EXPRESSIONS & EQUATIONS: 6 TH GRADE Erin Craig, Chelsea
Keen, Krista Milroy, Becki Schwindt, & Joana Wu
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STRAND 1 Apply and extend previous understandings of arithmetic
to algebraic expressions.
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PROGRESSIONS STRAND I K-5 Grade6 th Grade7 th -8 th Grade Since
Kindergarten, students have been writing numerical expressions such
as: 2 + 3 3 + 6 + 7 4 x (2 x 3) 8 x 5 + 8 x 2 Students begin
working with the Order of Operations in 3 rd grade. In 5 th grade,
they write and interpret numerical expressions using brackets and
parentheses. In Grade 5 they used whole number exponents to express
powers of 10 In Grade 6 they start to incorporate whole number
exponents into numerical expressions, for example when they
describe a square with side length 50 feet as having an area of 50
square feet They use the any order, any grouping property
(combination of the commutative and associative properties) to see
the expression 7 + 6 + 3 as (7 + 3) + 6 allowing them to quickly
evaluate it. Start working systematically with the square root and
cube root symbols Work with estimates of very large and very small
quantities Move toward an understanding of the idea of a function
In 7 th grade, students will solve numeric and algebraic
expressions, equations, and inequalities with rational numbers
applying the properties of operations. In 8 th grade, students will
be working with integer exponents. Apply and extend previous
understandings of arithmetic to algebraic expressions
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PROGRESSIONS STRAND I K-5 Grade6 th Grade7 th -8 th Grade As
they start to solve word problems algebraically, students use more
complex expressions. As word problems get more complex, students nd
greater benet in representing the problem algebraically by choosing
variables to represent quantities, rather than attempting a direct
numerical solution, since the former approach provides general
methods and relieves demands on working memory As students move
from numerical to algebraic work, the multiplication and division
symbols x and are replaced by the conventions of algebraic
notationstudents learn to use a dot for multiplication, or 3x
instead of 3 x x As students start to build a unified notion of the
concept of function, they are able to compare proportional
relationships in different ways By 8 th grade, students have the
tools to solve an equation which has a general linear expression on
each side of the equal sign Apply and extend previous
understandings of arithmetic to algebraic expressions
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Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011).
Developing essential understanding of expressions, equations, and
functions for teaching mathematics in grades 6-8. Reston, VA: The
National Council of Teachers of Mathematics, Inc. 3 + 3 + 3 + 3 + 3
+ 3 + 33 x 7 8 + 8 + 8 + 8 + 8 Multiplication is Shorthand for
Repeated Addition 8 x 5 2 + 2 + 2 + 2 2 x 4 3 x 3 x 3 x 3 x 3 x 3 x
3 3 7 8 x 8 x 8 x 8 x 8 Exponents are Shorthand for Repeated
Multiplication 8 5 2 x 2 x 2 x 2 2 4 Vocabulary: Numeric
expression, Repeated multiplication, Power, Exponent, Base,
Squared, Cubed Begin with what students know and what is concrete.
Squares (draw arrays) and (cubes make models) Discuss with your
group.. Is (2+5) 3 = 2 + 5 3 6.EE.A.1 Write and evaluate numerical
expressions involving whole-number exponents.
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Literature Connection
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Write an expression that is equivalent to 64 using each of the
following numbers and symbols once in the expression. (_ 2 means to
the exponent of 2) http://www.smarterbalanced.org 7 7 7 _2_2 + ( )
IN THE END STUDENTS WILL BE EXPECTED TO ANSWER QUESTIONS SUCH
AS
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Connections: Students will continue to use skills for
evaluating numeric expressions using the order of operations.
Students will continue to apply knowledge of operations with whole
numbers. Common Mistakes/Misconceptions: Students often will
evaluate 8 as 8 * 2 instead of 8 * 8. Students often get confused
by the vocabulary. May have trouble understanding which base the
exponent applies to Essential Questions: Why do people use
exponents? What do they mean? When evaluating expressions, why do
we evaluate exponents before we add and subtract? What exponent is
described by each of the following words? Square? Cube? Why is
something raised to the second power referred to as being squared?
What about cubed?
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6.EE.A.2 Write, read, and evaluate expressions in which letters
stand for numbers.
6.EE.A.2A Write expressions that record operations with numbers
and letters standing for numbers. SAMPLE PROBLEMS Express the
calculation Subtract y from 5 as 5 y Write an expression that is
equivalent to 64 using each of the following numbers and symbols
once in the expression. 7, 7, 7, 2 - as exponent, +, , and ( )
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6.EE.A.2B coefficient variables
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6.EE.A.2C 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.
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THE TRUTH ABOUT PEDMAS Approaches to teaching order of
operations do not involve students developing significant
understanding of why order should be followed (no transfer)
Developing the hierarchy-of-operators triangle and its application
encourages conceptual thinking and understanding Mathematics
Teaching in the Middle School Vol. 16, No. 7, March 2011
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WRITING A PEMDAS STORY Goal is to identify the basic
operations, show the need for establishing an order to approach
expressions with multiple operations, and explore possible problems
that might arise w/out such an order PEMDAS story activity helps
students realize that numbers and manipulation of quantities in
math are often descriptions of real-world events Mathematics
Teaching in the Middle School Vol. 5, No. 9, May 2000
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6.EE.A.3 Apply the properties of operations to generate
equivalent expressions.
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6.EE.A.3 EXAMPLE 1
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6.EE.A.3 EXAMPLE 2
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MISCONCEPTIONS Thinking that subtraction and division are
commutative Thinking that subtraction and division are associative
When speaking, mixing up the order of the problem Example: the
problem is 15 3 and the student verbally says, Three divided by
15.
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6.EE.A.4 THE STANDARD: Identify when two expressions are
equivalent (i.e. when the two expressions name the same number
regardless of which value is substituted into them). Example: y + y
+ y = 3y
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6.EE.A.4 - MISCONCEPTIONS The equal sign means you have to
solve or do an operation. The answer has to go on the right.
Simplify means making the problem easier. Errors such as n 2 + 5 as
n 7 (instead of n + 3) OR 5x x = 5 Ask students to prove it what if
x=1? Not thinking in a relational manner. Source: Knuth, E.,
Alibali, M., Hattikudur, S., McNeil, N., Stephens, A. (2008). The
importance of equal sign understanding in the middle grades.
Mathematics Teaching in the Middle School.
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6.EE.A.4 TEACHING STRATEGIES: Goal is to develop Conceptual
Understanding and Relational Thinking True/False and Open Sentences
with the use of variables Have students write their own
relationships ___ + ___ = ___ + ___ Have students look at
simplified equations that have errors and fix the mistakes
explaining their thinking Explain how to fix this simplification.
Give reasons. (2x +1) (x + 6) = 2x + 1 x + 6 Visual Representations
such as: Algebra Balance Scales Weight Logic Relation Take the
visual representations and progress to having the students write
equations to explain the relationships Image Source: Van de Walle,
J., Karp, K., Bay-Williams, J. (2013).
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6.EE.A.4 - GAMES Whats in the Bag? Equation War Online Tools
Algebra Scales
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ASSESSMENT Which of the following expressions are equivalent?
Why? If an expression has no match, write 2 equivalent expressions
to match it. a. 2(x + 4) b. 8 + 2x c. 2x + 4 d. 3(x + 4) (4 + x) e.
x + 4 Source: Illustrative Mathematics
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MISCONCEPTIONS Misunderstanding/misreading of the expression.
For example, knowing the operations that are being referenced with
notation like 4x, 3(x + 2y) is critical. The fact that x means xxx,
means x times x times x, not 3x or 3 times x 4x means 4 times x or
x+x+x+x, not forty-something. When evaluating 4x when x = 7,
substitution does not result in the expression meaning 47. Use of
the x notation as both the variable and the operation of
multiplication can complicate this understanding.
http://katm.org/wp/wp- content/uploads/2012/06/6FlipBookedite
d22.pdf
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STRAND 2 Reason about and solve one-variable equations and
inequalities.
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PROGRESSIONS STRAND II Reason about and solve one-variable
equations and inequalities K-5 Grade6 th Grade7 th -8 th Grade
Students have been writing numerical equations and simple equations
involving one operation with a variable Students start the
systematic study of equations and inequalities and methods of
solving them As word problems grow more complex in grades 6- 7,
analogous arithmetical and algebraic solutions show the connection
between the procedures of solving equations and the reasoning
behind those procedures
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6.EE.B.5 Understand solving an equation or inequality as a
process of answering a question: which values from a specified set,
if any, make the equation or inequality true? Use substitution to
determine whether a given number in a specified set makes an
equation or inequality true. Example: Billy has 27 marbles and his
friend gave Billy all of his marbles, now Billy has 100 marbles,
how many marbles did his friend give him? Students write the
equation: 27 + n = 100
http://www.brainpop.com/math/dataanalysis/inequalities/preview.wemlwww.brainpop.com/math/dataanalysis/inequalities/preview.weml
Students are being asked to solve equations, based on prior
knowledge and reasoning, requiring them to understand the meaning
of the equation as well demonstrate their thinking with the use of
a scale or model to demonstrate understanding of the question being
asked. www.commoncorestandards.org
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RESEARCH Algebraic thinking is present at every grade level
progressing from patterns to generalizations. At middle school
level, algebra becomes more abstract and symbolic in comparison to
K-5 levels. Methods used to compute and the structures of our
number system should be generalized, for example a+b = b+a All
areas of mathematics involve generalizing and formalizing
connecting all content areas of mathematics together in some way.
Van de Walle, NCTM, CCSSO 2010
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ASSESSMENT Sample assessment question: Which of the following
is a solution to n 5 = 19? A. n=10 B. n=11 C. n=24 D. n=16 Now, can
you write a problem with possible solutions like this one? Formulas
A formula is an algebraic rule for evaluating some quantity. A
formula is a statement. Example 7. Here is the formula for the area
A of a rectangle whose base is b and whose height is h.
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6.EE.B.6 Variables are tools for expressing mathematical ideas
clearly and concisely. They are the basis for the transition from
arithmetic to algebra and have many different meanings, depending
on context and purpose. Understanding Algebra requires knowing what
variables are and using them as tools to indicate relationship.
Students need to learn both to use that language to show their
ideas and to understand the meaning of someone else's mathematical
representation Using variables permits writing expressions whose
values are not known or vary under different circumstances.
Traditionally schools have taken to the unknown meaning of
variables although the idea of a changing quantity is more powerful
for understanding relationships mathematically and in real world
situations. Varying (functions, linear relationships) R=d/t 25-29
The language and use of Mathematics: The meaning and use of
variable Glenda Lappan NCTM news bulletin, January 2000
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Hisaki is making sugar cookies for a school bake sale. He has 3
1/2 cups of sugar. The recipe calls for 3/4 cup of sugar for one
batch of cookies. Which equation can be used to find b, the total
number of batches of sugar cookies Hisaki can make? A 3 x = b B 3 =
b C 3 + b = D 3 b = Let b represent any number. Use the numbers,
operation symbols, and letter below to create an expression that
represents the following: 5 more than the product of 3 and the
number b ( Not all objects will be used) 35 b + x
http://www.smarterbalanced.org
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Ricardo has 8 pet mice. He keeps them in two cages that are
connected so that the mice can go back and forth between the cages.
One of the cages is blue and the other is green. Show all the ways
that 8 mice can be in the two cages. Stevens, C. Ana Developing
Students Understandings of Variables Mathematics teaching in the
Middle School September 2005 Vol. 11 No 2 g+b=8
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MISCONCEPTIONS: Students view variables as abbreviations or
labels rather than letters that stand for quantities Assign values
to letters based on their positions in the alphabet Unable to
operate with algebraic letters as varying quantities rather than
specific values Different letter within a number sentence must
represent different numerical values a=ac=r Always SometimesNever
Stevens, C. Ana Developing Students Understandings of Variables
Mathematics teaching in the Middle School Setpember 2005 Vol. 11 No
2
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6.EE.B.7 Solve real-world and mathematical problems by writing
and solving equations of the form x + p = q and px = q for cases in
which p, q, and x are all nonnegative rational numbers.
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6.EE.B.7 - EXAMPLE
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6.EE.B.8
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RESEARCH Symbolism, especially involving equality and
variables, must be well understood conceptually for students to be
successful in mathematics, particularly algebra. (p. 258)
Exploration of representations in a variety of ways strengthens
understanding of the same relationship from different points of
view. Van de Walle, 2013
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INSTRUCTIONAL APPROACH The skill of solving the equation must
be developed conceptually before it is developed procedurally.
Provide multiple situations in which students must determine if
there is a single solution or multiple solutions, creating the need
to use different types of equations. Provide practice in the use of
positive and negative numbers. Provide practice in writing
equations by giving students an equation and having them write a
problem and by having them write an equation from a word problem.
CCSS, Arizona DOE, Ohio DOE, North Carolina DOE
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ASSESSMENT GRADE6COMMONCOREMATH.WIKISPACES.HCPSS.ORG 10 divided
by y is greater than or equal to 2
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MISCONCEPTIONS STRAND II Misunderstanding or misreading of the
expression Lack of understanding of the operations being used Use
of x as both the variable and the operation of multiplication
complicates understanding (e.g. 6x x 5 = 600) Multiplication symbol
variable
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STRAND 3 Represent and analyze quantitative relationships
between dependent and independent variables.
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PROGRESSIONS STRAND III Represent and analyze quantitative
relationships between dependent and independent variables K-5
Grade6 th Grade7 th -8 th Grade Learning to use symbols, such as
parentheses, to write equations Write simple equations Interpret
numerical expressions without evaluating them Generate two
numerical patterns using two given rules As they work with
equations, students begin to develop a dynamic understanding of
variables Students can use tables and graphs to develop an
appreciation of varying quantities Begin the systematic study of
equations and inequalities and how to solve them This prepares
students for work with functions in later grades Arithmetical and
algebraic solutions show the connection between procedures of
solving equations and the reasoning behind the procedures
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STRAND 3 Represent and analyze quantitative relationships
between dependent and independent variables. Progression: 3.OA.9
4.OA.C 5.OA.B 6.RP 8.F 6.EE.C 7.RP 8.EE.B
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6.EE.C.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. Integrates with Science BIG TIME
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6.EE.C.9 - MISCONCEPTIONS Confusion about which variable is the
dependent and which variable is the independent D.R.Y. M.I.X.
Students may think of the variable as a place holder for one exact
number, rather than a representation of multiple, possible infinite
values (Van de Walle, et al., 2013).
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6.EE.C.9 TEACHING STRATEGIES Look at the relationships between
the dependent and independent variables. Sketch or look at the
shape of graphs and describe the story they would tell. Include
fractions and decimals with the variables. Use real-world examples
and integrate different representations of data. Make tables with
input (independent variable) and output (dependent variable) to
help develop thinking about the functional relationship. It is a
good idea to include a column for my thinking as well. Source:
Markworth, K. (2012). Growing patterns: Seeing beyond counting.
Image Source: Van de Walle, J., et al., (2013).
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ASSESSMENT We would like students to make connections across
multiple representations. Students should be able to translate
freely across a variety of representations. When given a
representation to start with, students should be able to develop a
different representation using the same information. Physical
materials or drawings Tables Words Symbols Graphs
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RESEARCH SOURCES CCSS for Mathematics University of Arizona
Progression Document Zimba Schematic Van de Walle BrainPOP
Connected Mathematics Teachers Pay Teachers Illustrative
Mathematics www.learnzillion.com www.learnzillion.com
www.graniteschools.org www.graniteschools.org