Lesson objective: Understand that multiplication and division are inverse operations. Extend understanding that multiplication can be used to solve division problems. Division problems have an unknown factor.

Students bring prior knowledge of products of whole numbers being interpreted as a number of groups or a number of objects from Grade 3, Unit 1; 3.OA.A.1. This prior knowledge is extended to products and number of groups being used to solve division situations as students work to find the unknowns. A conceptual challenge students may encounter is that factors can be used in any order in multiplication, but this cannot be applied to division. Knowing how to find the unknown in division requires making sense of the problem and understanding multiplication.

The concept is developed through work with equal grouping, which helps students visually see the connection between multiplication and division.

This work helps students deepen their understanding of operations because using the relationship between multiplication and division can help them solve problems involving either operation.

Students engage in Mathematical Practice 1 (Make sense of problems and persevere in solving them) as they understand they are finding an unknown factor. Using manipulatives to represent doubling and halving, students will want to find as many possibilities through their rearrangement of the arrays. They should persevere to find multiple answers and can be asked, "Are those all the possibilities? How do you know? Are there other options that won't work?"

Students engage in Mathematical Practice 7 (Look for and make use of structure) as they understand that they are finding an unknown factor. They draw pictures or arrays, equal groups, or number lines and write equations to represent the situation. Giving students a multiplication equation to represent the problem can help them start to make sense of how they can find the unknown factor. Doubling and halving factors, breaking apart numbers and sharing into equal groups are strategies that will help them to solve these problems. Be sure to ask if there is a better way to solve the problem if you see students still using additive reasoning to count multiple tally marks or share larger numbers into groups. Using number sense and multiplicative reasoning, they can be more efficient in finding answers to division situations.

**Key vocabulary:**

- array
- doubling
- equal groups
- equation
- half
- inverse operations
- product
- quotient
- unknown factor

**Special materials needed:**

- base ten blocks to reason with, such as breaking apart numbers
- manipulatives such as square tiles, counters, or cubes