Navigating through Algebra in Grades 3-5
Lesson alignment with
Common Core State Standards for Mathematics
These lessons may need to be adapted or modified to align with the CCSS.
NCTM National Council of Teachers of Mathematics
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Navigating through Algebra in Grades 3-5
Lesson alignment with Common Core State Standards for Mathematics
Chapter
Lesson Title,
Page Number and Goal
Student Behaviors
(Mathematical Practices) to Look For…
CCSS Cluster/Standard
Third Grade
Chapter 1: Patterns
Hundred-Board Wonders
page 9
Goals: Given a rule, students
will explore number patterns
using a hundred board.
Solve problems involving the four operations, and identify
and explain patterns in arithmetic.
3.OA.9 Identify arithmetic patterns (including patterns in the
addition table or multiplication table), and explain them using
properties of operations.
For example, observe that 4 times a number is always even,
and explain why 4 times a number can be decomposed into
two equal addends.
This tasked could be modified for third by making the patterns
for the students then have them identify the pattern.
Chapter 2: Variables and
Equations
The Variable Machine
page 39
Solve problems involving the four operations, and identify
and explain patterns in arithmetic.
3.OA.8 Solve two-step word problems using the four
operations. 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.3
Goals: The student will*explore the idea of variable as a
symbol that can stand for any
3 This standard is limited to problems posed with whole numbers and having
number of a set of numbers;
*substitute numbers for variables whole-number answers; students should know how to perform operations in
the conventional order when there are no parentheses to specify a particular
(letters) to discover unknown
order.
values.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
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.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Chapter 2: Variables and
Equations
I Spy Patterns
page 48
Goals: The student will*partition the given array into
different parts;
*translate visual patterns into
numerical expressions;
Explore how equivalent
numerical expressions represent
the commutative and associative
properties of operations.
Understand properties of multiplication and the
relationship between multiplication and division.
3.OA.5 Apply properties of operations as strategies to
multiply and divide.2
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also
known. (Commutative property of multiplication.) 3 × 5 × 2
can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 =
10, then 3 × 10 = 30. (Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7
as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
6. Understand division as an unknown-factor problem. For
example, find 32 ÷ 8 by finding the number that makes 32
when multiplied by 8.
2
Chapter 3: Functions
That’s Odd!
page 61
Goals: The student will*observe various patterns in an
array;
*represent observed visual
patterns as numerical patterns;
*represent a numerical pattern as
a functional relationship.
Mathematical Practice focus is in bold:
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.
Students need not use formal terms for these properties.
Solve problems involving the four operations, and identify
and explain patterns in arithmetic.
3.OA.9 Identify arithmetic patterns (including patterns in the
addition table or multiplication table), and explain them using
properties of operations.
For example, observe that 4 times a number is always even,
and explain why 4 times a number can be decomposed into
two equal addends.
(This lesson could be used in 3rd or 4th depending on focus.)
Mathematical Practice focus is in bold:
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.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Fourth Grade
Chapter 1: Patterns
Hundred-Board Wonders
page 9
Goals: Given a rule, students
will explore number patterns
using a hundred board.
Chapter 1: Patterns
Watch Them Grow
page 12
Goals: Students will*observe patterns and
relationships;
*make conjectures about
patterns and test those
conjectures;
*discuss verbalize, generalize,
and represent patterns and
relationships.
Chapter 1: Patterns
Calculator Patterns
page 15
Goals: Students will*explore patterns using a
Generate and analyze patterns.
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.
Generate and analyze patterns.
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.
Generate and analyze patterns.
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
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
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.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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calculator;
*translate patterns into
numerical patterns.
the terms appear to alternate between odd and even numbers.
Explain informally why the numbers will continue to alternate
in this way.
Chapter 2: Variables and
Equations
Catch of the Day!
page 41
Goals: The student will*work with variables as they
determine the number of each
kind of fish caught;
*record algebraically the
statements of the results of their
“catch.”
Use the four operations with whole numbers to solve
problems.
4.OA.3 Solve multistep word problems posed with whole
numbers and having whole-number 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.
Chapter 2: Variables and
Equations
Building Houses
page 51
Use the four operations with whole numbers to solve
problems.
4.OA.3 Solve multistep word problems posed with whole
numbers and having whole-number 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.
Goals: The student will*verbalize the numerical
relationship in each problem;
*translate each relation into an
algebraic equation.
Chapter 3: Functions
Triangle-Rule Machine
page 58
Generate and analyze patterns.
4.OA.5 Generate a number or shape pattern that follows a
given rule. Identify apparent features of the pattern that were
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.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
1. Make sense of problems and persevere in
solving them.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Goals: The student will*investigate the perimeter of
figures composed of equilateral
triangles arranged in a row;
*describe (verbally and
symbolically) the “rule,” or
function, that will produce the
perimeter for any given
arrangement of triangles;
*make a connection to the idea
of function when describing the
rule for a pattern with numbers.
Chapter 3: Functions
That’s Odd!
page 61
Goals: The student will*observe various patterns in an
array;
*represent observed visual
patterns as numerical patterns;
*represent a numerical pattern as
a functional relationship.
Chapter 3: Functions
Squares Cubes
page 64
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.
Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
4.MD.3 Apply the area and perimeter formulas for rectangles
in real world 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.
This lesson uses triangles not rectangles as noted in
standard so lesson could be used as an extension of
standard or modify task and use rectangles.
Generate and analyze patterns.
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.
(This lesson could be used in 3rd or 4th depending on focus.)
Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
4.MD.3 Apply the area and perimeter formulas for rectangles
in real world and mathematical problems.
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.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Goals: The student will*investigate the functional
relationship between the length
of the sides of a square and its
perimeter and area;
*describe the functional
relationship between the length
of the sides of a square and(a) their perimeters and the
sides increase in length,
(b) their areas as the sides
increase in length;
*describe the functional
relationship between the length
of the sides of cubes and their
volumes as the sides increase in
length. (5th Grade)
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.
(This lesson could be used in 4th or 5th depending on focus.)
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.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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Fifth Grade
Chapter 1: Patterns
Tiling a Patio
page 18
Goals: Students will –
*observe patterns and
relationship;
*make conjectures about
patterns and test those
conjectures;
*discuss, verbalize, and
represent patterns and
relationships.
Chapter 1: Patterns
The Ups and Downs of Patterns
page 27
Goals: The students will identify
and analyze situations with
constant or varying rates of
change.
Chapter 1: Patterns
Graphic Stories
page 31
Goals: The students will –
Analyze patterns and relationships.
5.OA.3 Generate two numerical patterns using two given
rules. Identify apparent relationships between corresponding
terms. Form ordered pairs consisting of corresponding terms
from the two patterns, and graph the ordered pairs on a
coordinate plane.
For example, given the rule “Add 3” and the starting number
0, and given the rule “Add 6” and the starting number 0,
generate terms in the resulting sequences, and observe that
the terms in one sequence are twice the corresponding terms
in the other sequence. Explain informally why this is so.
Mathematical Practice focus is in bold:
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.
5.OA.3 Generate two numerical patterns using two given
rules. Identify apparent relationships between corresponding
terms. Form ordered pairs consisting of corresponding terms
from the two patterns, and graph the ordered pairs on a
coordinate plane.
For example, given the rule “Add 3” and the starting number
0, and given the rule “Add 6” and the starting number 0,
generate terms in the resulting sequences, and observe that
the terms in one sequence are twice the corresponding terms
in the other sequence. Explain informally why this is so.
Mathematical Practice focus is in bold:
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.
5.OA.3 Generate two numerical patterns using two given
rules. Identify apparent relationships between corresponding
terms. Form ordered pairs consisting of corresponding terms
from the two patterns, and graph the ordered pairs on a
coordinate plane.
Mathematical Practice focus is in bold:
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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*explore relationship between
variables;
*interpret relationship expressed
in a line graph.
For example, given the rule “Add 3” and the starting number
0, and given the rule “Add 6” and the starting number 0,
generate terms in the resulting sequences, and observe that
the terms in one sequence are twice the corresponding terms
in the other sequence. Explain informally why this is so.
Chapter 1: Patterns
What’s the Best Deal?
page 33
5.OA.3 Generate two numerical patterns using two given
rules. Identify apparent relationships between corresponding
terms. Form ordered pairs consisting of corresponding terms
from the two patterns, and graph the ordered pairs on a
coordinate plane.
For example, given the rule “Add 3” and the starting number
0, and given the rule “Add 6” and the starting number 0,
generate terms in the resulting sequences, and observe that
the terms in one sequence are twice the corresponding terms
in the other sequence. Explain informally why this is so.
Goals: The student will –
*create a table of values for a
given pattern;
*graph the given table of values;
Discuss the shape of the graph
(straight lines, curved lines) and
what the graphs tell about the
“growth: of the pattern (i.e., the
function)
Chapter 3: Functions
Squares Cubes
page 64
Goals: The student will*investigate the functional
relationship between the length
of the sides of a square and its
perimeter and area;
*describe the functional
relationship between the length
of the sides of a square and(a) their perimeters and the
sides increase in length,
Geometric measurement: understand concepts of volume
and relate volume to multiplication and to addition.
5.MD.3 Recognize volume as an attribute of solid figures and
understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said
to have “one cubic unit” of volume, and can be used to
measure volume.
b. A solid figure which can be packed without gaps or
overlaps using n unit cubes is said to have a volume of n
cubic units.
5.MD.4 Measure volumes by counting unit cubes, using cubic
cm, cubic in, cubic ft, and improvised units.
5.MD.5 Relate volume to the operations of multiplication and
addition and solve real world and mathematical problems
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.
Mathematical Practice focus is in bold:
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.
Mathematical Practice focus is in bold:
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.
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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(b) their areas as the sides
increase in length;
*describe the functional
relationship between the length
of the sides of cubes and their
volumes as the sides increase in
length. (5th Grade)
involving volume.
a. Find the volume of a right rectangular prism with wholenumber side lengths by packing it with unit cubes, and
show that the volume is the same as would be found by
multiplying the edge lengths, equivalently by multiplying
the height by the area of the base. Represent threefold
whole-number products as volumes, e.g., to represent the
associative property of multiplication.
b. Apply the formulas V = l × w × h and V = b × h for
rectangular prisms to find volumes of right rectangular
prisms with whole-number edge lengths in the context of
solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid
figures composed of two non-overlapping right
rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real
world problems.
(This lesson could be used in 4th or 5th depending on focus.)
DRAFT: North Carolina Department of Public Instruction - Navigating through Algebra in Grades 3-5
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