# 1. Understand that quotients are shares or groups (C)

## About this lesson

Lesson objective: Understand that quotients can be interpreted as equal groups or shares. Extend understanding that division problems find an unknown factor based on the relationship between multiplication and division.

Students bring prior knowledge of representing multiplication with repeated addition and skip counting from Grade 2, Unit 13; 2.OA.C.4 and that multiplication is the total number of groups or objects in a group from Grade 3, Unit 1; 3.OA.A.1. This prior knowledge is extended to quotients as students use arrays and equal groups to explore division situations. A conceptual challenge students may encounter is thinking that the order of the numbers in a division operation are interchangeable as they are in multiplication.

The concept is developed through work with arrays and equal groups, which gives students a visual representation of the problem situation including sharing or making groups.

This work helps students deepen their understanding of operations because understanding equal groups in multiplication can help students see that quotients are also found by using equal groups and sharing.

Students engage in Mathematical Practice 1 (Make sense of problems and persevere in solving them) as they practice dividing and sharing manipulatives into different equal groups or rows. Using doubling and halving, breaking apart numbers, and skip counting, they explore with even and odd numbers, and understand the meaning of the parts of a division sentence.

Students engage in Mathematical Practice 7 (Look for and make use of structure) as they form distinguishable groups, rows, or jumps to decompose 48 into parts, shares, or groups in different combinations. This supports students as they begin to understand and leverage the relationship between multiplication and division. Earlier in Grade 3, students worked to make some number of groups of some size. Now they work from a total, making either some number of equal groups or groups of a particular size.

**Key vocabulary:**

- array
- divide
- equal groups
- quotient
- share
- unknown factor

**Special materials needed (optional):**

- manipulatives such as square tiles, counters, or cubes