Grade 6 Mathematics, Quarter 2, Unit 2.1
Identify, Write, and Evaluate Expressions
Overview
Number of instructional days:
15
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Using mathematical vocabulary (sum, term,
product, factor, quotient, coefficient), identify
parts of an expression.
Make sense of problems and persevere in solving
them.
•
Write expressions using variables.
•
Evaluate expressions and formulas using the
order of operations.
•
Using whole number exponents, write and
evaluate numerical expressions.
•
Translate real-world problems into algebraic
expressions.
•
Understand the meaning of a mathematical
expression.
•
Use the order of operations appropriately.
•
Check for accuracy, asking, “Does the answer
make sense?”
Reason abstractly and quantitatively.
•
Given a situation, represent it symbolically
with variables.
•
Understand the meaning of the variables.
Attend to precision.
•
State the meaning of variables and symbols
(operations).
•
When necessary, use appropriate labels in
formulas.
•
Calculate accurately and efficiently.
•
State the meaning of the symbols they choose.
Essential questions
•
What is the difference between a numerical
expression and an algebraic expression?
•
How does a whole number exponent affect the
base number?
•
What is a variable and what does it represent?
•
How can you evaluate real-world problems
using the order of operations?
•
Why is it important to use appropriate
mathematical language?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 13 Grade 6 Mathematics, Quarter 2, Unit 2.1
Identify, Write, and Evaluate Expressions (15 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Expressions and Equations
6.EE
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.1
Write and evaluate numerical expressions involving whole-number exponents.
6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
a.
Write expressions that record operations with numbers and with letters standing for
numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
b.
Identify parts of an expression using mathematical terms (sum, term, product, factor,
quotient, coefficient); view one or more parts of an expression as a single entity. For
example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as
both a single entity and a sum of two terms.
c.
Evaluate expressions at specific values of their variables. Include expressions that arise
from formulas used in real-world problems. Perform arithmetic operations, including
those involving whole-number exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations). For example, use the
formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of
length s = 1/2.
Common Core Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 14 Grade 6 Mathematics, Quarter 2, Unit 2.1
2
Identify, Write, and Evaluate Expressions (15 days)
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In third grade, students worked with orders of operations. In fifth grade, students wrote and interpreted
numerical expressions using brackets and parentheses. They also used whole number exponents to denote
powers of 10.
Current Learning
In sixth grade, students start to incorporate whole number exponents into numerical expressions and to
translate words to algebraic expressions. They evaluate variable expressions and formulas that have whole
number exponents and that incorporate order of operations. Students use appropriate vocabulary. This is a
PARCC major cluster and a critical area according to the CCSS. Later in the year, students will work with
inequalities on a number line.
Future Learning
Students will use distributive properties in later units of study in sixth grade. In seventh grade, they will
solve more complex numeric and algebraic expressions, and equations and inequalities with rational
numbers, applying the properties of operations. They will also simplify general linear expressions with
rational coefficients. In eighth grade, students will work with integer exponents.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 15 Grade 6 Mathematics, Quarter 2, Unit 2.1
Identify, Write, and Evaluate Expressions (15 days)
Additional Findings
According to the PARCC Model Content Frameworks, “applying and extending previous understandings
of arithmetic to algebraic expressions is a major cluster” (p. 30).
According to A Research Companion to Principles and Standards for School Mathematics, “instruction
should connect with student experience and build on the resources and strengths present in the
conceptions they bring” (p. 133).
According to Curriculum Focal Points, students “write mathematical expressions and equations that
correspond to given situations, evaluate expression and use expression and formulas to solve problems”
(p. 18).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 16 Grade 6 Mathematics, Quarter 2, Unit 2.2
Variables in the Real World
Overview
Number of instructional days:
15
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Use variables to represent numbers.
Reason abstractly and quantitatively.
•
Use variables to write expressions and
equations.
•
Understand the meaning of a mathematical
expression.
•
Write an inequality (< or >) and represent it on
a number line.
•
Take a given situation and represent it
symbolically.
•
Use graphs, tables, and equations to analyze
independent and dependent variables.
•
Make sense of quantities and relationships.
•
Solve real-world problems using inequalities
and equations.
Model with mathematics.
•
Apply the mathematics they know to solve
everyday problems.
•
Use graphs, tables, and number lines.
•
Analyze relationships to draw conclusions.
Use appropriate tools strategically.
•
Recognize usefulness and limitations of
particular visual tools.
•
Use prior knowledge of paper and pencil
models, concrete models and technology to
deepen understanding of concepts.
•
Using a table or graph, how can you analyze
the relationship between a dependent and an
independent variable?
Essential questions
•
What is the difference between an equation, an
expression, and an inequality?
•
How do you differentiate between a dependent
and an independent variable?
•
•
How do you determine which quantity is
dependent or independent?
How do you model an inequality on a number
line?
•
What is an example of a real-world problem
that can illustrate an inequality?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 17 Grade 6 Mathematics, Quarter 2, Unit 2.2
Variables in the Real World (15 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Expressions and Equations
6.EE
Reason about and solve one-variable equations and inequalities.
6.EE.6
Use variables to represent numbers and write expressions when solving a real-world or
mathematical problem; understand that a variable can represent an unknown number, or,
depending on the purpose at hand, any number in a specified set.
6.EE.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.9
Use variables to represent two quantities in a real-world problem that change in relationship to
one another; write an equation to express one quantity, thought of as the dependent variable, in
terms of the other quantity, thought of as the independent variable. Analyze the relationship
between the dependent and independent variables using graphs and tables, and relate these to
the equation. For example, in a problem involving motion at constant speed, list and graph
ordered pairs of distances and times, and write the equation d = 65t to represent the
relationship between distance and time.
Common Core Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 18 Grade 6 Mathematics, Quarter 2, Unit 2.2
4
Variables in the Real World (15 days)
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
Clarifying the Standards
Prior Learning
In grades 4 and 5, students generated and analyzed patterns that follow a given rule. They also studied the
relationship between ordered pairs. In fifth grade, students wrote and interpreted numerical expressions
without evaluating them.
Current Learning
The use of a variable with inequalities is a new concept for sixth grade. Students are beginning to work
with algebraic expressions and inequalities and methods of solving them. They are reasoning about and
solving single-variable equations and inequalities. Students represent and analyze quantitative
relationships between dependent and independent variables by using tables and graphs. These are major
clusters according to PARCC and a critical area according to the CCSS.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 19 Grade 6 Mathematics, Quarter 2, Unit 2.2
Variables in the Real World (15 days)
Future Learning
In seventh grade, students will solve multi-step real life problems. Students in seventh grade will
understand that rewriting an expression in different forms in a problem context can shed light on the
problem and how quantities in it are related. They will have to graph the solution set of an inequality
(< , < or > , >) and interpret it in the context of the problem.
Additional Findings
According to Curriculum Focal Points, “students know that the solutions of equations are the values of
the variables that make the equation true. They solve simple one-step equations by using number sense,
operations, and the idea of maintaining equality on both sides of the equations.” (p. 18)
According to the PARCC Model Content Frameworks, “the use of a variable with inequalities is a new
concept for sixth grade. Students are beginning to work with algebraic expressions and inequalities and
methods of solving them. They are reasoning about and solving one-variable equations and inequalities.
Students are representing and analyzing quantitative relationships between dependent and independent
variables.” (p. 30)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 20 Grade 6 Mathematics, Quarter 2, Unit 2.3
Simplifying Algebraic Expressions Through
Properties of Operations
Overview
Number of instructional days:
12
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Reason abstractly and quantitatively.
•
•
Apply the properties of operations (associative
of addition, associative of multiplication,
identity, commutative of addition, commutative
of multiplication and the distributive property),
to expressions.
Use the properties of operations to generate
equivalent expressions.
•
Understand the meaning of a mathematical
expression.
•
Take a given situation and represent it
symbolically.
•
Know and flexibly use properties of
operations.
Identify two equivalent expressions.
Look for and make use of structure.
•
Look at a problem and identify the structure.
•
See complicated problems as simpler problems
put together.
•
What is an example of each property
(associative of addition, associative of
multiplication, identity, commutative of
addition, commutative of multiplication and the
distributive property)?
•
How can you show that two expressions are
equivalent?
Essential questions
•
What is equivalence?
•
What is an example of two equivalent algebraic
expressions?
•
How do the properties of operations help to
simplify expressions?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 21 Grade 6 Mathematics, Quarter 2, Unit 2.3
Simplifying Algebraic Expressions Through
Properties of Operations (12 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Expressions and Equations
6.EE
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.3
Apply the properties of operations to generate equivalent expressions. For example, apply the
distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x;
apply the distributive property to the expression 24x + 18y to produce the equivalent
expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent
expression 3y.
6.EE.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same
number regardless of which value is substituted into them). For example, the expressions y + y
+ y and 3y are equivalent because they name the same number regardless of which number y
stands for.
Common Core Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 22 Grade 6 Mathematics, Quarter 2, Unit 2.3
Simplifying Algebraic Expressions Through
Properties of Operations (12 days)
Clarifying the Standards
Prior Learning
In fifth grade, students wrote and interpreted numerical expressions without evaluating them. Students in
K–5 have extensive experience working with the properties of operations with whole numbers, decimals,
and fractions.
Current Learning
In grade 6, students are starting to use properties of operations to manipulate algebraic expressions and
produce different but equivalent expressions for different purposes. PARCC indicates this is a major
cluster, to apply previous understandings of arithmetic to algebraic expressions. It is a critical area
according to the CCSS.
Future Learning
In seventh grade, students will apply properties of operations as strategies to add, subtract, factor, and
expand linear expressions with rational coefficients. They will also understand that rewriting an
expression in different forms can shed light on the problem and how the quantities in it are related.
Additional Findings
According to Curriculum Focal Points, “students use the distributive property to show that two
expressions are equivalent. They also illustrate properties of operations by showing that two expressions
are equivalent in a given situation” (p. 18).
According to the PARCC Model Content Frameworks, “as word problems become more complex in
grades 6 and 7, analogous and arithmetical and algebraic solutions show the connection between the
procedures of solving equations and the reasoning behind those procedures” (p. 7).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 23 Grade 6 Mathematics, Quarter 2, Unit 2.3
Simplifying Algebraic Expressions Through
Properties of Operations (12 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 24