SUPPOSE ,UIS IS INCHES TALLER THAN -ARCO
(378) . #ESAR IS INCHES TALL
(379) AND THE DIFFERENCE OF #ESARS AND -ARCOS HEIGHT IS INCHES (OW TALL IS ,UIS . . ( 45"/%"3%*;&%5&4513"$5*$& &YT BOE. &9&3$*4&4. )0.&803, ,&: . . )*/54"/%)0.&803,)&-1 GPS&YT BUDMBTT[POFDPN. 4,*--13"$5*$& 70$"#6-"3: #OPY AND COMPLETE THE STATEMENT. 4HE PROCESS OF FINDING ALL POSSIBLE VALUES THAT MAKE A MATHEMATICAL SENTENCE TRUE IS !N IS A MATHEMATICAL SENTENCE FORMED BY PLACING AN EQUAL SIGN BETWEEN TWO EXPRESSIONS 3%% %8!-0,% . 40-7*/(&26"5*0/4 3OLVE THE EQUATION USING MENTAL MATH. POQ GPS&YTo. P . Y . X . Z . M . C . K . ]z . R . Y . X . Z . X . C . A . N. . . K . BUDMBTT[POFDPN. $IBQUFS7BSJBCMFTBOE&RVBUJPOT.
(380) $)&$,*/(40-65*0/4 4ELL WHETHER THE GIVEN VALUE IS A SOLUTION. 3%% %8!-0,% . X X . X X . POQ GPS&YTo. X X . X X . B B . D D . (.6-5*1-&$)0*$& 7HICH OF THE FOLLOWING IS A SOLUTION OF X 6 . 7 . 8 . 9 . &3303"/"-:4*4 ! STUDENT SAYS THAT A SOLUTION OF X IS . %XPLAIN WHY THE STUDENTS SOLUTION IS INCORRECT 53"/4-"5*/(&26"5*0/4 7RITE THE EQUATION AS A QUESTION IN WORDS . 4HEN SOLVE THE EQUATION T . T . T . T . ]z . *%&/5*':*/(40-65*0/4 4ELL WHICH OF THE GIVEN VALUES IS A SOLUTION OF . THE EQUATION X X X
(381)
(382) . X X X
(383)
(384) . X X X
(385)
(386) . X X X
(387)
(388) . X X X
(389)
(390) . X X X
(391)
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