Module 1Version 1.3.1
Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10
This 25 -day module begins the year by building on students’ fluency with addition and their knowledge of arrays. In Topic A, students initially use repeated addition to find the total from a number of equal groups (2.OA.4). As students notice patterns, they let go of longer addition sentences in favor of more efficient multiplication facts (3.OA.1). Lessons in Topic A move students’ Grade 2 work with arrays and repeated addition a step further by developing skip-counting rows as a strategy for multiplication. Arrays become a cornerstone of the module. Students use the... Read More
This -day module begins the year by building on students’ fluency with addition and their knowledge of arrays. In Topic A, students initially use repeated addition to find the total from a number of equal groups (2.OA.4). As students notice patterns, they let go of longer addition sentences in favor of more efficient multiplication facts (3.OA.1). Lessons in Topic A move students’ Grade work with arrays and repeated addition a step further by developing skip-counting rows as a strategy for multiplication. Arrays become a cornerstone of the module. Students use the language of multiplication as they understand what factors are and differentiate between the size of groups and the number of groups within a given context. In this module, the factors , , , , and provide an entry point for moving into more difficult factors in later modules.
The study of factors links Topics A and B; Topic B extends the study to division. Students understand division as an unknown factor problem and relate the meaning of unknown factors to either the number or the size of groups (3.OA.2, 3.OA.6). By the end of Topic B, students are aware of a fundamental connection between multiplication and division that lays the foundation for the rest of the module.
In Topic C, students use the array model and familiar skip-counting strategies to solidify their understanding of multiplication and practice related facts of and . They become fluent enough with arithmetic patterns to add or subtract groups from known products to solve more complex multiplication problems (3.OA.1). They apply their skills to word problems using drawings and equations with a symbol to find the unknown factor (3.OA.3). This culminates in students using arrays to model the distributive property as they decompose units to multiply (3.OA.5).
In Topic D, students model, write, and solve partitive and measurement division problems with and (3.OA.2). Consistent skip-counting strategies and the continued use of array models are pathways for students to naturally relate multiplication and division. Modeling advances as students use tape diagrams to represent multiplication and division. A final lesson in this topic solidifies a growing understanding of the relationship between operations (3.OA.7).
Topic E shifts students from simple understanding to analyzing the relationship between multiplication and division. P ractice of both operations is combined—this time using units of —and a lesson is explicitly dedicated to modeling the connection between them (3.OA.7). Skip-counting, the distributive property, arrays, number bonds, and tape diagrams are tools for both operations (3.OA.1, 3.OA.2). A final lesson invites students to explore their work with arrays and related facts through the lens of the commutative property as it relates to multiplication (3.OA.5).
Topic F introduces the factors and , familiar from skip-counting in Grade 2. Students apply the multiplication and division strategies they have used to mixed practice with all of the factors included in Module (3.OA.1, 3.OA.2, 3.OA.3). Students model relationships between factors, analyzing the arithmetic patterns that emerge to compose and decompose numbers, as they further explore the relationship between multiplication and division (3.OA.3, 3.OA.5, 3.OA.7).
In the final lesson of the module, students apply the tools, representations, and concepts they have learned to problem solving with multi-step word problems using all four operations (3.OA.3, 3.OA.8). They demonstrate the flexibility of their thinking as they assess the reasonableness of their answers for a variety of problem types.
The Mid-Module Assessment follows Topic C. The End-of-Module Assessment follows Topic F.
Notes on Pacing for Differentiation
If pacing is a challenge, consider the following modifications and omissions.
Consolidate Lessons and , both of which are division lessons sharing the same objective. Include units of and units of in the consolidated lesson.
Omit Lessons and . Lesson uses the tape diagram to provide a new perspective on the commutative property, a concept students have studied since Lesson . Lesson introduces the significant complexity of the distributive property with division. The concepts from both lessons are reinforced within Module .
Focus Standards for Mathematical Practice
MP.1 Make sense of problems and persevere in solving them. Students model multiplication and division using the array model. They solve two-step mixed word problems and assess the reasonableness of their solutions.
MP.2 Reason abstractly and quantitatively. Students make sense of quantities and their relationships as they explore the properties of multiplication and division and the relationship between them. Students decontextualize when representing equal group situations as multiplication and when they represent division as partitioning objects into equal shares or as unknown factor problems. Students contextualize when they consider the value of units and understand the meaning of the quantities as they compute.
MP.3 Construct viable arguments and critique the reasoning of others. Students represent and solve multiplication and division problems using arrays and equations. As they compare methods, they construct arguments and critique the reasoning of others. This practice is particularly exemplified in daily Application Problems and in specific lessons dedicated to problem solving in which students solve and reason with others about their work.
MP.4 Model with mathematics. Students represent equal groups using arrays and equations to multiply, divide, add, and subtract.
MP.7 Look for and make use of structure. Students notice structure when they represent quantities by using drawings and equations to represent the commutative and distributive properties. The relationship between multiplication and division also highlights structure for students as they determine the unknown whole number in a multiplication or division equation.
- Fluency Practice
- Application Problem
- Concept Development
- Student Debrief
- Lesson 7: Objective: Demonstrate the commutativity of multiplication, and practice related facts by skip-counting objects in array models.
- Lesson 8: Objective: Demonstrate the commutativity of multiplication and practice related facts by skip-counting objects in array models.
- Lesson 9: Objective: Find related multiplication facts by adding and subtracting equal groups in array models.
- Lesson 10: Objective: Model the distributive property with arrays to decompose units as a strategy to multiply.
- Lesson 11: Objective: Model division as the unknown factor in multiplication using arrays and tape diagrams.
- Lesson 12: Objective: Interpret the quotient as the number of groups or the number of objects in each group using units of 2.
- Lesson 13: Objective: Interpret the quotient as the number of groups or the number of objects in each group using units of 3.
- Lesson 14: Objective: Skip-count objects in models to build fluency with multiplication facts using units of 4.
- Lesson 15: Objective: Relate arrays to tape diagrams to model the commutative property of multiplication.
- Lesson 16: Objective: Use the distributive property as a strategy to find related multiplication facts.
- Lesson 17: Objective: Model the relationship between multiplication and division.
- Lesson 18: Objective: Apply the distributive property to decompose units.
- Lesson 19: Objective: Apply the distributive property to decompose units.
- Lesson 20: Objective: Solve two-step word problems involving multiplication and division and assess the reasonableness of answers.
- Lesson 21: Objective: Solve two-step word problems involving all four operations and assess the reasonableness of answers.
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