4th Grade Mathematics - Investigations
Unit 4: Multi-Digit Number Division
Teacher Resource Guide
2012-2013
In Grade 4, instructional time should focus on three critical areas:
1. Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find
quotients involving multi-digit dividends;
Students apply their understanding of models for multiplication (equal-sized groups, arrays, area models) and models
for division, place value, and the distributive property as they discuss and use efficient methods to estimate and
compute products and quotients. They develop fluency with efficient procedures for multiplying whole numbers,
understand and explain why the procedures work based on place value and properties, and use them to solve
problems. Students interpret remainders based upon the context of the problem.
2. Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and
multiplication of fractions by whole numbers;
Students recognize that two different fractions can be equal (12/4 = 6/2), and they develop methods for generating
and recognizing equivalent fractions. Students extend understandings about how fractions are built from unit
fractions (3/4 = ¼ + ¼ + ¼), and use the meaning of multiplication to multiply a fraction by a whole number.
3. Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides,
perpendicular sides, particular angle measures, and symmetry.
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing
two-dimensional shapes, students understand properties of two-dimensional objects and solve problems involving
symmetry.
4th Grade
2012-2013
Page 1
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
TRIMESTER 3
TRIMESTER 2
TRIMESTER 1
4th Grade Mathematics 2012-2013
Unit
1: Multiplication and Division Concepts
2: Multi-Digit Multiplication
3: Measurement/Geometry
4: Multi-Digit Division
5: Fractions
Time Frame
Test By
6 weeks
8/27 – 10/5
October 5
6 weeks
10/8 – 11/16
November 16
4 weeks
11/19 – 12/21
December 21
7 weeks
1/2- 2/22
February 22
7 weeks
2/25-4/19
April 19
6 weeks
4/22 – 5/29
May 29
6: Decimal Fractions
Math Wiki: http://dmps-mathematics.wikispaces.com
4th Grade
2012-2013
Page 2
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
Big Ideas
Essential Questions
Estimation is helpful in understanding if an answer is reasonable.
When you know what the question is asking you know how to use the remainder. Sometimes
the remainder is part of the answer, sometimes it changes the answer and sometimes the
remainder can be ignored.
Identifier
STANDARDS
4.NBT.6
4.NBT.1
4.OA.3
4.OA.2
4th Grade
Why do we estimate?
How does the remainder affect the answer?
Standards
Mathematical Practices
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.
1) Make sense of problems and persevere in
solving them.
Recognize that in a multi-digit whole number, a digit in one place represents ten
times what it represents in the place to its right.
Solve multistep word problems posed with whole numbers and having wholenumber 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.
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
2012-2013
2) Reason abstractly and quantitatively.
3) Construct viable arguments and critique the
reasoning of others.
4) Model with mathematics.
5) Use appropriate tools strategically.
6) Attend to precision.
7) Look for and make use of structure.
8) Look for and express regularity in
repeated reasoning.
Page 3
Unit 4: Multi-Digit Division
Identifier
4.NBT.6
STANDARDS
4.NBT.1
4.OA.3
January 2-February 22 (7 weeks)
Standards
Bloom’s
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.
Remember (1)
Find (whole number
quotients &
remainders)
Understand (2)
Illustrate and explain
(the calculation using
equations, arrays, or
area models)
Apply (3)
Solve (multi-step word
problems w/ whole
numbers & interpret
remainders)
Recognize that in a multi-digit whole number, a digit in one place represents ten
times what it represents in the place to its right.
Solve multistep word problems posed with whole numbers and having wholenumber 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.
Understand (2)
Evaluate (5)
4.OA.2
4th Grade
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.2
2012-2013
Skills
Represent (problems
w/ equations using a
letter for the unknown
quantity)
Concepts
remainders
dividend
divisor
quotient
equations
rectangular
arrays
area models
equations
unknown
quantity
mental
computation
estimation
rounding
Assess
(reasonableness of
answer)
Page 4
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
Instructional Strategies for ALL Students
Critical Reading Prior to Instruction:
Teaching Student-Centered Mathematics Grades 3-5, Van de Walle & Lovin, Pearson, 2006, p. 121-128 (building resource, SPED)
Real-world context – For students to reach the level of rigor intended for the operations of multiplication and division in the Iowa Core, they must develop understanding of the
operations within real-world contexts. The tendency in the United States is to have students solve a lot of problems in a single class period. The focus of these lessons seems to
be on how to get answers. In Japan, however, a complete lesson will often revolve around one or two problems and the related discussion (Reys & Reys, 1995). A lesson built
around word problems focusing on how students solve the problem is ideal. When the teacher focuses the discussion of the word problems on students’ strategies, attention is
turned toward the reasoning and knowledge of the number system necessary to solve the problems.
Division Problem Types – There are two types of division problems: Measurement and Partitive. In Measurement problems, the group size is known and the number of groups is
unknown. For example: If 18 plums are to be packed 6 to a bag (group size known), then how many bags are needed (number of groups unknown)? In Partitive problems, the
number of groups is known and the group size is unknown. For example: If 18 plums are shared equally into 6 bags (number of groups known), then how many plums will be in
each bag (group size unknown)? Students need to recognize both types of problems as division situations, should be able to model and solve each type of problem, and should
know the units of the result: Is it 3 bags or 3 plums per bag? (See p. 7 in Unit One’s guide for further explanation of the problem types.)
Use of models to solve division problems – It is essential for students to learn methods for computation that are meaningful. The primary methods for multi-digit division used
in 4th grade are the Rectangle Sections Method, the Partial Quotients Method, the Expanded Notations Method, and the Digit-by-Digit Method. In all four methods, students find
partial answers that are combined for the final quotient. This unit is the beginning of multi-digit division for students. The focus is one-digit divisors. Work with these models will
continue in 5th grade as students work to become more proficient with division up to two-digit divisors. Keep in mind that the Rectangular Sections Method, the Partial
Quotients Method and the Expanded Notation Method are based upon place value much like the Area Model for multi-digit multiplication. The Digit-by-Digit Method (traditional
algorithm) will be harder for students because it is based on digits instead of place value. The Partial Quotients Method and Digit-by-Digit Methods are efficient ways to divide
multi-digit numbers. With the new Iowa Core, 6th graders are expected to be fluent in whole number operations. Therefore, multi-digit multiplication and multi-digit division are
critical priorities for 4th and 5th grades.
Interpreting a remainder – More often than not, division does not result in a simple whole number. If there is no context given for a division problem, a remainder can only be
dealt with in two ways: It can either remain a quantity left over or be partitioned into fractions. In real contexts, remainders can have three additional effects on answers: The
remainder is discarded leaving a smaller whole-number answer, or the remainder can “force” the answer to the next highest whole number, or the answer is rounded to the
nearest whole number for an approximate result. Students should not be led to think of remainders as “R3” or “left over.” Remainders should be put in context and dealt with
accordingly. (See Real-world context above.)
Estimation (to assess reasonableness of an answer) – Approximate numbers and rounding are estimation strategies for multi-digit division. In our number system, some
numbers are “nice.” They are easy to think about and work with. Multiples of 10, 100, 5, and 25 are often referred to as nice numbers. Numbers such as 100, 700, and 50 are
easier to use than 94, 517, and 762. Multiples of 25 are nice because they are easily combined into multiples of 50 and 100. Flexible thought with numbers and many estimation
skills are related to the ability to substitute a nice number for one that is not so nice. This substitution can be helpful in mentally computing an estimate for a multi-digit
multiplication problem. Students have primarily been taught to round as an estimation strategy. A context for why it might be necessary to round is typically not emphasized. It
is ideal for students to know strategies for nice numbers and rounding and have many opportunities to use the strategies in real-world contexts (adapted from Van de Walle,
2006).
4th Grade
2012-2013
Page 5
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
Routines/Meaningful Distributed Practice
Distributed Practice that is Meaningful and Purposeful
Practice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and
distributed.



Meaningful: Builds on and extends understanding
Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
Distributed: Consists of short periods of systematic practice distributed over a long period of time
Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum.. The skills presented during
routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or
review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited
and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer.
Selection of the routine should be made based on informal teacher observation and formative assessments.
Skill
Standard
A digit in one place represents ten times what it represents in the place to its right.
4.NBT.1
Multiplication and Division fact fluency
Draw points, lines, line segments, rays, angles, parallel and perpendicular lines
(May be taught in whole group lesson and reinforced during MDP time)
Solve word problems involving distance, intervals of time, liquid volumes, masses of objects and money
3.OA.3
Whole number multiplication
Round multi-digit whole numbers to any place
4.NBT.5
4.NBT.3
Multi-digit Addition & Subtraction
4.NBT.4
Resource
4.G.1
4.MD.2
Other skills students need to develop based on teacher observations and formative assessments.
4th Grade
2012-2013
Page 6
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
Investigations Resources for Unit 4 - Multi-digit Division
Investigations
Standards
Addressed
4.NBT.6
CGI Problems
4.OA.3
Instructional Plan
Resource
Week 1 and Week 2
Unit 3: Multiple Towers and Division Stories
 Investigation 2 Division
Week 3 and Week 4
Unit 8: How Many Packages? How Many Groups?
 Investigation 3 Solving Division Problems
Do 3.5A before 3.5 (CC90-CC94, C61-C64)
Week 5 and Week 6
 Introduce “Using Groups of the Divisors” strategy.
 Solve CGI problems including problems in which the
remainder must be interpreted.
 Solve multi-step CGI problems.
Additional Focus needed on:
 Assess the reasonableness of answers
 Solve equations with variables for the unknown
quantity
4th Grade
Estimate before solving and explain reasoning
CGI Problems
2012-2013
Page 7
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
These lessons are intended to provide additional resources based on student need.
Lesson Progression
Lesson Focus
Lessons are listed in
intended order
Time Frame
Resource
Teacher Directions
Standards
Addressed
Multi-digit subtraction
Throughout the unit
(as warm up or MDP for
whole group review)
Wiki: Strategies for multi-digit
subtraction
These strategies may be reviewed during warm up
or MDP. They may also be taught during a small
group lesson.
4.NBT.4
Multi-step word
problems
Throughout the unit
Wiki: Multi-step word problems: without
multi-digit division
Include only addition, subtraction, multiplication
and simple division (division facts). Include multidigit division after the alternative algorithms are
taught.
4.OA.3
Understanding how to multiply and divide with
multiples of 10 is critical when using partial
quotients to solve multi-digit division problems.
4.NBT.1
(a few problem per week)
Teachers may need to write some problems
as well. Teacher-made problems can be
shared via a Discussion on the Wiki.
Dividing multiples of 10
4th Grade
3-5 days
Then continue
throughout the unit as
warm up or MDP
Wiki: Tack on Trailing Zeros
(multiplication) Lesson 22
Wiki: Tack on Trailing Zeros
(multiplication) Lesson 23
Wiki: Tack on Trailing Zeros
(multiplication) Lesson 24
Wiki: Divide by Multiplying
Wiki: Tack on Trailing Zeros
Wiki: Canceling Common Zeros
Wiki: Breaking up the Dividend
Wiki: Think Money
2012-2013
Page 8
Unit 4: Multi-Digit Division
January 2-February 22 (7 weeks)
Lesson Progression
Lesson Focus
Resource
Teacher Directions
Lessons are listed in
intended order
Partial Quotients
Wiki: Partial Quotients Examples
Wiki: Division Problem Bank
Teacher made problems
Explanation of Partial Quotients Method
Teacher-made problems can be shared via a Discussion on the Wiki.
Remainders
Wiki: Interpreting the Remainder Problem Bank
Multi-digit division
word problems
Wiki: Division Problem Bank
Teacher made problems
Standards
Addressed
4.NBT.6
Present word problems and allow students to solve
them using the Partial Quotients Method.
Students learn what to do with the remainder by
solving word problems. Read the paragraph about
remainders in the Instructional Strategies.
4.NBT.6
Present lots of problems in context. Include
multiplication and division. Students use an
algorithm/method of choice
4.NBT.6
4.OA.2
4.NBT.5
Present single and multi-step word problems daily.
Include all four operations and interpreting
remainders. Students use an algorithm/method of
choice. Whole group discussions on the various
ways students solved problems is critical.
4.OA.3
It is important to discuss the reasonableness of
answers. Read the Instructional Strategies for more
details.
4.OA.3
Teacher-made problems can be shared via a Discussion on the Wiki.
Multi-step word
problems
Wiki: Multi-step word problems
Reasonableness of
Answer
Throughout the unit
4th Grade
Teachers may need to write some problems as well. Teacher-made
problems can be shared via a Discussion on the Wiki.
2012-2013
Page 9