# Module 1

Version 1.3.0

### 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 $25$ -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 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 $2$ , $3$ , $4$ , $5$ , and  $10$ 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  $2$ and $3$ .  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  $2$ and  $3$ (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 $4$ —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  $5$ and $10$ , 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  $1$ (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  $12$ and , both of which are division lessons sharing the same objective. Include units of  $2$ and units of  $3$ 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 $7$ .  Lesson  introduces the significant complexity of the distributive property with division.  The concepts from both lessons are reinforced within Module $3$ .

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.

#### Distribution of Minutes

• Fluency Practice
• Application Problem
• Concept Development
• Student Debrief
1
 5 10 35 10
2
 15 5 30 10
3
 15 5 30 10
4
 14 6 30 10
5
 8 7 35 10
6
 8 7 35 10
7
 12 5 32 10
8
 6 10 34 10
9
 15 35 10
10
 11 5 34 10
11
 11 5 34 10
12
 13 5 32 10
13
 14 5 31 10
14
 12 5 33 10
15
 11 5 34 10
16
 14 5 31 10
17
 9 5 36 10
18
 9 5 36 10
19
 14 5 31 10
20
 9 8 33 10
21
 14 5 31 10