Paramount Unified School District
Educational Services
Grade 5 – Unit 4
Stage One – Desired Results
Unit 4: Expressions and Volume
In this unit, students will:

Write numerical expressions that represent verbal expressions and interpret numerical expressions in words without evaluating them

Discover how the use of grouping symbols changes the value of an expression

Evaluate numerical expressions and create real-life situations for expressions with grouping symbols

Engage in concrete exploration of volume in which they see volume as an attribute of solid figures and understand that cubic units are used to
measure it

Build rectangular prisms and count units to find the volume

Apply knowledge of area to solve problems involving volume

Develop an understanding of the relationship between volume and multiplicative reasoning which leads naturally to formulas for finding the volume of a
right rectangular prism

Discover that volume is additive when calculating the total volume of solid figures composed of non-overlapping, rectangular prisms.
Teacher Notes:

Students do not need to understand order of operations for this unit; the focus is on understanding that grouping symbols are calculated first when
evaluating expressions

Students have had experiences with area in grades 3 and 4 but grade 5 is the first time they explore volume
1
Unit 4 Overview: Expressions and Volume
Transfer Goals
1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution.
2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience.
3) Construct viable arguments and critique the reasoning of others using precise mathematical language.
Standards
OA.1 Use parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these symbols.
OA.2 Write simple expressions that record calculations
with numbers, and interpret numerical expressions without
evaluating them. For example, express the calculation “add
8 and 7, then multiply by 2” as 2 x (8 + 7). Recognize that 3
x (18932 + 921) is three times as large as 18932 + 921,
without having to calculate the indicate sum or product.
MD.3. Recognize volume as an attribute of solid figures and
understand concepts of volume measurement.
a. 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.
b. 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.
MD.4 Measure volumes by counting unit cubes, using cubic
cm, cubic in, cubic ft., and improvised units.
MD.5 Relate volume to the operations of multiplication and
addition and solve real-world and mathematical problems
involving volume.
a. Find the volume of a right rectangular prism with wholenumber 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.
b. Apply the formulas V = l x w x h and V = b x 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.
c. Recognize volume as additive. Find volumes of solid
figures composed of two non-overlapping right
rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve realworld problems
Meaning-Making
Understandings
Students will understand that…
 Numerical expressions
represent real-life and
mathematical situations
 In an expression, the way
numbers are grouped
affects the value
 Both multiplicative and
additive reasoning can be
applied to volume
Essential Questions
Students will keep considering…
 How do numerical expressions represent real-life and mathematical
situations?
 How would math be different if grouping symbols didn’t exist?
 How does multiplication relate to volume? And addition?
 How important is understanding area when determining the volume of a
figure?
Acquisition
Knowledge
Students will know…
Vocabulary:
 Expressions: Numerical
expression, verbal
expression, parentheses,
interpret, evaluate
 Volume: Volume,
rectangular prism, unit
cubes, cubic unit, height,
base, composite figure
Associative Property
Skills
Students will be skilled at and able to do the following…
 Write or identify a numerical expression that records a calculation
represented with words
 Interpret numerical expressions in words without evaluating them
 Evaluate numerical expressions with grouping symbols
 Determine the volume of a right rectangular prism with whole-number side
lengths by counting or packing unit cubes
 Apply the formulas V = l x w x h and V = b x h to solve real-world and
mathematical problems involving volume of right rectangular prisms
 Recognize that volume is additive when finding the volume of two nonoverlapping rectangular prisms
2
Paramount Unified School District
Grade 5 – Unit 4
Stage Two – Evidence of Learning
Educational Services
Unit 4: Expressions and Volume
Transfer is a student’s ability to independently apply understanding in a novel or unfamiliar situation. In mathematics, this requires that students use reasoning
and strategy, not merely plug in numbers in a familiar-looking exercise, via a memorized algorithm.
Transfer goals highlight the effective uses of understanding, knowledge, and skills we seek in the long run – that is, what we want students to be able to do
when they confront new challenges, both in and outside school, beyond the current lessons and unit. These goals were developed so all students can apply their
learning to mathematical or real-world problems while simultaneously engaging in the Standards for Mathematical Practices. In the mathematics classroom,
assessment opportunities should reflect student progress towards meeting the transfer goals.
With this in mind, the revised PUSD transfer goals are:
1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution.
2) Effectively communicate orally, in writing, and by using models (e.g., concrete, representational, abstract) for a given purpose and audience.
3) Construct viable arguments and critique the reasoning of others using precise mathematical language.
Multiple measures will be used to evaluate student acquisition, meaning-making and transfer. Formative and summative assessments play an important role in
determining the extent to which students achieve the desired results in stage one.
Formative Assessment
Summative Assessment
Aligning Assessment to Stage One
 What constitutes evidence of understanding for this lesson?
 Through what other evidence during the lesson (e.g. response to questions,
observations, journals, etc.) will students demonstrate achievement of the
desired results?
 How will students reflect upon, self-assess, and set goals for their future
learning?






Discussions and student presentations
Checking for understanding (using response boards)
Ticket out the door, Cornell note summary, and error analysis
Learn Zillion end-of-lesson assessments
“Check My Progress”, teacher-created assessments/quizzes
ST Math (curriculum progress, data reports, etc.)
 What evidence must be collected and assessed, given the desired results
defined in stage one?
 What is evidence of understanding (as opposed to recall)?
 Through what task(s) will students demonstrate the desired understandings?
Opportunities
 Unit assessments
 Teacher-created chapter tests or mid-unit assessments
 Challenge lessons
 Illustrative Mathematics tasks (https://www.illustrativemathematics.org/)
 Performance tasks
3
The following pages address how a given skill may be assessed. Assessment guidelines, examples and possible question types have been provided to
assist teachers in developing formative and summative assessments that reflect the rigor of the standards. These exact examples should not be used for
instruction or assessment, but can be modified by teachers. Note: Examples were pulled from SBAC Item Specifications and Engage NY released items.
Skill
Standard
Write or identify
a numerical
expression that
records a
calculation
represented
with words
OA.1
OA.2
Assessment Guidelines
Example

The student is prompted to select a
numerical expression, which includes up to
one set of grouping symbols, that represents
a calculation expressed with words.
Which expression correctly shows “12 times the sum of
5 and 7”?
A. 12 × 5 + 7
C. 12 × (5 + 7)
B. 5 + 7 × 12
D. 5 + (7 × 12)

The student is prompted to select a
numerical expression, which includes two
sets of grouping symbols, that represents a
calculation expressed with words.
Item difficulty may be adjusted via these
example methods:
 Expression contains one or two
operations outside the grouping
symbols.
 Expression contains whole numbers,
fractions, or decimals.
Which expression correctly shows the difference
between the product of 7 and 9 and the sum of 12 and
5?
A. 7 × (9 – 12) + 5 C. (7 × 9) – (12 + 5)
B. 7 × (9 + 12) + 5 D. (7 + 9) + (12 + 5)

A school spends $2.40 on every lunch it serves in the
cafeteria and $0.30 for each carton of milk.
• 250 people at the school get a lunch each day
• 120 people take a carton of milk
Create an expression using this information that shows
how much the school spends altogether on lunches and
milk each day.
A school spends $2.40 on every lunch it serves in the
cafeteria and $0.30 for each carton of milk.
• 250 people at the school get a lunch each day
• 120 people take a carton of milk
Possible
Question Type(s)
 Multiple Choice,
Single Correct
Response
 Equation/Numeric
 Multiple Choice,
Multiple Correct
Response
Which expression represents the amount of money the
school spends altogether on lunches and milk each day?
A. 250 x 2.40 + 120 x 0.30
B. 250 x 0.30 + 120 x 2.40
C. 250 x (2.40 + 0.30)
D. 120 x (2.40 + 0.30)
4
Skill
Standard
Assessment Guidelines
Example
Interpret
numerical
expressions in
words without
evaluating them
OA.1
OA.2
 The student is prompted to interpret a
numerical expression without evaluating it.
 Item difficulty may be adjusted via these
example methods:
 Expression contains zero, one, or two
sets of grouping symbols.
 Expression contains one or two
operations outside the grouping
symbols.
 Expression contains whole numbers,
fractions, or decimals.
Which statement describes the value of the expression 4
× (18,932 + 921)?
A. The value is 921 more than the product of 4 and
18,932.
B. The value is 18,932 more than the product of 4 and
921.
C. The value is 4 times as large as the sum of 18,932 and
921.
D. The value is 4 times as large as the product of 18,932
and 921.
 Multiple Choice,
Single Correct
Response

Examples with one set of grouping symbols:
 Equation/Numeric
Evaluate
numerical
expressions
with grouping
symbols
The student is presented with a numerical
expression that contains one or two nonnested sets of grouping symbols.
 Item difficulty may be adjusted via these
example methods:
 Expression contains one or two sets of
grouping symbols.
 Expression contains one or two
operations outside the grouping
symbols.
 Expression contains whole numbers,
fractions, or decimals.
 The student is presented with a numerical
expression that does not contain sets of
grouping symbols and is prompted to
identify the correct placement of
parentheses to equal a specific value.
Possible
Question Type(s)
Enter the value of 7 + (5 × 12).
Enter the value of 7 + (5 × 12) ─ 4.
Examples with two sets of grouping symbols:
Enter the value of (5 × 12) + (27 ÷ 9).
Enter the exact value of (6 × 23) + (28 + 38).
Enter the exact value of (2 ÷ 0.1) – (0.3 × 0.4).
Taryn must place parentheses around numbers in this
expression in order to make it equal 2.
30 ÷ 2 + 4 – 3
Which expression equals 2?
A. 30 ÷ (2 + 4 – 3) C. 30 ÷ 2 + (4 – 3)
B. 30 ÷ (2 + 4) – 3 D. (30 ÷ 2) + 4 – 3
 Multiple Choice,
Single Correct
Response
5
Skill
Determine the
volume of a
right
rectangular
prism with
whole-number
side lengths by
counting or
packing unit
cubes
Standard
Assessment Guidelines
Example
MD.3
MD.4
MD.5
 The student is
prompted to
determine the volume
of a right rectangular
prism with wholenumber side lengths
by counting unit
cubes.
The student is presented with the model of the bottom layer of a right
rectangular prism and the number of layers in the completed prism.
Elias is building a rectangular prism. The bottom layer of the rectangular prism is
shown. He builds a prism that has 4 layers.
Possible
Question Type(s)
 Equation/Numeric
Enter the volume, in cubic centimeters, of the rectangular prism.
Enter the volume, in cubic centimeters, of the completed rectangular prism.
The student is presented with a model of a completed right rectangular prism.
The rectangular prism shown is solid.
Enter the volume, in cubic centimeters, of the rectangular prism.
A rectangular box is completely filled with 48 same-sized cubes arranged as
shown. Julie opens the top of the box and sees 16 cubes.
Julie closes the top and then opens the right side of the box. How many cubes
should she see?
6
Skill
Standard
Assessment Guidelines
Example
Apply the
formulas V = l x
w x h and V = b
x h to solve realworld and
mathematical
problems
involving
volume of right
rectangular
prisms
MD.3
MD.4
MD.5
 The student is
prompted to apply the
formulas V = l x w x h
and V = b x h to solve
real-world and
mathematical
problems involving
rectangular prisms
The area of the base of this right rectangular prism is 18 square
centimeters and the height is 4 centimeters.
Enter the volume, in cubic centimeters, of this prism.
 The student is
prompted to
calculate the volume
of two nonoverlapping right
rectangular prisms of
given dimensions.
 The student is
prompted to identify
methods for finding
the volume of a right
rectangular prism.
Right rectangular prisms A and B are combined to create this model.
• The dimensions of Prism A are 4 by 3 by 20 millimeters.
• The dimensions of Prism B are 6 by 9 by 4 millimeters.
Possible
Question Type(s)
 Equation/Numeric
The edge lengths, in centimeters, of the right rectangular prism show are 4, 3,
and 6. Enter the volume, in cubic centimeters, of this prism.
 Equation/Numeric
Enter the combined volume, in cubic millimeters, of Prisms A and B.
The right rectangular prism shown has a length 6 centimeters, width 3
centimeters, and height 4 centimeters.
 Matching Tables
Determine whether each equation can be used to find the volume (V) of this
prism. Select Yes or No for each equation.
7
Paramount Unified School District
Grade 5 – Unit 4
Stage Three –Learning Experiences & Instruction
Educational Services
Unit 4: Expressions and Volume
Prior to planning for instruction, it is important for teachers to understand the progression of learning and how the current unit of instruction connects to
previous and future courses. Teachers should consider: What prior learning do the standards and skills build upon? How does this unit connect to essential
understandings of later content? How can assessing prior knowledge help in planning effective instruction? What is the role of activating prior knowledge in
inquiry?
Looking Back
Looking Ahead
In Grade 4, students:
In Grade 6, students will:

Interpreted a multiplication equation as a comparison, e.g. 35= 5×7 as a
statement that 35 is 5 times as many as 7 and 7 times as many as 5.

Write and evaluate numerical expressions involving whole-number
exponents

Represented verbal statements of multiplicative comparisons as
multiplication equations.

Write, read, and evaluate expressions in which letters stand for
numbers


Multiplied or divided to solve word problems involving multiplication
comparisons using drawings and equations with symbols for the
unknown number to represent the problem, distinguishing
multiplicative comparison from additive comparisons.
Write expressions that record operations with numbers and with letters
standing for numbers. For example, express the calculation “Subtract y
from 5” as 5 – y.

Identify parts of an expression using mathematical terms (sum, term,
product, factor, quotient, coefficient); view one or more parts of an
expression as a single entity. For example, describe the expression 2(8 +
7) as a product of two factors; view (8 + 7) as both a single entity and a
sum of two terms.

Find the volume of a right rectangular prism with fractional edge
lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as would be found
by multiplying the edge lengths of the prism. Apply the formulas V = l x
w x h and V = b x h to find volumes of right rectangular prisms with
fractional edge lengths in the context of solving real-world and
mathematical problems.

Solved multi-step word problems with whole numbers using the four
operations including problems in which remainders must be
interpreted.

Represented problems using equations with a letter standing for an
unknown quantity.

Applied area and perimeter formulas for rectangles in real world and
mathematical problems.
8
Transfer
Goals
Unit 4: Expressions and Volume
Timeframe: Jan. 9- Feb. 7
Course Textbook: McGraw Hill, My Math
ST Math Objectives:
 Using Parentheses
 Patterns and Relationships
 Volume
1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution.
2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience.
3) Construct viable arguments and critique the reasoning of others using precise mathematical language.
Understandings:
 Numerical expressions represent real-life and mathematical
situations
 In an expression, the way numbers are grouped affects the value
 Both multiplicative and additive reasoning can be applied to volume
Time
Skill
Learning Goal
7 days
Write or
identify a
numerical
expression
that records
a calculation
represented
with words
Use a model to represent a situation in a
word problem (e.g., cubes, bar diagram)
Write a numerical expression that
represents the situation depicted in the
model using minimally one operation and
two (if applicable)
Evaluate the expression(s) without
grouping symbols
Evaluate expressions involving grouping
symbols
Essential Questions:
 How do numerical expressions represent real-life and mathematical situations?
 How would math be different if grouping symbols didn’t exist?
 How does multiplication relate to volume? And addition?
 How important is understanding area when determining the volume of a figure?
Lesson/Activity/
Knowledge
Focus Questions
Teacher Notes
Resource
Chapter 7
Vocabulary
How do
In relation to order of
Lesson 1
Numerical
numerical
operations, students do
Numerical Expressions
expression
expressions
not need to know
Verbal
represent realGEMDAS/PEMDAS.
LearnZillion Lesson:
expression
life and
Emphasis in grade 5 is
“Write numerical
Parentheses
mathematical
on the use of grouping
expressions”
Evaluate
situations?
symbols and
(Quick Code AP7K3ZF)
How do grouping recognizing that
symbols affect the operations within
See Inquiry Question
parentheses should be
value of the
solved first.
See Investigation
equation?
Explore how grouping symbols affect the
value of an expression by comparing
expressions that are grouped differently
Lesson 3
Write Numerical Expressions
Create a real-life situation for an
expression with parentheses
LearnZillion Application
Tasks:
 “Create and Evaluate
Expressions” (Quick Code
4DP9G93)
 “Create a real-life
situation for an expression
with parentheses”(Quick
Code: GZR9MVN)
Investigation: Students explore how the use of grouping symbols
may change the value of an expression. Students place grouping
symbols in equations to make the equations true or they
compare expressions that are grouped differently. Examples:
 15 ─ 7 ─ 2 = 10
15 ─ (7 ─ 2) = 10
 3 x 125 ÷ .25 + 7 = 22
[3 x (125 ÷ .25)] + 7 = 22
 Compare 3 x 2 + .05 and 3 x (2 + .05)
 Compare 15 ─ 6 + 7 and 15 ─ (6 + 7)
What would
happen if we
didn’t have
grouping symbols?
Inquiry Question: In Caleb’s music class, there are 3
students who play cello and 3 students who play the flute.
Two times the students that play the cello and the flute
play the violin. How many students play the violin?
Teacher Notes: Students can solve using addition or
multiplication. Compare 3 + 3 x 2 vs. (3 + 3) x 2 vs. 3 + (3 x
2)—Ask, “How are they different”? Have students discover
the use of parentheses when evaluating expressions.
9
1 day
1 day
Time
Skill
Lesson/Activity/
Resource
Knowledge
Focus Questions
Teacher Notes
Independent Practice with Transfer Goals:
 See Illustrative Mathematics (Watch Out for Parentheses 1)
 Ty says that 16 - (4 + 4) = 8. Jade says it equals 12. Who do you agree with?
 Write a numerical expression with at least two operations so that when evaluated it equals 18.
 A school spends $2.40 on every lunch it serves in the cafeteria and $0.30 for each carton of milk.
 250 people at the school get a lunch each day
 120 people take a carton of milk
Create an expression using this information that shows how much the school spends altogether on lunches and milk each day.
Cumulative Review and Error Analysis of Unit 3 Extended Constructed Responses
Introduce students to the 4-point Extended-Constructed Response rubric. Use this opportunity to get students familiar with rubric.
Possible activities include evaluating their own work, peer feedback, whole-class discussion about displayed exemplars, reflecting on next steps, etc.
Use unit cubes to
find the volume of
a right rectangular
prism with whole
number side
lengths
2 days
Learning Goal
Use unit cubes to build
rectangular prisms
Make observations
about relationships
between length, width
and height; number of
cubes and volume
Inquiry Question:
Theresa is building a
rectangular prism. The bottom
layer of the rectangular prism is
shown.
She builds a prism that has 3
layers.
How many cubes will the prism
have altogether?
Chapter 12
Lesson 8
Use Models to Find Volume
Vocabulary
Volume
Rectangular
prism
Unit cube
Cubic unit
Height
Base
What is the
relationship between
the number of cubes
needed to build a
rectangular prism and
its volume, in cubic
units?
What is the pattern
you observe in your
completed table
between the length,
width, height, and
volume of each
prism? (pg. 949)
Grades 3-4 studied
both area and
perimeter but this is
the first time students
have worked with
volume.
For lesson 8, do not
focus on the formula
but on the concept of
volume and using unit
cubes.
Looking at your
completed table,
what is the
relationship between
area and volume?
(pg. 949)
10
3 days
Time
Skill
Apply the
formulas V = l x w
x h and V = b x h
to solve realworld and
mathematical
problems
involving volume
of right
rectangular
prisms
2 days
Recognize that
volume is additive
when finding the
volume of two
non-overlapping
rectangular
prisms
Learning Goals
Lesson/Activity/
Knowledge
Resource
Associative Property
Given the volume of
LearnZillion Lesson:
a prism, build models  “Understand that volume can be
that have this volume
measured by packing object with
unit cubes” (Quick Code 4AN6ST4)
Use area to solve for
Inquiry Question:
volume
Paco’s family filled a cooler with water and snacks
for a picnic (show image on pg. 956 Example 1). He
Use the two volume
formulas to calculate
thinks he can find the volume of the cooler because
various prisms
he knows the area of the base of the cooler.
Do you agree or disagree with Paco?
Explain how the
Associative Property
can be used to
mentally find the
volume of a prism
Use cubes to build
composite figures
Use multiplication
and addition to find
the volume of two
non-overlapping
rectangular prisms
Focus Questions
How is multiplication
related to finding the
volume of a rectangular
prism?
How are the two formulas
for calculating volume
related?
How is the Associative
Property used to find the
volume of a prism?
Lesson 9
Volume of Prisms
Inquiry Question
Lesson 10
Build Composite Figures (Hands-on)
Lesson 11
Volume of Composite Figures
(see Modeling the Math p g. 961B)
Vocabulary
Composite figure
Teacher Notes
Note: Learn Zillion
has several
lessons/videos to
support this content;
after lesson 8,
teachers can assess
student
understanding to
determine which
lessons they can use
to best meet the
needs of the
students.
How is finding the volume
of a composite figure
similar to finding the
volume of a rectangular
prism?
How is addition related to
finding the volume of a
composite figure?
Inquiry Question:
Sally uses Block A and Block B to create a model of a new
building. The dimensions of Block A are 3 in. by 3 in by 5 in. The
dimensions of Block B are 1 in. by 3 in. by 4 in. What do you think
the total area is of this figure? Provide students with cubes to
solve, if needed.
11
Time
Skill
Learning Goal
Lesson/Activity/
Resource
Knowledge
Focus Questions
Teacher Notes
1 day
Independent Practice with Transfer Goals:
 See Illustrative Mathematics (Box of Clay)
 How are area and volume alike and different?
 I packed 24 centimeter cubes inside of a container. What might the dimensions of the container be? What are other possible dimensions?
 Two prisms are joined together and have a combined volume of 60 cubic units. What could be the dimensions of the two prisms? (for
example: 2 x 4 x 3 = 24 and 9 x 2 x 2 = 36 so together they equal 60 cubic units)
3 days
Feb. 3-7
Unit 4 Assessment Review and Administration
For Review:
 Given 12 ÷ 4 + 2, where should the parentheses be in this equation to make it equal 2?
 What is a way to write an expression that is five times as much as 96 ÷ 3?
 How would you write the following equation in words? 32 = 2x + 7. Write a story problem that would represent the equation.


A rectangular prism has a volume of 36 cubic units and one of its dimensions is 3. What might the other dimensions be?
Looking at the dimensions of two different boxes with identical volumes, what do you notice about multiples and factors?
8 in x 2 in x 2 in
4 in x 2 in x 4 in
Common Core Practices
 Instruction in the Standards for Mathematical Practices
 Use of Talk Moves
 Writing in Math (e.g., notetaking)
 Use of manipulatives
 Use of technology
 Use of real-world scenarios
 Project-based learning
 Number Talks
12