Math 7( Period 4)

Simplifying & Comparing Fractions 5.4

Equivalent Fractions- just multiply the numerator and the denominator by the same number and you will get an equivalent ( equal fraction to the one you started with.

GOLDEN RULE OF FRACTIONS à do unto the numerator as you do unto the denominator

Simplifying fractions
(your parents & MS Baril call this reducing)

Two great ways:

1) Just divide both the numerator and the denominator by the GCF

2) Rewrite the numerator and the denominator into prime factorization (use factor trees or inverted division) Then simply cross out (cross cancel) each common factor on the numerator and the denominator (they cross out because each becomes 1  such as 4/4 =1)  You are left with the simplified fraction every time!

THE GCF METHOD

One of the reasons we learn the GCF is because it is the FASTEST way to simplify fractions in one easy step!

Just divide both the numerator and the denominator by the GCF

(The problem with this method is if you are not comfortable finding the GCF, you really can’t do this method easily)

The best reason to use this method is because it is the fastest. So imagine you have a “GCF Magical Voice” in your hear.. the voice tells you the GCF of the numerator and the denominator.. you simply use that GCF to divide both the top and the bottom of your fraction and your done… It’s a “gut feeling” – combined with your knowledge of the divisibility rules.. and YOU CAN DO IT!

THE PRIME FACTORIZATION METHOD:

This is sort of the GCF “Incognito” ( In disguise)  Rewrite the numerator and the denominator in prime factorization form (Use Factor Trees or Inverted Division to find the prime factorization, if necessary)

Then simply cross out each common factor on the top with the bottom. You are actually using the ID Property of Multiplication because each “cross out_ is really a quotient of 1 – again 4/4 = 1 You will be left with the simplified fraction every time.

If you actually multiplied together all the “cross outs” you would get the GCF—so you are using the GCF without ever computing it.

THE CROSS OUT METHOD:

You simply think of the first number that comes to your mind that “GOZ-into” both the numerator and the denominator and keep going until its simplified. If the numbers are both even—many students start dividing it in half. and then half again—if both are still even… This probably takes the longest, but in practice, most people use this method!

The problem with this method—you may think that a fraction is simplified but you miss a factor—this especially happens when the number is odd and you are always using 2 to divide.

The best reason to use this method –no one ever forget how to do this method—it comes rather naturally and there are no “precise” steps to do.  

Comparing Fractions

I. Benchmarks

0, ¼1/4, ½ 1/2 , 3/4 ¾ , and 1 (using your gut feeling).

How do you figure out which benchmark to use?

When the numerator is close to the denominator , the fraction is approaching 1 ( 9/11 or  45/55)

When you double the numerator and that is close to the denominator the fraction is close to ½ ½1/2. ( 4/9)

When the numerator is very far from the denominator the fraction is approaching zero ( 1/9 )

Also if one number is improper or a mixed number and the other is a proper fraction, then obviously the number greater than 1 will be bigger.

II. LCD—give them all the same denominator using the LCM as the LCD.

III. Use cross multiplication—sometimes several times to compare

IV. Change them to decimals ( my least favorite)

Comparing negative fractions

1) if one is positive and the other negative—the positive is always bigger… no matter what!

If both NEGATIVEà remember that the one closest to ZERO is bigger.

2) both proper fractions  if you have -3/4 and -1/4 then -1/4 is BIGGER because it is closer to ZERO ( It is to the right of -3/4 on the number line)

3) One is a proper fraction and the other is a improper (or mixed number) the proper negative fraction  will always be bigger because IT IS CLOSER to ZERO

-1/4 will always be bigger than – 5 1/4¼.¼

When cross multiplying with negative fractions—be careful Remember that these cross products are NEGATIVE so use integer concepts.

-3/4 vs – 5/6

Using cross products

(-3)(6) and (4)(-5)

-18 > -20 (integer concept)

-3/4 > -5/6