Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Common Core and Research- Understand concepts of volume and relate volume to multiplication and to addition
The major emphasis for measurement in Grade 5 is volume. Volume not only introduces a third dimension and thus a significant challenge to students’ spatial structuring,
but also complexity in the nature of the materials measured. That is, solid units are “packed,” such as cubes in a three-dimensional array, whereas a liquid “fills” threedimensional space, taking the shape of the container. The unit structure for liquid measurement may be psychologically one dimensional for some students.
“Packing” volume is more difficult than iterating a unit to measure length and measuring area by tiling. Students learn about a unit of volume, such as a cube with a side
length of 1 unit, called a unit cube.5.MD.3 They pack cubes (without gaps) into right rectangular prisms and count the cubes to determine the volume or build right rectangular
prisms from cubes and see the layers as they build.5.MD.4 They can use the results to compare the volume of right rectangular prisms that have different dimensions. Such
experiences enable students to extend their spatial structuring from two to three dimensions (see the Geometry Progression). That is, they learn to both mentally
decompose and recompose a right rectangular prism built from cubes into layers, each of which is composed of rows and columns. That is, given the prism, they have to be
able to decompose it, understanding that it can be partitioned into layers, and each layer partitioned into rows, and each row into cubes. They also have to be able to
compose such a structure, multiplicatively, back into higher units. That is, they eventually learn to conceptualize a layer as a unit that itself is composed of units of units—
rows, each row composed of individual cubes—and they iterate that structure. Thus, they might predict the number of cubes that will be needed to fill a box given the net of
the box.
Another complexity of volume is the connection between “packing” and “filling.” Often, for example, students will respond that a box can be filled with 24 centimeter cubes,
or build a structure of 24 cubes, and still think of the 24 as individual, often discrete, not necessarily units of volume. They may, for example, not respond confidently and
correctly when asked to fill a graduated cylinder marked in cubic centimeters with the amount of liquid that would fill the box. That is, they have not yet connected their
ideas about filling volume with those concerning packing volume. Students learn to move between these conceptions, e.g., using the same container, both filling (from a
graduated cylinder marked in ml or cc) and packing (with cubes that are each 1 cm3). Comparing and discussing the volume-units and what they represent can help students
learn a general, complete, and interconnected conceptualization of volume as filling three-dimensional space.
Students then learn to determine the volumes of several right rectangular prisms, using cubic centimeters, cubic inches, and cubic feet. With guidance, they learn to
increasingly apply multiplicative reasoning to determine volumes, looking for and making use of structure (MP7). That is, they understand that multiplying the length times
the width of a right rectangular prism can be viewed as determining how many cubes would be in each layer if the prism were packed with or built up from unit cubes. 5.MD.5a
They also learn that the height of the prism tells how many layers would fit in the prism. That is, they understand that volume is a derived attribute that, once a length unit is
specified, can be computed as the product of three length measurements or as the product of one area and one length measurement.
Then, students can learn the formulas V= L x W x H and V = B x H for right rectangular prisms as efficient methods
for computing volume, maintaining the connection between these methods and their previous work with
computing the number of unit cubes that pack a right rectangular prism. 5.MD.5b They use these competencies to find
the volumes of right rectangular prisms with edges whose lengths are whole numbers and solve real-world and
mathematical problems involving such prisms.
commoncoretools.wordpress.com.
Resources- TSCM pages 113-120
NCTM Focus in 5 pages 79-86
Sizing Up Measurement pages 122-144
1
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
The chart below highlights the key understandings of this unit 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
Essential Questions
Volume can be found by counting unit cubes.
How can volume be counted?
Volume is filling of 3-dimensional space
What is volume?
Multi-digit whole numbers can be calculated using a
standard algorithm.
What is the most efficient method for calculating
multi-digit whole numbers?
When multiplying three of more numbers, the
product is always the same regardless of their
grouping. (a x b) x c = a x (b x c)
How can we multiply large numbers?
What are effective methods for finding volume of a
rectangular prism?
Volume is additive
V= L x W x H and V = B x H are effective methods for
finding the volume of a rectangular prism.
Does volume change when we change the
measurement materials used?
How are volume and area related?
Key Vocabulary
Students should understand the concepts involved and
be able to recognize and/or demonstrate them with
words, models, pictures, or numbers. The terms below
are for teacher reference only and are not to be
memorized by the students.
measurement
attribute
volume
solid figure
right rectangular prism
unit
unit cube
gap
overlap
cubic units (cubic cm, cubic in, cubic ft)
nonstandard cubic units
edge lengths
height
area of base
standard algorithm
multiply
factor
product
associative property of multiplication
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Standards for Mathematical Practice
Connections to this Cluster
1. Make sense of problems and persevere
in solving them
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.1)
2. Reason abstractly and quantitatively
Students consider the units involved. (MP.2)
3. Construct viable arguments and critique
the reasoning of others
Students explain their reasoning as they practice partitioning figures into layers and each layer into rows and
each row into cubes. (MP.3)
4. Model with mathematics
5. Use appropriate tools strategically
Students pack the figures with unit cubes (MP.5)
6. Attend to precision
Students need to present solutions to multi-step problems in the form of valid chains of reasoning, using
symbols such as equal signs appropriately (for example, students will be awarded 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 multi-step problems and present corrected solutions. (MP.6)
7. Look for and make use of structure
Students decompose and recompose geometric figures to make sense of the special structure of volume
(MP.7).
Students pack the figures with unit cubes (MP.5) and connect this structure to multiplicative reasoning (MP.7)
8. Look for and express regularity in
repeated reasoning
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.1)
Students solve problems by applying the generalized formulas (MP.8)
3
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Geometric measurement: understand concepts of volume and
relate volume to multiplication and to addition.
Common Core State Standards
5.MD.3 Recognize volume as an
attribute of solid figures and
understand concepts of volume
measurement.
 A cube with side length 1
unit, called a “unit cube,” is
said to have “one cubic
unit” of volume, and can be
used to measure volume.
 A solid figure which can be
packed without gaps or
overlaps using n unit cubes
is said to have a volume of
n cubic units.
5.MD.4 Measure volumes by
counting unit cubes, using cubic
cm, cubic in, cubic ft, and
improvised units.
Instructional Strategies and Resource Support
NCTM Focus in 5 pages 79-86
Sizing Up Measurement pages 122-144
SMP.7 Look for and make use
of structure
SMP.3 Construct viable
arguments and critique the
reasoning of others
Compare the volume of
the rectangular prisms
Given boxes of various sizes, students predict the number of cubes that can be packed in the
boxes. Students then pack boxes and count the number of cubes it takes to pack the boxes.
Work with a partner. Use cm cubes to build a rectangular prism with a volume of 24cm³. Build as
many different rectangular prisms with a volume of 24cm³ as you can. Record your findings
using pictures, numbers, and words.
Students are given a net and asked to predict the number of cubes
required to fill the container formed by the net. In such tasks, students
may initially count single cubes or repeatedly add the number in each
layer to find the total number of unit cubes. In folding the net to make
the shape, students can see how the side rectangles fit together and
determine the layers.
Work with a partner. On a sheet of cm grid paper draw 4 patterns for different sized open
boxes. Cut out the patterns, fold up the sides, and tape them together. Fill each box with cm
cubes to find the volume. Record your findings in a chart and describe any patterns or
relationships that you notice.
Box
Length of Base
Width of Base
Height of Box
Sample Formative
Assessments
You have been asked to
design the packaging
for a new breakfast
cereal. The
manufacturer would
like you to design three
different sized boxes…
Draw a possible design
for each box. Label the
dimensions and
calculate the volume.
Which box do you think
would be the best
seller in your local
supermarket? Why?
Volume
A
B
Students decompose and recompose geometric figures to make sense of the special structure of
volume (SMP.7). Students explain their reasoning as they practice partitioning figures into layers
and each layer into rows and each row into cubes. (SMP.3)
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Perform operations with multi-digit whole numbers and with
decimals to hundredths.
Common Core State Standards
5.NBT.5 Fluently multiply multidigit whole numbers using the
standard algorithm. (2-digit by
3-digit)
SMP.1 Make sense of
problems and persevere in
solving them
SMP.8 Look for and express
regularity in repeated reasoning
Instructional Strategies and Resource Support
TSCM pages 113-120
NCTM Focus in Grade 4 pages 7-25
Students in Grade 4 Multiply a whole number of up to four digits by a one-digit whole number,
and multiply two two-digit numbers, using strategies based on place value and the properties of
operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or
area models. In Grade 5, students will build on that understanding to become fluent in
multiplying multi-digit whole numbers. See NCTM Focus in Grade 4 book pages 7-25 for ideas of
how to move students toward procedural fluency. (Corrected array or area model, partial
product, rectangular sections, expanded notation, rectangular rows)
Multiply.
4358 x 5
2 x 8325
348 x 30
Students should estimate for reasonableness of answers before solving each problem.
Students look for and correct misconceptions or common errors in other students’ work.
(Example-When students only see each factor as a single digit numeral, they will not understand
the magnitude of the numbers they are multiplying and may drop the zero when using the
standard algorithm.)
*In this unit 5.NBT.5 will focus
on operations with whole
numbers only. Operations with
decimals will be introduced in
January. These standards will be
finalized in May, but should be
practiced throughout the year
to provide opportunities for
students to develop proficiency
with these operations.
Sample Formative
Assessments
24 x 65
Use Number Talks to keep students flexibly thinking about numbers.
Students will be assessed on 2-digit by 3-digit on the PBA and 3-digit by 4-digit on the EOY
assessment.
Students look for regularity in their work with multiplication and division use their
understanding of the structure (SMP.8) to make sense of their solutions and understand the
approaches of other students (SMP.1)
*See excerpt from NCTM article on page 7-9 of this document for ways to move students from
conceptual understanding to the standard algorithm.
5
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Geometric measurement: understand concepts of volume and relate
volume to multiplication and to addition.
Common Core State Standards
Instructional Strategies and Resource Support
5.MD.5 . Relate volume to the operations of
multiplication and addition and solve real
world and mathematical problems involving
volume.
 Find the volume of a right rectangular
prism with whole-number side lengths
by packing it with unit cubes, and show
that the volume is the same as would be
found by multiplying the edge lengths,
equivalently by multiplying the height
by the area of the base. Represent
threefold whole-number products as
volumes, e.g., to represent the
associative property of multiplication.
 Apply the formulas V = l × w × h and V =
b × h for rectangular prisms to find
volumes of right rectangular prisms with
whole-number edge lengths in the
context of solving real world and
mathematical problems.
 Recognize volume as additive. Find
volumes of solid figures composed of
two non-overlapping right rectangular
prisms by adding the volumes of the
non-overlapping parts, applying this
technique to solve real world problems.
NCTM Focus in 5 pages 79-86
Sizing Up Measurement pages 122-144
SMP.5 Use appropriate tools strategically
Students need to present solutions to multi-step problems in the form of valid chains of
reasoning, using symbols such as equal signs appropriately (for example, students will be
awarded 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 multi-step
problems and present corrected solutions. (SMP.6)
SMP.6 Attend to precision
SMP.7 Look for and make use of structure
Have students outline a rectangle of their choice on graph paper. Ask students to build
rectangular prisms that have their chosen base and are 1 unit tall, 2 units tall, 3 units tall, and 4
units tall. Record the volumes of these prisms in a table.
Area of the base: _______ square units
Height (units)
1 2 3 4 20 100
Volume (cubic units)
Ask students how the volume changes as the height changes and what relationship they notice.
Using their chosen base, have students predict the volume of the prism that has various heights
such as 20 or 100 units, without continuing the table. Have students repeat with other bases.
Sample Formative
Assessments
Brooks needs more
sand to fill his
sandbox. His
sandbox measures
7 feet by 6 feet by
2 feet. He already
put 40 cubic feet of
sand in the
sandbox. How
many more cubic
feet of sand does
Brooks need to add
for the sandbox to
be full?
Give students word problems such as: A cereal box has the dimensions of 5” x 2” x 12”. What is
the volume? A cereal box has the dimensions of 5” x 4” x 12”. What is the volume? How do you
know? A toy box has a base of 20 square inches and a volume of 100 cubic inches. What is the
height of the box?
Students might design a science station for the ocean floor that is composed of several rooms
that are right rectangular prisms and that meet a set criterion specifying the total volume of the
station. They draw their station (e.g., using an isometric grid, SMP7) and justify how their design
meets the criterion (SMP1).
Students pack the figures with unit cubes (SMP.5) and connect this structure to multiplicative
reasoning (MP.7). They solve problems by applying the generalized formulas (SMP.8)
SMP.8 Look for and express regularity in
repeated reasoning
6
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Multi-digit Multiplication
Multiplication moves from the first year (in Grade 4) where
the approach of the standard algorithm is developed and
explained using visual models (diagrams) to the second year
(in Grade 5) where the approach of the standard algorithm
continues to be deepened and then is used fluently.
The major issue for multi-digit multiplication is what to
multiply by what and how the place values of the digits in the
factors affect the place values of the partial products. An
array or area model can help students understand these
issues in terms of how the partial products are recorded.
Figure 4 is modified slightly from the NBT Progression
document and shows area models, distributive property
equations addressing place value, and methods for recording
the standard algorithm with 1-digit multipliers. These help
students understand how each partial product comes from a
multiplication of a kind of unit in one number times a kind of
unit in the other number. The Methods A and B can be
abbreviated to Method C where the partial products are
written within the product space in one row rather than as
separate rows that show the place values, but this method is
more complex .Methods A and B are conceptually clearer.
7
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
Figure 5 shows how the place values in several methods for recording
the standard algorithm for 2-digit by 2-digit multiplication relate to the
place values in an area or array model and to each other. Methods D, F,
and G write all four partial products; while Method E abbreviates the
products of a given number into one row as did Method C. However, in
Method E, partial products can be seen as diagonals written as for
Method E in Figure 2 in addition (e.g., 24 ones is a small 2 in the tens
place and a 4 in the ones place and likewise for the 54 tens, 12 tens, 27
hundreds). Writing these below allows students to see these products,
and it puts all of the carries (regroupings) in the correct place. In
another variation of this abbreviated method, shown in Figure 6 on the
left as Method H, the 1 carried above the tens column is from 30 x 4 =
120, so it is actually 1 hundred and not 1 ten. It is confusing to have it in
the tens column. Furthermore, having the carries above disconnects
them from the rest of their product, so the steps and meanings of the
digits can get confused. This method also alternates multiplying and
adding, increasing its difficulty even further. This should not be an
emphasized method but might be discussed if students bring it into the
classroom. Method C also has a variation in which the carries are
written above instead of below, which changes the problem, and makes
it difficult to see the products because they are separated physically.
Multi-digit multiplication can be a challenging visual-spatial task. Some
students find it difficult to multiply without an area or array model. They
prefer to make a quick area sketch, write the products inside, and then
add up the products outside as in Area Method F in Figure 5. Such a
written method might be a little too long for fluency with the standard
algorithm because it involves a drawing, though it shows place values
clearly, can generalize, and is efficient. But it also seems better even in
Grade 5 to allow some students to use Method F with accuracy than to
use the more abstract partial products Method D if students are likely to
make errors with this more abstract recording.
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
October-November (20 days)
Unit # 3: Whole Number Multiplication and Volume
The lattice method of multiplication is hundreds of years old and has become popular in some
instructional programs. In Figure 5 we can see how this Lattice Method G is related to the area model. The
products of each digit in each factor are written within the area model squares. Diagonals are drawn
within each such square to locate the tens and ones of each partial product coming from the multiplication
of two single digits. These diagonals inside the whole area square have place values that move from ones
to thousands from the bottom right to the bottom left and then up to the top left. These place values are
derived from the patterns for multiplying place values (e.g., tens times tens is hundreds).We label the
place values of the factors and diagonals in the lattice multiplication so that we can see how the place
values in each partial product relate and align. If this method is used in the classroom, it is important to
emphasize these place values, so that students understand what they are doing, and are not rotely
memorizing a procedure. That is not a CCSS-M approach.
Method I in Figure 6 is a “helping step” version of Method D developed by a class of Grade 4 students from
a low SES school. These students recognized that many of them were making mistakes using Method D,
and they developed this method to help eliminate these mistakes. By writing out the tens and the ones in
each factor, they could see the number of zeros, and thus use the patterns involving tens and hundreds
more easily. (The partial products Method D and the Area Method F also show the place values in the
factors). They wrote the biggest product first so they could correctly align the other products under it,
which also has the additional advantage of showing the approximate size of the full product rapidly. They
wrote out the factors of each partial product because some students were not systematic in the order in
which they multiplied, and by writing the factors for each partial product, they could check on whether all
partial products were included. These steps also supported student efforts to explain each step in the
method, initially relating it to an area or array model but eventually omitting this model. Recording all of
these steps may be too extensive for fluency with the standard algorithm, but students did stop recording
particular steps as they no longer needed their support, thus moving toward the greater fluency of
Method D.
Method D can be undertaken from left to right, as can Method I, so Method I can collapse to the left to
right version of Method D. Area Method F and Method D are the conceptually clearest methods, and
Method D is fast enough for fluency.
9