Download Grade 4 Unit 4 Pacing - Paramount Unified School District

Paramount Unified School District
Educational Services
Grade 4 – Unit 4
Stage One – Desired Results
Unit 4: Multi-Digit Multiplication
In this unit, students will:

use place value understanding and visual representations to solve multiplication problems with multi-digit numbers

multiply single-digit numbers by multiples of 10, 100, and 1,000 and two-digit multiples of 10 by two-digit multiples of 10 and reason between arrays and
written numerical work to see the role of place value units in multiplication

decompose numbers into base ten units to find products of single-digit by multi-digit numbers

multiply two-digit by two-digit numbers using understanding of place value and of the area model to empower them to multiply by larger numbers (

connect the partial products appearing in the area model to the distributive property

use estimation to check for reasonableness before computing

apply understanding of addition, subtraction and multiplication to solving multi-step problems

apply algebraic thinking by using a symbol to represent an unknown quantity
Common Misconceptions:
Students may:

under generalizes the results of multiplication by powers of 10. To find products like 3 × 50 = 150 or 30 × 50 = 1,500, she must “work the product
out” using a long method of computation (see example).

can state and give example s of properties of multiplication but does not apply them to simplify computations. For example, the student labors
to find the product 12 × 15 because he does not realize that he could instead perform the equivalent but much easier computation, 6 × 30.

misapplies the procedure for multiplying multi-digit numbers by ignoring place value. For example, student multiplies correctly by ones digit
but ignores the fact that the 3 in the tens place means 30. Thus, 30 × 60 = 1,800 (see example).

knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written
as a multiplication problem). For example, the student cannot explain why he should multiply or connect multiplication to actions with
materials.
See http://www.westada.org/cms/lib8/ID01904074/Centricity/Domain/207/Misconceptions_Error%202.pdf

1
Unit 4 Overview: Multi-Digit Multiplication
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
NBT.1 Recognize that in a multi-digit
whole number, a digit in one place
represents ten times what it
represents in the place to its right.
NBT.3 Use place value understanding
to round multi-digit whole numbers to
any place.
NBT.5 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.
OA.3 Solve multi-step word problems
posed with whole numbers and having
whole answers using addition,
subtraction, multiplication, and
division; including problems in which
remainders must be interpreted.
Represent these problems using
equations with a letter standing for
the unknown quantity.
Meaning-Making
Understandings
Students will understand that…
 Place value and properties play a
critical role in number operations
 A variety of models and strategies
can be used to multiply multi-digit
numbers
 Estimation is important to use when
problem solving because it helps to
check the reasonableness of a
solution
Essential Questions
Students will consider….
 How do arrays and area models develop an understanding of multi-digit
multiplication?
 What is the role of place value and properties of operations in multi-digit
multiplication?
 How do different strategies for solving multi-digit multiplication relate to
each other (e.g., area model, partial products, distributive property?
 Why is estimation important to use when solving problems?
Acquisition
Knowledge
Students will know…
 Vocabulary: Multiple, mental
math, array, area model, product,
partial products, estimate, round,
about how many, approximately,
reasonable, operation, variable
 Distributive Property
 Associative Property
 Basic facts for multiplication
 Inverse relationship
 Symbols can represent the
unknown
Skills
Students will be skilled at and able to…
 Use basic facts and patterns in zeros to mentally multiply a number by a
multiple of ten
 Recognize that the digit in one place represents ten times what it
represents in the place to its right
 Illustrate multiplication problems using place value, rectangular arrays and
area models
 Estimate numbers and assess the reasonableness of answers
 Multiply a number up to 4-digits by a 1-digit number using partial products
and distributive property
 Multiply a 2-digit number by a multiple of ten using the Associative Property
 Multiply a 2-digit number by a 2-digit number using partial products and
distributive property
 Solve multi-step word problems involving addition, subtraction and
multiplication
 Represent word problems using equations with a letter symbolizing the
unknown quantity
2
Paramount Unified School District
Grade 4 – Unit 4
Stage Two – Evidence of Learning
Educational Services
Unit 4: Multi-Digit Multiplication
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
Use basic facts and
patterns in zeros to
mentally multiply a
number by a
multiple of ten
Standard
Example
NBT.1
The student is prompted to explain the difference
between the values of the same digit in different place
values.
 In each item, the numbers presented differ by a
factor of 10 (e.g., 8 and 80, or 1725 and 17,250).
 Items should be equally distributed across these
number bands: small numbers (up to 1,000),
medium numbers (from 1,000 up to 100,000), and
large numbers (from 100,000 up to 1,000,000).
Select the statement that explains how the
values of the numbers 420 and 4200 are
different.
A. 4200 is 1000 times as large as 420.
B. 4200 is 100 times as large as 420.
C. 4200 is 10 times as large as 420.
D. 4200 is 1 time as large as 420.
NBT.5
The student is prompted to multiply two whole numbers
 Item difficulty can be adjusted by using:
 One factor is a multiple of 10, 100, or 1000
Enter the product.
Recognize that the
digit in one place
represents ten times
what it represents in
the place to its right
Multiply a number
up to 4-digits by a 1digit number using
partial products and
distributive property
Assessment Guidelines
x
Possible
Question Type(s)
 Multiple Choice,
Single Correct
Response
 Equation/Numeric
5327
4
4
Skill
Multiply a 2-digit
number by a 2-digit
number using partial
products and
distributive property
Standard
NBT.5
Assessment Guidelines
The student is presented with a multiplication
expression in which properties of operations have
been used as strategies for multiplication, with one
unknown number
Example
36 x 94 = (30 + 6) x (___ + 4)
36 x 94 = 2700 + ___ + 540 + 24
Possible
Question Type(s)


Equation/Numeric
Multiple Choice,
Single Correct
Response
The student is presented with a multiplication
Which expression is equal to 36 x 94?
expression in the stem and expressions reflecting use
A. (30 x 90) + (6 x 4)
of the distributive property or decomposition of factors B. (30 x 6) + (90 x 4)
in the answer choices.
C. (30 x 6) x 94 + (30 + 6) x 4
D. (30 x 90) + (30 x 6) + (90 x 6) + (90 x 4)
The student is presented with a multiplication problem Which strategy for multiplying 36 and 94
and four vertically recorded partial solutions.
should result in the correct product?
The student is presented with a multiplication
expression in the stem and expressions reflecting
use of the distributive property or decomposition of
factors in the answer choices.
Which expression is equal to 36 × 94?
A. (30 × 90) + (6 × 4)
B. (30 + 6) × (90 + 4)
C. (30 + 6) × 94 + (30 + 6) × 4
D. (30 × 90) + (30 × 6) + (90 × 6) + (90 × 4)
The student is presented with a multiplication
expression in which properties of operations have
been used as strategies for multiplication, with one
unknown number.
Enter the unknown number that makes the
equation true.
36 × 94 = (30 + 6) × (□ + 4)
5
Paramount Unified School District
Grade 4 – Unit 4
Stage Three –Learning Experiences & Instruction
Educational Services
Unit 4: Multi-Digit Multiplication
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 3, students:
 Used place value understanding to round whole numbers to nearest 10 or
100.
In Grade 5, students will:
 Recognize that in a multi-digit number, a digit in one place represents 10
times as much as it represents in the place to its right and 1/10 of what it
represents in the place to its left.

Multiplied one-digit whole numbers by multiples of 10 in the range 10-90
(e.g., 9 × 80, 5 × 60) using strategies based on place value and properties
of operations.

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 = ?

Applied properties as strategies to multiply and divide.

Solved two-step word problems using the four operations.

Represented problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental
computation and estimation strategies including rounding.

Use place value understanding to round decimals to any place.

Fluently multiply multi-digit whole numbers using the standard algorithm.

Add, subtract, multiply, and divide decimals to hundredths, using
concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method and explain the
reasoning used.

Use parentheses, brackets, or braces in numerical expressions, and
evaluate expressions with these symbols.
6
Transfer
Goals
Unit 3: Multi-Digit Multiplication
Timeframe: Nov. 14-Jan. 19
Course Textbook: McGraw Hill,
My Math
ST Math Objectives:
 Multi-digit Multiplication
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.
1 day
Understandings: Students will understand that…
Essential Question(s): Students will consider….
 How do arrays and area models develop an understanding of multi-digit multiplication?
 Place value and properties play a critical role in number
operations
 What is the role of place value and properties of operations in multi-digit
multiplication?
 A variety of models and strategies can be used to multiply multi How do different strategies for solving multi-digit multiplication relate to each other
digit numbers
(e.g., area model, partial products, distributive property?
 Estimation is important to use when problem solving because it
helps to check the reasonableness of a solution
 Why is estimation important to use when solving problems?
Time
Skills
Learning Goal
Lesson/Activity/Resource
Knowledge
Focus Questions
Teacher Notes
for Lessons
Why does the product of a
Use basic facts and
Use basic facts to
Vocabulary:
In grade 3, students
Inquiry Question
multiple of 10 always have a
patterns in zeros to
mentally multiply a
Multiple
multiplied one-digit
5 friends go to a carnival.
zero
in
the
ones
place?
mentally multiply a
number
Mental math
whole numbers by
They each buy 3 tickets.
number by a
multiples of 10 in the
How many total tickets do
What
is
the
relationship
Observe
patterns
in
multiple of ten
range 10-90.
they have?
between a digit in one place
problems to mentally
and the place to its right?
multiply by a multiple
Recognize that the
LearnZillion video:
What if they had 30 tickets
of
ten
digit in one place
 Multiply by powers
each? And 300 tickets
represents ten times
of 10 (TP6RXQ4)
each?
What
do
you
Use place value to
what it represents in
observe is happening to
multiply by multiples of
the place to its right
the total of tickets?
10s, 100s and 1000s
Investigation
Look at these problems. See if you can find any patterns that
Chapter 4
will help you solve for d).
Lesson 1
a) 4 x 8= 32
Multiples of 10, 100, and
b) 4 x 80 = 320
1,000
c) 4 x 800 = 3,200
d) 4 x 8000 =
(patterns may include adding the zero when multiplying by a
number that is ten times larger as well as the number of digits in
the product.)
7
1 day
Time
Skills
Learning Goal
Knowledge
Focus Questions
for Lessons
Teacher Notes
Vocabulary
Estimate
Round
About how many
Approximately
Reasonable
Why is it important
to use estimation
when solving
problems?
LearnZillion video:
 “Use estimation
to check
reasonableness of
products” (LZ70)
Vocabulary
Array
Area model
Product
Partial products
What is the role of
place value when
using arrays and
area model to
multiply larger
numbers?
Independent practice with transfer goals: Illustrative Mathematics Thousands and Millions of 4th Graders
https://www.illustrativemathematics.org/content-standards/4/NBT/B/5/tasks/1808
Estimate products of a
2-digit number by a 1digit number
1 day
Estimate numbers
and assess the
reasonableness of
answers
Illustrate
multiplication
problems using
place value (unifix
cubes), rectangular
arrays and area
models
4 days
Lesson/Activity/
Resource
Use cubes to create
arrays that represent a
multiplication problem
Inquiry Question
Lena loves playing video games
at the arcade. She played 4
games and scored 23 points
each time. About how many
points did she score in total?
See Lesson 2 for additional
practice
Inquiry Lesson:
Arrays and Area Model
Use the area model to
represent a
multiplication problem
Use the area model to
multiply larger
numbers
Multiply a number
Use the distributive
up to 4-digits by a 1- property to multiply
digit number using
partial products and Use partial products to
distributive
multiply
property (2 digit by
1 digit)
Distributive Property
Inquiry Lesson:
Distributive Property and
Partial Products
LearnZillion videos:
 Use an array to
multiply a twodigit number by a
one digit number
(GENCU7W)
 Multiply multidigit whole
numbers by single
digit whole
numbers using an
When multiplying
area model
multi-digit numbers,
(4VFHCJK)
how do arrays, area
 Use place value
model, partial
understanding to
products and
multiply three and
distributive property
four digit numbers
relate to each
(9HNTY63)
other?
8
Time
Skills
4 days
1 day
1 day
Illustrate multiplication
problems using place
value (unifix cubes),
rectangular arrays and
area models
Learning Goal
Estimate products
Select a multiplication
strategy to solve a word
problem (e.g., area model,
partial products or
distributive property)
Represent word
Represent the product of a
problems using
equations with a letter word problem using a
letter
symbolizing the
unknown quantity
Lesson/Activity/
Resource
Exploration
Adrian has four boxes of
action figures. There are 12
in each box. Alex has 21
action figures in each of his
3 boxes. Who has more
action figures? How do you
know?
Knowledge
Vocabulary
Area model
Partial products
Distributive
Property
Focus Questions
for Lessons
What is the best
strategy for solving
multi-digit
multiplication? Why?
Teacher Notes
Students do not
need to solve multidigit multiplication
problems using the
standard algorithm.
They can use any of
the strategies they
have been learning
about to solve.
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.
Estimate numbers and Estimate the product of a
Vocabulary
What is the role of
The standard states
Inquiry Question
assess the
3-digit number by a 1-digit Lena goes back to the
Area model
place value when
that students
reasonableness of
number
Partial
products
using
arrays,
area
should multiply a
arcade. She played 4 games
answers
model, partial
number up to 4Use area model to multiply but this time she scored 275
products and the
digits. The textbook
a 3-digit number by a 1points each time! She thinks Distributive
Illustrate
Property
distributive
property
does not include
digit number
she has about 800 total
multiplication
to multiply larger
these kind of
points. Do you agree or
Use distributive property
disagree with Lena? Why?
problems using place
numbers?
problems so
and partial products to
value (unifix cubes),
teachers will need
multiply a 3-digit number
LearnZillion lesson:
rectangular arrays and by a 1-digit number
How can I apply my
to incorporate them
“Multiplying larger
area models
understanding of
during instruction
Use distributive property
numbers with arrays and
place
value
when
and practice (see
and partial products to
base-ten blocks”
Multiply a number up
multiplying factors
SBAC example).
multiply a 3-digit number
(XC2886R)
to 4-digits by a 1-digit
with
zeros?
(containing a 0) by a 1-digit
Inquiry Question
number using partial
LearnZillion video:
number
Diller Elementary is collecting
products and
"Use place value
See Lessons 9 and 11 for additional
money to donate to a charity.
distributive property
understanding to
$1023 is collected each month.
practice
multiply three and
DEPENDING ON YOUR STUDENTS, YOU CAN
How much money is collected in
four digit numbers"
REPEAT THIS SAME PROGRESSION TO
4 months?
(3W5DT6U)
MULTIPLY 4-DIGIT BY 1-DIGIT NUMBERS OR
STUDENTS CAN SELECT WHICH STRATEGY
THEY WOULD LIKE TO USE TO SOLVE 4-DIGIT
BY 1-DIGIT PROBLEMS.
9
1 day
1 day
Time
Skills
Lesson/Activity/
Knowledge
Focus Questions
Resource
for Lessons
Independent practice with transfer goals/Questions to Ask:
 How many different ways can you solve 289 x 8?
 How does the order of the digits in the factors impact the product? (e.g., 452 x 7 compared to 425 x 7)
 How are the values of the numbers 420 and 4200 different?
Multiply a 2-digit
number by a
multiple of ten using
the Associative
Property
4 days
Estimate numbers
and assess the
reasonableness of
answers
Illustrate
multiplication
problems using
place value,
rectangular arrays
and area models
1 day
Multiply a 2-digit
number by a 2-digit
number using
partial products and
distributive property
Learning Goal
Decompose 2-digit
numbers that end in zero
into a number times 10
Multiply a 2-digit number
by a multiple of ten using
the Associative Property
Estimate the product of a
2-digit number by a 2-digit
number
Use area model to
multiply a 2-digit number
by a 2-digit number
(decompose both factors)
Use distributive property
and partial products to
multiply a 2-digit number
by a 2-digit number
(decompose both
factors)*
Inquiry Question
A hummingbird flies 12
miles per hour. If it flies a
total of 40 hours, how far
did it fly?
Associative Property
Chapter 5
Lesson 1
Multiply by Tens
Inquiry Question
A concert ticket costs
$48. About how much
will tickets cost for a
group of 22 people?
Lesson 2
Estimate Products
LearnZillion lesson:
“Using place value strategies
to multiply two double digit
numbers” (Quick Code:
4XF9HY4)
LearnZillion lesson:
“Multiplying larger numbers
using the area model” (Quick
Code: 62G8AZ6)
Vocabulary
Estimate
Round
About how many
Approximately
Reasonable
Teacher Notes
How can we use
decomposition and
the Associative
Property to multiply
larger numbers?
Joey is finding 67 x
40. Explain why he
can think of 67 x 40
as 67 x 4 x 10?
How can you use
area model,
distributive property
and partial products
to multiply two 2digit numbers?
What is the role of
place value when
using area model,
distributive property
and partial products
to multiply two 2digit numbers?
See Lessons 3-4 for additional
practice
LearnZillion video:
 “Solve 2- by 2digit multiplication
problems using
partial products”
(LZ22)
For LearnZillion
lessons: only use
examples with 2digit by 2-digit and
4-digit by 1-digit
examples (not 4 digit
by 2-digit)
*Note: Students
should decompose
both factors when
solving using the
area model although
the textbook only
decomposes one
factor
Independent practice with transfer goals/LearnZillion Application Task:
“Zoo Donation: Using place value strategies to solve multi-digit multiplication problems” (Quick Code: S75ZE5R)
10
Time
Skills
Learning Goal
Jan. 9-10
2 days
Knowledge
Focus Questions
for Lessons
Teacher Notes
Review after Winter Break
Solve multi-step
word problems
involving addition,
subtraction and
multiplication
3 days
Lesson/Activity/
Resource
Represent word
problems using
equations with a
letter symbolizing
the unknown
quantity
Solve a one-step word
problem
Solve a multi-step word
problem
(teacher adds a second
step to the original onestep problem)
Solve a multi-step word
problem (see Challenge
Lesson)
Use a letter to represent
the unknown sum
Use a letter to represent
an unknown within the
context of the story
Inquiry Question
At the pet store, each small
dog weighs 35 pounds. If
there are 4 small dogs, how
many pounds do they weigh
together? If there are 6 large
dogs and each large dog
weighs 60 pounds, how much
do they weigh together? What
if I wanted to know the total
weight of small dogs and large
dogs—how could I do this?
Vocabulary
Operation
Variable
How can I use
equations to model
real-world problems?
How can you solve
problems with more
than one operation?
Students should
have experiences
in which the
unknown appears
in different places
in a story or in an
equation.
How can the unknown
be represented in an
equation?
See Lesson 5 for additional
practice
Challenge Lesson:
School Fundraising
Review and Administer Unit 4 Assessment
Review:
 How many different ways can you solve 94 x 64?
Jan. 17-19
 Create two multiplication sentences that could create a product between 500 and 600?
3 days
 Is the product of 29 x 34 over or under 900? Explain how you know.
 Think of an example in life when you might multiply two numbers? An example is, when might you multiply two two-digit numbers? Or a
three-digit number by a one-digit number?
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
11