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January 15, 2013
Prime Factorizaon
Algebra with Career Applicaons
Blitzer 5.1
Prime Factorizaon, GCF, LCM
Jan 20­4:24 PM
Greatest Common Factor
• Factors are the numbers that mulply together to give you a number. • Example: 3 and 5 are factors of 15
• Greatest Common Factor (GCF) is the number that divides into a set of given numbers without a remainder.
• Also called Greatest Common Divisor
• A prime number is a number that is divisible by 1 and itself. • Examples of primes are 2, 3 , 5, 7, 11, 13, 17, 19, 23, 29, 31, …
• Prime factorizaon is factoring a number unl you have nothing le but a product of prime numbers
• Example: Jan 20­4:24 PM
Find GCF using Prime Factorizaon
• Prime factor your number
• Take all of the number that are in all prime factorizaon to the lowest power on that number. • Mulply to get GCF
Example: Find the GCF for 72, 75, 90.
What number do all three have in common? Look at each number and what is the lowest exponent on all of the 2s, etc. Answer: Jan 20­4:24 PM
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Applicaon of GCF
We can use the GCF for problems like #92 on p. 236. A choral director needs to divide 180 men and 144 women into all male and all female singing groups so that each group has the same number of people. What is the largest number of people that can be placed in each singing group? Find the prime factorizaon of 180 and 144. Jan 20­4:24 PM
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Least Common Mulple
• Mulples are number found by mulplying the number by an integer. • Example: 28 is a mulple of 7
• Least Common Mulple (LCM) is the number that can be divided by the given numbers with no remainder
• Similar to Least Common Denominator
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Applicaon of LCM
Find LCM using Prime Factorizaon
• Prime factor your number
• Take all of the number that are in any prime factorizaon to the highest power on that number. • Mulply to get LCM
Example: Find the LCM for 72, 75, 90.
We can use the LCM for problems like #90 on p. 236. There are two species of insects, Magicicada septenndecim and Magicicada tredecim, that live in the same environment. They have a life cycle of exactly 17 and 13 years, respecvely. For all by their last year, they remain in the ground feeding on the sap of tree roots. Then, in their last year, they emerge en masse from the ground as fully formed cricket‐like insects, taking over the forest in a single night. How oen will both species of cidadas emerge in the same year?
What are all of the prime numbers in any of the three? Look at each number and what is the highest exponent on all of the 2s, etc. Answer: Jan 20­4:24 PM
Assignment: Blitzer p. 235 # 25, 29, 33, 37, 41, 45, 49, 51, 53, 57, 59, 63, 65, 67, 90 ­ 98
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Assignment: Blitzer p. 235 # 25, 29, 33, 37, 41, 45, 49, 51, 53, 57, 59, 63, 65, 67, 90 ­ 98
Find the prime factorization of each composite number. 25. 75
29. 105
33. 663
37. 1440
41. 3675
Find the greatest common divisor of the numbers. 45. 42 and 56
49. 60 and 108
51. 72 and 120
53. 342 and 380
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Algebra with Career Applicaons
Blitzer 5.2
Order of Number, Order of Operaons, Absolute Value
Jan 21­7:19 AM
Number Line
The Number line represents all of the real numbers. Numbers to the le are smaller than the numbers on the right. Where would you put the numbers 3, ‐6, 0, ¾ ?
Jan 22­5:21 PM
Less Than, Greater Than
> means greater than. In other words: BIGGER NUMBERS > smaller numbers
means greater than or equal to . In other words: ³
12 ³ 9 because 12 is greater than 9
9 ³ 9 because 9 equals 9
< means less than. In other words: smaller numbers < BIGGER NUMBERS
£ means less than or equal to. In other words: 2 £ 9 because 2 is less than 9
9 £ 9 because 9 equals 9
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Absolute Value
Absolute value is the distance of a number from zero on the number line. Since distance is always posive (or zero), absolute value is posive (or zero). January 15, 2013
Order of Operaons
• Perform all operaons within groupings symbols (Parentheses, absolute values, square roots, numerators, denominators, etc.)
• Evaluate all Exponenal expressions
• Do all Mulplicaons and Divisions in the order in which they occur, working le to right
• Do all Addion and Subtracon in the order in which they occur, working le to right
Please Excuse MyDear AuntSally
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Examples
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Where do we use integers?
On August 28, 2002, the NASDAQ composite average opened at 1347 points. During the day it lost 33 points. On August 29 it gained 22 points and on August 30, it lost 21 points. What was the closing NASDAQ composite average on August 30, 2002?
Source: A Survey of Mathemacs with Applicaons
Jan 22­5:21 PM
. Angel, Abbot, Runde. p. 226 Jan 22­5:21 PM
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Where do we use integers?
Where do we use order of operaons?
The World Standard Time Zone chart used by airlines and the US Navy is below. • Find the difference in me between Tokyo and New York. • Find the difference in me between Los Angeles and Rio De Janero. • The cost, in dollars, for Crescent City Tours to provide a tour for x people can be determined by the expression 220 + 2.75 x. Determine the cost of a tour for 75 people. Source: A Survey of Mathemacs with Applicaons
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Where do we use order of operaons?
. Angel, Abbot, Runde. p. 292 Jan 22­5:21 PM
Assignment: Blitzer p. 247 # 5 ­ 18 all, 53 ­ 65 odd, 81 ­ 105 odd
The rate of growth of grass in inches per week depends on a number of factors, including rainfall and temperature. For a certain area, this can be approximated by 0.2R2 +0.003RT +0.0001T 2, where R is the weekly rainfall, in inches, and T is the average weekly temperature, in degrees Fahrenheit. Find the amount of growth of grass for a week in which the rainfall is 2 in. and the average temperature was 70oF. Source: A Survey of Mathemacs with Applicaons. Angel, Abbot, Runde. p. 292 Assignment: Blitzer p. 247 # 5 ­ 18 all, 53 ­ 65 odd, 81 ­ 105 odd
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Even Answers p. 247 # 5 – 18 all, 53 – 65 odd, 81 – 105 odd
Algebra with Career Applicaons
Blitzer 5.3 Raonal Numbers
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Our Number System
Number System
Natural Numbers: 1, 2, 3, … (also called counng numbers)
Whole Numbers: 0, 1, 2, 3, …
Integers: …, – 3, – 2, –1, 0, 1, 2, 3, …
Raonal Numbers: –1/7, 0, 3/5, 117/8, …
Irraonal Numbers: Real Numbers
Natural Numbers
Whole Numbers
Irraonal
Numbers
Raonal Numbers
Raonal Numbers are numbers that can be wrien as a rao of two integers, provided that the denominator is not zero. In other words, fracons are raonal numbers.
Raonal numbers can be fracons, improper fracons, mixed numbers, or decimals that either end or repeat. Integers
Raonal Numbers
Jan 24­2:53 PM
Is every number a fracon?
• Ancient Greeks thought that natural numbers were a gi from God
• They thought raos of these numbers made up all numbers • Pythagorean School – secret society
•
•
Pythagoras (580 BCE – 500 BCE)
What is the length of the diagonal of a square with side length one unit?
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Converng Fracons
Write as a fracon in reduced form. Write as a mixed number.
Write as an improper fracon. Write as a decimal. Jan 24­2:53 PM
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Converng Decimals
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Use the process to write as a rao of two integers: 3.59595959…
• Set n = to the decimal • Count the number of repeang digits
• Mulply the equaon by 10^(# repeang digits)
• Subtract
• Solve for n
• Give an answer with no decimals
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• 100n = 359.59595959…
n = 3.59595959…
99n = 356
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Use the process to write as a rao of two integers: 5.321111111…
• n = 5.321111111…
• Set n = to the decimal • 1 repeang • Count the number of repeang digits
• Mulply the equaon by • 10n = 53.21111111…
n = 5.32111111…
10^(# repeang digits)
9n = 47.89
• Subtract
• Solve for n
• Give an answer with no decimals
• n = 3.59595959…
• 2 repeang Ordering Fracons
Use > or <.
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When do we use fracons? • Measurement – metric
When do we use fracons? • To help proofread her new book, Chris assigns three students to proofread ¼, 1/5, and ½ of the book, respecvely. She decides to proofread the rest of the book herself. If the book has 540 pages, how many pages must Chris proofread herself? • Measurement ‐ customary
Source: A Survey of Mathemacs with Applicaons. Abbot, Angel, Runde. p. 240
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Blitzer p. 260 # 49 ­ 55 odd, 101 ­ 106, 115 ­ 124 all
When do we use fracons? Follow the instrucons given on the box of oatmeal, determine the amount of water and oats that are needed to make 1 ½ servings of oatmeal. Directions:
Boil water or milk and salt (if desired)
2. Stir in oats.
Servings
1
2
Water (or milk)
1 cup
1 ¾ cup
Oats
½ cup
1 cup
Salt (optional)
Dash
1/8 tsp
3. Stirring occasionally, cook over medium heat for 5 minutes. Source: A Survey of Mathemacs with Applicaons. Abbot, Angel, Runde. p. 241
Blitzer p. 260 # 49 ­ 55 odd, 101 ­ 106, 115 ­ 124 all
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