Paramount Unified School District
Educational Services
Grade 4 – Unit 3
Stage One – Desired Results
Unit 3: Understanding Multiplication; Perimeter and Area
In this unit, students will:

Use drawings and equations to solve contextual problems involving multiplicative comparisons

Reason between arrays and written numerical work students to see the role of place value units in multiplication

Discover how the Associative Property can be used to solve problems mentally and more efficiently

Use models to develop understanding of perimeter and area of rectangles

Use repeated reasoning to discover why the formulas for perimeter and area make sense

Apply understandings and formulas to the solution of real-world and mathematical problems

Determine the unknown when provided with the width, length and/or perimeter/area and use symbols to represent the unknown

Generate rectangles with the same perimeter and different areas and rectangles with the same area and different perimeters
Common Misconceptions:
Students may:

know how to multiply but do not know when to multiply (other than because he was told to do so, or because the computation was written as a
multiplication problem).

know the Commutative Property of Multiplication but fails to apply it to simplify the “work” of multiplication.

see multiplication and division as discrete and separate operations (conception of the operations does not include the fact that they are linked as inverse
operations).

not understand the Distributive Property and does not know how to apply it to simplify the “work” of multiplication. For example, student has
reasonable ability with multiplication facts but cannot multiply 12 × 8 or 23 × 6.

confuse area and perimeter.

See http://www.westada.org/cms/lib8/ID01904074/Centricity/Domain/207/Misconceptions_Error%202.pdf
1
Unit 3 Overview: Understanding Multiplication; Perimeter and Area
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
Standards
OA.1 Interpret a multiplication equation
as a comparison, e.g. 35= 5x7 as a
statement that 35 is 5 times as many as 7
and 7 times as many as 5. Represent
verbal statements of multiplicative
comparisons as multiplication equations.
OA.2 Multiply or divide to solve word
problems involving multiplication
comparisons, e.g. by using drawings and
equations with symbols for the unknown
number to represent the problem,
distinguishing multiplicative comparison
from additive comparisons.
NBT.5 Multiply a whole number of up to
four digits by a one-digit whole number
(up to 100), 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.
NBT.6 Find whole-number quotients and
remainders with up to four-digit dividends
and one-digit divisors (up to 100), using
strategies based on place value, the
properties of operations, and/or the
relationships between multiplication and
division. Illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models
MD.3 Apply area and perimeter formulas
for rectangles in real world and
mathematical problems.
Meaning-Making
Understandings
Students will understand that…
 Creating models helps to develop
an understanding of the meaning
of operations
 A variety of strategies can be used
to solve multiplication problems
 Perimeter and area help us solve
problems in real-world situations
Essential Questions
Students will consider….
 How do arrays and area models develop an understanding of x and ÷?
 How can we use place value and properties as strategies to solve problems?
 How is comparing two groups related to multiplication?
 How are area and perimeter alike? different?
 How can understanding area and perimeter help us to solve problems in
the real world?
Acquisition
Knowledge
Students will know…
Vocabulary: Comparison, dividend,
divisor, quotient factor, product, fact
family, decompose, multiple,
perimeter, area, square units
 Properties: Commutative, Identity,
Zero, Associative
 Related facts (fact families)
 Phrases such as “times as many,”
“___ times more,” and “times as
much” which indicate a
multiplication comparison problem
 Perimeter formula of a rectangle
P = (2 × l) + (2 × w) and a square P = 4s
 Area formula L × W = A
Skills
Students will be skilled at and able to…
 Interpret a multiplication equation as a comparison and represent verbal
statements of multiplicative comparisons as equations
 X and ÷ to solve word problems involving comparisons using drawings and
equations with a symbol for the unknown to represent the problem
 Distinguish between multiplicative comparisons and additive comparisons
 Use and explain strategies for solving multiplication including the inverse
relationship and properties of operations
 Solve real-world and mathematical problems by applying area and perimeter
formulas for rectangles (including squares)
 Determine the unknown when provided with the width, length and/or
perimeter/area and use a symbol or letter to represent an unknown number
 Relate perimeter and area of rectangles
 Show rectangles with the same perimeter and different areas and rectangles
with the same area and different perimeters
2
Paramount Unified School District
Grade 4 – Unit 3
Stage Two – Evidence of Learning
Educational Services
Unit 3: Understanding Multiplication; Perimeter and Area
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?
 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






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.)





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 cannot be used for instruction or assessment, but can be modified by teachers.
Skill
Interpret a
multiplication
equation as a
comparison and
represent verbal
statements of
multiplicative
comparisons as
equations
X and ÷ to solve
word problems
involving
comparisons using
drawings and
equations with a
symbol for the
unknown to
represent the
problem
Standard
OA.1
Assessment Guidelines

The student is prompted to solve a contextual
problem involving multiplicative comparison.
 Numbers should fit in the parameters of up to
4-digit by 1-digit
 All quantities should be whole numbers.
 Problems may involve measurements

Item difficulty can be adjusted by:
 Using multiplication facts in the context
 Using non-math facts in the context

The student is presented with a contextual problem
involving multiplicative comparison that solves for
an unknown factor. The unknown is the multiplier
that describes how many times more one quantity is
than the other.
OA.2
Example
A cat has 4 times as many toys as a puppy.
The puppy has 12 toys. How many toys
does the cat have?
Possible
Question Type(s)
 Equation/Numeric
A puppy has 4 toys. A cat has 36 toys. How
many times more toys does the cat have
than the puppy?
Distinguish between
multiplicative
comparisons and
additive
comparisons
4
Skill
Use and explain
strategies for solving
multiplication
including the inverse
relationship and
properties of
operations
Standard
NBT.5
Assessment Guidelines



Solve real-world and
mathematical
problems by applying
area and perimeter
formulas for
rectangles (including
squares)
MD.3



The student is prompted to multiply two whole
numbers
Item difficulty can be adjusted by having students
solve when:
 One factor is a multiple of 10, 100, or 1000.
 One or more partial products result from
multiplying 5 by an even digit (e.g., multiplying
5 by 4 gives 20, but 5 by 40 gives 200 – the
extra 0 seems to violate the pattern of “when
you multiply ones by tens, just add a zero on
the end”).
 Factors contain digits that are easier to multiply
(e.g., multiplying by 2 or 5 is typically easier
than multiplying by 6, 7, or 8).
The student is presented with a non-contextual
multiplication problem.
The student uses the area formula and/or perimeter
formula to solve a problem in a mathematical or
real-world context.
 Items may describe rectangles (in pure math
context) or rectangular shapes (in a real-world
context)-e.g., described as “rectangular” as in a
rectangular garden, a rectangular kitchen, etc.
 The dimensions should be whole numbers with
units listed.
Item difficulty can be adjusted via these example
methods:
 How “friendly” the numbers are to work with
 Including a visual diagram with labeled sides
The student is presented with the dimensions of a
rectangle.
Example
Enter the product.
Possible
Question Type(s)

Equation/Numeric

Equation/Numeric
5327
× 4
How many different ways can you solve
289 by 8?
5
Skill
Standard
Determine the
unknown when
provided with the
width, length and/or
perimeter/area and use
a symbol or letter to
represent an unknown
number
MD.3
Assessment Guidelines


Relate perimeter and
area of rectangles
Show rectangles with
the same perimeter
and different areas and
rectangles with the
same area and different
perimeters
The student uses the area formula and/or
perimeter formula to solve a problem in a
mathematical or real-world context.
 Items may describe rectangles (in pure math
context) or rectangular shapes (in a real-world
context). The shapes presented in real-world
contextual items must be described as
“rectangular” (e.g., a rectangular garden, a
rectangular kitchen, etc.).

The dimensions, areas, and perimeters
should be whole numbers with units listed.
Item difficulty can be adjusted via these example
methods:
 How “friendly” the numbers are to work with
 Including a visual diagram with labeled sides

The student is presented with one dimension and
either the area or perimeter of a rectangle and
must find the unknown side length.

The student is presented with the area or
perimeter of a rectangle.
The dimensions for three rectangular gardens are
shown. Decide whether each garden has a
perimeter equal to 100 meters. Select Yes or No
for each garden.
Example
Use the diagram of the rectangle to solve the
problem.
Possible
Question Type(s)

Equation/Numeric

Matching Tables
The perimeter of the rectangle is 192 inches.
What is the length, in inches, of the
unknown side?
The dimensions for three rectangles are
shown. Decide whether each rectangle has an
area equal to 100 square feet. Select Yes or
No for each rectangle.
6
Paramount Unified School District
Grade 4 – Unit 3
Stage Three –Learning Experiences & Instruction
Educational Services
Unit 3: Understanding Multiplication; Perimeter and Area
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
In Grade 3, students:
 Interpreted products of whole numbers, e.g., interpret 5 × 7 as the total
number of objects in 5 groups of 7 objects each.
 Interpreted whole-number quotients of whole numbers (e.g., interpret 56
÷ 8 as the number of objects in each share when 56 objects are partitioned
equally into 8 shares, or as a number of shares when 56 objects are
partitioned into equal shares of 8 objects each.)
 Used multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities,
e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.
 Determined the unknown whole number in a multiplication or division
equation relating three whole numbers. For example, determine the
unknown number that makes the equation true in each of the equations 8
× ? = 48, 5 = □ ÷ 3, 6 × 6 = ?.
Looking Ahead
In Grade 5, students will:
 Fluently multiply multi-digit whole numbers using the standard algorithm.

Use parentheses, brackets, or braces in numerical expressions, and
evaluate expressions with these symbols.

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 × (8 + 7).
Recognize that 3 × (18932 + 921) as three times as large as 18932 + 921,
without having to calculate the indicated sum or product.


Relate volume to the operations of multiplication and addition
Solve real-world and mathematical problems involving volume
 Solved problems involving perimeters of polygons
 Found the perimeter given the side lengths and found an unknown side
length

Exhibit rectangles with the same perimeter and different areas or with
the same area and different perimeters
7
Unit 3: Understanding Multiplication; Perimeter
and Area
Timeframe: Oct. 17 – Nov. 10
Course Textbook: McGraw Hill,
My Math
ST Math Objectives:
Transfer
Goals
 Division Concepts
 Multiplication Concepts I and II
 Applying Area and Perimeter
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: Students will understand that…
 Creating models helps to develop an understanding of the meaning of
operations
 A variety of strategies can be used to solve multiplication problems
 Perimeter and area help us solve problems in real-world situations
Time
Skills
Use and explain strategies
for solving multiplication
including the inverse
relationship
Learning Goal
Use arrays to
represent the
inverse relationship
1 day
Write all the related
facts for an array
Write all the related
facts for a set of
numbers
Essential Question(s): Students will consider….
 How do arrays and area models develop an understanding of x and ÷?
 How can we use place value and properties as strategies to solve problems?
 How is comparing two groups related to multiplication?
 How are area and perimeter alike? different?
 How can understanding area and perimeter help us to solve problems in the
real world?
Lesson/Activity/Resource
Knowledge
Inquiry Question: Alisha
Vocabulary
has 15 Hot Wheels cars in
Related fact (fact
her collection. She wants to
family)
display them in equal rows
and columns. How can she
arrange her cars? What
number sentences can be
written to represent her
arrangement?
Focus Questions
for Lessons
How can
understanding
multiplication help
you to divide?
Teacher Notes
This is review from
grade 3.
Chapter 3 Lesson 1
Relate Multiplication and
Division
REMINDER: Teachers should use the textbook as a resource when needed for additional practice.
8
Time
Skills
Learning Goal
Use drawings and
equations to solve
contextual
problems involving
multiplicative
comparisons
Use arrays and bar
diagrams to recognize
the comparisons of
two groups as
multiplication
Compare
multiplicative
comparisons and
additive comparisons
4 days
Distinguish
between
multiplicative
comparisons and
additive
comparisons
Use a symbol or
letter to represent
an unknown
number
Use and explain
various strategies
for solving
multiplication
using properties of
operations
Use a symbol or
letter to represent
an unknown
number
Use comparisons to
multiply
Use comparisons to
solve word problems
in which a symbol or
letter is used as the
unknown
Identify Commutative,
Identify and Zero
Properties
Use these properties
to find the unknown
when solving
problems
Discover how the
Associative Property
allows us to mentally
calculate by grouping
known facts
Lesson/Activity/Resource
Inquiry Question: Teresa bought a
jacket that cost 3 times as much
as one pair of shoes. If shoes are
priced equally and cost $48 for 6
pairs, how much did Teresa’s
jacket cost?
LearnZillion Lesson:
“Understand multiplicative
comparison by comparing it to
additive comparison” (TK7FB37)
https://learnzillion.com/assignments/
TK7FB37
Knowledge
Vocabulary
Comparison
Phrases such as
“times as many,”
“___ times
more,” and
“times as much”
which indicate a
multiplication
comparison
problem
Focus Questions
for Lessons
What are some
ways to visually
represent
comparison
problems?
What is the
difference between
an additive
comparison and a
multiplicative
comparison?
Teacher Notes
Use arrays to
introduce
comparison (see pg.
149 #1-7) before
using the bar
diagram. This serves
as a nice transition
from using arrays
with area and also
when relating x and
÷.
Lesson 3
Multiplication as Comparison (see
Teacher Notes)
Lesson 4
Compare to Solve Problems
Check My Progress
Lesson 5
Multiplication Properties and Division
Rules
Inquiry Question: There are 6
video games in each value pack.
There are 7 value packs in each
box. If Raul buys 2 boxes for his
collection, how many video
games will he have? Solve the
problem. Explain how you
solved it and why you chose to
solve the problem in this way.
Lesson 6
Associative Property of Multiplication
(See Model the Math pg. 167B)
Properties:
Commutative
Identity
Zero
Associative
How do properties
and rules help to
solve multiplication
and division
problems?
Why doesn’t
grouping numbers
in different ways
change the value?
How does the
Associative Property
of Multiplication
make it easier to
multiply?
Only focus on
multiplication
properties (not
division rules)—
lesson 5 and 6 can
be taught together
since they both
address properties
LearnZIllion.com
video:
 "The
Commutative
Property"
(YMB5XA4)
9
3 days
1 day 1 day
Time
Skills
Learning Goal
Lesson/Activity/Resource
Knowledge
Focus Questions
for Lessons
Teacher Notes
Independent practice with transfer goals
 Illustrative Mathematics Task: Comparing Money Raised https://www.illustrativemathematics.org/content-standards/4/OA/A/2/tasks/263
Cumulative Review and Error Analysis of Unit 2 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.
Find the distance
Solve real-world and
See Inquiry Lesson
Vocabulary
When do we
LearnZillion video:
around a square
mathematical problems
Perimeter
measure perimeter  “Find the
by applying the perimeter Find the distance
in real-world
perimeter of a
Perimeter formula
around
a
rectangle
formula for rectangles
Chapter 13
situations?
rectangle”
of a rectangle
Use the patterns in
(including squares)
Lesson 1
(LZ3742)
P = (2 × l) + (2 × w) What is the most
the numbers to
Measure Perimeter
efficient way to find
Perimeter formula
discover the formula
the perimeter of a
of a square P = 4s
for perimeter
figure?
Determine the unknown
Given one of the sides
Inquiry Lesson:
when provided with the
and the perimeter,
Part 1: Albert walked around the park. If the park is square and one side of the park
width, length and/or
find the missing value
measures 2 miles, how far did Albert walk? (Students may solve using addition or
perimeter
Use a symbol (or
multiplication)
variable) to represent
Use a symbol or letter to
the unknown value
Part 2: What if the park was rectangular—one side measures 2 and the other side 4. How far
represent an unknown
would Albert walk? (Students should observe that they can only use addition since the park is
number
rectangular and not square).) Ask questions that allow students to observe the patterns in
the numbers they add [e.g., 2 + 4 + 2 + 4—I add 2 for of the same number, I add 2 for each
side that has the same measure, etc.) Students should discover the formula for perimeter (l +
l + w + w= 2l + 2w). Give students more problems to practice applying this strategy.
Part 3: What if Albert walks a distance of 12 miles? What could the park look like? (Students
may draw different variations. Ask questions where they have to explain why this is
possible.) Discuss how you can find the unknown given the perimeter. Give students
additional practice calculating the unknown value. Have them experiment using a symbol or
variable to represent the unknown.
10
Time
Skills
Learning Goal
Solve real-world and
mathematical problems
by applying the perimeter
formula for rectangles
(including squares)
Create arrays with
unit square tiles and
count the tiles that
cover the length and
width of the
rectangles
Draw the arrays on
grid paper and count
the tiles that cover the
rectangle
Determine the unknown
when provided with the
width, length and/or
area/perimeter
4 days
Use the relationship
between the numbers
to discover the
formula for area
Relate perimeter and area
of rectangles
Show rectangles with the
same perimeter and
different areas and
rectangles with the same
area and different
perimeters
Lesson/Activity/
Resource
See Inquiry Lesson
Lesson 3
Hands On: Model Area
LearnZillion video:
“Use area models to find the
area of rectangles” (LZ2374)
Lesson 4
Measure Area
(see Model the Math pg. 845B)
Given one of the sides
and the area, find the
missing value
Use a symbol (or
variable) to represent
the unknown value
Discover that
Investigation
rectangles can have
Your family is building a
the same perimeter
vegetable garden with a
and different area
perimeter of 14 meters. What
are the possible rectangular
Observe which
sizes for designing the
rectangles have the
vegetable garden? What do
smallest/largest area you notice about the area of
and why
each design?
Discover that
rectangles can have
the same area and
different perimeters
Lesson 5
Relate Area and Perimeter
Knowledge
Vocabulary
Area
Unit square
Square unit
Area formula
L×W=A
Focus Questions
for Lessons
What is the
relationship
between area and
square units?
Teacher Notes
LearnZillion videos:
 “Find the area of a
rectangle using the
standard formula”
(LZ2535)
How does
 “Find the area of a
multiplication relate
rectangle” (LZ3743)
to area?
 “Apply area and
perimeter formulas”
(LZ3744)
Inquiry Lesson:
Part 1: Students create arrays with unit square tiles and count
the tiles that cover the rectangles. Then, they draw the arrays
on grid paper.
Part 2: What if Albert wanted to measure the space inside the
rectangular park with sides that measure 2 miles and 4 miles?
How could we represent this using an array? What is the area
of the park? What do you to observe the relationship between
the sides and the area? Give more examples for students to
What is the
practice. Students should discover that you multiply the sides
difference between
to calculate the area.
area and
perimeter?
Part 3: If Albert measures the space inside the park and it
equals 12 square miles, what could be the values of the sides?
(Students may draw different variations. Ask questions where
they have to explain why this is possible.) Discuss how you can
find the unknown given the area. Give students additional
practice calculating the unknown value. Have them
experiment using a symbol or variable to represent the
unknown.
11
Learning Goal
Lesson/Activity/
Resource
Knowledge
Focus Questions
for Lessons
Teacher Notes
1 day
Skills
Challenge Lesson: Mrs. Potts’ Secret Garden (Area and Perimeter)
1 day
Time
Independent practice with transfer goals:
Illustrative Mathematics Task: Karl’s Garden https://www.illustrativemathematics.org/content-standards/4/MD/A/3/tasks/876
(Note: This task can be solved using invented strategies or properties, e.g. distributive. Students do not need to know 2-digit by 2-digit multiplication in
order to be successful at this task. If necessary, the teacher may change the dimensions of the gardens to fit the needs of the class.)
Nov.8-10
3 days
Review and Administer Unit 3 Assessment
Review:
 Mark’s recipe calls for three times as many potatoes as carrots. If Mark uses two cups of carrots, how many cups of potatoes will he use?
 Anna is 8 years old. Her mom is five times older than she is and her grandmother is eight times older than Anna. What multiplication sentences can
be written to represent the relationship between Anna’s age and her mom’s age? Between Anna’s age and her grandmother’s age? How old are
Anna’s mother and grandmother?
 A rectangular figure has a perimeter of 35 cm. What could the lengths of the sides be? Give two possibilities.
 A construction worker laid 54 square feet of hardwood in the rectangular family room of a new house. What could possible perimeters of the room
be?
Common Core Practices
 Instruction in the Standards for Mathematical Practices
 Use of Talk Moves
 Writing in math (e.g. math notes, prompts, journals)
 Use of manipulatives
 Use of technology
 Use of real-world scenarios
 Project-based learning
 Number Talks
12