Lesson objective: Understand that fractions can be interpreted as the division of the numerator by the denominator, and that a remainder can be partitioned to share items equally.
Students bring prior knowledge of interpreting division and remainders from 3.OA.1.2. This prior knowledge is extended to reporting remainders as fractions as students wrestle with fair share situations where a remainder is present. Students need to decide what is appropriate to do with a remainder in a given situation. A conceptual challenge students may encounter is thinking there is no way to share a remainder when it is not a whole share, as well as the challenge of naming the remainder. In this lesson the remainder is a fraction of 1 object in the original amount.
The concept is developed through work with a visual representation to support student understanding of seeing fractions as division.
This work helps students deepen their understanding of operations because they will access their current understanding of division and build their knowledge of fractional remainders.
Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they use a variety of numbers and situations to ponder division scenarios where they will need to represent fractional remainders.
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 number into which you are dividing in a division problem. 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 number 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.

remainder: The amount left over when values are divided into equal shares. In the division equation 16 ÷ 3 = 5 R1 the remainder is 1.