Grade 3 Mathematics Module 3, Topic A, Overview
The Properties of Multiplication and Division
Topic A begins by revisiting the commutative property. Students study familiar facts from Module 1 to identify known facts using units of 6, 7, 8, and 9 (3.OA.5, 3.OA.7). They realize that they already know more than half of their facts by recognizing, for example, that if they know 2 × 8, they also know 8 × 2 through commutativity. This begins a study of arithmetic patterns that becomes an increasingly prominent theme in the module (3.OA.9). The subsequent lesson carries this study a step further; students apply the commutative property to relate 5 × 8 and 8 × 5, and then add one more group of 8 to solve 6 × 8 and, by extension, 8 × 6. The final lesson in this topic builds fluency with familiar multiplication and division facts, preparing students for the work ahead by introducing the use of a letter to represent the unknown in various positions (3.OA.3, 3.OA.4).
Grade 3 Mathematics Module 3, Topic B, Overview
Multiplication and Division Using Units of 6 and 7
Topic B introduces units of 6 and 7, factors that are well suited to Level 2 skip-counting strategies and to the Level 3 distributive property strategy, already familiar from Module 1. Students learn to compose up to, then over the next decade. For example, to solve a fact using units of 7 they might count 7, 14, and then mentally add 14 + 6 + 1 to make 21. This strategy previews the associative property using addition and illuminates arithmetic patterns as students apply count-bys to solve problems (3.OA.9). In the next lesson, students apply the distributive property (familiar from Module 1) as a strategy to multiply and divide. They decompose larger unknown facts into smaller known facts to solve. For example, 48 ÷ 6 becomes (30 ÷ 6) + (18 ÷ 6), or 5 + 3 (3.OA.5, 3.OA.7). Topic B’s final lesson emphasizes word problems, providing opportunities to analyze and model. Students apply the skill of using a letter to represent the unknown in various positions within multiplication and division problems (3.OA.3, 3.OA.4, 3.OA.7).
Grade 3 Mathematics Module 3, Topic C, Overview
Multiplication and Division Using Units of 8
Topic C anticipates the formal introduction of the associative property with a lesson on making use of structure to problem solve. Students learn the conventional order for performing operations when parentheses are and are not present in an equation (3.OA.8). With this knowledge in place, the associative property emerges in the next lessons as a strategy to multiply using units up to 8 (3.OA.5). Units of 6 and 8 are particularly useful for presenting this Level 3 strategy. Rewriting 6 as 2 × 3 or 8 as 2 × 4 makes shifts in grouping readily apparent (see example below), and also utilizes familiar factors 2, 3, and 4 as students learn the new material. The following strategy may be used to solve a problem like 8 × 5:
8 × 5 = (4 × 2) × 5
8 × 5 = 4 × (2 × 5)
8 × 5 = 4 × 10
In the final lesson of Topic C, students relate division using units up to 8 with multiplication. They understand division as both a quantity divided into equal groups and an unknown factor problem for which—given the large size of units—skip-counting to solve can be more efficient than dividing (3.OA.3, 3.OA.4, 3.OA.7).
Grade 3 Mathematics Module 3, Topic D, Overview
Multiplication and Division Using Units of 9
Topic D introduces units of 9 over three days, exploring a variety of arithmetic patterns that become engaging strategies for quickly learning facts with automaticity (3.OA.3, 3.OA.7, 3.OA.9). Nines are placed late in the module so that students have enough experience with multiplication and division to recognize, analyze, and apply the rich patterns found in the manipulation of these facts. As with other topics, the sequence ends with interpreting the unknown factor to solve multiplication and division problems (3.OA.3, 3.OA.4, 3.OA.5, 3.OA.7).
Grade 3 Mathematics Module 3, Topic E, Overview
Analysis of Patterns and Problems Solving Including Units of 0 and 1
In Topic E, students begin by working with facts using units of 0 and 1. From a procedural standpoint, these are simple facts that require little time for students to master; however, understanding the concept of nothing (zero) is among the more complex, particularly as it relates to division. This unique combination of simple and complex explains the late introduction of 0 and 1 in the sequence of factors. Students study the results of multiplying and dividing with those units to identify relationships and patterns (3.OA.7, 3.OA.9). The topic closes with a lesson devoted to two-step problems involving all four operations (3.OA.8). In this lesson, students work with equations involving unknown quantities and apply the rounding skills learned in Module 2 to make estimations that help them assess the reasonableness of their solutions (3.OA.8).
Grade 3 Mathematics Module 3, Topic F, Overview
Multiplication of Single-Digit Factors and Multiples of 10
In Topic F, students multiply by multiples of 10 (3.NBT.3). To solve a fact like 2 × 30, they first model the basic fact 2 × 3 on the place value chart. Place value understanding helps them to notice that the product shifts one place value to the left when multiplied by 10: 2 × 3 tens can be found by simply locating the same basic fact in the tens column.
In the subsequent lesson, place value understanding becomes more abstract as students model place value strategies using the associative property (3.NBT.3, 3.OA.5). 2 × 30 = 2 × (3 × 10) = (2 × 3) × 10. The final lesson focuses on solving two-step word problems involving multiples of 10 and equations with unknown quantities (3.OA.8). As in Lesson 18, students estimate to assess the reasonableness of their solutions (3.OA.8).
The Properties of Multiplication and Division
Topic A begins by revisiting the commutative property. Students study familiar facts from Module 1 to identify known facts using units of 6, 7, 8, and 9 (3.OA.5, 3.OA.7). They realize that they already know more than half of their facts by recognizing, for example, that if they know 2 × 8, they also know 8 × 2 through commutativity. This begins a study of arithmetic patterns that becomes an increasingly prominent theme in the module (3.OA.9). The subsequent lesson carries this study a step further; students apply the commutative property to relate 5 × 8 and 8 × 5, and then add one more group of 8 to solve 6 × 8 and, by extension, 8 × 6. The final lesson in this topic builds fluency with familiar multiplication and division facts, preparing students for the work ahead by introducing the use of a letter to represent the unknown in various positions (3.OA.3, 3.OA.4).
Grade 3 Mathematics Module 3, Topic B, Overview
Multiplication and Division Using Units of 6 and 7
Topic B introduces units of 6 and 7, factors that are well suited to Level 2 skip-counting strategies and to the Level 3 distributive property strategy, already familiar from Module 1. Students learn to compose up to, then over the next decade. For example, to solve a fact using units of 7 they might count 7, 14, and then mentally add 14 + 6 + 1 to make 21. This strategy previews the associative property using addition and illuminates arithmetic patterns as students apply count-bys to solve problems (3.OA.9). In the next lesson, students apply the distributive property (familiar from Module 1) as a strategy to multiply and divide. They decompose larger unknown facts into smaller known facts to solve. For example, 48 ÷ 6 becomes (30 ÷ 6) + (18 ÷ 6), or 5 + 3 (3.OA.5, 3.OA.7). Topic B’s final lesson emphasizes word problems, providing opportunities to analyze and model. Students apply the skill of using a letter to represent the unknown in various positions within multiplication and division problems (3.OA.3, 3.OA.4, 3.OA.7).
Grade 3 Mathematics Module 3, Topic C, Overview
Multiplication and Division Using Units of 8
Topic C anticipates the formal introduction of the associative property with a lesson on making use of structure to problem solve. Students learn the conventional order for performing operations when parentheses are and are not present in an equation (3.OA.8). With this knowledge in place, the associative property emerges in the next lessons as a strategy to multiply using units up to 8 (3.OA.5). Units of 6 and 8 are particularly useful for presenting this Level 3 strategy. Rewriting 6 as 2 × 3 or 8 as 2 × 4 makes shifts in grouping readily apparent (see example below), and also utilizes familiar factors 2, 3, and 4 as students learn the new material. The following strategy may be used to solve a problem like 8 × 5:
8 × 5 = (4 × 2) × 5
8 × 5 = 4 × (2 × 5)
8 × 5 = 4 × 10
In the final lesson of Topic C, students relate division using units up to 8 with multiplication. They understand division as both a quantity divided into equal groups and an unknown factor problem for which—given the large size of units—skip-counting to solve can be more efficient than dividing (3.OA.3, 3.OA.4, 3.OA.7).
Grade 3 Mathematics Module 3, Topic D, Overview
Multiplication and Division Using Units of 9
Topic D introduces units of 9 over three days, exploring a variety of arithmetic patterns that become engaging strategies for quickly learning facts with automaticity (3.OA.3, 3.OA.7, 3.OA.9). Nines are placed late in the module so that students have enough experience with multiplication and division to recognize, analyze, and apply the rich patterns found in the manipulation of these facts. As with other topics, the sequence ends with interpreting the unknown factor to solve multiplication and division problems (3.OA.3, 3.OA.4, 3.OA.5, 3.OA.7).
Grade 3 Mathematics Module 3, Topic E, Overview
Analysis of Patterns and Problems Solving Including Units of 0 and 1
In Topic E, students begin by working with facts using units of 0 and 1. From a procedural standpoint, these are simple facts that require little time for students to master; however, understanding the concept of nothing (zero) is among the more complex, particularly as it relates to division. This unique combination of simple and complex explains the late introduction of 0 and 1 in the sequence of factors. Students study the results of multiplying and dividing with those units to identify relationships and patterns (3.OA.7, 3.OA.9). The topic closes with a lesson devoted to two-step problems involving all four operations (3.OA.8). In this lesson, students work with equations involving unknown quantities and apply the rounding skills learned in Module 2 to make estimations that help them assess the reasonableness of their solutions (3.OA.8).
Grade 3 Mathematics Module 3, Topic F, Overview
Multiplication of Single-Digit Factors and Multiples of 10
In Topic F, students multiply by multiples of 10 (3.NBT.3). To solve a fact like 2 × 30, they first model the basic fact 2 × 3 on the place value chart. Place value understanding helps them to notice that the product shifts one place value to the left when multiplied by 10: 2 × 3 tens can be found by simply locating the same basic fact in the tens column.
In the subsequent lesson, place value understanding becomes more abstract as students model place value strategies using the associative property (3.NBT.3, 3.OA.5). 2 × 30 = 2 × (3 × 10) = (2 × 3) × 10. The final lesson focuses on solving two-step word problems involving multiples of 10 and equations with unknown quantities (3.OA.8). As in Lesson 18, students estimate to assess the reasonableness of their solutions (3.OA.8).