LESSON 3.1. Your Turn

Transcription

1 MODULE Rational Numbers Are You Ready?. 9 7 = 9 7 =. 7 =. = =. 9 = 9 = or. = = = =. = = = 7. = = 7 = = 9 7 = 9 7 = 9. = =. = = =. = = = =. = = = ( 7 - ) ( - ) + ( ) ( - ) - ( ) LESSON. Your Turn Dividing into _- _-9 _-7 _- _- _- Second appearance of Because the number repeats during the division -. _ 7, or -.77

2 ... Dividing into _- Second appearance of Because the number repeats during the division. _, or. _.. 9. _- _ = - + _.7. _- _- _- -.7; terminating = -. 7 = 7 + = 7.. Dividing into _- Second appearance of Because the number repeats during the division 7. _, or 7. ; repeating Guided Practice _... _-.; terminating _ _- 9 _-9 -.9; terminating. Simplify the fraction: = =.. Dividing into _- Second appearance of Because the number repeats during the division. _, or. ; repeating Dividing into _-9 _-9 Second appearance of Because the number repeats during the division. _, or. ; repeating Dividing into 7 _- 7 Second appearance of 7 Because the number 7 repeats during the division. 7 _, or ; repeating _.. 9. _-7 _- -.; terminating _. 7.. _-.; terminating.. 7. _-7 7 _-7 _- _- First appearance of _-7 9 _- Second appearance of Because the number repeats during the division -. _, or -. ; repeating _. 9.. _- _-.; terminating

3 . - = - + = _- 7 _- _- First appearance of _- Second appearance of Because the number repeats during the division - 7 ; -. _ or = + 9 _.9 9. _- 9 _-9 9 ; _- _- _- - ; -. = 9 = - + = -. 7 = 7 + = 7.. _- _- ; 7.. = + = _- 7 _- First appearance of _- _-77 Second appearance of Because the number repeats during the division 97 ;. _ 7 or = - + = -.. _- _- First appearance of _-9 Second appearance of Because the number repeats during the division - ; -. _ or -.. = + =.. _-9 First appearance of _- Second appearance of Because the number repeats during the division. _, or. lbs of apples = _.7. _- 7 _- _- _- -.7 lb = -. Disagree; the definition of a rational number is a number that can be written as the ratio of two integers with a denominator not equal to zero, and is a well-defined ratio of two integers. Tom did 7 not divide long enough to correctly determine that the quotient is a repeating decimal.

4 Independent Practice basketball players 9. football players =.. Dividing into _- _- Second appearance of Because the number repeats during the division ;. _, or. ; repeating hockey players. lacrosse players =.. _- ;.; terminating polo players. football players =.. Dividing into _- 7 _- Second appearance of Because the number repeats during the division ;. _ or. ; repeating lacrosse players. rugby players = = =.. Dividing into _- Second appearances of Because the number repeats during the division ;. _, or. ; repeating football players. soccer players = = ; ; terminating. Agree. Sample answer: There are players on the lacrosse team, and dividing the number of any other team by will simply move the decimal point one digit to the left. Therefore, the ratio of any team over the lacrosse team will be a decimal that terminates one place to the right of the decimal point.. a. - 7 = = - 9 b. _.7 9. _- 7 _- _- _- -.7 c. Sample answer: 7 is very close to. Therefore, You could estimate that the water level changes by inches per month. The total change in the water level at the end of the -month period would be approximately - inches.. integer; terminating 7. Ben is taller, because Ben s height of is equal to or. ft, while Marcus height of 7 is equal to 7 or.9 ft. The first store has the better deal because they offer or.7 of a bushel for $9 while the second store offers only or. of a bushel for $9. Focus on Higher Order Thinking 9. When the number is the denominator in a fraction its decimal form is simply the numerator. In all other cases concerning numbers to, the division process stops when either the remainder is or when the digits begin to repeat. When the numbers,,, or are in the denominator, the decimal form of a fraction will terminate. When the numbers,, 7, or 9 are in the denominator, the decimal form of a fraction will be a repeating decimal.. Julie made a higher score on her math test, since her math test score of is equal to a repeating decimal of approximately.9 while her science test score of 9 is equal to a terminating decimal of.9. Sample answer: The difference in scores cannot be determined by simply comparing the numerators of the two fractions because the denominators are not the same. For Julie to compare her scores she needs to divide the denominators into their respective numerators until one of the quotients is found to be greater than the other.. No; although the digits in the decimal appear to follow a pattern, a repeating decimal must have the same combination of digits that repeat, such as.

5 LESSON. Your Turn. + = (-.) = = ( - ) = = (- ) = = ( - ) = The overall change is cups (-) + (- ) + (- ) (. ) ( - ) + Guided Practice (-.) = = = (- ) = ( - ) = = = ; $ = ; $ (-) + (-.).7 + (-7.) -( ) = - + = (. -. ) (-) + (-.). + (-.) + (-) + (-) -. + (- = - ) ( - ) = - = -

6 . - + = - ( - ) = (- = -9 ) 7. Start at - and move units to the right. The sum is. Independent Practice. The opposite of +9 is $. + $. = $. - $. = $ m + (-.) = -.7 m. mi + (- mi ) = mi = mi.. m + (-. m ) = m above sea level. + - = - = pts. January: Income - Expenses $, - $,9. - ( $,9. - $, ) -$. February: Income - Expenses $, - $,. -($,. - $,) -$. Kameh lost $. in January, and $. in February.. June: Income - Expenses $, - $,. $.77 July: Income - Expenses $, - $,9. $. August: Income - Expenses $, - $,. $7. Kameh gained $.77 in June, $. in July, and $7. in August.. First, sum all the values in the Income column. Then, sum all the values in the Expenses column. Subtract the total expenses from the total income. Finally, add the $ profit from December (not shown in the table) to find the total profit or loss of the bakery by the end of August. Income = $, + $, + $, + $, $, + $, = $,9 Expenses = $,9. + $,. + $, + $,. + $,9. + $,. = $,9.9 Profit = $,9 - $,9.9 + $ = $ is the opposite or additive inverse of.. a. 7 ( ) + (-) = 7 - = pts b. ( ) + (-) = - = - pts c ( ) + ( ) + ( ) + ( ) + ( ); or ( ) + ( ) Focus on Higher Order Thinking 9. The sum of two negative rational numbers is always negative. The sum of a negative rational number and a positive rational number is negative if the absolute value of the negative number is greater than that of the positive number.. Sample answer: The student might have subtracted the absolute values of the numbers.. Yes:. and -. are opposites and -. and. are opposites, so the expression [. + (-.)] + (-. +. ) can be viewed as the sum of two opposites, which is always. LESSON. Your Turn = = = (-.) = ( - ) = Guided Practice (-) = = -

7 = = = (-.) (. -. ) ( - ) = + =. - ( - ) = + = ( - ) (- ) m + (. m ) = -. m. m below sea level (-) = -7 7, or 7., yards. Change in height = Starting height - ending height ft - (- ft ) = ft + ft = ft (-.) = -7.; $7.. Explain that she is supposed to start at positive on the number line, then move places to the left because she is subtracting a positive number. She will end on the number -, which is the answer. Independent Practice. -. C - 7. C = -. C ft -. ft +. ft = -9.7 ft; The diver is 9.7 ft below the surface.. -9, ft - (-, ft ) =, ft 9. 9, ft - (-,9 ft ) =, ft 7, ft - (-, ft ) = 9, ft Mars has the greater difference by: 9, ft - (, ft ) =,77 ft. a. - F + 7 F - F b. 7 F - F c. - F + 7 F - F 7 F - F - F 7 F - 7 F = F 7 F - F = F. a. -$. + (-$.7) = -$. b. -$. + (-$) = -$. c. $., since -$. + $. =. a. No,. in. =. in. b.. in. - in. =. in. more needed. a. ft - 7. ft +. ft b. ft - 7. ft +. ft ft +. ft - 7. ft 7. ft - 7. ft = ft; because he moved the same distance backward and forward.. a. Yes, $. + $.9 + $.99 = $. < $ b. $ - $. = $.7 left over Focus on Higher Order Thinking. The Commutative Property of Addition (CPA) could be used to simplify the two terms that already have a common denominator first: + ) (- + ) (- ) ( = (- 7 + ) (- ( -7 + ( - ) ) + ( - ) ( - ) + ( - ) ( - ) + ( - ) ( - ) + ( - ) ( - + ( - ) ) ( - ) = - ) + ( - ) by CPA. Lowest: -. F + (- F) = -. F Highest: -9. F + (- F) = -. F 7. Sample answer: Yes, because both numbers are rational numbers, each can be written as the ratio of two integers, for example a c and. Both fractions b d could be given a common denominator and then one could then be subtracted from the other. The result would be a fraction, which is a rational number.. No. Sample answer: It is possible for the difference of two negative numbers to be negative: [ - - ( - ) = - ], but it is also possible for the difference to be positive: [ - - ( - ) = ].

8 LESSON. Your Turn ( -. ) = ( -. ) = ( - 7) = 7 = 7 = 7 = 7. ( - ) ( - 7) ( - ) = - 7 = - 7 = - 7. (- ) (- ) ( = ). ( ) ( - 9 ) ( Guided Practice.. = = ) = - 9 = - = ( - ) = ( - ) = - = - = (- = ) - = - = ( - ) = - - = = = =. (-) = -. -.( ) = (-.) =... ( ) =. 9. -(-.) =. -.( ) = -7. )( ) ) ( ). = ( = ( = =. - 7 (- ) (- 7 = ) (- 7 ) (- 7 ) = (- 7 ) = - 7 = - = -. (- ) = (- ) = - = - = - 7

9 . (- ) ( ) (- = 7) 7 = 7 = 7. (-.) = -, or a $ change in price.. (-) = -,, or a $, change. 7. Sample answer: Count the number of times there is a negative sign. If there are an even number of negative signs, then the final product will be positive. If there is an odd number of negative signs, then the final product will be negative. Independent Practice. a. (-9.9) Note that the change in her bank account balance does not depend on the initial amount. b. + (-9.9) = - 9. =.; $. 9. Sample answer: Start at, then move. units to the left (because. is negative in this case) times. You are now on -. Then, because is negative in this case, we want to move to the opposite of - which is.. (- = - ) ( ) = - = - = - - min. At the same rate, the watch will be minutes behind after weeks.. (-) = -97 ft. The change in depth is -97 ft. Therefore, the submarine will be 97 below sea level (below the surface).. + ( - ) (. ) = + ( -. ) =. cups left. Matthew is incorrect; Sample answer: Matthew should have said that multiplying by two negatives is like multiplying the opposite of a positive twice. The opposite of a positive twice brings you back to a positive.. ( - ) = -7 min. Therefore, she will be late by 7 minutes, or hour and minutes.. Total score is: ( ) + (. ) + 7 ( -. ) + ( -. ) = = -. =. pts Focus on Higher Order Thinking. Temperature at kilometers = Temp at ground level + change in temp = + (-.) = + (-) = - C 7. a b c d If the product of two numbers is positive, then the two numbers must have the same sign; either they are both positive or both negative. If the sum is negative, then at least one of the numbers must be negative. Therefore the two integers that add to -7 and multiply to must both be negative. The negative paired factors of are: - and -, - and -, and - and -. Of those choices, only - and - add to -7. LESSON. Your Turn.. - = -. = =.. = = - = = -.. The diver's change in elevation was -. feet per minute = = = -. = - = - = - = -

10 - 9. = - = - = - = - Guided Practice = = = = - 7 = = - 7 = -. (- = ) - = - = - = - = = 9 = 7 = - 7 = = - = = = -, = - change in water level 9. = change in days = -. L day = -.7 L, or -.7 L per day day total change in price. = change in days = -$.7 day = -$9. per day on average total change in altitude. number of minutes = -. mi. min = -. mi min = -. mile per minute. First, find the sign of the numerator, which is negative. Next, find the sign of the denominator, which is negative. Therefore, the quotient will be positive because the numerator and denominator both have the same sign. Independent Practice. - = -. (- ).. = - + = - = - = - = = = - - = = - = - + = = 7.. (-.) = = -. = -.. = -.. = -. = -. = = 9 = 9 -. = - = - = =. =.. total hours number of days = h 7 days = h day total time lost: h days = 9 hours. day Alex usually runs hours per week, so divide by 7 to find that he runs hours per day. Since he is unable to run for days, his time is decreased by 9 hours, or -9.. total change in yards number of runs - = + yd 9 run = - 7 yd 9 run = - yd 9 runs = - yd 9 run = - 7, or -, yards per run 9

11 . -. C + (-7. C) + (-. C) + (-7. C ) =. C = -. C per day total profit. a. number of days = $,7 7 days = $ per day b. $ 7 days = $, day total change c. number of days = - $9 7 days = -$7 per day. total meters descended number of seconds = 99. m s =. m/s 7. When converting the division equation into a multiplication problem, he forgot to multiply by the reciprocal and instead multiplied by the fraction in the denominator. The correct answer is given by: - = - = year m ( 99 ) years = -.7 year m 7 years = -.9 m Focus on Higher Order Thinking 9. Sample answer: The average change in temperature per day would be given by - if the temperature were to drop of F over days. - F d = - 7 F d = - 7 F d, or - F d -.7 F d On average, the temperature changed by -.7 F every day.. Yes. By definition, the result of dividing an integer by a non-zero integer is a rational number.. Yes. The result of dividing an integer by a non-zero integer always results in a rational number, by definition. LESSON. Your Turn. Find the total commercial time. = 7 Find the total entertainment time. - 7 = Find the length of each entertainment segment. = Each entertainment segment is minutes long.. Find the number of cups of sugar in the bag. 9. Find the number of -cup portions in the bag batches can be made from the bag of sugar. Find the cost of batch..9.9 The cost of the sugar is $.9 per batch.. Convert the percent to a decimal. % =.% =. Find the worth after year. 7. = =. Find the worth after years = 9.9 Find the worth after years = 7.7 The stock is worth $,7.7. Guided Practice..h or. miles per hour =. miles. miles or. miles per hour = hours.. inches inches. inches -..7 inches. Sample answer: Using a calculator to solve a problem that involves complicated arithmetic can help you avoid errors. It can also help you to check solutions to any problems you solved by hand. Independent Practice. Find the total weight. 7. =. Find the weight each climber carries.. =. Each climber carries. kg.

12 . Find the available width on the page. - = Find half the width. = He should put the picture inches from each side of the page.. Find the amount of cereal needed for all the children. = = = Compare the total needed with the amount in the box. < Yes, there is enough Oaties for all the children. The amount needed is cups and that is less than the amount in the box, cups. 7. Find half of the distance that the referee walked. = 7 Find how far that distance is from the goal line. - 7 = 9 The referee is 9 yards from the nearest goal line.. Donovan s score was 9 = 7%, Marci s score was ( 7 + )% = %. 9. Find the number Marci answered correctly. % = = Find how many more that Marci answered than Donovan. - 9 = Marcie answered more questions correctly than Donovan.. Sample answer: Donovan got about out of questions right, or about %. Since Marci scored % more, that is about 9%. 9% is. So Marci answered about -, or more questions correctly than Donovan.. Yes, -.7 is a reasonable estimate. 9 is about and. is about. and 7. ( - ) = Sample answer: approximately -.7; Use a calculator. Divide -9 by 7, multiply the quotient by., then round the product.. Sample answer: Yes; Focus on Higher Order Thinking. Find the time of the descent (-.) =. Find the time for the ascent = Find the distance of the ascent (-79 ) 9 = Find the rate of the ascent. = The diver s rate of change in elevation during the ascent was ft/min.. Sample answer: () Convert the mixed number 7 to the decimal 7., find the sum of 7. and.9, then multiply the result by.7. () Convert the mixed number 7 to the decimal 7.. Then use the Distributive Property so that (7. +.9).7 = (7.)(.7) + (.9)(.7). Multiply both 7. and.9 by.7, and add the products. I would use the first method because there are fewer steps and so fewer chances to make errors.. Sample answer: You need to know how many gallons of paint you need to paint a wall. Measure the length and width of the wall with a yardstick, then find the area. Use the calculator to divide the area by the number of square feet a gallon of the paint covers. Round up rather than down to the nearest gallon so you have enough paint. MODULE Ready to Go On?. = + _.. _- _-. =

13 . = + = _- _- _- first _- second Because the number repeats during the division.9 _, or = + =.. _- _- _- _-9 _- _ = = = - 7 = =. _ 9.7. _-97 _ Add the initial stock price with the increase from the second day. $. + $. = $9.9 Convert the percent decrease to a decimal. - = -.7%, or -.7 Multiply the price on the second day times the percent decrease and then subtract the result from the price on the second day to find the final stock price. $ $.7 $9.9 - $.7 = $9. The final stock price is $9.. Yes, this is reasonable; price on day $, price on day $. So, the price on day $9.. Sample answer: You can use negative numbers to represent temperatures below zero or decreases in prices.. + (- ) = + =. - = = - + (- 7 ) = - = (-.9) =. +.9 = ( ) 7 = - 7 = - 7 = - 7 = -, or (-.)( ) =.. = 7.