Download Dividing Multi-Digit Whole Numbers

Grade 4 Mathematics, Quarter 2, Unit 2.2
Dividing Multi-Digit Whole Numbers
Overview
Number of instruction days:
6–8
(1 day = 90 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated

2 Reason abstractly and quantitatively.
Use understanding of place value, properties of
operations, and the relationship between
multiplication and division to find whole
number quotients with up to four-digit
dividends and one-digit divisors.

Solve word problems involving multiplicative
comparison.

Solve multi-step word problems involving
whole numbers and having whole number
answers.

Represent problems using an equation with a
letter standing for an unknown quantity.

Make sense of quantities and their relationships
in problem situations, not just how to compute
them.

Decontextualize a given situation and represent
it symbolically using equations, area models,
and/or rectangular arrays.
8 Look for and express regularity in repeated
reasoning.

Notice repeated calculations as a solution to
division problems.

Use models to examine patterns and generate
the patterns’ algorithm.
Essential Questions

How can you explain division using equations,
area models, and/or rectangular arrays?

How can you represent the unknown when
solving word problems?

What are some strategies for solving division
problems?

How can you explain this problem?
Providence Public Schools
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Grade 4 Mathematics, Quarter 2, Unit 2.2
Version 4
Dividing Multi-Digit Whole Numbers (6–8 days)
Standards
Common Core State Standards for Mathematical Content
Number and Operations in Base Ten2
2
4.NBT
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit
divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
Operations and Algebraic Thinking
4.OA
Use the four operations with whole numbers to solve problems.
4.OA.2
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using
drawings and equations with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive comparison.1
1
4.OA.3
See Glossary, Table 2.
Solve multistep word problems posed with whole numbers and having whole-number answers
using the four operations, including problems in which remainders must be interpreted.
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.
Common Core State Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
8
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying
attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2)
with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the
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Dividing Multi-Digit Whole Numbers (6–8 days)
Grade 4 Mathematics, Quarter 2, Unit 2.2
Version 4
regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 +
x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a
problem, mathematically proficient students maintain oversight of the process, while attending to the
details. They continually evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In Grade 3, students understood division as the sharing of objects. They used division within 100 to solve
word problems involving equal groups, arrays, and measurement quantities. They also understood
division as an unknown-factor problem. By the end of third grade, students fluently divided within 100
using the relationship between multiplication and division. They also solved two-step word problems with
the four operations. Division was a major focus in Grade 3 according to the PARCC framework.
Current Learning
In Grade 4 students find whole number quotients and remainders with up to four-digit dividends and onedigit divisors. Division strategies are based on place value, properties of operations, and the relationship
between multiplication and division. Students solve word problems involving multiplicative comparisons
(see CCSS Glossary, Table 2). Students solve multi-step word problems with division and represent an
unknown quantity in a division equation with a letter. Division is a major focus in Grade 4 according to
the PARCC framework.
Future Learning
In Grade 5 students will extend their knowledge of division to finding the quotients and remainders of
problems with four-digit dividends and two-digit divisors. Estimation becomes relevant when extending
to two-digit divisors. Fifth-grade students will also divide decimals to the hundreds place. They will
interpret a fraction as the division of the numerator by the denominator, and they will also divide
fractions.
Additional Findings
“As we indicated earlier, relatively little research is available to shed light on how students think about
multi-digit division or what learning activities might be of the most help to them. Sample teaching lessons
have been proposed, and preliminary results suggest that students can construct their own procedures that,
over time, approximate standard algorithms. As with multiplication, however, the best that educators can
do at this point is to examine some alternative algorithms that are likely to support students’ efforts to
develop proficiency with multi-digit divisors.” (Adding It Up, p. 210)
In grades 3-5, helping students develop meaning for whole-number multiplication and division should
become a central focus. By creating and working with representations (such as diagrams or concrete
objects) of multiplication and division situations, students can gain a sense of the relationships among the
operations. (Principles and Standards for School Mathematics, p. 34)
According to Progressions for the Common Core State Standards, K: Counting and Cardinality; K–5,
Operations and Algebraic Thinking, students in Grade 4 learn to compare multiplicatively. The language
in Multiplication Compare problems can be difficult. In this document, a tape diagram illustrates a
compare problem used in Table 3. (p. 29)
Providence Public Schools
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Grade 4 Mathematics, Quarter 2, Unit 2.2
Version 4
Dividing Multi-Digit Whole Numbers (6–8 days)
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.

Find whole number quotients with up to four-digit dividends and one-digit divisors.

Solve word problems with multiplication or division involving multiplicative comparison.

Solve multi-step word problems involving whole numbers and having whole number answers.

Represent problems using an equation with a letter standing for an unknown quantity.
Instruction
Learning Objectives
Students will be able to:

Use understanding of place value, properties of operations and the relationship between multiplication
and division to find whole number quotients with up to four-digit dividends and one-digit divisors.

Solve word problems with multiplication or division involving multiplicative comparison.

Solve multi-step word problems involving whole numbers and having whole number answers.

Represent problems using an equation with a letter standing for an unknown quantity.

Estimate reasonableness of answer.

Demonstrate understanding of the concepts and skills in this unit.
Resources
enVision Math Grade 4, Pearson Education, Inc., 2009

Topic 4, Division Meanings and Facts, Teacher Edition

Topic 8, Dividing by 1-Digit Divisors, Teacher Edition
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
Grade 4 Mathematics, Quarter 2, Unit 2.2
Version 4
Also see Section I, Supplemental Materials
Lesson 8.3A, Estimating Quotients for Greater Dividends
Lesson 8.3B, Using Objects to Divide: Division as Repeated Subtraction
Lesson 8.3C, Division as Repeated Subtraction
Lesson 8.8A, Dividing 4-Digit by 1-Digit Numbers

Teacher Resource Masters

Student Edition
Investigations in Numbers, Data, and Space, Grade 4, Pearson Education, Inc., 2008

Implementing Investigations in Grade 4- Implementation Guide

Unit 3; Multiple Towers and Division Stories, Teacher Edition
Investigation 2: Solving Division Problems

Unit 8; How Many Packages? How Many Groups?, Teacher Edition
Investigation 3: Solving Division Problems

Also see Section I, Supplemental Materials
Session 3.5A, Dividing 4-Digit Numbers

Teacher Resources Binder
Pearson Online Success Net - www.pearsonsuccessnet.com/snpapp/login/login.jsp
Implementing Investigations Site - http://investigations.terc.edu/
Exam View Assessment Suite
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for ample resources to refer to
when planning your unit and individual lessons.
Materials
Connecting cubes, color tiles, centimeter grid paper, array cards, place value blocks, counters, pennies,
300 chart
Instructional Considerations
Key Vocabulary
dividend
factor
quotient
multiple
divisor
Planning for Effective Instructional Design and Delivery
Division is not new to grade 4. In fourth grade, students build on their third grade work with division
within 100. It was a major focus at that grade level. When planning for the unit, take this into
consideration.
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Grade 4 Mathematics, Quarter 2, Unit 2.2
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Dividing Multi-Digit Whole Numbers (6–8 days)
This standard calls for students to explore division through various strategies. After developing an
understanding of using arrays to divide, students begin to use a more abstract model for division. This
model connects to a recording process that will be formalized in the 5th grade.
At the beginning of Lesson 4-1, use a cues, questions, and advance organizers strategy to activate
students’ prior knowledge of division. Make connections to the meaning of division by asking students a
constructing support question. For example, you can prompt their prior knowledge by asking, “How are
multiplication and division related?” Have students share their examples. Connect this discussion to the
key vocabulary for this unit.
When planning for this unit, refer to the Operations and Algebraic Thinking Progression,
http://commoncoretools.files.wordpress.com, page 23. Here you will find a table with examples of
multiplication and division situations.
In this unit the focus is only on whole number division, division with remainders will be taught in the
third quarter.
In Lesson 8-5 (TE, page 177), use identifying similarities and differences to review and practice concepts.
Students will compare factors and multiples using a Venn diagram.
Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are
aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and
Investigations Activities offer additional practice for student learning and support small group
differentiated instruction.
Use teacher created common tasks as formative assessments to monitor student progress and
understanding of critical content and essential questions. Use data from formal and informal assessments
to guide your instruction and planning.
For planning considerations, read through the teacher editions for suggestions about scaffolding
techniques, using additional examples, and differentiated instruction as suggested by the envision and
Investigations resources, particularly the Algebra Connections and Teacher Notes section.
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Grade 4 Mathematics, Quarter 2, Unit 2.2
Version 4
Notes
Providence Public Schools
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Grade 4 Mathematics, Quarter 2, Unit 2.2
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Dividing Multi-Digit Whole Numbers (6–8 days)
Providence Public Schools