Download Muskingum Valley ESC Standards-Based Mathematics Course of

Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – Number, Number Sense and Operations Standard
Students demonstrate number sense, including 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
A. Use scientific notation to
express large numbers and
numbers less than one.
1. Demonstrate an understanding of place value using powers of 10 and write large
numbers in scientific notation. (Grade 7)
Number and Number Systems
1. Use scientific notation to express large numbers and small numbers between 0 and 1.
B. Identify subsets of the
real number system.
1. scientific notation – a form of
writing numbers as the product of
a power of 10 and a decimal
number greater than or equal to 1
and less than 10
3. Describe differences between rational and irrational numbers, such as use
technology to show that some numbers (rational) can be expressed as terminating or
repeating decimals and others (irrational) as non-terminating and non-repeating
decimals. (Grade 7)
Number and Number Systems
2. Recognize that natural numbers, whole numbers, integers, rational numbers and
irrational numbers are subsets of the real number system.
C. Apply properties of
operations and the real
number system and justify
when they hold a set of
numbers.
Meaning of Operations
F. Explain the effects of
operations on the magnitude
of quantities.
10. Recognize that a quotient may be larger than the dividend when the divisor is a
fraction, such as 6 ÷ ½ = 12. (Grade 6)
4. Explain and use the inverse and identity properties and use inverse relationships
(addition/subtraction, multiplication/division, squaring/square roots) in problemsolving situations.
2. integers – set of whole
numbers and their opposites
rational numbers – number that
can be expressed as a ratio of two
integers
irrational numbers – a number
that cannot be represented by a
ratio of two integers
5. Explain the meaning and effect of adding, subtracting, multiplying and dividing
integers, such as how adding two integers can result in a lesser value. (Grade 7)
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
Eighth Grade – Number, Number Sense and Operations Standard (continued)
Students demonstrate number sense, including 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
G. Estimate, compute and
solve problems involving
real numbers, including
ratio, proportion and
percent, and explain
solutions.
Computation and Estimation
5. Determine when an estimate is sufficient and when an exact answer is needed in
problem situations, and evaluate estimates in relation to actual answers, such as very
close, less than, greater than.
6. Estimate, compute and solve problems involving rational numbers, including ratio,
proportion and percent, and judge the reasonableness of solutions.
5. Recognize and identify perfect squares and their roots. (Grade 5)
H. Find the square root of
perfect squares, and
approximate the square root
of non-perfect squares.
Computation and Estimation
7. Find the square root of perfect squares, and approximate the square root of nonperfect squares as consecutive integers between which the root lies, such as √130 is
between 11 and 12.
2. Explain the meaning of exponents that are negative or 0. (Grade 7)
I. Estimate, compute and
solve problems involving
scientific notation, square
roots and numbers with
integer exponents.
Meaning of Operations
3. Apply order of operations to simplify expressions and perform computations
involving integer exponents and radicals.
Computation and Estimation
8. Add, subtract, multiply, divide and compare numbers written in scientific notation.
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
Eighth Grade – 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
A. Solve increasingly
complex non-routine
measurement problems and
check for reasonableness of
results.
Analyze problem situations involving measurement concepts, select appropriate
strategies and use an organized approach to solve narrative and increasingly complex
problems.(Grade 7)
Use Measurement Techniques and Tools
6. Solve and determine the reasonableness of the results for problems involving rates
and derived measurements, such as velocity and density, using formulas, models and
graphs.
B. Use formulas to find
surface area and volume for
specified three-dimensional
objects accurate to a
specified level of precision.
Use Measurement Techniques and Tools
C. Apply indirect
measurement techniques,
tools and formulas, as
appropriate, to find
perimeter, circumference
and area of circles, triangles,
quadrilaterals and composite
shapes and to find volume
of prisms, cylinders and
pyramids.
Use Measurement Techniques and Tools
3. Use appropriate levels of precision when calculating with measurements.
3. absolute error
relative error
4. Derive formulas for surface area and volume and justify those using geometric
models and common materials. For example, find:
a. the surface area of a cylinder as a function of its height and radius;
b. that the volume of a pyramid (or cone) is one-third of the volume of a prism (or
cylinder) with the same base area and height.
5. Determine surface area for pyramids by analyzing their parts.
9. Demonstrate understanding of the concepts of perimeter, circumference and area by
using established formula for triangles, quadrilaterals, and circles to determine the
surface area and volume of prisms, pyramids, cylinders, spheres and cones. (Note:
Only volume should be calculated for spheres and cones).
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
Eighth Grade – Measurement Standard (continued)
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
1. Compare and order the relative size of common U.S. customary units and metric
units, such as mile and kilometer, gallon and liter, pound and kilogram.
2. Use proportional relationships and formulas to convert units from one measurement
system to another, such as degrees Fahrenheit to degrees Celsius.
Use Measurement Techniques and Tools
7. Apply proportional reasoning to solve problems involving indirect measurements or
rates.
E. Estimate and compute
various attributes, including
length, angle measure, area,
surface area and volume, to
a specified level of
precision.
F. Write and solve realworld, multi-step problems
involving money, elapsed
time and temperature, and
verify reasonableness of
solutions.
7. similar figures
trigonometry ratios
Use Measurement Techniques and Tools
3. Use appropriate levels of precision when calculating with measurement.
8. Find the sum of the interior and exterior angles of regular convex polygons with and
without measuring the angles with a protractor.
10. Use conventional formulas to find the surface area and volume of prisms, pyramids
and cylinders and the volume of spheres and cones to a specified level of precision.
Use Measurement Techniques and Tools
6. Solve and determine the reasonableness of the results for problems involving rates
and derived measurements, such as velocity and density, using formulas, models and
graphs.
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
Eighth Grade – 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
B. Describe and apply the
properties of similar and
congruent figures; and
justify conjectures involving
similarity and congruence.
Characteristics and Properties
C. Recognize and apply
angle relationships in
situations involving
intersecting lines,
perpendicular lines and
parallel lines.
Characteristics and Properties
1. Make and test conjectures about characteristics and properties, such as sides, angles,
symmetry, of two-dimensional figures and three-dimensional objects.
3. Use proportions in several forms to solve problems involving similar figures (partto-part, part-to-whole, corresponding sides between figures).
2. Recognize the angles formed and the relationship between the angles when two lines
intersect and when parallel lines are cut by a transversal.
D. Use coordinate geometry Characteristics and Properties
to represent and examine the 1. Make and test conjectures about characteristics and properties, such as sides, angles,
symmetry, of two-dimensional figures and three-dimensional objects.
properties of geometric
figures.
Spatial Relationships
4. Represent and analyze shapes using coordinate geometry, such as given three
vertices and the type of quadrilateral, find the coordinate of the fourth vertex.
Visualization and Geometric Models
E. Draw and construct
6. Draw nets for a variety of prisms, pyramids, cylinders and cones.
representations of two- and
three-dimensional geometric
objects using a variety of
tools, such as straightedge,
compass and technology.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
6. a two-dimensional
representation of a threedimensional figure for net
templates of various solids
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – 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
F. Represent and model
transformations in a
coordinate plane and
describe the results.
Transformations and Symmetry
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
5. Draw the results of translations, reflections, rotations and dilations of objects in the
coordinate plane, and determine properties that remain fixed, such as lengths of sides
remain the same under translations.
1. Make and test conjectures about characteristics and properties, such as sides, angles,
symmetry, of two-dimensional figures and three-dimensional objects.
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
Eighth Grade - 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
B. Identify and classify
functions as linear or
nonlinear, and contrast their
properties using tables,
graphs or equations.
Use Patterns, Relations and Functions
C. Translate information
from one representation
(words, table, graph or
equation) to another
representation of a relation
or function.
Use Patterns, Relations and Functions
D. Use algebraic
representations, such as
tables, graphs, expressions,
functions and inequalities,
to model and solve problem
situations.
Use Algebraic Representation
2. Generalize patterns and sequences by describing how to find the nth term.
3. Identify functions as linear or nonlinear based on information given in a table, graph
or equation.
3. linear equation – an equation
that when graphed, forms a
straight line
1. Relate the various representations of a relationship, must include relate a table to
graph, description and symbolic form.
4. Extend the uses of variables to include covariants where y depends on x.
5. Use physical models to add and subtract monomials and polynomials, and to
multiply a polynomial by a monomial.
7. Use symbolic algebra (equations and inequalities), graphs and tables to represent
situations and solve problems.
8. Write, simplify and evaluate algebraic expressions (including formulas) to generalize
situations and solve problems.
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
Eighth Grade - 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 Algebraic Representation
F. Solve and graph linear
equations and inequalities.
Use Algebraic Representation
6. Describe the relationship between the graph of a line and its equation, including
being able to explain the meaning of slope as a constant rate of change and y-intercept
in real-world problems.
6. If the slope and the y-intercept
of a line are known, the equation
can be written in standard form.
7. Use symbolic algebra (equations and inequalities), graphs and tables to represent
situations and solve problems.
9. Solve linear equations and inequalities graphically, symbolically and using
technology.
G. Solve quadratic
equations with real roots by
graphing, formula and
factoring.
H. Solve systems of linear
equations involving two
variables graphically and
symbolically.
I. Model and solve problem
situations involving direct
and inverse variation.
Use Algebraic Representation
12. graph of a quadratic equation
is a symmetric curve (parabola)
12. Solve simple quadratic equations graphically, such as y = x² - 16.
Use Algebraic Representation
10. Solve 2 x 2 systems of linear equations graphically and by simple substitution.
11. Interpret the meaning of the solution of a 2 by 2 system of equations, such as point,
line no solution.
Analyze Change
14. Differentiate and explain types of changes in mathematical relationships, such as
linear vs. non-continuous, direct variation vs. inverse variation.
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
Eighth Grade - 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
J. Describe and interpret
rates of change from
graphical and numerical
data.
Use Algebraic Representation
13. Compute and interpret slope, midpoint and distance given a set of ordered pairs.
Analyze Change
15. Describe and compare how changes in an equation affect the related graphs, such as
for a linear equation changing the coefficient of x affects the slope and changing the
constant affects the intercepts.
16. Using graphing calculators or computers to analyze change, such as interest
compounded over time as a nonlinear growth pattern.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
16. simple interest, compound
interest
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – 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.
B. Evaluate different
graphical representations of
the same data to determine
which is the most
appropriate representation
for an identified purpose.
Data Collection
1. Use, create and interpret scatterplots and other types of graphs as appropriate.
Data Collection
2. Evaluate different graphical representations of the same data to determine which is
the most appropriate representation for an identified purpose, such as line graph for
change over time, circle graph for part-to-whole comparison, scatterplot for
relationship between two variants.
2. graphs that compare – single
bar graphs, double bar graphs
graphs that show clusters of data
information – line plots,
histograms, box-and-whisker
plots, stem-and-leaf plots, double
stem-and-leaf plots
3. Differentiate between discrete and continuous data and appropriate ways to represent
graphs that show change over
each.
time – single line graphs, multiple
line graphs
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.
Statistical Methods
5. Explain the mean’s sensitivity to extremes and its use in comparison with the median 5. outlier – an outlier can skew the
information of a set of data when
and mode.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
finding the mean
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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – 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
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.
Statistical Methods
E. Evaluate the validity of
claims and predictions that
are based on data by
examining the
appropriateness of the data
collection and analysis.
Statistical Methods
F. Construct convincing
arguments based on analysis
of data and interpretation of
graphs.
Statistical Methods
4. Compare two sets of data using measures of center (mean, mode, median) and
measures of spread (range, quartile, interquartile range, percentiles).
8. Describe how the relative size of a sample compared to the target population affects
the validity of predictions.
6. Make conjectures about possible relationship in a scatterplot and approximate line of
best fit.
9. Construct convincing arguments based on analysis of data and interpretations of
graphs.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
135
Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – 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.
J. Compute the probabilities
of compound events,
independent events and
simple dependent events.
Statistical Methods
7. Identify different ways of selecting samples, such as survey response, random
sample, representative sample and convenience sample.
7. random sample – members of
population have an equal chance
of being selected for the survey
representative sample – has
about the same proportions as the
population being studied
Probability
10. Calculate the number of possible outcomes for a situation, recognizing and
accounting for when items may occur more than once or when order is important.
convenience sample – that
portion of the population that are
available for the survey. This
method may have biased results.
7. Design an experiment to test a theoretical probability and explain how the results
may vary. (Grade 6)
8. Make predictions based on theoretical probabilities, design and conduct an
experiment to test the predictions, compare actual results to predicted results and
explain differences. (Grade 7)
Probability
11. Demonstrate an understanding that the probability of either of two disjoint events
occurring can be found by adding the probabilities for each and that the probability of
one independent event following another can be found by multiplying the probabilities.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
136
Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – 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.
B. Apply mathematical knowledge and skills routinely in other
content areas and practical situations.
C. Recognize and use connections between equivalent
representations and related procedures for a mathematical
concept; such as, zero of a function and the x-intercept of the
graph of the function, apply proportional thinking when
measuring, describing functions and comparing probabilities.
D. Apply reasoning processes and skills to construct logical
verifications or counter-examples to test conjectures and to
justify and defend algorithms and solutions.
E. Use a variety of mathematical representations flexibly and
appropriately to organize, record and communicate
mathematical ideas.
Note: Letters/numbers correlate to the Ohio Academic Content Standards book.
Specific grade-level
indicators have not
been included for the
mathematical
processes standard
because content and
processes should be
interconnected at the
indicator level.
Therefore,
mathematical
processes have been
embedded within the
grade-level indicators
for the five content
standards.
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
137
Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003
Eighth Grade – Mathematical Processes Standard (continued)
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
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.
Specific grade-level
indicators have not
been included for the
mathematical
processes standard
because content and
processes should be
interconnected at the
indicator level.
Therefore,
mathematical
processes have been
embedded within the
grade-level indicators
for the five content
standards.
Other Stated Verbs in the Indicators:
use
derive
simplify
apply
demonstrate
evaluate
explain
make
differentiate
estimate
test
create
compute
draw
construct
solve
relate
calculate
find
generalize
order
extend
Problem-Solving Strategies
 choose a method of computation
 draw a diagram
 guess and check
 look for a pattern
 make a list
 make a model
 solve a simple problem
 use logical reasoning
 work backward
 eliminate possibilities
 reasonable answers
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