Multiplication and Division, Part 2

Students deepen their understanding of multiplication and division, including their properties and extending their study of factors to include all units from 0 to 10, as well as multiples of 10 within 100.

Math

Unit 3

3rd Grade

Unit Summary


The scope and sequence for 3rd Grade Math was adjusted in August 2022. The scaled bar and picture graphs (3.MD.3) topic has been moved from this unit to Unit 2. 

Unit 3 extends the study of factors from 2, 3, 4, 5, and 10, which students explored in Unit 2, to include all units from 0 to 10, as well as multiples of 10 within 100. To work with these more challenging factors, students will rely on skip-counting (a Level 2 strategy) and converting to an easier problem (a Level 3 strategy dependent on the properties of operations). They then will apply their understanding of all four operations to two-step word problems as well as arithmetic patterns at the end of the unit.

Topic A begins by reminding students of the commutative property they learned in Unit 2, as well as introducing them to the distributive and associative properties, which they will rely upon for many of the strategies they learn for the larger factors. In order to be able to use these properties, they need to understand how to compute with a factor of 1, which they explore along with 0, as well as understand how to use parentheses. They’ll then explore the factors of 6, 7, 8, and 9 in Topics B and C. In Unit 2, when students were first introduced to multiplication and division with the easier factors of 2-5 and 10, students could count all of the objects to find the product, a Level 1 strategy, and used skip-counting as their fluency developed. But, because of the increased difficulty of 6–9 facts, students will not only rely on skip-counting (a Level 2 strategy), but also convert to an easier problem (a Level 3 strategy). Converting to an easier problem is dependent on the properties of operations (e.g., to find $$6\times7$$, think of $$5\times7$$ and add a group of 7 is dependent on the distributive property). Thus, students will work with the properties extensively throughout the unit, with their understanding of them and notation related to them growing more complex and abstract throughout the unit. In Topic D, students will multiply one-digit numbers by multiples of 10 and by two-digit numbers using the associative property. Finally, students solve two-step word problems involving all four operations, assessing the reasonableness of their answer, and identify arithmetic patterns and explain them using the properties of operations.

In Unit 3, students deepen their understanding of multiplication and division, including their properties. "Mathematically proficient students at the elementary grades use structures such as…the properties of operations…to solve problems" (MP.7) (Standards for Mathematical Practice: Commentary and Elaborations for K–5, p. 9). Students use the properties of operations to convert computations to an easier problem (a Level 3 strategy), as well as construct and critique the reasoning of others regarding the properties of operations (MP.3). Lastly, students model with mathematics with these new operations, solving one- and two-step word problems using them (MP.4). 

Students’ understanding of multiplication and division will further develop in Unit 4, when students study area. Students will also use their understanding of these operations in Unit 7 when they apply them in the context of measurement word problems. In subsequent years, students will depend on their conceptual understanding and fluency with these operations to apply and extend them in a variety of ways—everything from multi-step multiplicative comparison words problems in Grade 4 to polynomial multiplication and division in Algebra 2, and lots in between. Thus, this unit sets the seal on major work of Grade 3 as well as deeply important foundational work upon which students will rely for years to come.

Pacing: 26 instructional days (23 lessons, 2 flex days, 1 assessment day)

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Assessment


The following assessments accompany Unit 3.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 16.

Post-Unit

Use the resources below to assess student mastery of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.

Unit Prep


Intellectual Prep

Unit Launch

Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.

Intellectual Prep for All Units

  • Read and annotate "Unit Summary" and "Essential Understandings" portion of the unit plan. 
  • Do all the Target Tasks and annotate them with the "Unit Summary" and "Essential Understandings" in mind. 
  • Take the Post-Unit Assessment.

Unit-Specific Intellectual Prep

Read the following table that includes models used throughout the unit.

Equal groups

Example: 4 equal groups of 3 stars

Array

Example: 4 rows of 3

Tape diagram

Example: There are 4 bags with 3 plums in each bag. How many plums are there in all?

Concrete or pictorial base ten blocks

Example: Find $$2 \times 60$$ using base ten blocks. 

Essential Understandings

  • Multiplication problems can be solved using a variety of strategies of increasing complexity, including making and counting all of the quantities involved in multiplication or division (Level 1 strategy), repeated counting on by a given number (Level 2), and using the properties of operations to compose and decompose unknown facts into known ones (Level 3). 
  • Unknown numbers in a count sequence can be found by using mental strategies. "For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14 + 7 = 14 + 6 + 1 = 20 + 1 = 21" (OA Progression, p. 25). 
  • Two, five, and ten are often helpful values to use when converting to an easier equivalent problem to solve. For example, to solve $$6\times8$$, one might think of the friendly fact of $$5\times8$$ and then add another $$8$$.
  • Making sense of problems and persevering in solving them is an important practice when solving word problems. Key words do not always indicate the correct operation.

Materials

  • Base ten blocks (24 tens, 2 hundreds per student or small group) — Students might not need these depending on their reliance on concrete materials. Students can also use Paper base ten blocks cut into units instead. See Lesson 17 Anchor Task 1 Notes for more information.
  • Optional: Blank Multiplication Chart (1 per student) — See Lesson 1 for more information.
  • Multiplication Chart (1 per student)

Vocabulary

parentheses

To see all the vocabulary for Unit 3, view our 3rd Grade Vocabulary Glossary.

Unit Practice


Word Problems and Fluency Activities

Access daily word problem practice and our content-aligned fluency activities created to help students strengthen their application and fluency skills.

Lesson Map


Topic A: Introduction to The Properties of Operations

Topic B: Multiplication and Division by 6 and 7

Topic C: Multiplication and Division by 8 and 9

Topic D: Multiplication and Division by Values Greater than 10

Topic E: Two-Step Word Problems and Patterns in Arithmetic

Common Core Standards


Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

Number and Operations in Base Ten

  • 3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Operations and Algebraic Thinking

  • 3.OA.A.3 — Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
  • 3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
  • 3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
  • 3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • 3.OA.D.8 — Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
  • 3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Foundational Standards

Number and Operations in Base Ten

  • 2.NBT.A.1
  • 2.NBT.A.2
  • 3.NBT.A.2

Operations and Algebraic Thinking

  • 2.OA.A.1
  • 2.OA.C.3
  • 2.OA.C.4
  • 3.OA.A.1
  • 3.OA.A.2
  • 3.OA.B.5
  • 3.OA.B.6

Future Standards

Number and Operations in Base Ten

  • 4.NBT.B.5
  • 4.NBT.B.6

Number and Operations—Fractions

  • 4.NF.B.4
  • 5.NF.B.3
  • 5.NF.B.4
  • 5.NF.B.5
  • 5.NF.B.6
  • 5.NF.B.7

Operations and Algebraic Thinking

  • 4.OA.A.1
  • 4.OA.A.2
  • 4.OA.A.3
  • 4.OA.B.4

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

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Unit 2

Multiplication and Division, Part 1

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Unit 4

Area