Download Throughout this cluster, students will develop their use of the 8

Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Maryland College and Career-Ready Standards
In this unit, students will build on their understanding of multiplication and division from grade 3. In the first part of the unit, students will
extend their understanding of the base-ten system by recognizing that the value of each place is ten times the value of the place to the
immediate right. Students will develop an understanding of multiples and factors, while applying their understanding of multiplication from
grade 3. This understanding lays a strong foundation for generalizing strategies learned in previous grades. Students will be able to
develop, discuss, and use efficient, accurate, and generalizable computational strategies involving multi-digit numbers. These concepts
and the terms “prime” and “composite” are new to grade 4, so they are introduced early in the year to give students ample time to
develop and apply this understanding.
Students will also focus on building conceptual understanding of multiplication of larger numbers and division with remainders. Students
should make connections between their conceptual understanding of place value developed in unit 1 with their understanding of
multiplication and division. Place value models and area of rectangles can provide visuals and context for further developing these
understandings. A strong understanding of the connection between multiplication and division with smaller numbers will help students
understand division of larger numbers and make sense of remainders. Students should be able to apply alternative algorithms that make
sense based on place value so that they can explain the reasonableness of their computation.
These strategies will lay a strong foundation based in understanding of number and operation for when they are ready for the
multiplication and division traditional algorithms in later grades.
Research
In the base-ten system, the value of each place is 10 times the value of the place to the immediate right. Because of this, multiplying by 10
yields a product in which each digit of the multiplicand is shifted one place to the left. Students need to develop an understanding of the
concepts of number theory such as prime numbers and composite numbers.
For students to develop an understanding of procedures for multiplying multi-digit numbers, area models and arrays are powerful
representations. These models allow students to see why and how we multiply each place in one number by each place in another
number. This is the big idea behind multi-digit multiplication, students should have opportunities to build and work with concrete
manipulatives to build this understanding. It illustrates the distributive property of multiplication over addition. The distributive property uses
known products to find unknown products. For example, 8 x 7 can be thought of as (8 x 5) + (8 x 2) = 40 + 16 = 56. An array shows the
partial products resulting from separately multiplying the digits in the different place value positions. (Focus in Grade 4: Teaching with
Curriculum Focal Points)
To explore division with remainders, students should use familiar division models to solve non-basic fact division equations, such as 44 ÷ 7 =
___. After understanding the meaning of remainders, through the use of manipulatives and pictorial representations, students should be
instructed to develop a sense of the “closest basic fact under” concept. For example, they might think 7 x 6 = 42 and 7 x 7 = 49. Since 49 is
greater than 44, students realize that when 44 objects are divided equally among 7 groups, the largest whole number of objects that can
be placed in each group is 6, and 2 objects are left over. Consequently, 7 x 6 = 42 is the “closest basic fact” of 7 “under” 44. (Focus in
Grade 4: Teaching with Curriculum Focal Points)
1
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
The chart below highlights the key understandings of Unit 2 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

Students will understand that:





Place value helps
determine relationships
between numbers
Estimation can be used
to determine the
reasonableness of an
answer.
Mathematical properties
show number
relationships and can be
used to compute flexibly.
Problems can be
represented and solved
accurately using a variety
of strategies.
Problems can be solved
using multiplication and
division.
How does the value of a digit change when it
is multiplied and divided by 10?

How can patterns, strategies, and formulas
help solve problems accurately?

When is estimation more useful than finding a
precise answer?

Why is place value understanding important in
multiplication and division?

What is the relationship between multiplication
and division?

How can we use mathematical properties
and/or rules to solve problems?

What are factors? How are factors of a
number determined?

What does it mean for a number to be
classified as either prime or composite?

How are remainders and divisors related?

What is the meaning of a remainder in a
division problem?

How do compatible numbers aid in dividing
whole numbers?

How can an understanding of representing
multi-digit multiplication help in solving area
problems?
Key Vocabulary
Commutative property
Composite
Distributive property
(of multiplication over
addition)
Dividend
Divisor
Equation
Even
Expression
Factor
Factor pairs
Multiple
Odd
Pattern
Place value
Prime
Product
Quotient
Remainder
Rule
Sequence
Term
Value
Variable
Background Reading
Focus in Grade 4:
Teaching with
Curriculum Focal Points
Teaching StudentCentered Mathematics
– Grades 3-5
Putting the Practices
into Action:
Implementing the
Common Core
Standards for
Mathematical Practice
K-8
2
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. The
mathematical practices in the shaded boxes should be emphasized during instruction of this unit due to how well they connect with the
content standards in this unit.
Standards for Mathematical
Practice
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
3. Construct viable arguments and
critique the reasoning of others
4. Model with mathematics
Connections to this Cluster
Solve problems in which the solution is not immediately evident.
To determine and articulate what the problem is asking:

Ask students to restate the problem in their own words.

Have students turn to a partner to state the problem.

Discuss familiar problems (When have we seen something like this before? What did we do?)
To self-monitor progress and change directions when necessary:

Have students talk or write about how they got “stuck” and then “unstuck” when solving a problem.

Think aloud to show students how to change course when needed.
To demonstrate perseverance in problem-solving and identify different ways to solve a problem:

Make a classroom list of possible strategies.

Acknowledge those who modify their thinking and persevere to get to the solution and have students show
and talk about how they solved problems.

Encourage students to show at least two ways to solve a problem.
Write an equation for a situation and be able to explain how the equation relates to the situation presented. Solve
the equation outside of the context of the problem, and then connect the solution back to the situation presented.
To make sense of quantities and their relationships in problem situations:

Represent a given multiplication and division situations problem with the equation using a variable for the
unknown.

Write a situation problem that matches a specific equation.
Explain why a number is prime or composite, based on knowledge of factors. Justify a numeric pattern based on
following a rule, and analyze and critique the reasoning of others.

Provide multiple opportunities for students to explain, explore, record, and try student-invented strategies.

Model how effective labeling communicates math reasoning.

Discuss student representations and solution methods.
-Ask students to restate and try peer methods for solving problems.
-Help students make mathematical connections between different representations/solution methods.
-Evaluate the efficiency of the strategies based on the probability of an error occurring.
Write equations for various problem situations and solve problems about the situations. Represent the unknown with
a variable.
 Represent problem situations in multiple ways including numbers, words (mathematical language), drawing
pictures, using objects, and creating equations
 Connect the different representations and explain the connections.
 Evaluate their results in the context of the situation and reflect on whether the results make sense.
3
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in
repeated reasoning
Use manipulatives, area models, arrays, number lines (for estimation), and drawings that represent mathematical
situations.

Consider the available tools (including estimation) when solving a mathematical problem and decide when
certain tools might be helpful.

Explain how tools can assist them in seeing patterns and relationships with numbers.
Create accurate drawings and representations of mathematical situations.
Use specific math vocabulary to communicate mathematical ideas.
Compute accurately.

Develop and display anchor charts with precise math vocabulary.

Orally rephrase student explanations using appropriate vocabulary.
Look closely to discover a pattern or structure and apply the properties of operations to solve problems.

Use properties of operations to perform and explain calculations (partial products model) with multi-digit
numbers; provide opportunities in which students explore mathematical properties; and ask students to
construct arguments to prove mathematical properties.

Complete and describe number patterns with whole numbers.

Explore patterns in different ways using hundred charts (prime and composite numbers) and multiplication
charts.

Use ratio tables to show the relationship between two pieces of data and to explore patterns and functions.
such as in the following problem:
Zoe was painting flowers on the classroom mural. Every flower had 6 petals. How many petals were on the mural
after 8 flowers had been painted?
Notice repetitive actions in computation and make generalizations about rules and “short-cuts” to get to answers
more quickly.

Use models to explain calculations and understand how algorithms work.

Use models to examine patterns and generate alternative algorithms.
4
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Generalize place value understanding for multi-digit whole
numbers.
Maryland College and
Career-Ready Standards
4.NBT.A.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.
For example,
recognize that 700
divided by 70 equals
10 by applying
concepts of place
value and division.
This standard strengthens student’s understanding of place value related to multiplying and
dividing by multiples of 10. Students reason about the magnitude of digits in a number.
In the base-ten system, the value of each place is 10 times the value of the place to the
immediate right. For example, in the number 2,443 the digit 4 in the hundreds place is worth
ten times the digit 4 in the tens place. (400 is ten times 40, or conversely, 40 is ten times less
than 400).
In third grade, students explored multiplying a single-digit number by a multiple of ten
(3.NBT.3). For example, when multiplying 8 x 10, the product of 8 x 1 is shifted one place to
the left indicating 8 tens or 80. In fourth grade, students build on this understanding when
interpreting and comparing the value of digits in multi-digit numbers.
4.NBT.A.1a
How is the digit 7
in the number 3,784
different from and
similar to the digit 7
in the number
7,843?
4.NBT.A.1b
How are the
numbers 24,019 and
20,914 alike? How
are they different?
Grade 4 expectations
in this domain are
limited to whole
numbers less than or
equal to 1,000,000.
SMP
7. Look for and make use of
structure.
Formative
Assessments
Instructional Strategies and Resource Support
4.NBT.A.1c
(Progressions for the Common Core State Standards in Mathematics (draft), The
Common Core Standards Writing Team, April 2012, p. 12)
The amount of jelly
beans in each jar is
a five -digit number
with the following
digits:
2
9
7
5
3
5
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Generalize place value understanding for multi-digit whole numbers.
Maryland College and CareerReady Standards
4.NBT.A.1
Recognize that in a multidigit whole number, a
digit in one place
represents ten times what
it represents in the place
to its right.
For example, recognize
that 700 divided by 70
equals 10 by applying
concepts of place value
and division.
Grade 4 expectations in
this domain are limited to
whole numbers less than
or equal to 1,000,000.
SMP
7. Look for and make use of
structure.
Instructional Strategies and Resource Support
Students need varied opportunities to build multi-digit
numbers in order to solidify understanding of the relationship
between places in greater numbers. Base-ten blocks, Digiblocks, and place value disks are useful tools to help build
numbers and compare values of digits.
When comparing digits in 2 different numbers, students
should be able to reason about the magnitude of the digits
in relation to multiples of ten. For example, when comparing
5,368 and 2,643, students should recognize that the “6” in
5,368 is worth 10 times less than the “6” in 2,643. Additionally,
the “3” in 5,368 is 100 times the value of the “3” in 2,643.
Formative Assessments
4.NBT.A.1d
Lauren worked at Slide and
Splash Water Park over the
summer. During her first week,
she recorded the number of
people who visited the park
daily.
Complete her log using the
clues that follow.
Day of the
Week
Number
of Visitors
Thursday
Friday
Saturday
Sunday
2,945
4,163
Teaching Student-Centered Mathematics – Grades 3-5
“Numbers Beyond 1,000” - pp. 47-51
Activity 2.8 “What Comes Next?” (p.48)
Lexi has 25 ten-dollar bills; Abbey has 25 hundred-dollar bills. Do they
have the same amount of money? How many 10-dollar bills are equal to
15 hundred-dollar bills? Show your thinking. What if there was a
thousand dollar bill, how many of those would be equal to 25 hundreddollar bills?
6
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Maryland College and CareerReady Standards
Gain familiarity with factors and multiples.
4.OA.4 Find all factor pairs
for a whole number in the
range 1-100. Recognize
that a whole number is a
multiple of each of its
factors. Determine whether
a given whole number in
the range 1-100 is a
multiple of a given onedigit number. Determine
whether a given whole
number in the range 1-100
is prime or composite.
NOTE:
The focus of this standard is
not necessarily to become
fluent in finding all factor
pairs, but to use student’s
understanding of the concept
and language to discuss the
structure of multiples and
factors.
SMP
3. Construct viable arguments
and critique the reasoning of
others.
7. Look for and make use of
structure.
Instructional Strategies and Resource Support
This standard requires students to demonstrate understanding of factors
and multiples of whole numbers. A factor is a number that is multiplied by
another number to create a product. Three and six are factors of 18. A
multiple is the product of a whole number and any other whole number.
Twenty is a multiple of four, because 4 x 5 = 20.
Students should understand the process of finding factor pairs.
Examples:
 Use visual models, such as an array with tiles or grid paper or a
rectangular area model
 Factor tree
 Factor “rainbow”
Formative Assessments
4.OA.B.4a
4 is a factor of two different
numbers. What else might be
true about both of the numbers?
4.OA.B.4b
a.
b.
Factors of 18 – 1, 2, 3, 6, 9, 18
c.
Multiples can be thought of as the result of skip-counting by each of the
factors. When skip-counting, students should be able to identify the
number of factors counted, e.g. 4, 8, 12, 16, 20, 24 – there are six 4’s in 24.
Example:
4.OA.B.4c
Use counters to determine if 51
is prime or composite.
Multiples – 1, 2, 3, 4, … 18
2, 4, 6, 8, 10, 12, 14, 16, 18
3, 6, 9, 12, 15, 18
6, 12, 18
9, 18
18
4.OA.B.4d
Students should be able to determine when they have made all possible
arrays and listed all factor pairs for a number between 1 and 100.
Teaching Student-Centered Mathematics – Grades 3-5
Make a list of the first ten
multiples of 4.
Which of the numbers in
the list are multiples of 8?
Describe the pattern you
see where the multiples of
8 appear in the list.
Which numbers in the list
are multiples of 3?
pp. 63-64
Evan is getting ready to make
s’mores. He bought 2 bags of
marshmallows.
The bags each contain 36
marshmallows. Evan wants
everyone to have the same
amount of marshmallows and
not have any left over.
7
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Maryland College and CareerReady Standards
Gain familiarity with factors and multiples.
4.OA.4 Find all factor pairs
for a whole number in the
range 1-100. Recognize
that a whole number is a
multiple of each of its
factors. Determine whether
a given whole number in
the range 1-100 is a
multiple of a given onedigit number. Determine
whether a given whole
number in the range 1-100
is prime or composite.
NOTE:
The focus of this standard is
not necessarily to become
fluent in finding all factor
pairs, but to use student’s
understanding of the concept
and language to discuss the
structure of multiples and
factors.
Instructional Strategies and Resource Support
Formative Assessments
Through exploration with multiples of a number between 1 and 100, the
following generalizations should emerge, as well as others:
 All even numbers are multiples of 2.
 All numbers that can be halved twice, with a whole number result,
are multiples of 4.
 All numbers with a 0 or 5 in the ones place are multiples of 5.
 All numbers with a 0 in the ones place are multiples of 10.
 All numbers that are multiples of both 2 and 3 are also multiples of
6.
Additionally, this standard refers to prime and composite numbers. Prime
numbers have exactly two factors, the number one and itself. For
example, the number 13 has the factors of 1 and 13. Composite numbers
have more than two factors. For example, 9 has the factors 1, 3, and 9.
A common misconception is that the number 1 is prime, when in fact; it is
neither prime nor composite. Another common misconception is that all
prime numbers are odd numbers. However, the number 2 (an even
number) has only 2 factors: 1 and 2.
To explore prime and composite numbers, build rectangular arrays and
determine which numbers have more than two arrays.
Examples:
Rectangular Arrays for 8
Rectangular Arrays for 7
SMP
3. Construct viable arguments
and critique the reasoning of
others.
7. Look for and make use of
structure.
1 x 7 OR 7 x 1
Prime: Exactly 2 factors:
(1 and 7)
1 x 8 OR 8 x 1
2 x 4 OR 4 x 2
Composite: More than 2
factors: (1, 2, 4, and 8)
8
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Generate and analyze patterns.
Maryland College and CareerReady Standards
4.OA.5 Generate a number or
shape pattern that follows a
given rule. Identify apparent
features of the pattern that were
not explicit in the rule itself. For
example, given the rule “add 3”
and the starting number 1,
generate terms in the resulting
sequence and observe that the
terms appear to alternate
between odd and even
numbers. Explain informally why
the numbers will continue to
alternate in this way.
NOTE:
Although there are shape
patterns in arrays, the focus of
this unit is number patterns.
4.OA.5 will appear again in Unit
7 – Geometry and Patterns,
where the focus will be on
identifying shape patterns.
SMP
3. Construct viable arguments
and critique the reasoning of
others.
7. Look for and make use of
structure.
Instructional Strategies and Resource Support
Formative Assessments
Teaching Student-Centered Mathematics – Grades 3-5
pp. 299-303
Students need many opportunities creating and extending number
patterns. Numerical patterns allow students to reinforce basic facts,
develop fluency with operations, and to be more analytical about
number sequences and relationships.
What’s Next and Why?
(TSCM, Grades 3-5, p. 299)
Show students five or six numbers from a number pattern. Then, have
them extend the pattern for several more terms and explain the rule for
generating the pattern.
Examples:
1, 2, 4, 8, 16 … double the previous number
2, 5, 11, 23 … double the previous number and add 1
1, 2, 4, 7, 11 … successively add 1, then 2, then 3, and so on
0, 1, 5, 14, 30 … add the next square number
2, 2, 4, 6, 10, 16 … add the preceding two numbers
Make up your own pattern or challenge students to create their own
number pattern rule.
Have students skip count starting at a number other than zero. Have
them make conjectures and then determine why they are true.
Example:
3, 8, 13, 18, 23, 28
When skip counting by 5 starting at 3, the ones place alternates
between 3 and 8. This happens because 5 is half of 10, so a given term is
ten more than 2 terms before it. When you add 10 to a number, the
ones place stays the same.
4.OA.C.5a
__
__
16
__
__
Complete a number pattern above.
What is the rule for your pattern?
Identify at least two features of
the pattern that are not part of the
rule.
4.OA.C.5b
Skip-count by 5’s beginning
with the number 6.
____ , ____ , ____ , ____ , ____ , ____
, ____ , ____
What features do you notice in
the pattern?
Students should be able to identify common terms of two patterns and
identify features of those common terms. For example, if skip counting
by 4 and 8 starting at zero, all multiples of 8 are also multiples of 4.
9
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use the four operations with whole numbers to solve problems.
Maryland College and CareerReady Standards
4.OA.3 Solve multistep word
problems posed with whole
numbers and having wholenumber answers using the four
operations, including problems
in which remainders must be
interpreted. Represent these
problems using equations with a
letter standing for the unknown
quantity. Assess the
reasonableness of answers using
mental computation and
estimation strategies including
rounding.
NOTE:
This is the first time students are
expected to interpret
remainders based upon the
context. All four operations will
be addressed in Unit 5 – solving
measurement problems, and the
standard will be finalized at the
end of the year.
SMP
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
6. Attend to precision.
7. Look for and make use of
structure.
Instructional Strategies and Resource Support
This standard has multiple areas of focus:
A. Solving multistep problems using a variety of strategies
B. Using compatible numbers (numbers that sum to 10 or 100) or
rounding in order to estimate a solution that is reasonable
C. Interpreting remainders in division problems
A. Students often believe that there is only one way to solve a problem
and that a problem only has one solution. They need many
opportunities to explore problems that can be solved using a variety of
strategies and do not have an obvious answer. Encourage students to
share problem-solving approaches in order to develop communication
of reasoning and to assess efficiency of strategies.
B. Problems should be structured so that all acceptable estimation
strategies will arrive at a reasonable answer. Students need many
opportunities solving multistep story problems. One aspect of fluency in
whole-number multiplication is the ability to use estimation to
approximate products and to determine the reasonableness of exact
results. Encourage students to use rounding, patterns in powers of 10,
and basic multiplication facts to accomplish this.
Examples:
1. To estimate 427 x 8 , round 427 to 400, then multiply 400 x 8.
2. Extend facts that involve the patterns with zeros, e.g. 6 x 7, 60 x 7,
60 x 70.
Focus in Grade 4: Teaching with Curriculum Focal Points
Estimation and fluency in multiplication -- p. 20
SMP: Present solutions to multi-step problems in the form of valid chains of
reasoning, using symbols such as equals signs appropriately (for example, rubrics
award less than full credit for the presence of nonsense statements such as
12x2=24÷4=6 even if the final answer is correct), or identify or describe errors in
solutions to multi-step problems and present corrected solutions.
Formative Assessments
4.OA.A.3a
There were 5 baseball players
taking batting practice. They
hit 85 balls all together. If
each player hit the same
number of baseballs, how
many baseballs did 1 player
hit? ______
4.OA.A.3b
There are five 4th grade
classes in the gym for a
science program about the
Chesapeake Bay. Each class
has 26 students. During the
program, the students need to
be in groups of 6 to complete
Bay-themed activities.
4.OA.A.3c
Lauren opened her dog-walking
business over the summer and
charged $7 for an hour walk.
She walked a total of 57 dogs
during June, July, and August.
If Lauren walked the same
amount of dogs each month,
how much money did she earn
in July?
(PARCConline.org – evidence tables)
10
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use the four operations with whole numbers to solve problems.
Maryland College and Career-Ready
Standards
4.OA.3 Solve multistep word
problems posed with whole
numbers and having wholenumber answers using the four
operations, including problems in
which remainders must be
interpreted. Represent these
problems using equations with a
letter standing for the unknown
quantity. Assess the
reasonableness of answers using
mental computation and estimation
strategies including rounding.
NOTE:
This is the first time students are
expected to interpret remainders
based upon the context. All four
operations will be addressed in Unit
5 – solving measurement problems,
and the standard will be finalized at
the end of the year.
Instructional Strategies and Resource Support
C. More often than not, division does not result in a simple whole
number. In some division word problems, a leftover quantity … or,
remainder … needs to be interpreted. Remainders may have
several different effects on answers.
Focus in Grade 4: Teaching with Curriculum Focal Points
Interpreting remainders – pp. 22-23
It is useful to include, in instruction, story problems with remainders
and explore the different ways that students handle these in context.
The following examples illustrate various interpretations of remainders:

You have 30 pieces of candy to share fairly with 7 children.
How many pieces of candy will each child receive?
Left over amount: 4 pieces of candy and 2 left over

The rope is 25 feet long. How many 7-foot jump ropes can be
made?
Discarded amount: 3 jump ropes and 4 feet left over

A ferry can hold 8 cars. How many trips will it have to make
to carry 25 cars across the river?
Pushed up to next whole number: 4 trips

Six children are planning to share a bag of 50 pieces of
bubble gum. About how many pieces will each child get?
Rounded result: About 8 pieces of bubble gum for each child
SMP
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
6. Attend to precision.
7. Look for and make use of structure.
Formative Assessments
Students will need to write equations with a letter standing for the
unknown quantity to represent these problems. Additionally, they
need to be able to recognize that more than one equation may
accurately represent a problem.
11
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use place value understanding and properties of operations to perform multidigit arithmetic.
Maryland College and CareerReady Standards
4.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.
Instructional Strategies and Resource Support
Formative
Assessments
Focus in Grade 4: Teaching with Curriculum Focal Points pp. 13-18
4.NBT.B.5a
Teaching Student-Centered Mathematics – Grades 3-5
Expanded Lesson: Area Model for Multiplication, pp. 129-130
Strategies for Multiplication pp. 113-118
Students should use a variety of strategies to multiply numbers. Developing flexibility in
breaking numbers apart allows students to have a stronger understanding of the
importance of place value and the distributive property in multiplying multi-digit
numbers. Children may use area models, base-10 blocks, compensation strategies,
etc. to multiply whole numbers, then use words and diagrams to explain their thinking.
Using a variety of strategies enables students to develop fluency with multiplication, as
well as transferring their understanding to division. The standard algorithm for
multiplication is an expectation in the 5th grade.
The average Maryland blue
crab measures 8 inches
long. If you placed 153
crabs end to end in a line,
how long would the line be?
4.NBT.B.5b
Solve the following problem in
2 different ways.
63 x 47
NOTE:
Grade 4 expectations in this
domain are limited to whole
numbers less than or equal
to 1,000,000.
Students build on their understanding of multiplication from 3 rd grade. They move
beyond multiplying a one-digit number by a multiple of 10 to multiplication of a onedigit number by a whole number of up to four digits and to multiply two two-digit
numbers. Students benefit from building arrays/area models using base-ten blocks to
multiply greater numbers. Examples:
SMP
6 rows
4 x 632
6 x 10 = 60
20 rows
3. Construct viable arguments
and critique the reasoning of
others.
6. Attend to precision.
7. Look for and make use of
structure.
4.NBT.B.5c
Find the product of each.
13 per row
10 per row
How are both of your strategies
alike?
20 x 13 = 260
Can be broken apart
in many ways
including:
10 x 13 = 130
10 x 13 = 130
130 + 130 = 260
SMP
Students use strategies and base explanations on properties of operations,
the relationship between multiplication/division, and place value.
(PARCConline.org – evidence tables)
4 x 623
What do you notice about the
problems? What conclusions
can you make?
4.NBT.B.5d
At Camden Yards, the seats in
section 68 are located by third
base. There are 27 rows in this
section. If each row has 14
seats, how many seats are
there? Solve this problem in
more than 1 way.
12
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use place value understanding and properties of operations to perform
multi-digit arithmetic.
Maryland College and CareerReady Standards
4.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.
Instructional Strategies and Resource Support
Formative
Assessments
The distributive property over addition allows numbers to be decomposed into
base-10 units, then have units be computed and combined. By decomposing the
factors into like base-10 units and applying the distributive property, multi-digit
computations are simplified to single-digit multiplications and products of numbers
with multiples of 10, 100, and 1,000.
Students can connect
using an area model to
thinking developed
previously with base-10
block arrays. This problem
can be seen as 8 rows of
549 decomposed into
8 x 500 + 8 x 40 + 8 x 9.
Students will apply
multiplication patterns
when multiplying by
multiples of 10.
NOTE:
Grade 4 expectations in
this domain are limited to
whole numbers less than
or equal to 1,000,000.
SMP
3. Construct viable
arguments and critique the
reasoning of others.
6. Attend to precision.
7. Look for and make use of
structure.
Example:
36 x 94 = (30 + 6) x (90 + 4)
= (30 + 6) x 90 + (30 + 6) x 4
= 30x90 + 6x90 + 30x4 + 6x4
= 2,700 + 540 + 120 + 24
= 3,384
13
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use place value understanding and properties of operations to perform
multi-digit arithmetic.
Maryland College and CareerReady Standards
4.NBT.6 Find wholenumber quotients and
remainders with up to fourdigit dividends and onedigit divisors, using
strategies based on place
value, the properties of
operations, and/or the
relationship between
multiplication and division.
Illustrate and explain the
calculation by using
equations, rectangular
arrays, and/or area
models.
NOTE:
Grade 4 expectations in
this domain are limited to
whole numbers less than or
equal to 1,000,000.
SMP
3. Construct viable arguments
and critique the reasoning of
others.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity
in repeated reasoning
Instructional Strategies and Resource Support
Focus in Grade 4: Teaching with Curriculum Focal Points
pp. 22-23
Students begin to build on their understanding of basic division facts, using
problems that do not “come out even.” This will help students develop an
understanding of remainders. Students can use familiar models including
base-10 blocks and pictures to solve non-basic fact division equations.
Students strengthen understanding of how to use multiplication to solve
division problems. They can think of division as an “unknown factor” problem.
For example: 35 ÷ 8 = m
Think 8 x m = 35.
Students move towards thinking of the closest multiple of 8 that is under 35.
(8 x 4 = 32) and then determine how much more is needed to get to 35.
(8x4) + 3 = 35
Therefore, 35 ÷ 8 = 4 r 3.
After students have had ample experience with “near facts” (quotient and
divisor are both less than 10, but there is a remainder), then move to division
problems with quotients more than 9. The processes then evolve into
invented strategies for division. Students apply their understanding of
multiplication patterns with multiples of 10.
For example: 47 ÷ 3
Possible solution: I know that 3 x 10 = 30. 47 – 30 = 17. I know that 3 x 5 = 15. If
I take 15 from 17, I’m left with 2.
I have 10 groups of 3 and 5 groups of 3 OR (10 + 5) x 3 with 2 left over. My
answer is 15 r 2.
Teaching Student-Centered Mathematics pp. 121-122
Formative
Assessments
4.NBT.B.6a
Kim is making sandwich
platters for a party. The table
below shows the total number
of each type of sandwich she
made. She has small, medium,
and large trays to hold the
sandwiches. What
combination of all three
different-size trays can be used
to hold all of the sandwiches
that Kim made?
4.NBT.B.6b
At the bakery, Michael is
decorating cupcakes with
mini-pumpkin candies. He has
46 pumpkins to place on 6
cupcakes. As he decorates,
Michael places one candy at a
time on each cupcake and
continues in this way until all
candies are used.
4.NBT.B.6c
Solve 92 ÷ 4 in 2 different
ways.
Which way was most efficient
for you? Why?
SMP
Students use strategies and base explanations on properties of
operations, the relationship between multiplication/division, and
place value.
(PARCConline.org – evidence tables)
14
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Use place value understanding and properties of operations to perform
multi-digit arithmetic.
Maryland College and CareerReady Standards
4.NBT.6 Find wholenumber quotients and
remainders with up to fourdigit dividends and onedigit divisors, using
strategies based on place
value, the properties of
operations, and/or the
relationship between
multiplication and division.
Illustrate and explain the
calculation by using
equations, rectangular
arrays, and/or area
models.
NOTE:
Grade 4 expectations in this
domain are limited to whole
numbers less than or equal to
1,000,000.
SMP
3. Construct viable arguments
and critique the reasoning of
others.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity
in repeated reasoning
Instructional Strategies and Resource Support
Formative Assessments
Computing quotients of multi-digit numbers and one-digit divisors relies on
the same understandings as for multiplication. For example, recognizing that
56 ÷7 is related to 560 ÷7 and 5600 ÷7. Students should also be able to
reason that 5600 ÷ 7 means partitioning 56 hundred into 7 equal groups, so
there are 8 hundreds in each group.
Another general method for multi-digit division computation is decomposing
the dividend into like base-10 units and finding the quotient unit-by unit. This
requires an understanding of the distributive property.
Progressions for the Common Core State Standards in Mathematics (draft, 2012)
Students should remain flexible in their understanding of breaking numbers
apart when dividing. Not all approaches will be the same, but solid place
value understanding should be applied. Examples of solutions for 537 ÷ 8.
There are 60 8’s in 537.
537 – 480 = 57
There are 7 8’s in 57.
57 – 56 = 1
60 + 7 = 67 8’s with 1 left
8 ) 537
- 400
137
80
57
56
1
8 x 50
8 x 10
67 8’s
8x 7
67 r 1
15
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
Maryland College and CareerReady Standards
4.MD.3 Apply the area
and perimeter formulas
for rectangles in realworld and mathematical
problems. For example,
find the width of a
rectangular room given the
area of the flooring and the
length, by viewing the area
formula as a multiplication
equation with an unknown
factor.
NOTE:
This standard provides the
context of area and
perimeter of rectangles to
use for problem-solving.
Students are first introduced
to formulas in this unit and
make sense of the formulas
using their prior work with
area and perimeter.
SMP
1. Make sense of problems
and persevere in solving
them
2. Reason abstractly and
quantitatively
8. Look for and express
regularity in repeated
reasoning
Instructional Strategies and Resource Support
Students consider perimeter and area of rectangles by building on
previous 3rd grade understanding. Fourth graders, through
constructing arrays and applying features of rectangles, reason to
develop and understand the formula for the area of a rectangle as
A = l x w.
The formula is based on the understanding that a rectangle, whose
sides have a width of “w” units and a length of “l” units, can be
partitioned into w rows of unit squares with l squares in each row.
The product l x w gives the number of unit squares in the rectangle.
Students use the relationship between multiplication and division to
solve problems in which the length of a missing side needs to be
found. Given the area and one side, students can divide to find the
missing side.
Teachers should emphasize that it is not important to designate a
specific number as the width in a rectangle because of the
commutative property of multiplication.
Examples of Area Problems:
*Jacob’s desk is 32 inches by 18 inches. Ella’s desk is 24 inches x 24
inches. Whose desk is larger? What could we do to give both
students the same amount of work space on their desks?
*Determine how many square inches are in one square foot.
*The area of a rectangle is 36 square centimeters. The length is 5
centimeters longer than the width. What is the perimeter of the
rectangle?
Additional resources – Measuring Area:
Formative
Assessments
4.MD.A.3a
Sarah is painting a mural on
her bedroom wall. She gets to
choose which wall she would
like to paint. The first wall is
17 feet by 8 feet. The second
wall is 14 feet by 11 feet.
Which wall has more space for
the mural? Show your work.
4.MD.A.3b
The area of Drew’s rectangular
kitchen is 120 square feet. The
width of the kitchen is 8 feet.
What is its length? Include a
labeled diagram of Drew’s
kitchen.
4.MD.A.3c
In Mia’s garden, she plants
carrots in one section and all of
the green vegetables in another
section. The total area of Mia’s
garden is 120 square feet. Use
the diagram below to help you
find the area of the carrot
section.
green vegetables
carrots
Teaching Student-Centered Mathematics Grades 3-5 pp. 261-264
Expanded lesson: Fixed Areas (Based on Activity 9.7) pp. 288-289
Focus in Grade 4: Teaching with Curriculum Focal Points pp. 60-67
16
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 2: Multiplication and Division - 28 Days
In Quarter 1, students will be expected to solve all equal groups situations and arrays/areas situations.
Students will be expected to represent the highlighted problem situations with equations on assessments.
17