Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Targeted Content Standard(s):
Student Friendly Learning Targets
5.NBT.6 Find whole-number quotients of whole numbers with up to
four-digit dividends and two-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.
I can…
 Divide multi-digit whole numbers.
 Explain how various models can be used
to represent and solve problems
involving division situations.
Targeted Mathematical Practice(s):
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure.
Look for an express regularity in repeated reasoning
Supporting Content Standard(s): (optional)
5.OA.1 Use parentheses, brackets, or braces in numerical expressions,
and evaluate expressions with these symbols.
Explanation of Rigor: (Fill in those that are appropriate.)
Conceptual:
Students will model division with
multi-digit numbers using
manipulatives, area models/arrays,
and equations.
Procedural:
Students will use place value
strategies and properties of
operations to divide multi-digit whole
numbers.
Application:
Students will apply their knowledge
of the area model of division to solve
real-life problems.
Vocabulary:
Divide
Divisor
Dividend
Tens
Hundreds
Thousands
Area model (length, width, area)
Array
Equation
Parentheses
Partial Quotient
Quotient
Evidence of Learning (Assessment):
Pre-Assessment: Division-Subtraction Relationship, Division-Multiplication Relationship
Formative Assessment(s): Dividing with Base 10 Blocks – Observation Checklist
Array/Area Model for Division-Student Practice Sheet
Partial Quotients Division Worksheet
Column division vs. Partial quotients Formative Assessment
Summative Assessment: 5.NBT.6 Base Ten Summative Assessment
Multi-Digit Division Summative Assessment
1
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Self-Assessment: 5.NBT.6 Multi-Digit Division Summative Self-Assessment
Lesson Segments:
1. Assess student understanding of division/subtraction and division/multiplication relationships
2. Manipulative, area and array models for division
3. Solve real-world problems
4. Connecting area models and equations
5. Partial Quotients
6. Column Division
7. Assessment
2
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Procedures:
Segment 1
Approximate Time Frame:
20-30 minutes
Focus:
Assess students’ ability to connect
area and multiplication.
Lesson Format:
Resources:
Division-Subtraction Relationship
Division-Multiplication Relationship
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Mathematical Practice Look Fors:
 MP4: Model with mathematics. Students should be
making connections between the visual models and
equations that they are using to represent
multiplication of multi-digit numbers.
 MP7: Look for and make use of structure. Student’s
visual models should represent the base 10 structure
of numbers.
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Differentiation for Remediation:
Have students model multiplication with either base 10
blocks or digi-blocks before making area models.
Differentiation for English Language Learners:
English language learners would benefit from being pretaught the vocabulary: “area model” and “array” before
beginning this segment.
Differentiation for Enrichment:
Have students create real world situation word problems
for each of the problems.
Potential Pitfall(s):
Students may have difficulty connecting visual models to
justify procedures for multiplying multi-digit numbers.
Independent Practice (Homework):
Procedure:
Teacher Notes/Reflections:
Have students complete the Division-Subtraction and DivisionMultiplication Tasks independently.
These documents are pre-assessments for this lesson. Students
have used area models in the previous lesson for multiplication
and in Grade 4 for division by 1-digit numbers. Students will
build upon this experience to use area models and arrays for
division of multi-digit numbers.
3
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 2
Approximate Time Frame:
90-100 min.
Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Focus:
Students will create manipulative,
area and array models for division.
Math Practice Look For(s):
MP4: Model with mathematics. Students will use base
ten blocks to connect place value concepts to model the
process of division.
MP5: Use appropriately tools strategically. Look for
students to strategically use the base 10 blocks to
represent division. Students should know when to break
up the 100 into 10 tens to represent the division
situation.
MP7: Look for and make use of structure. Students can
see and understand how numbers are put together as
parts and wholes.
Resources:
Base 10 Blocks
Tape to make a Workmat (optional)
Dividing with Base 10 Blocks Worksheet
Dividing with Base 10 Blocks Observation
Checklist
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Differentiation for Remediation: Students may benefit
from being reminded how to write numbers in expanded
notation to determine where to segment the diagram
into sections.
Differentiation for English Language Learners: Permit
ELLs to work with partners who speak the same language
and discuss their representations in their native
language.
Differentiation for Enrichment: Students who are using
mental math or multiplication to solve problems should
be asked to justify their calculations using visual models.
They should also be asked to connect the written
equations (using parentheses) to the visual models.
Students can solve problems such as 403 ÷ 31, where it
requires regrouping in the dividend. Students have to
keep in mind that a rectangular array must be formed
with the dividend. Example:
4
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Potential Pitfall(s):
Some students may not choose to use hundreds, but only
work with tens and ones.
Students may have difficulty representing the portions of
the diagram with equations. They may not know where
to divide the diagram into portions.
Students may need to be reminded that the dividend
should form a rectangular array.
Independent Practice (Homework):
1) Write an equation for the following base ten blocks:
Answer: 396 ÷ 33 = 21
2) Draw a representation for 231 ÷ 21 =
5
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Procedure:
Teacher Notes/Reflections:
1. Group students in partners or triads. Give each pair/triad 5
hundreds, 30 tens, 30 ones and a workmat.
2. Have the partners collaboratively build the number 320 out of
base 10 blocks.
3. Next, tell them they will need to divide these into equal
groups of 20. (If students can do this mentally, have them
create the visual model to justify their calculation.)
4. Observe how students are arranging their groups. See if any
students arrange their models in an array. Listen to students
discussions as they collaborate with their partners to hear
their strategies. (Observation Checklist)
5. If any students have arranged their manipulatives into an
array, have them share their work. If not, model for students
how you can arrange the base 10 blocks into an array using
the workmat. (See diagram- The green represents the divisor
20 shown with 2 tens. The blue represents the dividend 320
shown as 2 hundreds and 12 tens. Start with the divisor and
then lay the blocks for the dividend to line them up along the
divisor blocks. Finally ask what block would be multiplied by
10 to equal 100? What block would be multiplied by 1 to
equal 10? Lay the blocks down accordingly.
quotient
divisor
dividend
*Note: If the students have not used base ten blocks to model
multiplication, they may need help understanding that the
width of the divisor is the width of the dividend and the length
of the quotient is the length of the dividend.
*Students need to think about how to arrange the dividend
into a rectangular array based upon the number of equal
groups that must be created.
6. Once the students model the problem with base ten blocks,
they can connect it to an area picture (not drawn to scale).
Ask them how this picture connects back to the area model of
multiplication.
10 +
20
200
6
120
6
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
7. Have students repeat the process with their partners and then
ask them to try to solve 440 ÷ 22 using the base ten blocks
Teacher Notes/Reflections:
*Note: The teacher may initially model how to place the
divisor (green) on the workmat first and allow the students to
explore how to place the dividend (blue).
on the workmat. Once they have it placed they can
determine the quotient (red).
quotient
divisor
dividend
8. Once the students model the problem with base ten blocks,
they can connect it to an area picture (not drawn to scale).
20
20 +
2
400
40
9. Again observe and listen using the observation checklist with
the groups. Have students share out and then try to solve 132
÷ 12 with the blocks.
7
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 3
Approximate Time Frame:
45-50 minutes
Focus:
Lesson Format:
Resources:
Whole Group
Small Group
Independent
5.NBT.6 Base Ten Assessment
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Students will apply their knowledge
of base ten blocks and division to
solve real world problems.
Math Practice Look For(s):
Differentiation for Remediation:
MP1: Make sense of problems and persevere in solving
them. Students will be able to plan a solution that makes
sense of the problem by connecting their prior
knowledge of dividing with base ten blocks.
Allow students to use base ten blocks to help solve the
problem.
MP3: Construct viable arguments and critique the
reasoning of others. Students will cite evidence and
develop a logical argument.
Define length, width, and area to assist with the garden
problem.
Differentiation for English Language Learners:
Differentiation for Enrichment:
Have the students create another possible solution using
224 as the dividend. Then either use base ten blocks to
model the solution or draw a diagram of the base ten
blocks.
Potential Pitfall(s):
Independent Practice (Homework):
In the garden problem, students need to find various
unknowns within the dividend and quotient. They need
to analyze the problem and determine a strategy to
figure out the various problems which includes
eliminating unnecessary information.
Procedure:
Teacher Notes/Reflections:
1. Either assign partners or small groups to work together to
solve the assessment.
8
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 4
Approximate Time Frame:
135-150 minutes
Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Focus:
Students will connect arrays to area
models and use equations to represent
the division situations.
Resources:
 Array/Area Model for DivisionTeacher Notes
 Array/Area Model for DivisionStudent Practice Sheet
 Division Array/Area Model
Formative Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
Differentiation for Remediation:
MP2: Reason abstractly and quantitatively. Students need to
reason to appropriate divisors to use when dividing using the
array model. Students need to be able to use all of the
operations to solve the problems.
You may want to list the multiples of the divisor
from 1-10 instead of just the multiples of 2, 5, and
10 in the help box.
MP4: Model with mathematics. Students will use the area
model to bridge the understanding of division with base ten
blocks to partial quotients.
For students that need a visual that is drawn to
scale, they may begin with a smaller dividend and
draw it to scale on centimeter grid paper.
Differentiation for English Language Learners:
MP7: Look for and make use of structure. Students can see
and understand how the dividend can be broken into smaller
parts.
Differentiation for Enrichment:
Potential Pitfall(s):
Independent Practice (Homework):
Some people prefer to draw a large rectangle first and divide it
in sections as they solve the problem. Some prefer to build the
sections, as they solve it. However, the rectangles do not need
to be drawn to scale which may lead to some confusion.
Encourage the students to create multiple
solutions using the area model and then describe
which one was the most efficient.
A fish tank at the Shedd Aquarium holds 6,358
gallons of water. If every fish needs 13
gallons of water to survive, how many fish can
be safely put into the tank?
Use an array model to solve the problem.
Or you can use the Division Array/Area Model
Formative Assessment.
9
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Procedure:
1. Review the Array/Area Model for Division-Teacher notes
prior to the lesson. It clearly defines how to model the
method and offers examples that can be used with the
students in the Student Practice Sheet.
2. You may choose to watch the Teacher Tube video listed
in the notes by yourself or as a class.
3. The benefits of this division method are that it is a visual
representation that connects the multiplication area
model and it allows the students to use different factors,
which provides differentiation.
4. Prior to solving the problem, it may be beneficial to
create a “help box” that offers multiples of the divisor.
2, 5, and 10 were chosen for the “help box” because
they are basic factors that can later transfer to mental
math.
Work out the following problem on the board.
Help Box
Problem: 494 ÷ 13
13 x 2 = 26
13 x 5 = 65
Example 1:
13 x 10 = 130
Step 1: Draw a rectangle to represent the area model and record
the divisor on the left. Eventually the area of the rectangle will
be equal to the dividend.
Step 2: Choose a multiple of 2, 5, or 10 so that 13 x that multiple
= a number less than 494. Possible dialogue: “What would be
the best number to choose to multiply by 13 to get an answer
close to 494? 2, 5, or 10?”
10
494
- 130
13
130
364
10
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Step 3: Choose a multiple of 13 that gets close to 364 without
going over. Possible dialogue: “I chose 10 last time and it
equaled 130, this time I am going to chose 20 so I can get closer
to 364.”
10
20
260
130
13
494
Teacher Notes/Reflections:
364
-130 -260
364
104
Step 4: Continue to choose a multiple of 13 until you no longer
have a remainder or that the remainder is less than the divisor.
Answer:
10
13
20
260
130
494
364 104
39
5
2
1
65
26 13
13
-130 -260 - 65 -26 -13
364
104
39
13
0
Step 5: Then add 10 + 20 + 5 + 2 + 1 = 38.
11
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Teacher Notes/Reflections:
Therefore, 494 ÷ 13 = 38
5. Hand out the Array/Area Model for DivisionStudent Practice Sheet. Continue to solve practice problems
either as a class, small groups, or partners until the students feel
comfortable. Refer to the Array/Area Model for DivisionStudent Teacher Notes to promote dialogue. Remember that
students can solve these problems multiple ways. Allow them to
discuss and share their thoughts.
For example, in the earlier problem, one could have solved it like
this:
13
10
10
10
5
130
130
130
65 39
3
OR
13
30
8
104
104
12
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 5
Approximate Time Frame:
90-100 min.
Focus:
Lesson Format:
Resources:
Use any division worksheet used in
the past, but have the students use
partial quotients method to solve
the problems.
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Students will use the partial quotients
method to divide multi-digit whole
numbers.
Math Practice Look For(s):
MP2: Reason abstractly and quantitatively. Students
need to reason to appropriate divisors to use when
dividing using the partial quotients method. Students
need to be able to use all of the operations to solve the
problems.
MP6: Attend to precision. The students are using place
value concepts to help them calculate accurately and
efficiently.
Differentiation for Remediation: Initially, provide
students with a completed area model of the same
product so they can see the direct connection to the
partial product method.
Differentiation for English Language Learners:
Students will need clarification of divisor, dividend, and
quotient.
Differentiation for Enrichment:
MP7: Look for and make use of structure. Students can
see and understand how the dividend can be broken into
smaller parts.
Potential Pitfall(s):
Independent Practice (Homework):
Initially, students will use 2 or 10 as the only choice when
dividing the dividend into smaller parts. Therefore,
students need ample time to solve the problems in
multiple ways so they can become more efficient in
choosing factors.
Have students create a word problem for 368 ÷ 23. Then
write out the steps to solve it using number and words
such as the bookshelf problem.
Procedure:
Teacher Notes/Reflections:
The partial quotient method is an algorithm similar to the
area model in the fact each step gets us closer to the
quotient. Each step along the process allows the students to
think of the values of the numbers.
13
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
NOTE: While students proceed throughout the lesson encourage
them to discuss the connections to the area model. Also encourage
them to solve the problem using various multiples of the divisor.
As they discuss with each other, they will become more efficient in
their problem solving.
Teacher Notes/Reflections:
1. This example, 494 ÷ 13, is the same problem used in the
previous Area Model lesson so you may want to post the
diagram on your board and refer back to it so students can
connect this process to it.
Pose the following problems:
A used bookstore has 494 books that need to be placed on 13
shelves. How many books are on a shelf?
1. Explain that 10 books
are on each shelf, thus
130 books are dispersed
all together.
2. There are still enough
books left to put at least
20 more books on each
shelf, 260 more books are
put on the shelf.
3. Continue to put books
on each shelf until all the
books are put onto the
shelves. Then determine
how many books are on a
shelf. 38 books/shelf.
Encourage a discussion
among the students,
could they have put
the books on the shelf
differently?
14
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Problem: Your family bought a new couch for $2469 and signed up
for a payment plan through the store. The minimum amount to pay
is $32 per month. How many months will it take to pay it off, if they
only pay the minimum amount each month?
While the students continue to complete sample problems, have
them work in pairs. Use one paper and pencil and pass the paper
back and forth to solve the problem. Each time they receive the
paper, they try to divide by the largest number possible. Then pass
it back to their partner to solve it. They continue to pass it back
and forth until there is no remainder or the remainder is less than
the divisor. During each step, the partner double- checks the work.
15
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 6
Approximate Time Frame:
Lesson Format
90-100 min.
Whole Group
Small Group
Independent
Focus:
Modeled
Guided
Collaborative
Assessment
Resources:
Column division vs. Partial quotients
Formative Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Students will learn the column
division strategy to divide multi-digit
whole numbers.
Math Practice Look For(s):

MP6: Attend to precision. The students are using
place value concepts to help them calculate
accurately and efficiently.
Differentiation for Remediation: Allow students to use
the area model or partial quotients method.
Differentiation for English Language Learners:
Differentiation for Enrichment:

MP3: Construct viable arguments and critique the
reasoning of others. Students will cite evidence and
develop a logical argument.
Potential Pitfall(s):
Independent Practice (Homework):
Students will rely on mnemonics instead of place value
understanding to solve problems using the column
division strategy.
Procedure:
Independent Practice (Homework):
1. Either assign partners or small groups to work together to
solve the assessment.
2. Listen to the students’ discussion and address any
misconceptions. Help students focus on place value while
analyzing the work.
3. Once the students discover that “Katie’s way” of solving
the problem is acceptable, teach more problems using the
column method/aka traditional method.
4. Be sure to stress the place value when solving division.
For example: When solving the problem, 855 ÷ 19, it is
misleading to say, “There are 4 groups of 19 in 85?”
Instead state, “There are 40 groups of 19 in 855.”
16
Grade 5
Lesson Title: Multi-Digit Division
Unit 1: Whole Number Computation and Applications
Time Frame: 7-12 days
Essential Question: How do you choose different division strategies to divide multi-digit numbers?
Segment 7
Approximate Time Frame:
40-50 minutes
Focus:
Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Assess students conceptual and
procedural understanding of division
of multi-digit whole numbers.
Resources:
Multi-Digit Division Summative
Assessment
Self-Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
Differentiation for Remediation:
MP5: Use appropriate tools strategically. Students can
decide what is the best method for them to solve the
multi-digit division problem.
Differentiation for English Language Learners:
Differentiation for Enrichment:
MP7: Look for and make use of structure. Through the
use of place value concepts and partial quotients,
students can understand the structure behind dividing
multi-digit whole numbers.
MP8: Look for an express regularity in repeated
reasoning. After students have fully developed
conceptual understanding of multi-digit division, they can
fluently use a standard algorithm to solve a problem.
Potential Pitfall(s):
Independent Practice (Homework):
Procedure:
Decide if you would like to assess the students
individually or with partners.
Afterwards, have the students self-assess themselves.
17