DOMAIN I. COMPETENCY.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill. Apply ratio ad proportio to solve real-world problems. A ratio is a compariso of umbers. If a class had boys ad 4 girls, the ratio of boys to girls could be writte oe of ways: :4 or to 4 or 4 The ratio of girls to boys is: 4:, 4 to or 4 Ratios ca be reduced whe possible. A ratio of cats to 8 dogs would reduce to :, to or. Note: Read ratio questios carefully. Give a group of 6 adults ad 5 childre, the ratio of childre to the etire group would be 5:. A proportio is a equatio i which a fractio is set equal to aother. To solve the proportio, multiply each umerator times the other fractio's deomiator. Set these two products equal to each other ad solve the resultig equatio. This is called crossmultiplyig the proportio. 4 x = is a proportio. 5 60 To solve this, cross multiply. (4)(60) = (5)( x) 40 = 5x 6 = x
x + x + 4 = 5 is a proportio. To solve, cross multiply. 5( x + ) = (x + 4) 5x + 5 = 6x + 8 7 = x x + = 8 x 4 is aother proportio. To solve, cross multiply. ( x )( x ) + 4 = 8( ) x x 8 = 6 x x 4 = 0 ( x 6)( x+ 4) = 0 x = 6 or x = 4 Proportios ca be used to solve word problems wheever relatioships are compared. Some situatios iclude scale drawigs ad maps, similar polygos, speed, time ad distace, cost, ad compariso shoppig. Which is the better buy, 6 items for $.9 or 8 items for $.69? Fid the uit price. 6 = 8 = 9. x 69. x 6x =.9 8x =.69 x = 0.5 x = 0.5 Thus, 6 items for $.9 is the better buy.
A car travels 5 miles i.5 hours.. How far will it go i 6 hours? Write a proportio comparig the distace ad time. miles hours.5x = 750 x = 00 5 5. = x 6 Thus, the car ca travel 00 miles i 6 hours. The scale o a map is 4 ich = 6 miles. What is the actual distace betwee two cities if they are iches apart o the map? Write a proportio comparig the scale to the actual distace. scale actual 4 = 6 x x = 6 4 x = 9 4 x = Thus, the actual distace betwee the cities is miles. Skill. Solve real-world problems that ivolve percets, decimals, fractios, or umbers expressed i scietific ad expoetial otatio. Percet = per 00 (writte with the symbol %). Thus 0% 0 = =. 00 0 Decimals = deci = part of te. To fid the decimal equivalet of a fractio, use the deomiator to divide the umerator as show i the followig example. Fid the decimal equivalet of 7 0. Sice 0 caot divide ito 7 evely 7 0.7 0 =
The expoet form is a shortcut method to write repeated multiplicatio. Basic form: b, where b is called the base ad is the expoet. b ad are both real umbers. b implies that the base b is multiplied by itself times. Examples: 4 = = 8 = = 8 4 ( ) = ( ) ( ) ( ) ( ) = 6 Key expoet rules: 4 = ( ) = 6 For a ozero, ad m ad real umbers: m ( m ) ) a a = a + Product rule m a ( m) ) = a Quotiet rule a ) a a m = a a m Whe 0 is raised to ay power, the expoet tells the umbers of zeroes i the product. 0 7 = 0,000,000 Cautio: Uless the egative sig is iside the paretheses ad the expoet is outside the paretheses, the sig is ot affected by the expoet. 4 ( ) 4 implies that - is multiplied by itself 4 times. implies that is multiplied by itself 4 times, the the aswer is egated.
Scietific otatio is a more coveiet method for writig very large ad very small umbers. It employs two factors. The first factor is a umber betwee ad 0. The secod factor is a power of 0. This otatio is a shorthad for expressig large umbers (like the weight of 00 elephats) or small umbers (like the weight of a atom i pouds). Recall that: 0 = (0) Te multiplied by itself times. 0 0 = Ay ozero umber raised to power of zero is. 0 = 0 0 = 0 0 = 00 0 = 0 0 0 = 000 (kilo) 0 = 0 (deci) 0 = 00 (ceti) 0 = 000 (milli) 6 0 =,000,000 (micro) Write 46,68,000 i scietific otatio. ) Itroduce a decimal poit ad decimal places. 46,68,000 = 46,68,000.0000 ) Make a mark betwee the two digits that give a umber betwee -9.9 ad 9.9. 4 6,68,000.0000 ) Cout the umber of digit places betwee the decimal poit ad the mark. This umber is the -the power of te. 7 So, 46,68,000 = 4.668 0
Write 0.0097 i scietific otatio. ) Decimal place is already i place. ) Make a mark betwee ad 9 to get a oe umber betwee -9.9 ad 9.9. ) Move decimal place to the mark ( hops). 0.00 97 Motio is to the right, so of 0 is egative. Therefore, 0.0097.97 0 = Word problems ivolvig percets ca be solved by writig the problem as a equatio, the solvig the equatio. Keep i mid that of meas multiplicatio ad is meas equals. The Ski Club has 85 members. Eighty percet of the members are able to atted the meetig. How may members atted the meetig? Restate the problem. What is 80% of 85? Write a equatio. = 0.8 85 Solve. = 68 Sixty-eight members atted the meetig. There are 64 dogs i the keel. Forty-eight are collies. What percet are collies? Restate the problem. 48 is what percet of 64? Write a equatio. 48 = 64 Solve. 48 64 = = 4 = 75% 75% of the dogs are collies.
The auditorium was filled to 90% capacity. There were 558 seats occupied. What is the capacity of the auditorium? Restate the problem. 90% of what umber is 558? Write a equatio. 0.9 = 558 Solve. = 558.9 = 60 The capacity of the auditorium is 60 people. Shoes cost $4.00. Sales tax is 6%. What is the total cost of the shoes? Restate the problem. What is 6% of 4? Write a equatio. = 0.06 4 Solve. =.5 Add the sales tax to the cost. $4.00 + $.5 = $44.5 The total cost of the shoes, icludig sales tax, is $44.5. A alterative method would be to multiply $4.00 by.06. $4.00 x.06 = $44.5 (cost icludig sales tax) COMMON EQUIVALENTS = 0.5 = 50% = 0.= % = 0.5 = 5% 4 5 = 0. = 0% = 0.667 = 6 6 % = 0.5= % 8 0 = 0. = 0% = 0.6667 = 66 % 5 = 0.8 = 8 % 6 = 0.75 = 7 % 8 5 = 0.65 = 6 % 8 7 = 0.875 = 87 % 8 =.0 = 00%
Skill. Apply umber cocepts icludig primes, factors, ad multiples to build umber sequeces. A umeratio system is a set of umbers represeted a by a set of symbols (umbers, letters, or pictographs). Sets ca have differet bases of umerals withi the set. Istead of our base 0, a system may use ay base set from o up. The positio of the umber i that represetatio defies its exact value. Thus, the umeral has a value of te whe represeted as 0. Early systems, such as the Babyloia, used positio i relatio to other umerals or colum positio for this purpose sice they lacked a zero to represet a empty positio. A base of uses oly 0 ad. Thus, 9 i Base 0 becomes 00 i Base. Decimal Biary Coversio Decimal Biary Place Value 0 0 4 00 8 000 9+4 = (Base 0) becomes 00 + 00 = 0 (Base ). Fractios, ratios ad other fuctios alter i the same way. Computers use a base of but combie it ito 4 uits called a byte to fuctio i base 6 (hexadecimal). A base of 8 (octal) was also used by older computers. Prime umbers are umbers that ca oly be factored ito ad the umber itself. Whe factorig ito prime factors, all the factors must be umbers that caot be factored agai (without usig ). Iitially umbers ca be factored ito ay factors. Check each resultig factor to see if it ca be factored agai. Cotiue factorig util all remaiig factors are prime. This is the list of prime factors. Regardless of which way the origial umber was factored, the fial list of prime factors will always be the same.
Factor 0 ito prime factors. Divide by as may times as you ca, the by, the by other successive primes as required. Factor 0 ito ay factors. 5 6 Now factor the 6. 5 These are all prime factors. Factor 0 ito ay factors. 0 Now factor the 0. 5 These are the same prime factors eve though the origial factors were differet. Factor 40 ito prime factors. Factor 40 ito ay factors. 4 0 Now factor both 4 ad 0. 4 6 5 Now factor both 4 ad 6. 5 These are prime factors. 4 This ca also be writte as 5