Download Muskingum Valley ESC Standards-Based Mathematics Course of

Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Number, Number Sense and Operations Standard
Students demonstrate number sense, include an understanding of number systems and operations and how they relate to one another. Students
compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
C. Apply properties of
operations and the real
number system and justify
when they hold for a set of
numbers.
Number and Number Systems
D. Connect physical, verbal
and symbolic
representations of integers,
rational numbers and
irrational numbers.
Number and Number Systems
E. Compare, order and
determine equivalent forms
of real numbers.
F. Explain the effects of
operations on the magnitude
of quantities.
G. Estimate, compute and
solve problems involving
real numbers, including
ratio, proportion and
percent, and explain
solutions.
(Gr. 9#1) Identify and justify whether properties (closure, identity, inverse,
commutative and associative) hold for a given set and operations; such as, even
integers and multiplication.
(Gr.10#1) Connect physical, verbal and symbolic representations of irrational numbers;
such as, construct √2 as a hypotenuse or on a number line.
Meaning of Operations
(Gr.10#2) Explain the meaning of the nth root.
Number and Number Systems
(Gr.9#2) Compare, order and determine equivalent forms for rational and irrational
numbers.
Meaning of Operations
(Gr.9#3) Explain the effects of operations, such as multiplication or division, and of
computing powers and roots on the magnitude of quantities.
Computation and Estimation
(Gr.9#4) Demonstrate fluency in computations using real numbers.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
(9#4) Elements of computational
fluency:
 efficiency
 accuracy
 fluency
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Number, Number Sense and Operations Standard (continued)
Students demonstrate number sense, include an understanding of number systems and operations and how they relate to one another. Students
compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
I. Estimate, compute and
solve problems involving
scientific notation, square
roots and numbers with
integer exponents.
Computation and Estimation
(Gr.9#5) Estimate the solutions for problem situations involving square and cube roots.
(9#5) Always use ≈ instead of =
when numbers are approximate.
(Gr.10#4) Approximate the nth root of a given number greater than zero between
consecutive integers when n is an integer; such as, the 4th root of 50 is between 2 and 3.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Measurement Standard
Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
D. Use proportional
reasoning and apply indirect
measurement techniques,
including right triangle
trigonometry and properties
of similar triangles, to solve
problems involving
measurements and rates.
Measurement Units
(Gr.9#1) Convert rates within the same measurement system; such as, miles per hour to
feet per second; kilometers per hour to meters per second.
Use Measurement Techniques and Tools
(Gr.9#2) Use unit analysis to check computations involving measurement.
(Gr.9#3) Use the ratio of lengths in similar two-dimensional figures or threedimensional objects to calculate the ratio of their areas or volumes respectively.
(Gr.9#4) Use scale drawings and right triangle trigonometry to solve problems that
include unknown distances and angle measures.
(Gr.9#5) Solve problems involving unit conversion for situations involving distances,
areas, volumes and rates within the same measurement system.
(9#4) scale drawing – drawing
that is similar but is either larger
or smaller than the original object
– example of a scale drawing is a
map.
(Gr.10#5) Determine the measures of central and inscribed angles and their associated
major and minor arcs.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Geometry and Spatial Sense Standard
Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures
and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve
problems.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
A. Formally define
geometric figures.
Characteristics and Properties
(Gr.10#1) Formally define and explain key aspects of geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median, altitude, midsegment);
c. points of concurrency related to triangles (centroid, incenter, orthocenter and
circumcenter);
d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector,
segment, inscribed angle).
midsegment = midline
Midsegment Theorem: The
segment connecting the midpoints
of two sides of a triangle is
parallel to the third side and is
half as long.
(Gr.10#2) Recognize and explain the necessity for certain terms to remain undefined,
such as point, line and plane.
Transformations and Symmetry
(Gr.10#6) Identify the reflection and rotation symmetries of two- and threedimensional figures.
Visualization and Geometric Models
(Gr.10#10) Solve problems involving chords, radii and arcs within the same circle.
(10#10) chord – line segment that
joins two points on a circle
radii – (plural of radius)
arc – part of a curve between any
two points
C. Recognize and apply
angle relationships in
situations involving
intersecting lines,
perpendicular lines and
parallel lines.
Characteristics and Properties
(Gr.8#2) Recognize the angles formed and the relationship between the angles when
two lines intersect and when parallel lines are cut by a transversal.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Geometry and Spatial Sense Standard (continued)
Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures
and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve
problems.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
E. Draw and construct
representations of two- and
three-dimensional geometric
objects using a variety of
tools, such as straightedge,
compass and technology.
Spatial Relationships
(Gr.10#4) Construct right triangles, equilateral triangles, parallelograms, trapezoids,
rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic
geometry software.
(Gr.10#5) Construct congruent or similar figures using tools, such as compass,
straightedge and protractor or dynamic geometry software.
Transformations and Symmetry
(Gr.10#7) Perform reflections and rotations using compass and straightedge
constructions and dynamic geometry software.
F. Represent and model
transformations in a
coordinate plane and
describe the results.
G. Prove or disprove
conjectures and solve
problems involving twoand three-dimensional
objects represented within a
coordinate system.
(10#4-5) Provide experiences with
all tools to construct various
figures. ex. compass, straightedge,
protractor and dynamic geometry
as appropriate.
Transformations and Symmetry
(Gr.10#8) Derive coordinate rules for translations, reflections and rotations of
geometric figures in the coordinate plane.
(10#7) Use graph paper.
(10#8) Distance between an
object and its translated, reflected
or rotated image can be shown
and calculated if images are
placed in a coordinate plane.
Visualization and Geometric Models
(Gr.9#3) Analyze two-dimensional figures in a coordinate plane; such as, use slope and
distance formulas to show that a quadrilateral is a parallelogram.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Geometry and Spatial Sense Standard (continued)
Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures
and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve
problems.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
H. Establish the validity of
conjectures about geometric
objects, their properties and
relationships by
counterexample, inductive
and deductive reasoning,
and critiquing arguments
made by others.
Characteristics and Properties
(Gr.10#3) Make, test and establish the validity of conjectures about geometric
properties and relationships using counterexample, inductive and deductive reasoning,
and paragraph or two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and quadrilaterals;
d. test a conjecture using basic constructions made with a compass and straightedge
or technology.
Visualization and Geometric Models
(Gr.10#10) Solve problems involving chords, radii and arcs within the same circle.
I. Use right triangle
trigonometry relationships
to determine lengths and
angle measures.
Characteristics and Properties
(Gr.9#1) Define the basic trigonometric ratios in right triangles: sine, cosine and
tangent.
(Gr.9#2) Apply proportions and right triangle trigonometric ratios to solve problems
involving missing lengths and angle sizes in similar figures.
(9#2) mnemonic device:
SOHCAHTOA
sine = opposite over hypotenuse
cosine = adjacent over hypotenuse
tangent = opposite over adjacent
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series - Patterns, Functions and Algebra Standard
Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze,
model and solve problems using various representations, such as tables, graphs and equations.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
A. Generalize and explain
patterns and sequences in
order to find the next term
and the nth term.
Use Patterns, Relations and Functions
(9#2) tabular = table
(Gr.9#2) Generalize patterns using functions or relationships (linear, quadratic and
exponential), and freely translate among tabular, graphical and symbolic
representations.
Use a function table to help
identify the pattern in a sequence.
(9#1) domain – set of values of
the independent variable of a
function
B. Identify and classify
functions as linear or
nonlinear, and contrast their
properties using tables,
graphs or equations.
Use Patterns, Relations and Functions
(Gr.9#1) Define function with ordered pairs in which each domain element is assigned
exactly one range element.
range – possible values for y in a
function
function – relation when the
value of x has a value of y
(Gr.9#3) Describe problem situations (linear, quadratic and exponential) by using
tabular, graphical and symbolic representations.
(Gr.10#1) Define function formally and with f(x) notion.
(Gr.10#2) Describe and compare characteristics of the following families of functions:
square root, cubic, absolute value and basic trigonometric functions; such as, general
shape, possible number of roots, domain and range.
C. Translate information
from one representation
(words, table, graph or
equation) to another
representation of a relation
or function.
Use Patterns, Relations and Functions
(Gr.9#2) Generalize patterns using functions or relationships (linear, quadratic and
exponential), and freely translate among tabular, graphical and symbolic
representations.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series - Patterns, Functions and Algebra Standard (continued)
Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze,
model and solve problems using various representations, such as tables, graphs and equations.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
D. Use algebraic
representations, such as
tables, graphs, expressions,
functions and inequalities,
to model and solve problem
situations.
Use Algebraic Representation
(Gr.9#7) Use formulas to solve problems involving exponential growth and decay.
(Gr.9#11) Add, subtract, multiply and divide monomials and polynomials (division of
polynomials by monomials only.
(Gr.9#12) Simplify rational expressions by eliminating common factors and applying
properties of integer exponents.
(Gr.10#3) Solve equations and formulas for a specified variable; such as, express the
base of a triangle in terms of the area and height.
(Gr.10#4) Use algebraic representations and functions to describe and generalize
geometric properties and relationships.
(Gr.10#5) Solve simple linear and nonlinear equations and inequalities having square
roots as coefficients and solutions.
(Gr.10#6) Solve equations and inequalities having rational expressions as coefficients
and solutions.
(Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic
exponential or square root functions.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series - Patterns, Functions and Algebra Standard (continued)
Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze,
model and solve problems using various representations, such as tables, graphs and equations.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
E. Analyze and compare
functions and their graphs
using attributes, such
as,rates of change, intercepts
and zeros.
Use Patterns, Relations and Functions
F. Solve and graph linear
equations and inequalities.
Use Algebraic Representation
(Gr.9#4) Demonstrate the relationship among zeros of a function, roots of equations
and solutions of equations graphically and in words.
(Gr.9#5) Describe and compare characteristics of the following families of functions;
linear, quadratic and exponential functions; such as, general shape, number of roots,
domain, range, rate of change, maximum or minimum.
(Gr.9#6) Write and use equivalent forms of equations and inequalities in problem
situations; such as, changing a linear equation to the slope-intercept form.
(9#6) y = mx + b
m is the slope
b is the y-intercept
(Gr.9#8) Find linear equations that represent lines that pass through a given set of
ordered pairs, and find linear equations that represent lines parallel or perpendicular to
a given line through a specific point.
(Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic,
exponential or square root functions.
G. Solve quadratic
equations with real roots by
graphing, formula and
factoring.
Use Algebraic Representation
(Gr.10#8) Graph the quadratic relationship that defines circles.
(Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic,
exponential or square root functions.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series - Patterns, Functions and Algebra Standard (continued)
Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze,
model and solve problems using various representations, such as tables, graphs and equations.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
H. Solve systems of linear
equations involving two
variables graphically and
symbolically.
Use Algebraic Representation
(Gr.9#9) Solve and interpret the meaning of 2 by 2 systems of linear equations
graphically, by substitution and by elimination, with and without technology.
(Gr.10#7) Solve systems of linear equalities.
(10#7) A system of linear
equations can have one solution,
many solutions or no solution.
(Gr.10#11) Solve real-world problems that can be modeled, using systems of linear
equations and inequalities.
I. Model and solve problem
situations involving direct
and inverse variation.
Analyze Change
(Gr.9#13) Model and solve problems involving direct and inverse variation using
proportional reasoning.
(9#13) direct variation y = kx
inverse variation y = k
x
k is variation constant
(Gr.9#14) Describe the relationship between slope and the graph of a direct variation
and inverse variation.
J. Describe and interpret
rates of change from
graphical and numerical
data.
Use Algebraic Representation
(Gr.10#9) Recognize and explain that the slopes of parallel lines are equal and the
slopes of perpendicular lines are negative reciprocals.
Analyze Change
(Gr.9#15) Describe how change in the value of a constant in a linear or quadratic
equation affects the related graphs.
(9#15) slope = change in y
change in x
(Gr.10#12) Describe the relationship between slope of a line through the origin and the
tangent function of the angle crated by the line and the positive x-axis.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Data Analysis and Probability Standard
Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate
inferences, predictions and arguments that are based on data.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
A. Create, interpret and use
graphical displays and
statistical measures to
describe data; such as, boxand-whisker plots,
histograms, scatterplots,
measures of center and
variability.
Data Collection
(Gr.9#1) Classify data as univariate (single variable) or bivariate (two variables) and as
quantitative (measurement) or qualitative (categorical) data.
(Gr.9#2) Create a scatterplot for a set of bivariate data, sketch the line of best fit and
interpret the slope of the line of best fit.
(Gr.10#2) Represent and analyze bivariate data using appropriate graphical displays
(scatterplots, parallel box-and-whisker plots, histograms with more than one set of
data, tables, charts, spreadsheets) with and without technology.
(Gr.10#3) Display bivariate data where at least one variable is categorical.
(Gr.10#4) Identify outliers on a data display; such as, use the interquartile range to
identify outliers on a box-and-whisker plot.
Statistical Methods
(Gr.9#3) Analyze and interpret frequency distributions based on spread, symmetry,
skewness, clusters and outliers.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Data Analysis and Probability Standard (continued)
Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate
inferences, predictions and arguments that are based on data.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
C. Compare the
characteristics of the mean,
median and mode for a
given set of data, and
explain which measure of
center best represents the
data.
D. Find, use and interpret
measures of center and
spread, such as mean and
quartiles, and use those
measures to compare and
draw conclusions about sets
of data.
E. Evaluate the validity of
claims and predictions that
are based on data by
examining the
appropriateness of the data
collection and analysis.
Data Collection
(10#1) mean, median, mode
(Gr.10#1) Describe measures of center and the range verbally, graphically and
algebraically.
Statistical Methods
(Gr.10#6) Interpret the relationship between two variables using multiple graphical
displays and statistical measures; such as, scatterplots, parallel box-and-whisker plots
and measures of center and spread.
Statistical Methods
(Gr.9#4) Describe and compare various types of studies (survey, observation,
experiment), and identify possible misuses of statistical data
Statistical Methods
F. Construct convincing
arguments based on analysis
of data and interpretation of
graphs.
(Gr.9#6) Make inferences about relationships in bivariate data, and recognize the
difference between evidence of relationship (correlation) and causation.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Data Analysis and Probability Standard (continued)
Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate
inferences, predictions and arguments that are based on data.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
G. Describe sampling
methods and analyze the
effects of method chosen on
how well the resulting
sample represents the
population.
H. Use counting techniques,
such as permutations and
combinations, to determine
the total number of options
and possible outcomes.
I. Design an experiment to
test a theoretical probability,
and record and explain
results.
Statistical Methods
(Gr.9#5) Describe characteristics and limitations of sampling methods, and analyze the
effects of random versus biased sampling; such as, determine and justify whether the
sample is likely to be representative of the population.
(Gr.10#5) Provide examples and explain how a statistic may or may not be an attribute
of the entire population; such as, intentional or unintentional bias may be present.
(10#5) Bias (intentional or
unintentional) may not be
representative of the population if
a large number of responders did
not tell the truth.
ex. amount of time students spend
on homework
Probability
(9#7) Fundamental Counting
Principle – one can find the total
number of ways that two or more
separate things can happen by
multiplying the number of ways
each thing can occur
(Gr.9#7) Use counting techniques and Fundamental Counting Principle to determine
the total number of possible outcomes for mathematical situations.
Probability
(Gr.9#8) Describe, create and analyze a sample space and use it to calculate
probability.
(9#8) sample space – all the
possible outcomes of an
experiment
ex. roll of a number cube – there
are 6 possible outcomes
A list or tree diagram can be used
to show all possible outcomes of
an experiment.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Data Analysis and Probability Standard (continued)
Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate
inferences, predictions and arguments that are based on data.
Ohio Benchmarks
Grade-Level Indicators
Resources/Vocabulary
Grades 8-10
J. Compute the probabilities
of compound events,
independent events and
simple dependent events.
K. Make predictions based
on theoretical probabilities
and experimental results.
Probability
(Gr.9#9) Identify situations involving independent and dependent events, and explain
differences between, and common misconceptions about, probabilities associated with
those events.
(9#9) dependent event –
occurrence of one event will have
an effect on the probability that a
second event will occur
(Gr.10#7) Model problems dealing with uncertainty with area models (geometric
probability).
independent event – occurrence of
one event has no effect on the
probability that a second event
will occur
Probability
Illustrate situations with concrete
objects to assist students in
distinguishing between dependent
and independent events.
ex. spinner
(Gr.9#10) Use theoretical and experimental probability, including simulations or
random numbers, to estimate probabilities and to solve problems dealing with
uncertainty; such as, compound events, independent events and simple dependent
events.
(Gr.10#8) Differentiate and explain the relationships between the probability of an
event and the odds of an event, and compute one given the other.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Integrated Series – Mathematical Processes Standard
Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and
communicate mathematical ideas. The benchmarks for mathematical processes articulate what students should demonstrate in problem-solving,
representation, communication, reasoning and connections at key points in their mathematics program.
Ohio Benchmarks
Grade-Level
Resources/Vocabulary
Grades 8-10
Indicators
A. Formulate a problem or mathematical model in response to a specific
need or situation, determine information required to solve the problem,
choose method for obtaining this information and set limits for acceptable
solution.
Specific grade-level
indicators have not
been included for the
mathematical
B. Apply mathematical knowledge and skills routinely in other content
processes standard
areas and practical situations.
because content and
processes should be
C. Recognize and use connections between equivalent representations and
related procedures for a mathematical concept; such as, zero of a function interconnected at the
indicator level.
and the x-intercept of the graph of the function, apply proportional
Therefore,
thinking when measuring, describing functions and comparing
mathematical
probabilities.
processes have been
D. Apply reasoning processes and skills to construct logical verifications
embedded within the
or counter-examples to test conjectures and to justify and defend
grade-level indicators
algorithms and solutions.
for the five content
standards.
E. Use a variety of mathematical representations flexibly and
appropriately to organize, record and communicate mathematical ideas.
F. Use precise mathematical language and notations to represent problem
situations and mathematical ideas.
G. Write clearly and coherently about mathematical thinking and ideas.
H. Locate and interpret mathematical information accurately, and
communicate ideas, processes and solutions in a complete and easily
understood manner.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
compare: to determine how two things are alike and/or different;
the common/critical attributes must be identified
Compare is involved in ALL of the following:
analyze: to investigate by breaking it down so as to more clearly
understand the impact to the situation
describe: to analyze into its parts but less detailed than explain
determine: to reach a decision after a thorough investigation; to
find the cause of and then to solve or set limits to a situation
identify: to show or prove the sameness of
interpret: a student must first analyze and then make an
inference; this is more subjective than an evaluation
predict: to state what one believes will happen (based on data)
recognize: to examine closely and identify the common/critical
attributes
Other Stated Verbs in the Indicators:
justify
define
connect
construct
explain
derive
order
test
demonstrate
apply
estimate
generalize
approximate
add
convert
subtract
use
multiply
solve
divide
simplify
write
find
graph
model
classify
create
display
differentiate
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