Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Background Information and Research
By fifth grade, students should understand that division can mean equal sharing or partitioning of equal groups or arrays. They should also understand that it is the same as
repeated subtraction, and since it’s the inverse of multiplication, the quotient can be thought of as a missing factor. In fourth grade, students divided 4-digit dividends by 1digit divisors. They also used contexts to interpret the meaning of remainders. Division is extended to 2-digit divisors in fifth grade, but fluency of the traditional algorithm is
not expected until sixth grade. Division models and strategies that have been used in previous grade levels, such as arrays, number lines, and partial quotients, should
continue to be used in fifth grade as students deepen their conceptual understanding of this division.
There are two common types of division problems.
 The first type is called repeated subtraction. You know the total amount and the amount in each
group and are looking for the number of groups. For example, how many packages of 6 ping pong
balls can be made from 42 ping pong balls? You know that there are 7 groups of 6 in 42, so you know
7 packages can be made.
 The second type is called sharing division. You know the total amount and the number of groups and
you need to find out how many are in each group. For example, Jody has 42 apples. She puts them
into 6 equally sized groups. How many are in each group? There are 7 in each group.
Division strategies in Grade 5 involve breaking the dividend apart into like base-ten units and applying the
distributive property to find the quotient place by place, starting from the highest place. (Division can also be
viewed as finding an unknown factor: the dividend is the product, the divisor is the known factor, and the quotient is the unknown factor.) Students continue their fourth
grade work on division, extending it to computation of whole number quotients with dividends of up to four digits and two-digit divisors. Estimation becomes relevant when
extending to two-digit divisors. Even if students round appropriately, the resulting estimate may need to be adjusted.
Teaching Student-Centered Mathematics (Grades 3-5) pg. 124 – 128
Focus in Grade 5 Pages 9-26
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
The chart below highlights the key understandings of this cluster along with important questions that teachers should pose to promote these understandings. The chart also
includes key vocabulary that should be modeled by teachers and used by students to show precision of language when communicating mathematically.
Enduring Understandings
•
•
•
•
There are two common situations where division may
be used: fair sharing (given the total amount and the
number of equal groups, determine how many/much in
each group) and measurement (given the total amount
and the amount in a group, determine how many
groups of the same size can be created).
Some division situations will produce a remainder, but
the remainder will always be less than the divisor. If the
remainder is greater than the divisor, that means at
least one more can be given to each group (fair sharing)
or at least one more group of the given size (the
dividend) may be created.
The quotient remains unchanged when both the
dividend and the divisor are multiplied or divided by the
same number.
The properties of multiplication and division help us
solve computation problems easily and provide
reasoning for choices we make in problem solving.
Essential Questions
•
•
•
•
•
How can estimating help us when solving division
problems?
What strategies can we use to efficiently solve
division problems?
How can I effectively explain my mathematical
thinking and reasoning to others?
How can I effectively critique the reasoning of
others?
How can identifying patterns help determine
multiple solutions?
Key Vocabulary
Dividend
Divisor
Quotient
Remainder
Properties
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards.
Specific connections to this cluster and instructional strategies are provided in the following chart.
Standards for Mathematical Practice
Cluster Connections and Instructional Strategies
1. Make sense of problems and persevere
in solving them
Students solve problems by applying their understanding of operations with whole numbers, including the order of
operations. Students seek the meaning of a problem and look for efficient ways to solve it.
2. Reason abstractly and quantitatively
Students demonstrate abstract reasoning to connect quantities to written symbols and create a logical
representation of the problem at hand. Students write simple expressions that record calculations with numbers and
represent numbers using place value concepts.
3. Construct viable arguments and critique
the reasoning of others
Students construct arguments using concrete referents, such as objects, pictures, and drawings. They explain
calculations based upon models and properties of operations and rules that generate patterns. They explain their
thinking to others and respond to others’ thinking.
4. Model with mathematics
Students use base ten blocks, drawings, and equations to represent place value and powers of ten. They interpret
expressions and connect them to representations.
5. Use appropriate tools strategically
Students select and use tools such as estimation, graph paper, and place value charts to solve problems with whole
number operations.
6. Attend to precision
Students use clear and precise language (math talk) in their discussions with others and in their own reasoning.
Students use appropriate terminology when referring to expressions, place value, and powers of ten
7. Look for and make use of structure
Students use properties of operations as strategies to add, subtract, multiply, and divide with whole numbers. They
explore and use patterns to evaluate expressions. Students utilize patterns in place value and powers of ten and
relate them to graphical representations of them.
8. Look for and express regularity in
repeated reasoning
Students use repeated reasoning to understand algorithms and make generalizations about patterns. Students
connect place value and properties of operations to fluently perform operations.
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Convert like measurement units within a given measurement
system.
Maryland College and Career-Ready
Standards
5.MD.1. Convert among
different-sized standard
measurement units within
a given measurement
system (e.g., convert 5 cm
to 0.05 m), and use these
conversions in solving
multi-step, real world
problems.
SMP.2 Reason abstractly
and quantitatively
SMP.3 Construct viable
arguments and critique the
reasoning of others
SMP.5 Use appropriate
tools strategically
Instructional Strategies and Resource Support
Sizing Up Measurement- pp. 1-52, 96-122, 183-205, 222-248
Students worked with both metric and customary units of length in second grade. In third grade, students work with metric
units of mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in
length, mass and volume using whole numbers. Early in the year, students worked on converting measurements in the
metric system. During this unit, students will work with converting customary units of measure and time.
Sample Formatives
Write <, >, = to
compare
measurements.
Students present solutions to multi-step problems in the form of valid chain of reasoning, using symbols such as equal signs
appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1+4=5+7=12,
even if the final answer is correct), or identify or describe errors in solutions to mult-step problems and present corrected
solutions. (SMP.3, 5,6)
Students should apply their understanding of converting within the metric system to converting within the customary
system.
Students should see the pattern and make the generalization that:

To change from a larger unit to a smaller unit, multiply by the appropriate number.

To change from a smaller unit to a larger unit, divide by the appropriate number.
Students should have securely held knowledge from 4th grade and will not be available on the reference sheet: 1 foot=12
inches, 1 yard=3 feet, 1 day =24 hours
SMP.6 Attend to precision
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Convert like measurement units within a given
measurement system.
Maryland College and Career-Ready
Standards
Instructional Strategies and Resource Support
Sample Formatives
5.MD.1. Convert among
different-sized standard
measurement units within
a given measurement
system (e.g., convert 5 cm
to 0.05 m), and use these
conversions in solving
multi-step, real world
problems.
SMP.2 Reason abstractly
and quantitatively
SMP.3 Construct viable
arguments and critique the
reasoning of others
SMP.5 Use appropriate
tools strategically
SMP.6 Attend to precision
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Perform operations with multi-digit whole numbers and with
decimals to hundredths.
Maryland College and Career-Ready Standards
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.
*5.NBT.6 is a milestone along the way to
reaching fluency with the standard
algorithm. In this unit 5.NBT.6 will focus on
operations with whole numbers only.
Operations with decimals will be introduced
in March and will be finalized in May and
June, but should be practiced throughout
the year to provide opportunities for
students to develop proficiency with these
operations.
SMP.3 Construct viable arguments and
critique the reasoning of others
SMP.5 Use appropriate tools strategically
SMP.6 Attend to precision
SMP.8 Look for and express regularity in
repeated reasoning
Instructional Strategies and Resource Support
Developing fluency with multi-digit division takes three years because students first develop
and explain the approach of the standard algorithm with visual models for dividing by onedigit numbers in Grade 4 and then extend the approach in Grade 5 to dividing by 2-digit
numbers where the difficulties of estimating complicate division. In Grade 6 the standard
algorithm is used fluently for one- and two-digit divisors.
Teaching Student-Centered Mathematics (Grades 3-5)
pg. 65 *The Broken Division Key - Activity 2.26 pg. 124 - *Figure 4.17 pg. 124 – 128
Focus in Grade 5 Pages 9-26
See Focus in Grade 5 book for complete explanation on using rectangular arrays and area
models as well as partial quotient methods.
See Excerpt from NCSM Journal-Fall/Winter 2012-2013 below for an explanation of methods
used that will teach the concept of division with multi-digit whole numbers and lead to the
standard algorithm. When students use the traditional algorithm, they often treat each digit in
the dividend separately and do not look at the value of the entire number. Encourage the
students to estimate prior to dividing, this helps them see what a reasonable quotient will be.
When students show an understanding of the concept, they will naturally work toward fluency
with the standard algorithm. Fifth grade students should begin working with the standard
algorithm, but are not expected to be fluent until 6th grade.
Students look for regularity in their work with multiplication and division use their
understanding of the structure (MP.8) to make sense of their solutions and understand the
approaches of other students (MP.3)
*On Summative Unit and Mid-Year, students will be allowed to solve using any method.
Students will also need to explain their thinking and/or the thinking of others. They will base
arithmetic explanations/reasoning on diagrams, connecting diagrams to a symbolic
method.(SMP.3, SMP.5, SMP.6)
Sample Formative
Assessments
Use the area
model to find
the quotient.
Show your
thinking.
Jeremy divided
this array of 336
stars into the
groups below.
What division
equation has he
represented?
What is the
division
sentence for the
visual model?
Savannah
bought a used
car for $9,900.
She needs to
make 36 equal
payments…
Use the diagram
to complete the
equation and
show your
thinking.
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Excerpt from NCSM Journal-Fall/Winter 2012-2013
Array and area models can support understanding of strategies for
division. Figure 7 shows Method A including both the recording of the
numerical calculations and the area model that corresponds to those
calculations as shown in the NBT Progression document. The full multiplier
of the divisor at each step in Method A is written above the dividend so
that students can see the place value and make a clear connection to the
place values in the area model. The full product is written at each step, and
the amount of the dividend not yet used is also written in full. After
experience connecting it to a drawing, Method A can also be undertaken
without a drawing as a standard algorithm when calculating quotients that
are whole numbers or decimals.
Another written method for this standard algorithm is also shown as
Method B in Figure 7. In this method, the zeros are not written in the
multipliers or in the partial products within the problem. Digits are brought
down within the problem one at a time. Method B makes it more difficult
to make the connections to the meaning of the computation, and for that
reason, we included conceptual language to communicate the underlying
meaning of each step of the calculation. But this method does show clearly
the single-digit calculations that are used, and these single-digit
calculations are in their place-value locations, as indicated in Method A.
Method B becomes important when a quotient has many places, for
example, when explaining why decimal expansions of fractions eventually
repeat (8.NS.1). Some students may be able to understand and explain
Method B right away, and many students can move to it from strategies
like Method A, because the move is small. There are also other written
methods for the standard algorithm that are variations in between the two
shown in Methods A and B. For instance, you could write the zeros on the
top but not within the subtractions, or vice versa. Students might develop
and use any of these variations.
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Unit #4 : Division of Whole Numbers and Measurement
Method A has a further advantage when dividing by 2digit numbers because it can involve the use of
reasonable estimates. The example shown in the NBT
Progression document is depicted in Figure 8 and
demonstrates how an estimate of the quotient can be
used and then adjusted. Underestimates can be
repaired in any place by subtracting another partial
product for that place (and adding another rectangle
to the area model when first using models). Such a
simple repair is an acceptable written method for the
standard algorithm. However, some introductions to
division allow students to dramatically underestimate
the quotient, and as a result, they then have many
partial products (e.g., multipliers of 10 + 10 + 10 + 10 +
10) that are used to adjust the estimate. Many extra
multipliers and partial products are not consistent with
the fluency expectations of the standard algorithm, so
students need to be encouraged to be brave and use a
multiplier as close as possible to the largest multiplier,
for the sake of efficiency. With Method B students
cannot continue on from a low estimate. They need to
be exact, which often means erasing their
underestimate or overestimate. Of course, a student
could leave the product and difference they already
found, cross out their low multiplier, increase it by 1 or
2, and take away that partial product as another step
as in Method A. (See the second variation in Figure 8).
Note that overestimates are still best fixed by erasing
and trying a lower partial quotient because repairs are
difficult to carry out correctly.
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