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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
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 4th grade to polynomial multiplication and division in Algebra 2, and lots in between. Thus, this unit sets the seal on major work of 3rd grade 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 understanding of the unit content and action plan for future units.
Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.
Unit Prep
Intellectual Prep
Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.
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.
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equal groups |
Example: 4 equal groups of 3 stars |
|
array |
Example: 4 rows of 3 |
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tape diagram |
Example: There are 4 bags with 3 plums in each bag. How many plums are there in all? |
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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.
Vocabulary
Unit Vocabulary
parentheses, $${\left( \right)}$$
To see all the vocabulary for Unit 3, view our 3rd Grade Vocabulary Glossary.
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 Table (1 per student) — See Lesson 1 for more information.
- Multiplication Table (1 per student)
Unit Practice
Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.
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
Operations and Algebraic Thinking
Foundational Standards
Number and Operations in Base Ten
Operations and Algebraic Thinking
Future Standards
Number and Operations in Base Ten
Number and Operations—Fractions
Operations and Algebraic Thinking
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
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