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```					                                                                                        Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples

Waterbury Public Schools
Unit Instructional Tool
Unit 2

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                    Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples

Grade Four Standards for Mathematical Practice
The K-12 Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. This page gives examples of what the practice
standards look like at the specified grade level.
Standards                                                                                            Explanations and Examples
Students are expected to:               In fourth grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a
1. Make sense of problems and problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking
persevere in solving them.              by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their
Students are expected to:.              Fourth graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the
2. Reason abstractly and                problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with
quantitatively.                         fractions and decimals. Students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts.
Students are expected to:               In fourth grade, students may construct arguments using concrete referents such as objects, pictures, and drawings. They explain their thinking and make connections
3. Construct viable arguments between models and equations. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you
and critique the reasoning of           get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.
others.
Students are expected to:               Students experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, making a
4. Model with mathematics.              chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use
all of these representations as needed. Fourth graders should evaluate their results in the context of the situation and reflect on whether the results make sense.
Students are expected to:               Fourth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they
5. Use appropriate tools                may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative
strategically.                          size of units within a system and express measurements given in larger units in terms of smaller units.
Students are expected to:               As fourth graders develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning.
6. Attend to precision.                 They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot.
Students are expected to:               In fourth grade, students look closely to discover a pattern or structure. For instance, students use properties of operations to explain calculations (partial products model).
7. Look for and make use of             They relate representations of counting problems such as tree diagrams and arrays to the multiplication principal of counting. They generate number or shape patterns that
Students are expected to:               Students in fourth grade should notice repetitive actions in computation to make generalizations. Students use models to explain calculations and understand how
8. Look for and express                 algorithms work. They also use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent
regularity in repeated                  fractions.
reasoning.

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                                                 Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples

Unit Title                         Pacing                    Standards

2 weeks
+           4.OA.1
2. Factors and Multiples                                                          4.OA.4
1 week re-
4.OA.5
teaching/enrichment

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                              Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples

Pacing: 2 weeks (plus 1 week for re-teaching/enrichment) – 10 days + 5 days re-teach/enrichment = 15 days total

Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.

Practices in bold are to be emphasized in the unit.
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.
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.
Domain and Standards Overview
Operations and Algebraic Thinking
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                            Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples

Days                                                                                                                                                                                                         CMT
*Priority and Supporting                                                                                                                                      **Background
(Assumes                                                           **Performance Objectives                    **Suggested Instructional Strategies          Resources                                         Concept
CCSS                                                                                                                                                  Knowledge
70 min/day)                                                                                                                                                                                                      Match
4            4.OA.1. Interpret a                    Conceptual:                                      1. Concrete—Pictorial—Abstract                      Math       Conceptual:                          Models for
multiplication equation as a            Students will recognize that any two           Representation                                   Expressions    Students will possess an           Operation
comparison, e.g., interpret               factors and their product can be read                                                                           understanding of number
    Have students use manipulatives to
35 = 5 × 7 as a statement                 as a comparison (e.g., 8 is the same as                                                                         sense, such as decomposing
show what x groups of y looks like            Unit 1
that 35 is 5 times as many as             4 sets of 2 or 2 sets of 4; 8 is 4 times as                                                                     numbers (35 is the same as 10
7 and 7 times as many as 5.                                                                  (e.g., 3 groups of 5; 4 groups of 7).        Overview
many as 2, or 2 times as many as 4).                                                                            + 10 + 10 + 5 or 30 + 5 or
Represent verbal statements                                                                  Solve for the product.                       143L-M          …) and the reasonableness of
 Students will recognize that
of multiplicative                         multiplication represents groupings of            Build arrays for equations using the          L4–5           answers.
comparisons as                            numbers, and identify that the first               Commutative Property of                                    Students will be able to
multiplication equations.                 factor in the equation represents the                                                            Unit 3         identify place value, including
Multiplication (e.g., with 5 groups of 7
number of groups and the second                    and 7 groups of 5, 5 rows with 7 chairs        L6            writing numbers in expanded
factor represents how many within                                                                               form to recognize grouping by
in each row looks different from 7 rows
each group. Students should be able to                                                                          place value.
make a comparison that 5 groups of 7               with 5 chairs in each row, but there are
 Students will understand the
is the same as 7 groups of 5. Both                 35 chairs in both sets).                                     base ten number operations of
products are 35.                                  In math journals, have the students                          addition.
 Students will understand that this                  create a representation (e.g., array or                   Procedural:
representation illustrates the                     picture illustrating the grouping) of the                  Students can skip count to
Commutative Property of                            process that they completed using                            compute.
Multiplication.                                                                                               Students can solve problems
manipulatives. Write the two equations
that are equivalent to the given
 Students can interpret a multiplication                                                                        Students can fluently use basic
equation. Solve for the product.                   representations, and solve for the
multiplication facts 0-9.
Representational:                                     product.
 Students can represent and solve
multiplication equations through the
use of models (e.g., arrays),
illustrations, and writing.
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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                                                        Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples
Days                                                                                                                                                                                                   CMT
*Priority and Supporting                                                                                                                                 **Background
(Assumes                                                           **Performance Objectives              **Suggested Instructional Strategies          Resources                                         Concept
CCSS                                                                                                                                             Knowledge
70 min/day)                                                                                                                                                                                                Match
    Students should be able to illustrate 2. In cooperative learning groups,                              Representational:
that 5 groups of 7 is the same producthave each group create either an equation                        Students can model skip
as 7 groups of 5.                     or a model. Then have the students rotate                          counting, repeated addition
through each group and construct an                                and basic multiplication facts
using manipulatives,
equivalent form of what is observed.
drawings, algorithms, and
3. Fact families—reinforce the principles                          journaling.
of related facts using triangle flash cards or
Memory-type games.
8            4.OA.4. Find all factor                Conceptual:                                Using a hundreds chart, color-code the              Math      Conceptual:                           Basic Facts
pairs for a whole                       Students will understand factor pairs as multiples for each of the numbers 2 through      Expressions   Students will understand
number in the range 1–                    two whole numbers that multiply         9. Identify patterns for multiples of single                     whole numbers as the
100. Recognize that a                     together to get one product.                                                                             counting numbers plus 0.
digit numbers.                                   Fluency Plan
whole number is a                       Students will understand that prime                                                                     Students will understand the
multiple of each of its                                                                                                               L 2-15
numbers have exactly one factor pair. Use the Sieve of Eratosthenes to find the                          Commutative Property of
factors. Determine                                                                                                                                 Multiplication.
 Students will understand that            prime numbers from 1 to 100. Note: The              Unit 1
whether a given whole                                                                                                                           Procedural:
composite numbers have more than
number in the range 1–                                                            video is for teacher information only. Do             L8
one factor pair.                                                                                       Students have fluency in basic
100 is a multiple of a                                                            NOT simply show it to the students.
 Students will understand multiples as a                                                                   multiplication and division
given one-digit number.
product of two given whole numbers.                                                                      facts 0-9
Determine whether a
Procedural:                                Students should work through the 100 chart                    Representational:
given whole number in
the range 1–100 is prime                Students can list the multiples of the   using the sieve method described in the                        Students are familiar with
or composite.                             numbers 2 through 9 up to 100.          video. Take several days for this activity.                      number line structure
 Students can create a list or chart of                                                                  Students have the ability to
factor pairs of whole numbers 1-100.                                                                     build arrays for basic facts
 Students can identify, from a list or
chart, which whole numbers are prime
or composite.
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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                                                  Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples
Days                                                                                                                                                                                                   CMT
*Priority and Supporting                                                                                                                                 **Background
(Assumes                                                           **Performance Objectives               **Suggested Instructional Strategies          Resources                                        Concept
CCSS                                                                                                                                             Knowledge
70 min/day)                                                                                                                                                                                                Match
Representational:
 Students can use tools such as number
lines, hundreds charts, arrays, or cubes
to model relationships between factors
and multiples.

3            4.OA.5. Generate a number              Conceptual:                                   Play “Count around the Room.” Give the            Math        Conceptual:                        Patterns
or shape pattern that follows           Students will understand that number        counting rule (e.g., count by 7s) as well as   Expressions     Students will possess an
a given rule. Identify                    and shape patterns follow a given rule.    the starting place (e.g., begin at 3).                           understanding of number
apparent features of the                Students will understand that there are                                                                      sense.
Students continue with a given pattern,          Unit 8
pattern that were not                     sometimes features of the pattern that                                                                    Students will know whether a
explicit in the rule itself. For                                                     each student giving a number of the pattern    Lesson 1 & 2
are not stated in the rule.                as you move around the room.                                     number is odd or even.
example, given the rule “Add 3”        Procedural:                                                                                                  Students will understand the
and the starting number 1,
 Students can complete a given number                                                                         relationship between the four
generate terms in the resulting
or shape pattern                                                                                            basic operations.
sequence and observe that the
(e.g., 3, 6, 9, ___ , ___ , 18).                                                                          Students have familiarity with
terms appear to alternate
between odd and even numbers.           Students can determine the rule of a                                                                         number and shape patterns.
Explain informally why the                given pattern (e.g., 3 , 6, 9, 12, 15,                                                                   Procedural:
numbers will continue to                  18, … The rule is to skip count by 3                                                                      Students have concrete
alternate in this way.                    or multiples of 3).                                                                                         understanding of basic math
 Students can generate a number                                                                               facts.
pattern that follows a rule and state                                                                     Students can solve math
the rule.                                                                                                   operations and sequences that
 Students can generate a shape pattern                                                                        include unknowns.
that follows a rule and state the rule.                                                                   Students can use mental math
 Students can identify and state any                                                                          strategies.
alternate features of the pattern that
are not stated in the rule.

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                                                 Unit 2
Waterbury Public Schools Grade 4 Mathematics
Unit Instructional Support Tool
Unit 2- Factors and Multiples
Days                                                                                                                                                                                                  CMT
*Priority and Supporting                                                                                                                                **Background
(Assumes                                                           **Performance Objectives                **Suggested Instructional Strategies         Resources                                       Concept
CCSS                                                                                                                                            Knowledge
70 min/day)                                                                                                                                                                                               Match
Representational:                                                                                          Representational:
 Students can demonstrate the ability to                                                                   Students can use
complete a given number or shape pattern                                                                 manipulatives, drawings,
using manipulatives, skip counting,                                                                      algorithms, and/or journaling
pictures, journals, etc.                                                                                 to complete various patterns.

Concepts                                                                    Skills                                   Bloom’s Taxonomy Levels
What Students Need to Know                                          What Students Need To Be Able To Do
Factor Pairs                                                        FIND (for a whole number 1-100)                                                   2

Whole Number                                                        RECOGNIZE (that it’s a multiple of each of its factors)                           2
DETERMINE                                                                         2
 (if it’s a multiple of a given one-digit number)
 (whether prime or composite)
Multiplicative Equations                                            INTERPRET (as multiplicative comparisons)                                         3
Multiplicative Comparisons                                          REPRESENT (as verbal statements of multiplicative equations)                      3

GENERATE (number or shape that follows a given rule)
Pattern                                                             IDENTIFY (features not explicit in rule itself)                                   3
2

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Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.                                                                                                                               Unit 2