Introduction to Dividing Fractions: Fifth Grade Mathematics

Dividing fractions involves finding how many times one fraction fits into another, formalized through the "Keep, Change, Flip" algorithm.

The "Keep, Change, Flip" Algorithm

To divide any two fractions:
1. KEEP the first fraction (the dividend) exactly as it is.

2. CHANGE the division sign to a multiplication sign.

3. FLIP the second fraction (the divisor) to its reciprocal.

4. Multiply straight across and simplify.

+--------------------------------------------------------------------------+
|                  KEEP, CHANGE, FLIP (KCF) ALGORITHM                      |
|                                                                          |
|       Problem:  (3/4) / (2/5)                                            |
|                                                                          |
|       [ KEEP ]    First fraction stays:       3/4                        |
|       [ CHANGE ]  Division becomes multiply:   *                         |
|       [ FLIP ]    Second fraction inverts:    5/2                        |
|                                                                          |
|       Calculation: (3/4) * (5/2) = (3 * 5) / (4 * 2) = 15/8 = 1 7/8      |
+--------------------------------------------------------------------------+

Common-Denominator Method for Division

An alternative conceptual method is finding a common denominator:
- (3/4) / (1/2) = (3/4) / (2/4).

- Once the denominators are equal, divide the numerators: 3 / 2 = 3/2 = 1 1/2.

- Both methods yield the exact same quotient.

Dividing Involving Mixed Numbers

When division problems involve mixed numbers:
1. Convert all mixed numbers to improper fractions first.

2. Apply Keep, Change, Flip.

3. Cross-cancel common factors.

4. Multiply and simplify.

Dividing Mixed Numbers: 2 1/4 / 1 1/2
  Improper fractions: 9/4 / 3/2
  Keep, Change, Flip: (9/4) * (2/3)
  Cross-cancel: (3/2) * (1/1) = 3/2 = 1 1/2.

Exercises and Solutions

Exercise 1: Applying Keep, Change, Flip. Calculate (5/6) / (2/3). Show each step of the Keep, Change, Flip algorithm, cross-cancel if possible, and write the answer in simplest form.

Solution 1: Keep the first fraction: 5/6. Change division to multiplication: *. Flip the divisor 2/3 to its reciprocal: 3/2. The expression is (5/6) * (3/2). Cross-cancel the 3 in the numerator and 6 in the denominator (both divisible by 3): 3 becomes 1; 6 becomes 2. Multiply remaining factors: (5 * 1) / (2 * 2) = 5/4. Convert 5/4 to a mixed number: 1 1/4.

Exercise 2: Ribbon Strips Problem. A seamstress has 3 3/4 yards of lace trim. Each doll dress requires 3/8 of a yard of lace. How many doll dresses can the seamstress trim?

Solution 2: Divide the total lace by the amount per dress: 3 3/4 / (3/8). First, convert 3 3/4 to an improper fraction: (3 * 4 + 3)/4 = 15/4. Apply Keep, Change, Flip: (15/4) * (8/3). Cross-cancel: 15 and 3 divide by 3 (15 becomes 5; 3 becomes 1). 8 and 4 divide by 4 (8 becomes 2; 4 becomes 1). Multiply remaining factors: (5 * 2) / (1 * 1) = 10/1 = 10. The seamstress can trim exactly 10 doll dresses.