# 7th grade math pacing guide by huanghengdong

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```									        7th Grade Math

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                 Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                          Objective                                  Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                *teacher word

Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
problem solving is important, takes significant time and repeated efforts, and requires reflection.
Essential Question - What are the specific strategies that have wide application in attacking problems and
can help in problem solving?
C.7.2                  NO.3.7.2                                     Solve with and without appropriate          with and without appropriate technology:      mental computation
Solve, with and without appropriate          technology, multi-step problems using a     *represent multi-step problems using          technology
technology, multi-step problems using a      variety of methods and tools                manipulatives (including 2 or more steps)     Distributive Property
variety of methods and tools                 Ex.                                         *solve problems mentally                      rational numbers
Ex.                                          objects, mental computation, paper and      *solve multi-step problems with paper and     like terms
objects, mental computation, paper and       pencil                                      pencil                                        problem solving
pencil                                                                                                                                 strategy

FIRST NINE WEEKS
1. Enduring Understanding - Real numbers are either rational or irrational.
1a. Essential Question - What kinds of numbers make up the set of real numbers?
F.7.3                  NO.1.7.6                                    Identify subsets of the real number system   *demonstrate that natural numbers are         real number system
Recognize subsets of the real number        (natural, whole, integers, rational, and     counting numbers beginning with 1 and         natural numbers
system (natural, whole, integers, rational, irrational numbers)                          that they build on one another                whole numbers
and irrational numbers)                                                                  *identify whole numbers as numbers            integers
whose value includes zero and all natural     rational numbers
numbers                                       irrational numbers
*integers are all whole numbers and their     perfect square
opposites                                     square root
*apply the definition of a rational number    undefined fraction
(any number that can be written as a/b
where a is any integer and b is any integer
except 0. Including any terminating
decimals, repeating decimals and perfect
squares)
*recognize that a fraction whose
denominator is zero is undefined
*identify irrational numbers as numbers
that have infinite but non-repeating
decimal representations (Ex. π; any
square root that is not a perfect square)
*identify subsets of the real number
system

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                               Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                           Objective                               Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                              *teacher word

Enduring Understanding - Successful problem solvers possess a set of core ways ofthat support numbers.
2. Enduring Understanding - Exponential and scientific notation are efficient beliefs expressing their work:
problem solving is important, takes significant time and repeated efforts, and requires reflection.
2a. Essential Question - What is the purpose of exponents?
F.7.3                  NO.1.7.2                                     A. Demonstrate, with and without           with and without appropriate technology:     scientific notation
Demonstrate, with and without                appropriate technology, an understanding   *write the number in expanded notation       exponents
appropriate technology, an understanding     of place value using powers of 10          Ex. 103 = 10×10×10                           place value
of place value using powers of 10 and        B. Write numbers greater than one in       *multiply and divide by multiples of 10      expanded notation
write numbers greater than one in            scientific notation                        *demonstrate the value of a decimal when     power
scientific notation                                                                     dividing or multiplying by a power           base
(multiple) of 10
*demonstrate that exponents are repeated
multiplication
*identify the whole number of a scientific
notation number as a value ≥ 1 and < 10

F.7.3           1      NO.1.7.3                                     Convert between scientific notation and    *multiply and divide by powers of ten to     scientific notation
Convert between scientific notation and      standard notation using numbers greater    convert between standard and scientific      standard notation
standard notation using numbers greater      than one                                   notation                                     powers of ten
than one

3. Enduring Understanding - Fractions, decimals, and percents can be used interchangeably.
3a. Essential Question - What determines an appropriate representation for a number?
F.7.3                  NO.1.7.5                                     Compare and represent integers,            *locate integers on a number line            integers
Compare and represent integers, fractions,   fractions, decimals and mixed numbers      *locate fractions on a number line           number line
decimals and mixed numbers and find their    and find their approximate location on a   *locate decimals on a number line
approximate location on a number line        number line                                *locate mixed numbers on a number line
*locate and compare the values of
integers, fractions, decimals and mixed
numbers on a number line

F.7.1                  M.13.7.6                                     A. Find the distance between two points    *plot two points on a number line        distance
Find the distance between two points on a    on a number line                           *find the distance between two points    midpoint
number line and locate the midpoint          B. Locate the midpoint                     *average two numbers on a number line to number line
locate the midpoint

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                   Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                              Objective                                Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                  *teacher word

F.7.1                  NO.1.7.4                                     A. Find decimals for mixed numbers &     *explain how a part relates to a whole   decimal
Enduring Understanding
Find decimal and percent equivalents for - Successful problem solvers possess a set of core beliefs that support
explain why they represent the same      *restate a mixed number as a whole       percent                         their work:
value                                    number + fraction
mixed numbers and explain why they
repeated efforts, denominator equivalents
problem solving is important, takes significant time and*convert fractions by dividing and requires reflection.
represent the same value                     B. Find percent equivalents of mixed                                              convert
numbers & explain why they represent the into numerator
same value                               *write a decimal as a mixed number
*express mixed numbers as percents >
100
*demonstrate that the percent can be
found by finding the fraction-to-decimal
equivalent and then multiply the decimal
by 100

F.7.1           1      NO.1.7.1                                        A. Show with and without models and         *explain how a part relates to a whole and   part-to-part
Relate, with and without models and pictures,   pictures the concept of ratio including     vice versa                                   part-to-whole
concepts of ratio, proportion, and percent,     ratios <1 and >100                          *match decimals and percents of the          ratio
including percents less than 1 and greater      B. Show with and without models and         same value                                   proportion
than 100                                        pictures the concept of percent including   *represent percents out of 100               percent
percents <1 and >100                        *describe a situation in which a percent
C. Show with and without models &           might be between 1 & 100
pictures proportion including proportions   *draw a picture or model a number as a
<1 and >100                                 decimal and percent
D. Relate with and without models and
pictures concepts of ratio, percent, and
proportions including percents <1 and
>100

F.7.3                  NO.3.7.4                                        A. Apply factorization to solve problems    *find multiples and factors of given         prime numbers
Apply factorization, LCM, and GCF to solve      using more than two numbers and explain     numbers                                      composite numbers
problems using more than two numbers            the solution                                *find the prime factorization of a number    prime factorization
and explain the solution                        B. Apply LCM to solve problems using        (Ex. tree diagram)                           LCM
more than two numbers and explain the       *identify numbers as prime or composite      GCF
solution                                    *find the LCM and GCF of more than two
C. Apply GCF to solve problems using        numbers
more than two numbers and explain the       *apply and explain solutions
solution

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                   Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                             Objective                                 Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                  *teacher word

F.7.1         MC-1     NO.3.7.6                                Solve, with and without appropriate     with and without appropriate technology: technology
OR-1     Enduring Understanding - Successful problem solvers possess a set of core beliefs that support
Solve, with and without appropriate     technology, real-world percent problems *solve percent problems                   percent                              their work:
technology, real-world percent problems Ex.                                     Ex.                                       interest
repeated efforts,
problem solving is important, takes significant time andWhat number is 50% of 80? and requires reflection.
Ex.                                     I=PRT                                                                             principal
I=PRT (whole unknown)                                                           80 is what percent of 50?                 rate
80 is 50% of what number?                 time
*use fractions, decimals and percents
interchangeably
*apply knowledge of percent to real-world
situations
Ex.
I = PRT

4. Enduring Understanding - Integers have magnitude and direction.
4a. Essential Question - How do operations with integers compare to operations with whole numbers?
F.7.3                  NO.2.7.4                                       A. Model and develop addition and           *model addition and subtraction of positive   integers
Model and develop addition, subtraction,       subtraction of integers                     and negative numbers using                    absolute value
multiplication and division of integers        B. Model and develop multiplication and     manipulatives                                 opposites
division of integers                        *model multiplication and division of         inverse operations
positive and negative numbers using
manipulatives
F.7.3           2      NO.2.7.3                                       A. Apply rules (conventions) for order of   *solve problems with integers using order     order of operations
Apply rules (conventions) for order of         operations to integers including            of operations                                 integers
operations to integers and positive rational   parentheses, brackets or exponents          *solve problems with positive rational        rational numbers
numbers including parentheses, brackets or     B. Apply rules (conventions) for order of   numbers using order of operations             exponents
exponents                                      operations to positive rational numbers     *use rules for order of operations            grouping symbols
including parentheses, brackets or          including:                                    (brackets,
exponents                                   •grouping symbols ( ), [ ]                    parentheses)
•fraction bar
•exponents
•multiplication & division left to right
•addition & subtraction left to right

rev. 05-2009                                                                                                                                                                                4 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                          Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                          Objective                               Task Analysis                    Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                         *teacher word

F.7.3                  NO.2.7.1                                       A. Apply the Distributive Property of       *solve using Distributive Property to  Distributive Property
multiplication over addition or solvers simplify computations of core or
Enduring Understanding - Successful problem subtraction possess a set with addition beliefs that support
Apply the Distributive Property of                                                                                                integers                their work:
multiplication over addition or subtraction to to simplify computations with integers      subtractions of integers               solve
problem solving is important, takes significant time and5(2+3)
simplify computations with integers, fractions B. Apply the Distributive Property of
repeated efforts, and requires reflection.
5(2+3)                  simplify
and decimals                                   multiplication over addition or subtraction 5(5)          5(2)+5(3)                evaluate
to simplify computations with fractions       25          10 + 15                  decompose
C. Apply the Distributive Property of                        25
to simplify computations with decimals        5(3-2)        5(3-2)
5(3)-5(2)         5(1)
15-10            5
5
*solve using Distributive Property to
subtraction to solve problems with
fractions
2(5 + 1/2)       2(5 - 1/2)
10 + 1           10 - 1
11              9

*solve using Distributive Property to
subtraction to solve problems with
decimals
5.03 (3)
5(3) + .03(3)
15 + .09
15.09

*use subtraction to make multiplication of
decimals easier (mental math)
5(10.98)       5(11) - 5(.02)
50+4.90          55 - .10
54.9            54.9

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                               Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                            Objective                              Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                              *teacher word

Enduring Understanding - Successful representations generalize patterns and relationships.
5. Enduring Understanding - Algebraicproblem solvers possess a set of core beliefs that support their work:

F.7.3           1
5a. Essential Question - Why are mathematical rulesand repeated efforts, and requires reflection.
problem solving is important, takes significant time necessary?
NO.3.7.1                                       Compute, with and without appropriate    with and without appropriate technology:     technology
Compute, with and without appropriate          technology, with integers and positive   *solve word problems using addition of       integers
technology, with integers and positive         rational numbers using real-world        positive integers, decimals, and fractions   positive rational
rational numbers using real-world situations   situations to solve problems             (including mixed numbers)                    numbers
to solve problems                                                                       *solve word problems using subtraction of    absolute value
positive integers, decimals, and fractions
(including mixed numbers)
*solve word problems using multiplication
of positive integers, decimals, and
fractions (including mixed numbers)
*solve word problems using division of
positive integers, decimals, and fractions
(including mixed numbers)
*explain solution

6. Enduring Understanding - Computational estimations produce approximate results.
6a. Essential Question - What determines a reasonable estimation for a given situation?
F.7.3           1      NO.3.7.3                                       A. Determine when an estimate is         *determine when an estimate is sufficient    estimate
Determine when an estimate is sufficient and   sufficient                               *use estimation strategies to check for      estimation
use estimation to decide whether answers       B. Use estimation to decide whether      reasonableness of solutions                  reasonable
are reasonable in problems including           answers are reasonable in problems
fractions and decimals                         including fractions and decimals

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                               Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                           Objective                                 Task Analysis                      Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                              *teacher word

Enduring Understanding - Successful problem are related.
7. Enduring Understanding - Powers and roots solvers possess a set of core beliefs that support their work:
7a. Essential question - How are powers and roots related?
problem solving is important, takes significant time and repeated efforts, and requires reflection.
F.7.3                  NO.3.7.5                                      A. Represent and solve problem situations   with and without appropriate technology:   absolute value
Represent and solve problem situations that   that can be modeled by and solved using     *write and solve problems using the        exponents
can be modeled by and solved using            concepts of absolute value with and         concept of absolute value                  square roots
concepts of absolute value, exponents and     without appropriate technology              *write and solve problems using            perfect squares
square roots (for perfect squares) with and   B. Represent and solve problem situations   exponents                                  technology
without appropriate technology                that can be modeled by and solved using     *write and solve problems using square
concepts of exponents with and without      roots
appropriate technology
C. Represent and solve problem situations
that can be modeled by and solved using
concepts of square roots (for perfect
squares) with and without appropriate
technology

rev. 05-2009                                                                                                                                                                        7 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                       Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                                Objective                                 Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                      *teacher word

SECOND NINE WEEKS
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
1. Enduring Understanding - Patterns and relationships can be expressedand requires numerically,
problem solving is important, takes significant time and repeated efforts, graphically, reflection.
symbolically, and verbally.
1a. Essential Question - What can be learned from an identified pattern/function?
F.7.3         MC-2     A.4.7.2                                          Identify and extend patterns in real-world   *use skills in A.4.7.1                         patterns
OR-1     Identify and extend patterns in real-world       situations                                   *find missing terms and numbers in a           term
situations                                                                                    pattern or a function table                    arithmetic sequence
*predict patterns in real-world situations     geometric sequence
*write a rule to extend a pattern to the nth
term

F.7.3                  A.4.7.3                                          Interpret and write a rule for a two-        *A.4.6.2                                       function table
Interpret and write a rule for a two-operation   operation function table                     *interpret and write the algebraic rule for
function table                                   Ex.                                          one- and two-operation function tables
Ex.                                              multiply by 2, add 1
F.7.3                  A.4.7.1                                          Create and complete a function table         *identify (x-value) as input and (y-value)     function table
Create and complete a function table             (input/output) using a given rule with two   as output                                      input/output
(input/output) using a given rule with two       operations                                   *apply a given rule with two operations to
operations                                                                                    create and complete a function table

F.7.3                  A.6.7.3                                          Create and complete a function table         *create and complete a function table          function table
Create and complete a function table             (input/output) using a given rule with two   using a given rule with two operations in      input/output
(input/output) using a given rule with two       operations in real-world situations          real-world situations
operations in real-world situations
F.7.1                  A.7.7.1                                          Use, with and without appropriate            *use tables to identify situations with        technology
Use, with and without appropriate technology,    technology, tables and graphs to compare     constant or varying rates of change (Ex..      rates
tables and graphs to compare and identify        and identify situations with constant or     d=rt)                                          constant rate of
situations with constant or varying rates of     varying rates of change                      *use graphs to identify situations with        change
change                                                                                        constant or varying rates of change            varying rates of
change

rev. 05-2009                                                                                                                                                                                   8 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                       Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                              Objective                                   Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                      *teacher word

Enduring Understanding - Successful problem solvers possess a set of verbally, graphically, numerically,
2. Enduring Understanding - Functional relationships can be expressed core beliefs that support their work:
and symbolicallyis important, takes significant time and repeated efforts, and requires reflection.
problem solving and are used to solve real-world problems.
2a. Essential Question - How can a function be identified?
F.7.3                  A.6.7.1                                         A. Use tables to represent linear equations   *use tables to represent the relationship in   linear equations
Use tables and graphs to represent linear       by plotting, with and without appropriate     a linear equation, with and without            technology
equations by plotting, with and without         technology, points in a coordinate plane      appropriate technology                         coordinate plane
appropriate technology, points in a             B. Graph linear equations by plotting, with   *use graphs to represent the relationship      x-axis
coordinate plane                                and without appropriate technology, points    in a linear equation on a coordinate plane,    y-axis
in a coordinate plane                         with and without appropriate technology        ordered pairs
origin

2b. Essential Question - What can be learned from studying functions?
F.7.3           1      A.5.7.2                                         A. Solve simple linear equations using        *solve linear equations                        linear equations
Solve simple linear equations using             integers                                      *graph solution points on a coordinate         integers
integers and graph on a coordinate plane        B. Graph simple linear equations in a         plane using a T-chart                          coordinate plane
use a T-chart                                                                                                                                x-axis
y-axis
origin
F.7.1                  G.10.7.1                                        Plot points in the coordinate plane           *identify four quadrants on coordinate         coordinate plane
Plot points in the coordinate plane                                                           plane                                          x-axis
*identify x and y axis                         y-axis
*use ordered pairs to plot points on           ordered pairs
origin
F.7.3           1      A.6.7.2                                         Represent, with and without appropriate       *same as A.6.7.1 with real-world               technology
Represent, with and without appropriate         technology, linear equations by plotting      applications                                   linear equations
technology, linear equations by plotting and    and graphing points in the coordinate                                                        coordinate plane
graphing points in the coordinate plane using   plane using all four quadrants given data                                                    quadrants
all four quadrants given data in a table from   in a table from a real-world situation                                                       origin
a real-world situation                                                                                                                       x-axis
y-axis

3. Enduring Understanding - Algebraic representations generalize patterns and relationships.
3a. Essential Question - Why are equations and inequalities useful?
F.7.3                  A.5.7.4                                         A. Write algebraic expressions using          *write and solve algebraic expressions         algebraic
Write and evaluate algebraic expressions        positive rational numbers                     with one or more variables using positive      expressions
using positive rational numbers                 B. Evaluate algebraic expressions using       rational numbers                               rational numbers
positive rational numbers                                                                    variables

rev. 05-2009                                                                                                                                                                                9 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                        Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                                 Objective                                 Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                       *teacher word

F.7.3           4      A.5.7.3                                 A. Translate phrases into algebraic     *express word phrases using variables, algebraic
positive numbers, rational core and expressions
Enduring Understanding - Successful problem solvers possess a set of numbers,beliefs that support
Translate phrases and sentences into    expressions                                                                                                                    their work:
algebraic expressions and equations     B. Translate sentences into equations
problem solving is important, takes significant time andoperations use of the Distributive requires reflection.
including parentheses and positive and
repeated efforts, and
including parentheses, positive numbers *explain the
equations
rational numbers
rational numbers and simplify algebraic and rational numbers                    Property to simplify like terms        coefficient
expressions by combining like terms     C. Simplify algebraic expressions by    *simplify algebraic expressions by     constant
combining like terms                    combining like terms                   variable
F.7.3                  A.5.7.1                                           A. Solve and graph one-step linear           *identify phrases that indicate inequalities   linear equations
Solve and graph one-step linear equations         equations using a variety of methods (Ex.    in real-world situations                       inequalities
and inequalities using a variety of methods       hands-on, inverse operations, symbolic)      *solve one-step linear equations and           inverse operations
(Ex. hands-on, inverse operations, symbolic)      with real-world application with and         inequalities, with and without appropriate     technology
with real-world application, with and without     without appropriate technology               technology, using a variety of methods         variables
appropriate technology                            B. Solve and graph one-step linear           *graph one-step equations and
inequalities using a variety of methods      inequalities on a number line
(Ex. hands-on, inverse operations,
symbolic) with real-world application with
and without appropriate technology

F.7.3           1      NO.2.7.2                                          A. Apply the addition, subtraction,          *know the properties of equality for           equations
Apply the addition, subtraction, multiplication   multiplication and division properties of    addition, subtraction, multiplication, and     integers
and division properties of equality to one-step   equality to one-step equations with          division and apply to one-step equations       properties of equality
equations with integers, fractions, and           integers                                     using integers                                 inverse operations
decimals                                          B. Apply the addition, subtraction,          *know the properties of equality for           one-step equations
multiplication and division properties of    addition, subtraction, multiplication, and
equality to one-step equations with          division and apply to one-step equations
fractions                                    using fractions
C. Apply the addition, subtraction,          *know the properties of equality for
multiplication and division properties of    addition, subtraction, multiplication, and
equality to one-step equations with          division and apply to one-step equations
decimals                                     using decimals

rev. 05-2009                                                                                                                                                                                   10 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                        Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                               Objective                                   Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                       *teacher word

THIRD NINE WEEKS
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
1. Enduring Understanding - Graphical representations and statistical measures influence interpretations
problem solving is important, takes significant time and repeated efforts, and requires reflection.
1a. Essential Question - How can the results of a statistical investigation be used to support an argument?

F.7.1           2      DAP.14.7.1                                       Identify different ways of selecting          *develop appropriate questions for             sample vs.
Identify different ways of selecting samples     samples and compose appropriate               surveys                                        population
and compose appropriate questions                questions                                     *identify survey response, random sample,      random sample
Ex.                                              Ex.                                           representative sample & convenience            survey response
survey response, random sample,                  survey response, random sample,               sample                                         representative
representative sample and convenience            representative sample and convenience                                                        sample
sample                                           sample                                                                                       convenience sample

F.7.1                  DAP.14.7.2                                       A. Explain that line graphs are appropriate   *identify that line graphs are used to graph   line graph
Explain which types of display are appropriate   for graphing change over time                 change over time                               circle graph
for various data sets (line graph for change     B. Explain that circle graphs are             *identify that circle graphs are used to       part-to-whole
over time, circle graph for part-to-whole        appropriate for graphing part-to-whole        graph part-to-whole comparisons                comparison
comparison, scatter plot for trends)             comparisons                                   *identify that scatter plots are used to       scatter plot
C. Explain that scatter plots are             graph trends                                   trends
appropriate graphing trends

F.7.3           1      DAP.15.7.2                                       A. Analyze, with and without appropriate      with and without appropriate technology: technology
Analyze, with and without appropriate            technology, a set of data by using and        *find the mean, median and mode for a      central tendencies
technology, a set of data by using and           comparing measures of central tendencies      set of data                                mean
comparing measures of central tendencies         (mean, median, mode)                          *compare the mean, median, mode for a average
(mean, median, mode) and measures of             B. Analyze, with and without appropriate      set of data                                median
spread (range, quartile, interquartile range)    technology, a set of data by using and        *find measures of spread (range, quartile, mode
comparing measures of spread (range,          interquartile range) for a set of data     measure of spread
quartile, interquartile range)                *compare measures of spread using a box- range
and-whiskers plot                          quartile
*analyze measures of central tendency      interquartile range
and measures of spread (including          outliers
outliers) for a set of data

F.7.1           1      DAP.15.7.1                                       Analyze data displays, including ways that *identify data display composition:               misleading
Analyze data displays, including ways that       they can be misleading                     •scale                                            scale
they can be misleading                                                                      •data sample population techniques                intervals
•misuse of central tendencies                     central tendencies

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Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                          Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                          Objective                               Task Analysis                    Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                         *teacher word

F.7.1         MC-3     DAP.14.7.3                                     A. Construct and interpret circle graphs   with and without appropriate technology: double bar graphs
OR-1     Enduring Understanding - with and without appropriate solvers *circle
Construct and interpret circle graphs, box- Successful problemtechnology possess a set of core beliefs that support                circle graphs         their work:
and-whisker plots, histograms, scatter         B. Construct and interpret box-and-
problem solving is important, takes significant time and repeated efforts, and            •explain part-to-whole correlation requires reflection.
box-and-whisker
plots and double-line graphs, with and without whisker plots with and without appropriate •demonstrate taking numbers from         plots
appropriate technology                         technology                                 fractions to decimals to percents to     histograms
C. Construct and interpret histograms with degrees                                  scatter plots
and without appropriate technology         •use a protractor to create angles       double line graphs
D. Construct and interpret scatter plots   •construct circle graph                  technology
with and without appropriate technology    •interpret data                          correlation
E. Construct and interpret double-line     *box-and-whisker plots                   trend line
graphs with and without appropriate        •represent each quartile as 25%          line of best fit
technology                                 •analyze and record the spread           stem-and-leaf plots
•interpret central tendencies            Venn diagrams
•find medians and extremes
•construct box-and-whisker plot & identify
outliers, if any
*histogram
•create a frequency table, place the
frequency of occurency on y-axis &
intervals of continuous data on x-axis
•construct histogram (similar to a bar
graph except bars touch)
•compare independent variables to
dependent variables
•interpret data
*scatter plot
•determine dependent (y) variable &
independent (x) variable
•plot points (x,y)
•decide the overall correlation
•interpret data
*double line graphs
•determine dependent (y) & independent
(x) variable
•identify the data sets (2 or more)

rev. 05-2009                                                                                                                                                                     12 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                      Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                               Objective                                  Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                     *teacher word

•plot points & connect data set
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
independently
compare changes over time
problem solving is important, takes significant time and•evaluate andkey efforts, and requires reflection.
repeated all graphs appropriately
*title, label &

2. Enduring Understanding - Sampling affects the relationship between experimental and theoretical
probability.
2a. Essential Question - How are experimental and theoretical probability related?
F.7.1           1      DAP.17.7.1                                      Demonstrate that probability can take any     *describe events that will not occur as       probability
Understand that probability can take any        value between 0 and 1 (events that are        probability 0                                 certainty
value between 0 and 1 (events that are not      not going to occur have probability 0,        *describe events that will certainly occur    outcome
going to occur have probability 0, events       events certain to occur have probability 1)   as probability of 1                           most likely
certain to occur have probability 1)                                                          *create situations in which the probability   least likely
falls between 0 & 1                           equally likely
F.7.1                  DAP.17.7.2                                    Design, with and without appropriate            with and without appropriate technology:      technology
Design, with and without appropriate          technology, an experiment to test a             *create theoretical probabilities and test    theoretical probability
technology, an experiment to test a           theoretical probability and explain how the     them (experiment)                             experimental
theoretical probability and explain how the   results may vary                                *explain how results vary                     probability
results may vary                              Ex.                                             *formulate reasons why theoretical and
Ex.                                           suggested materials for simulations are         experimental results vary
suggested materials for simulations are: two- two-color counters, a number cube, and
color counters, a number cube, and spinners spinners

3. Enduring Understanding - Linear relationships are characterized by a constant rate of change.
3a. Essential Question - Why are linear functions useful?
F.7.1           1      DAP.16.7.1                                      A. Make, with and without appropriate         *given a scatter plot determine a             technology
Make, with and without appropriate              technology, conjectures of possible           relationship between the dependent and        conjectures
technology, conjectures of possible             relationships in a scatter plot               independent variables                         scatter plot
relationships in a scatter plot and             B. Approximate the line of best fit (trend    *make a conjecture about possible             line of best fit
approximate the line of best fit (trend line)   line)                                         relationships between variables               trend line
*construct a line of best fit to most         dependent variable
appropriately represent the "average"         independent variable
relationship between dependent and            relationship
independent variables

rev. 05-2009                                                                                                                                                                                 13 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                    Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                               Objective                                 Task Analysis                       Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                   *teacher word

4. Enduring Understanding - Selectionproblem solvers possess atools and units depends on real-world
Enduring Understanding - Successful of standard measurement set of core beliefs that support their work:
situations.
problem solving is important, takes significant time and repeated efforts, and requires reflection.
4a. Essential Question - Why are scientific units and tools used to measure different attributes?
F.7.2           1      M.12.7.1                                         For real-world problems to the required     *state the common standard units             customary
Understand, select and use the appropriate       degree of accuracy:                         *apply the prefixes of the metric system     measurement
units and tools (metric and customary) to        A. Select and use the appropriate units     *select the appropriate tool to measure in   hours to minutes
measure length, weight, mass and volume to       and tools (metric and customary) to         applicable units (Ex. ruler, meter stick,    feet to inches
the required degree of accuracy for real-world   measure length                              gallon, liter, grams, pound, tons,           quarts to gallons
problems                                         B. Select and use the appropriate units     protractor, degrees, etc.)                   benchmark prefixes:
and tools (metric and customary) to         *apply the appropriate tool to measure       milli-, centi-, kilo-
Note: Use the appropriate tool to measure        measure weight                              length, weight, mass, and volume of a        length
degrees of angles. Students are given a          C. Select and use the appropriate units     variety of real-world objects                weight
protractor to use on the Benchmark test.         and tools (metric and customary) to         *use a variety of measurement tools          mass
There have been test questions concerning        measure mass                                (graduated cylinder, triple beam balance,    volume
the use of the protractor.                       D. Select and use the appropriate units     meter stick, protractor and others)          degree
and tools (metric and customary) to                                                      metric units
measure volume                                                                           gram/liter/meter
E. Select and use the appropriate tool to                                                pounds
measure angle degrees (protractor)                                                       gallons
yards
capacity
protractor

C.7.1                  M.13.7.2                                         A. Draw and measure distance to the         *know measuring begins with the first tic accurately
Draw and measure distance to the nearest         nearest 1/16 inch accurately                mark                                        precision
millimeter and 1/16 inch accurately.             B. Draw and measure distance to the         *be able to record distance to the nearest tic mark/origin
nearest millimeter accurately               1/16 inch
*be able to draw a line to the nearest 1/16
inch
*be able to record distance to the nearest
millimeter
*be able to draw a line to the nearest
millimeter

rev. 05-2009                                                                                                                                                                             14 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                   Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                           Objective                                  Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                  *teacher word

F.7.1           2      M.12.7.2                                    Compare units within the same system *compare standard units
Enduring Understanding - Successful problem solvers possess a set of
Understand relationships among units within                              core beliefs that support their work:
*compare metric units
the same system                                                                    seconds↔minutes↔
problem solving is important, takes significant time and repeated efforts, and requires reflection.
hours↔days
inches↔feet↔
yards↔miles
ounces↔cups↔
pints↔
quarts↔gallons
benchmark prefixes
kilo-↔deci-↔centi-
↔milli-
grams
liters
meters
C.7.1                  M.13.7.1                                     A. Solve real-world problems involving two   *explain time concepts (clock and              elapsed time
Solve real-world problems involving two or   or more elapsed times, counting forward      calendar; Ex. number of days in each
more elapsed times, counting forward and     and backward on a calendar                   month) in the context of counting forward
backward (calendar and clock)                B. Solve real-world problems involving two   and backward
or more elapsed times, counting forward      *solve real-world situations involving two
and backward on a clock                      or more elapsed times

5. Enduring Understanding - Relationships exist among the angles, sides, lengths, perimeters, areas, and
volumes of geometric figures.
5a. Essential Question - How are interior angles and the number of sides of the polygon related?
C.7.1         OR-1     G.8.7.4                                      A. Use paper or physical models to           *fold/cut paper to make physical models of     interior angles
Use paper or physical models to              determine the sum of the measures of         quadrilaterals & triangles                     quadrilaterals
determine the sum of the measures of         interior angles of triangles                 *demonstrate that the measures of the          triangles
interior angles of triangles and             B. Use paper or physical models to           sum of the interior angles of quadrilaterals   non-overlapping
quadrilaterals                               determine the sum of the measures of         is 360º                                        overlapping
interior angles of quadrilaterals            *demonstrate that the measures of the          diagonals
sum of the interior angles of triangles is
180º
*find a missing angle given two angles in a
triangle
*find a missing angle given three angles in

rev. 05-2009                                                                                                                                                                         15 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                        Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                               Objective                                  Task Analysis                          Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                       *teacher word

Enduring Understanding - Successful problem can change position and maintain the same attributes work:
6. Enduring Understanding - Geometric figures solvers possess a set of core beliefs that support their on a
coordinate plane. important, takes significant time and repeated efforts, and requires reflection.
problem solving is
6a. Essential Question - How does the movement of a geometric figure affect its attributes?
F.7.1                  G.10.7.2                                        Plot points that form the vertices of a       *identify vertices of polygons             vertices
Plot points that form the vertices of a         geometric figure and draw, identify and       *replicate (draw) polygons on a coordinate faces
geometric figure and draw, identify and         classify the figure                           plane using ordered pairs                  edges
classify the figure                                                                           *classify figures according to number of
vertices
C.7.1           1      G.9.7.2                                         A. Perform translations of two-dimensional    *fold/cut paper to perform translations of 2-   rotation
Perform translations and reflections of two-    figures using a variety of methods (paper     dimensional figures                             translation
dimensional figures using a variety of          folding, tracing, graph paper)                *trace to perform translations of 2-            reflection
methods (paper folding, tracing, graph paper)   B. Perform reflections of two-dimensional     dimensional figures                             two-dimensional
figures using a variety of methods (paper     *use graph paper to perform translations        figure
folding, tracing, graph paper)                of 2-dimensional figures
*fold/cut paper to perform reflections of 2-
dimensional figures
*trace to perform reflections of 2-
dimensional figures
*use graph paper to perform reflections of
2-dimensional figures

7. Enduring Understanding - Relationships exist among angles, sides and triangles.
7a. Essential question - How can properties of triangles be applied to solve problems?
F.7.1                  G.8.7.6                                         Develop the properties of similar figures     *categorize shapes                              similar figures
Develop the properties of similar figures       (ratio of sides and congruent angles)         *explain the concept of similarity and scale    congruent angles
(ratio of sides and congruent angles)                                                         *identify similar figures according to          ratio
properties of sides and angles                  scale

F.7.1         MC-1     M.13.7.5                                        Apply properties (scale factors, ratio, and   *identify corresponding parts of triangles      scale factors
OR-1     Apply properties (scale factors, ratio, and     proportion) of congruent or similar           *set up proportion formulas (scale factors,     ratio
proportion) of congruent or similar             triangles to solve problems involving         ratios) using variables to find a missing       proportion
triangles to solve problems involving           missing lengths and angle measures            length of a triangle                            congruent
missing lengths and angle measures                                                            *apply correlation properties to find a         similar
missing angle of a triangle                     correlation

rev. 05-2009                                                                                                                                                                            16 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                           Objective                                 Task Analysis                       Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                               *teacher word

FOURTH NINE WEEKS
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
1. Enduring Understanding - Constructions are time and repeated efforts, and requires reflection.
problem solving is important, takes significant based on properties of geometric figures.
1a. Essential Question - How are geometric properties used in constructions?
F.7.2           2      G.8.7.1                                      A. Identify, draw, classify and compare 2- *identify and model 2-dimensional             regular polygon
Identify, draw, classify and compare         D geometric figures using models and real- geometric figures including regular and       irregular polygon
geometric figures using models and real-     world examples                             irregular polygons                            two-dimensional
world examples                               B. Identify, draw, classify and compare 3- *classify 2-dimensional geometric figures     figures
D geometric figures using models and real- *compare 2-dimensional geometric figures      three-dimensional
world examples                             using real-world examples                     figures (pyramid,
*draw 2-dimensional geometric figures         cone, sphere,
*identify and model 3-dimensional figures     cylinder, cube, prism)
*compare geometric 3-dimensional figures      (rectangle, square,
using real-world examples                     parallelogram,
*draw 3-dimensional geometric figures         trapezoid, rhombus)
circle
ellipse
triangles (scalene,
isosceles, equilateral,
right, acute, obtuse)
pentagons,
hexagons, octagon,
decagon, n-gon

F.7.2           3      G.8.7.2                                      Investigate geometric properties and their   *draw convex and concave polygons           one-dimensional
Investigate geometric properties and their   relationships in one-, two-, and three-      *compare and contrast the characteristics   model
relationships in one-, two-, and three-      dimensional models, including convex and     of one-, two-, and three-dimensional        two-dimensional
dimensional models, including convex and     concave polygons                             models to see relationships among them      model
concave polygons                                                                                                                      three-dimensional
model
angles
convex polygon
concave polygon
classification of
polygons by sides
classification of
polygons by angles

rev. 05-2009                                                                                                                                                                           17 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                             Objective                                Task Analysis                      Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                               *teacher word

F.7.2                  G.8.7.3                                        A. Identify the pairs of angles formed and *identify vertical angles and know that they angles
Enduring Understanding - Successful between the solvers are congruent
Recognize the pairs of angles formed and the relationship problemangles of two possess a set of core beliefs that support                   right angle          their work:
the relationship between the angles            intersecting lines (vertical, adjacent, linear *identify supplementary angles and know acute angle
problem solving is important, takes significant time andthat they are congruent to 180º
pair, supplementary, perpendicular)
repeated efforts, and requires reflection.  obtuse angle
including two intersecting lines and
parallel lines cut by a transversal (vertical, B. Identify the pairs of angles formed and *identify complementary angles and know intersecting lines
supplementary, complementary,                  the relationship between parallel lines cut that they are congruent to 90º                   transversal
corresponding, alternate interior, alternate by a transversal (vertical, supplementary, *identify corresponding angles and know vertical angles
exterior angles and linear pair)               corresponding, alternate interior, alternate that they are congruent                         supplementary
exterior angles and linear pair)               *identify alternate interior angles/alternate angles
C. Identify the pairs of angles and the        exterior angles and know that they are        complementary
relationship that exists to form               congruent                                     angles
complementary angles                           *identify linear pairs and adjacent angles corresponding
*identify the relationship of the angles      angles
formed by a transversal and parallel lines alternate interior
angles
alternate exterior
angles
linear pair
alternate pair
parallel lines
interior
exterior
congruent

rev. 05-2009                                                                                                                                                                             18 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                 Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                              Objective                                Task Analysis                      Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                *teacher word

2. Enduring Understanding - Relationships exist among the angles, sides, lengths, perimeters, and two-
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
dimensional areas. important, takes significant time and repeated efforts, and requires reflection.
problem solving is
2a. Essential Question - How are the areas of rectangles, parallelograms, triangles, trapezoids, and circles
related?
F.7.2                  M.13.7.3                                       A. Develop and use strategies to solve      *use a variety of strategies to solve for   trapezoid
Develop and use strategies to solve problems   problems involving area of a trapezoid      area of a trapezoid                         altitude/height
involving area of a trapezoid and              B. Develop and use strategies to solve      *apply formula for area of trapezoid        base 1, base 2
circumference and area of a circle             problems involving circumference of a       *use a variety of strategies to solve for   circumference
circle                                      finding the circumference of a circle       area of a circle
C. Develop and use strategies to solve      *apply the formula for circumference of a   radius
problems involving area of a circle         circle                                      pi (π)
*use a variety of strategies to solve for
finding the area of a circle
*apply the formula for area of a circle

F.7.3                  M.12.7.3                                       A. Find different areas for a given         *given a perimeter, solve for area of       linear units
Find different areas for a given perimeter     perimeter                                   various shapes                              volume
and find a different perimeter for a given     B. Find a different perimeter for a given   *label using appropriate units              perimeter
area                                           area                                        *given an area, identify all perimeter      area
combinations using whole numbers            square units
*label using appropriate units

rev. 05-2009                                                                                                                                                                     19 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                     Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                             Objective                                  Task Analysis                         Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                    *teacher word

3. Enduring Understanding - Generalizations can be made to represent the relationship between area and
Enduring Understanding - Successful problem solvers possess a set of core beliefs that support their work:
volume. solving is important, takes significant time and repeated efforts, and requires reflection.
problem
3a. Essential Question - How do the dimensions of a geometric figure affect area, surface area, and
volume?
F.7.2           2      M.13.7.4                                       A. Derive and use formulas for surface       *use the formulas for finding the areas of     surface area
Derive and use formulas for surface area and   area of prisms and justify them using        common shapes: circles, squares,               volume
volume of prisms and cylinders and justify     geometric models and common materials        triangles and rectangles                       prisms
them using geometric models and common         B. Derive and use formulas for surface       *draw nets and use them to find the            cylinders
materials                                      area of cylinders and justify them using     formula for the surface area of a prism        nets
geometric models and common materials        *use the formula to find the surface area      derived formula
C. Derive and use formulas for volume of     of a prism
prisms and justify them using geometric      *use real world items to compare to nets
models and common materials                  *apply appropriate units to surface area
D. Derive and use formulas for volume of     *draw nets and use them to find the
cylinders and justify them using geometric   formula for the surface area of a cylinder
models and common materials                  *use the formula to find the surface area
of a cylinder
*use manipulatives to find a formula for
the volume of rectangular prisms
*apply the derived formula of volume to a
variety of rectangular prisms (including
common item i.e., matchbox, tissue box,
shoebox, etc.)
*identify the surface area parts of a
cylinder as a rectangle attached to two
circles at each end
*use manipulatives to find a formula for
the volume of cylinders
*apply the derived formula of volume to a
variety of cylinders (paper towel roll, soup
can, etc.)

rev. 05-2009                                                                                                                                                                         20 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                                   Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                            Objective                                  Task Analysis                        Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                                  *teacher word

3b. Essential Question - How can formulas be developed using models?core beliefs that support their work:
Enduring Understanding - Successful problem solvers possess a set of
F.7.2                  M.13.7.7                                Estimate and compute the area of more   *apply the formulas for finding the areas of regular two-
problem solving is of more              complex or significant time
Estimate and compute the areaimportant, takes irregular two-dimensional and repeated efforts, and requires reflection.
basic shapes: circles, quadrilateral,        dimensional shapes
complex or irregular two-dimensional    shapes by dividing them into more basic triangles                                    irregular two-
shapes by dividing them into more basic shapes                                  *divide irregular/complex figures into basic dimensional shapes
shapes                                                                          2-dimensional shapes                         area
*find the area of basic shapes and           complex
combine these areas to find the area of
the irregular figure

F.7.2                  G.8.7.5                                        Model and develop the concept that pi (π)   *measure the radius and diameter              pi (π)
Model and develop the concept that pi (π) is is the ratio of the circumference to the      *measure the circumference of circle          ratio
the ratio of the circumference to the diameter diameter of any circle                      *discover a relationship between the          circumference
of any circle                                                                              diameter and circumference (known as pi)      diameter
*apply the formula for finding the            radius
circumference of a circle                     chord

4. Enduring Understanding - Geometric relationships exist between two-dimensional and three-dimensional
figures.
4a. Essential Question - How are two-dimensional and three-dimensional figures related?
F.7.2           1      G.11.7.1                                      Build three-dimensional solids from two-     *match 2-dimensional patterns to 3-           rectangular prism
Build three-dimensional solids from two-      dimensional patterns (nets)                  dimensional solids                            pyramids
dimensional patterns (nets)                                                                *build 3-dimensional solids from a given 2-   cylinders
dimensional pattern (net)                     cones
3-dimensional solids
2-dimensional
patterns
nets

F.7.2           1      G.11.7.2                                      Construct a building out of cubes from a     *practice visualization skills using sets of cubes
Construct a building out of cubes from a      set of views (front, top, side)              cubes (front, back, right side, left side and
set of views (front, top, side)                                                            top)
*construct buildings using cubes

rev. 05-2009                                                                                                                                                                           21 of 22

Arch Ford/Northwest Arkansas Instructional Alignment
AR Department of Education
NCTM Focal    # of                                                                                                                                    Essential
Materials/
Point or   Times    CONTENT STANDARD/ Student Learning                       Objective                             Task Analysis                   Vocabulary
Resources
Connector   Tested           Expectations (SLE)                                                                                                   *teacher word

F.7.1           1      G.9.7.1                                      Examine the congruence, similarity, and *describe transformations-translations, congruence
Enduring Understanding - Successful symmetry of solvers possess arotations core beliefs that support
Examine the congruence, similarity, and line line or rotationalproblem objects       reflections, and set of                 similarity            their work:
or rotational symmetry of objects using      using transformations                   *identify symmetrical objects using     line symmetry
repeated
problem solving is important, takes significant time andtransformations efforts, and requires reflection.
transformations                                                                                                              rotational symmetry
*identify congruence/similarity in      transformations
relationship to transformations         translations
reflections
rotations



rev. 05-2009                                                                                                                                                               22 of 22
problem solving is important, takes significant time andtransformations efforts, and requires reflection.
transformations                                                                                                              rotational symmetry
*identify congruence/similarity in      transformations
relationship to transformations         translations
reflections
rotations



rev. 05-2009                                                                                                                                                               22 of 22

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