Lesson objective: Understand that the relationship between multiplication and division holds even when the division results in a remainder. (100 ÷ 40 = 2 R 20: therefore, 100 = 2 x 40 + 20).

Students bring prior knowledge of being able to solve multiplication and division problems by using arrays and equal groups from 3.OA.A.3. This prior knowledge is extended to form an understanding that the relationship between multiplication and division holds even when the division results in a remainder, as students determine which equation correctly matches a problem. A conceptual challenge students may encounter is not realizing that the remainder needs to be added to the product when showing the multiplicative inverse of the division problem. This unit will address this by modeling with illustrations that the remainder needs to be added to the product in order to return to the original factor.

The concept is developed through work with various picture representations, which explain why different equations do not match the given problem.

This work helps students deepen their understanding of operations because the relationship between multiplication and division is true even when the division equation results in a remainder.

Students engage in Mathematical Practice 8 (Look for and express regularity in repeated reasoning by evaluating the reasonableness of their results) as they determine the reasonableness of each equation when compared to the problem.

**Key vocabulary:**

- argument
- array
- dividend
- divisor
- equal groups
- equation
- inverse operations
- quotient
- relationship
- remainder

**Special materials needed:**

- colored squares for intervention