Lesson objective: Understand that fractions can be interpreted as the division of the numerator by the denominator, and that if the numerator is divided by the denominator, the quotient is located at the same place on the number line as the original fraction.
Students bring prior knowledge of interpreting quotients of whole numbers as a number of shares from 3.O.A.1.2. This prior knowledge is extended to their work with fractions as students decipher fairshare situations. A conceptual challenge students may encounter is thinking that fair shares cannot be made with numbers less than one.
The concept is developed through work with tape diagrams and number lines, which support student understanding of interpreting fractions as division.
This work helps students deepen their understanding of operations because they will be operating on fractions just as they learned to operate on whole numbers.
Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they work with tasks involving fair share situations. Students will begin to interpret fractions as division.
Key vocabulary:

denominator: the number of equalsize parts into which the whole has been partitioned. For example, in the fraction \({3 \over 8}\), 8 is the denominator. The denominator is written below the horizontal bar in a fraction. It is also the divisor.

dividend: the name for the quantity being divided or shared. For example, 36 is the dividend in the equation 36 ÷ 4 = 9.

division: a mathematical operation based on sharing or separating into equal parts.

divisor: the name for the quantity that divides another number. For example, in the equation 36 ÷ 4 = 9, the divisor is 4.

equal: exactly the same in value

numerator: the number of equal parts being considered. For example, in the fraction \({3 \over 8}\), 3 is the numerator. The numerator is written above the horizontal bar in a fraction. It is also the dividend.

quotient: The result of dividing one number by another number. For example, in the equation 36 ÷ 4 = 9, the quotient is 9.
Materials needed:
 Blank number lines
 Fraction bars or fraction pieces (optional)