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

What if we could look at a simplified form of both numbers? 1.Figure out the prime factors of both. 2.Do any factors cancel out? Cancel them. 3.Write the remaining factors. 4.Multiply the tops. 5.Multiply the bottoms.  That’s the simplest form of the fraction! Click to Advance

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: 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

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

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

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, Numbers ending in 5 or 0 can always be divided by 5 –Yes: 2345, 70, No: 37, 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

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 =(2)(3)(3)(11) Click to Advance

The mechanics of The “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 =(2)(3)(11) Click to Advance

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

Is a large number prime? What smaller primes do you have to check? 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 Here is a useful table of the squares of some small primes: 4 2 = = = = = = = =529 Click to Advance

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

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

Thank You For Learning about –Prime Factorization –Reducing Numeric fractions