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Reducing Numeric Fractions

Powerpoint 2007: Click on SLIDESHOW, then click on FROM BEGINNING Powerpoint 2003: Click on BROWSE, then click on FULL SCREEN. Reducing Numeric Fractions. Suggestion: Work with scratch paper and pencil as you go through this presentation. Turn Mystery into Mastery. Click to Advance.

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Reducing Numeric Fractions

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  1. Powerpoint 2007: Click on SLIDESHOW, then click on FROM BEGINNING Powerpoint 2003: Click on BROWSE, then click on FULL SCREEN Reducing Numeric Fractions Suggestion: Work with scratch paper and pencil as you go through this presentation. Turn Mystery into Mastery Click to Advance C2006 – DW Vandewater

  2. What if we could look at a simplified form of both numbers? • Figure out the prime factors of both. • Do any factors cancel out? Cancel them. • Write the remaining factors. • Multiply the tops. • Multiply the bottoms. • That’s the simplest form of the fraction! Click to Advance

  3. What is the Key Skill? • Prime Factorization! • (A big name for a simple process …) • Finding out how to write a number as the product of it’s prime factors. • Examples: • 6 = (2)(3) • 70 = (2)(5)(7) • 24 = (2)3(3) or (2)(2)(2)(3) • 11=(11) because 11 is prime Click to Advance

  4. Recognizing the Primes between 1 and 20 • 1 is not considered a prime number • 2 is the only even prime number • 3, 5, 7 are primes • (3)(3)=9, so 9 is not prime • 11, 13, 17, and 19 are prime • There are infinitely many primes above 20. • How can you tell if a large number is prime? Click to Advance

  5. Is it Prime? You can Find Out! • Use repeated division: • Start by finding the smallest prime number that divides evenly into the original number. • If you can find one, this division yields 2 factors: one CERTAIN prime and one that MAY be prime • Examples: • 36 divided by 2 is 18. • Therefore, 36=(2)(18) • 175 divided by 5 is 35. • Therefore, 175=(5)(35) • 147 divided by 3 is 49. • Therefore, 147=(3)(49) • 33 divided by 3 is 11. • Therefore, 33=(3)(11) • You still have to check the second factor for more primes Click to Advance

  6. Is it Prime?Tricks for recognizing factors 2, 3 and 5 • ANY even number can always be divided by 2 • Yes: 3418, 70, 122 No: 37, 120001 • Numbers ending in 5 or 0 can always be divided by 5 • Yes: 2345, 70, 41415 No: 37, 120001 • If the sum of a number’s digits divides evenly by 3, then the number always divides by 3 • Yes: 39, 120, 567 No: 43, 568 Click to Advance

  7. Finding all prime factors:The “Tree Root” Method • Write down a number • Break it into a pair of factors (use the smallest prime) • Try to break each new factor into pairs • Repeat until every number is prime • Collect the “dangling” primes 198 2 99 3 33 3 11 198=(2)(3)(3)(11) Click to Advance

  8. The mechanics ofThe “Tree Root” Method • Find the smallest prime number first • To get the other factor, divide it into the original number • Since 66 is positive, 2 must be a factor • Divide 2 into 66 to get 33 • Since 33’s digits add up to 6, 3 must be a factor • Divide 3 into 33 to get 11 • All the “dangling” numbers are prime, so we are done 66 2 33 311 66=(2)(3)(11) Click to Advance

  9. You can also use a linear approach Suggestion: Do your divisions in a work area to the right of the linear factorization steps. • 84=(2)(42) • =(2)(2)(21) • =(2)(2)(3)(7) • =22·3·7 (simpler notation) • 216=(2)(108) • =(2)(2)(54) • =(2)(2)(2)(27) • =(2)(2)(2)(3)(9) • =(2)(2)(2)(3)(3)(3) • =23·33 (simpler notation) Click to Advance

  10. Is a large number prime?What smaller primes do you have to check? Here is a useful table of the squares of some small primes: 42=16 52=25 72=49 112=121 132=169 172=289 192=361 232=529 • See where the number fits in the table above • Let’s use 151 as an example: • 151 is between the squares of 11 and 13 • Check all primes before 13: 2, 3, 5, 7, 11 • 2 won’t work … 151 is not an even number • 3 won’t work … 151’s digits sum to 7, which isn’t divisible by 3 • 5 won’t work … 151 does not end in 5 or 0 • 7 won’t work … 151/7 has a remainder • 11 won’t work … 151/11 has a remainder • So … 151 must be prime Click to Advance

  11. Practice:Let’s Reduce a Fraction • Here’s the problem -> • Find the factors of 1848 • Find the factors of 990 • Rewrite the fraction • Cancel matching factors • Rewrite the fraction • Multiply top & bottom • That is the simplest form Click to Advance

  12. More Practice • See if you can find the simplest forms.Do the work on paper and click to see the answer Click to Advance

  13. Thank You • For Learning about • Prime Factorization • Reducing Numeric fractions

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