Grade 7 Mathematics, Quarter 3, Unit 3.1
Development of Algebraic Expressions
Overview
Number of instructional days:
5
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Write algebraic expressions with wholenumber exponents or more than one variable.
Make sense of problems and persevere in solving
them.
•
Evaluate algebraic expressions with wholenumber exponents or more than one variable.
•
•
Evaluate algebraic expressions within an
equation.
Attend to precision.
•
Explain the meaning of a problem and looking
for entry points to its solution.
Calculate accurately and efficiently, expressing
numerical answers with a degree of precision
appropriate for the problem context.
Look for and express regularity in repeated
reasoning.
•
Notice if calculations are repeated and look
both for general methods and for shortcuts.
•
How do you write an algebraic expression to
represent a given situation?
Essential questions
•
How would you evaluate an algebraic
expression?
•
How are variables used in algebraic
expressions?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-33
Grade 7 Mathematics, Quarter 3, Unit 3.1
Final, July 2011
Development of Algebraic Expressions (5 days)
Written Curriculum
Grade-Level Expectations
M(F&A)–7–3 Demonstrates conceptual understanding of algebraic expressions by using letters to
represent unknown quantities to write algebraic expressions (including those with whole number
exponents or more than one variable); or by evaluating algebraic expressions (including those with whole
number exponents or more than one variable); or by evaluating an expression within an equation (e.g.,
determine the value of y when x = 4 given y = 5x3 – 2). (State)
Clarifying the Standards
Prior Learning
Beginning in grade 4, students used letters or symbols to represent unknown quantities to write simple
algebraic expressions with one operation. In grade 5, they progressed to writing expressions with two
operations. By sixth grade, they incorporated the order of operations into these expressions and began to
evaluate expressions with more than one variable. There is no GLE for these concepts in grades K–3.
Current Learning
Students continue to write and evaluate algebraic expressions, which now include whole-number
exponents and more than one variable. They continue to evaluate an expression within an equation (e.g.,
determine the value of y when x = 5 given y = 2x4 + 10). These concepts will be state-assessed.
Future Learning
In grade 8, students will evaluate and simplify algebraic expressions including those with square roots,
whole-number exponents, or rational numbers. There is no GSE for these concepts in grades 9–12.
Additional Research Findings
According to Principles and Standards for School Mathematics, “it is essential that [students] become
comfortable in relating symbolic expressions containing variables to verbal, tabular, and graphical
representations of numerical and quantitative relationships” (p. 223). The book also states, “An
understanding of the meanings and uses of variables develops gradually as students create and use
symbolic expressions, and relate them to verbal, tabular, and graphical representations” (p. 225).
A Research Companion to Principles and Standards for School Mathematics states “patterns of student
errors in their use of variables occur … when “the child experiences discomfort attempting to handle an
algebraic expression which represents a process that [he or she] cannot carry out … For example, in
arithmetic, a final answer is always in terms of a single numeric answer, whereas an algebraic ‘answer’
may express a process, such as 2 + 3a … An algebraic expression, such as 2 + 3a, represents both the
process by which the computation is carried out and also the product of that process. They attribute these
difficulties to traditional teaching in which rules for manipulation are given primary if not sole emphasis,
with the ‘forlorn hope’ that understanding will follow” (p. 139).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-34
Grade 7 Mathematics, Quarter 3, Unit 3.2
Understanding Equivalency Through Creating
and Solving Equations
Overview
Number of instructional days:
15
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Translate a problem-solving situation into an
equation.
Model with mathematics.
•
Solve whole-number multi-step linear
equations with all four operations, and with
addition and subtraction expressions on both
sides.
•
Demonstrate conceptual understanding of
equality by showing equivalence between two
expressions
.
•
Mathematically proficient students can apply
the mathematics they know to solve problems
arising in everyday life.
Look for and make use of structure.
•
Look for, develop, generalize and describe a
pattern orally, symbolically, graphically and in
written form.
Attend to precision.
•
Use clear definitions and state the meaning of
the symbols they choose, including using the
equal sign consistently and appropriately.
•
Strive for accuracy.
Reason abstractly and quantitatively.
•
Know and flexibly use different properties of
operations and objects.
•
When would it be better to solve a problem
with an equation rather than using a graph or
table?
Essential questions
•
What processes would you use when solving a
multi-step equation?
•
How do you maintain equivalence when
solving equations?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-35
Grade 7 Mathematics, Quarter 3, Unit 3.2
Final, July 2011
Understanding Equivalency Through
Creating and Solving Equations (15 days)
Written Curriculum
Grade-Level Expectations
M(F&A)–7–4 Demonstrates conceptual understanding of equality by showing equivalence between
two expressions (expressions consistent with the parameters of the left- and right-hand sides of the
equations being solved at this grade level) using models or different representations of the expressions,
solving multi-step linear equations of the form ax ± b = c with a ≠ 0, ax ± b = cx ± d with a, c ≠ 0, and
(x/a) ± b = c with a ≠ 0, where a, b, c and d are whole numbers; or by translating a problem-solving
situation into an equation consistent with the parameters of the type of equations being solved for this
grade level. (State)
Clarifying the Standards
Prior Learning
In grades 1–2, students learned about properties of equality by finding a missing value in an addition or
subtraction equation. In grade 3, students continued this concept and included multiplication. Grade 4
students solved one-step addition, subtraction, and multiplication equations with whole numbers. Grade 5
included division. Grade 4–5 students also prepared to solve multi-step equations by simplifying numeric
expressions using the order of operations. In grade 6, students learned to solve two-step equations
involving addition, subtraction, and multiplication of whole numbers in the form ax ± b = c, where a ≠ 0.
Current Learning
In grade 7, students show equivalence between two expressions. They solve multi-step linear equations
and translate problem-solving situations into equations involving all four operations (of the form
ax ± b = c where a ≠ 0, ax ± b= cx ± d where a, c ≠ 0, and (x/a) ± b = c where a ≠ 0, and where a, b, c,
and d are whole numbers). For the first time, students will solve equations with division and with addition
and subtraction expressions on both sides. All concepts in this unit are state-assessed.
Future Learning
Eighth-grade students will continue to solve multi-step equations and will solve for a variable in a variety
of formulas (e.g., d = rt and d/r = t). They will simplify both sides of an equation or inequality using field
properties and order of operations. In high school, problem solving with algebraic reasoning will
continue. By grade 12, students will solve functions and systems of equations.
Additional Research Findings
According to Principles and Standards for School Mathematics, most students will need extensive
experience in interpreting relationships among quantities in a variety of problem contexts before they can
work meaningfully with variables and symbolic expressions (pp. 225–227).
Relationships among quantities can often be expressed symbolically in more than one way, providing
opportunities for students to examine the equivalence of various algebraic expressions.
Students’ facility with symbol manipulation can be enhanced if it is based on extensive experience with
quantities in context.
C-36
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 7 Mathematics, Quarter 3, Unit 3.3
Linear and Non-Linear Representations
Overview
Number of instructional days:
8
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Look for and make use of structure.
•
•
•
•
Identify and extend patterns (linear and
nonlinear) by analyzing models, tables,
sequences, graphs, and problem situations.
Describe linear patterns using words and
symbols.
Write an expression or equation using words
and symbols.
Describe nonlinear patterns using words and
symbols.
•
Mathematically proficient students look
closely to discern a pattern or structure.
Make sense of problems and persevere in solving
them.
•
Analyze givens, constraints, relationships, and
goals.
•
Recognize and apply relationships between
equations, verbal descriptions, tables and
graphs.
Generalize a linear relationship using words
and symbols to find a specific case.
Reason abstractly and quantitatively
•
Represent a given situation symbolically and
manipulate the representing symbols.
Look for and express regularity in repeated
reasoning.
•
Notice the regularity, which might lead them to
a general formula/shortcut.
Essential questions
•
How do you determine the next term in a
sequence?
•
How do you determine the equation to express
a generalization?
•
How do you determine a missing term within a
sequence?
•
How can you recognize a linear relationship in
a table, graph, or equation?
•
How do you determine the nth term in a
sequence?
•
How can you recognize a non-linear
relationship in a table, graph, or equation?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-37
Grade 7 Mathematics, Quarter 3, Unit 3.3
Final, July 2011
Linear and Non-Linear Representations (8 days)
Written Curriculum
Grade-Level Expectations
M(F&A)–7–1 Identifies and extends to specific cases a variety of patterns (linear and nonlinear)
represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear
relationship using words and symbols; generalizes a linear relationship to find a specific case; or writes an
expression orsc equation using words orsc symbols to express the generalization of a nonlinear
relationship. (State)
Clarifying the Standards
Prior Learning
In kindergarten, students recognized patterns represented in pictures, sounds, colors, and letters. As they
progressed through grades 1–3, they extended and completed number patterns using models, tables, and
sequences. In grade 4, students described both linear and nonlinear patterns and wrote rules in words or
symbols. In grade 5, they applied their understanding to problem situations. Finding patterns in graphs
and describing nonlinear relationships was introduced in grade 6.
Current Learning
Beginning in grade 7, students use both words and symbols to generalize linear patterns (e.g., describing a
phone plan). They have the choice of using either words or symbols to describe nonlinear patterns.
Students generalize a linear relationship to find a missing or an nth term in a pattern. All concepts in this
unit of study and introduced at this grade level will be state-assessed. In the next unit, students will use
linear relationships to develop conceptual understanding of slope.
Future Learning
Grade 8 students will continue to generalize a nonlinear relationship using words or symbols. They will
generalize a common nonlinear relationship to find a specific case. In high school, they will use their
ability to generalize patterns for specific problem-solving situations.
Additional Research Findings
According to Principles and Standards for School Mathematics, the study of patterns and relationships in
the middle grades should focus on patterns that relate to linear functions, which arise when there is a
constant rate of change (p. 223).
According to Science for All Americans, “Patterns of change are of special interest in the sciences:
descriptions of change are important for predicting what will happen; analysis of change is essential for
understanding what is going on, as well as for predicting what will happen; and control of change is
essential for the design of technological systems. We can distinguish three general categories: (1) changes
that are steady trends, (2) changes that occur in cycles, and (3) changes that are irregular” (pp. 174–175).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-38
Grade 7 Mathematics, Quarter 3, Unit 3.4
Development of Linear Relationships
Overview
Number of instructional days:
12
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Demonstrate understanding of linear
relationships as a constant rate of change.
Make sense of problems and persevere in solving
them.
•
Recognize the relationship between slope and
rate of change.
•
•
Distinguish between constant and variable rates
of change in tables or graphs.
•
Describe how the change in the value of one
variable affects the other variable.
•
Determine the slope of a line informally from a
table or graph.
•
Describe the meaning of slope in concrete
situations (i.e., compare two hourly pay rates).
Recognize and apply relationships between
equations, verbal descriptions, tables and
graphs.
Construct viable arguments and critique the
reasoning of others.
•
They make conjectures and build a logical
progression of statements to explore the truth
of their conjectures.
Model with mathematics.
•
Interpret their mathematical results in the
contest of the situation and reflect on whether
the results make sense.
•
Analyze relationships mathematically to draw
conclusions.
•
Explain the concept of the slope of a line. How
does finding slope compare to finding the rate
of change between two variables in a linear
relationship?
•
If the cost for renting a kayak is represented by
the equation y = 8x + 5, where x represents the
number of hours, what could the slope of the
equation represent?
Essential questions
•
How can you distinguish between a constant
and a varying rate of change in a table or a
graph?
•
How can you determine the slope of a line from
a table or graph?
•
As the x-value increases, what happens to the
y-value?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-39
Grade 7 Mathematics, Quarter 3, Unit 3.4
Final, July 2011
Development of Linear Relationships (12 days)
Written Curriculum
Grade-Level Expectations
M(F&A)–7–2 Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as
a constant rate of change by solving problems involving the relationship between slope and rate of
change, by describing the meaning of slope in concrete situations, or informally determining the slope of
a line from a table or graph; and distinguishes between constant and varying rates of change in
concrete situations represented in tables or graphs; or describes how change in the value of one
variable relates to change in the value of a second variable in problem situations with constant rates of
change. (State)
Clarifying the Standards
Prior Learning
In grades 4–5, students began to develop a conceptual understanding of linear relationships by
identifying, describing, and comparing situations that represent constant rates of change (e.g., told a story
given a line graph about a trip). Students in grade 6 constructed and interpreted graphs and described the
slope of linear relationships (faster, slower, greater, or smaller) and described the effect of one variable on
another.
Current Learning
For the first time, students informally describe a rate of change as slope (e.g., whether the data is
increasing, decreasing, zero, or undefined). They distinguish between constant and variable rates of
change in tables or graphs. They continue to describe how the change in the value of one variable affects
the other variable and begin to describe the meaning of slope in concrete situations.
Future Learning
In grade 8, students will formally and informally determine slopes and intercepts. They will distinguish
between linear and nonlinear relationships. The study of slope will continue through grade 12 with
increasing complexity.
Additional Research Findings
According to Principles and Standards for School Mathematics, “students should develop a general
understanding of, and facility with, slope and y-intercept and their manifestation in tables, graphs, and
equations” (p. 224).
The book also states, “The study of patterns and relationships in the middle grades should focus on
patterns that relate to linear functions, which arise when there is a constant rate of change. Students
should problems in which they use tables, graphs, words, and symbolic expressions to represent and
examine functions and patterns of change” (p. 223).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-40
Grade 7 Mathematics, Quarter 3, Unit 3.4
Final, July 2011
Development of Linear Relationships (12 days)
A Research Companion to Principles and Standards for School Mathematics states that errors in students’
interpretation of algebraic statements “can be seen as instances of knowing arithmetic in a different way.
This view suggests that even at a basic level, children are being asked not only to generalize arithmetic
but also to re-conceptualize arithmetic. Only from an adult perspective, in which the re-conceptualization
has already occurred, does algebra appear as generalized arithmetic. Recognizing the complexity of the
situation, two research groups have recently called for the integration of algebraic thinking ‘or reasoning’
throughout the grades K–12 curriculum” (p. 140).
Notes About Resources and Materials
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-41
Grade 7 Mathematics, Quarter 3, Unit 3.4
Final, July 2011
Development of Linear Relationships (12 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-42