Download Algebra standard 9

Algebra standard 9.2.1 Understanding the concept of function, and identify important features of functions and other
relations using symbolic and graphical methods.
Benchmark
1.Understand the definition of a function
1. Use functional notation and evaluate a
function at a given point in its domain.
2. Distinguish between functions and
other relations defined symbolically,
graphically, or in tabular form.
3. Find the domain of a function
symbolically.
3. Find the domain of a function in a
real-world context.
4. Obtain information from graphs of
functions and other relations.
4. Draw conclusions from graphs of
functions and other relations.
5. Identify the vertex of the parabola
when the function is expressed in the
form
f(x)=ax2+bx+c.
5. Identify the vertex of the parabola
when the function is expressed in the
form
f(x)=a(x-h)2+k.
5. Identify the line of symmetry of the
parabola when the function is expressed
in the form
f(x)=ax2+bx+c.
5. Identify the line of symmetry of the
parabola when the function is expressed
in the form f(x)=a(x-h)2+k.
5. Identify the line of symmetry of the
parabola when the function is expressed
in factored form.
5. Identify the intercepts of the parabola
when the function is expressed in the
form
f(x)=ax2+bx+c.
5. Identify the intercepts of the parabola
Introduction of
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Practice
concept
Application
Example
when the function is expressed in the
form f(x)=a(x-h)2+k.
5. Identify the intercepts of the parabola
when the function is expressed in
factored form.
6. Identify intercepts from the graph of a
function.
6. Identify zeros from the graph of a
function.
6. Identify maxima from the graph of a
function.
6. Identify minima from the graph of a
function.
6. Identify intervals of increase from the
graph of a function.
6. Identify intervals of decrease from the
graph of a function.
7. Understand the concept of an
asymptote.
7. Identify asymptotes for exponential
functions using symbolic and graphical
methods.
7. Identify asymptotes for reciprocals of
linear functions
8. Make qualitative statements about the
rate of change of a function, based on its
graph.
8. Make qualitative statements about the
rate of change of a function, based on its
table of values.
9. Determine how translations affect the
symbolic forms of a function.
9. Determine how translations affect the
graphical forms of a function.
9. Know how to use graphing
technology to examine translations.
Algebra Standard 9.2.2 Recognize linear, quadratic, exponential and other common functions in real-world and
mathematical situations; represent these functions with tables, verbal descriptions, symbols and graphs; solve problems
involving these functions, and explain results in the original context.
Benchmark
1. Represent and solve problems in
various contexts using linear functions
1. Represent and solve problems in
various contexts using quadratic
functions
2. Represent and solve problems in
various context using exponential
functions, such as investment growth,
depreciation and populations growth.
3. Sketch graphs of linear functions
3. Sketch graphs of quadratic functions.
3. Sketch graphs of exponential
functions.
3. Translate between graphs, tables and
symbolic representations.
3. Know how to use graphic technology
to these functions.
4. Express the terms in a geometric
sequence recursively.
4. Express the terms in a geometric
sequence by giving an explicit (closed
form) formula.
4. Express the partial sums of a
geometric series recursively.
5. Recognize and solve problems that
can be modeled using finite geometric
sequences and series, such as home
mortgage and other compound interest
examples
5. Know how to use spreadsheets to
explore geometric sequences and series
in various contexts.
5. Know how to use calculators to
explore geometric sequences and series
in various contexts.
6. Sketch the graphs of common non-linear
functions.
6. Know how to use graphing technology to
examine non-linear functions.
Introduction of
concept
Practice concept
Applications
Example
Algebra Standard 9.2.3 Generate equivalent algebraic expressions involving polynomials and radicals; use algebraic properties to
evaluate expressions.
Benchmark
1. Evaluate polynomial and rational
expressions at specified points in their
domains.
1. Evaluated expressions containing
radicals specified points in their
domains.
1. Evaluate expressions containing
absolute values at specified points in
their domains.
2. Add polynomials.
2. Subtract polynomials.
2. Multiply polynomials.
2. Divide a polynomial by a polynomial
of equal or lower degree.
3. Factor common monomial factors
from polynomials.
3. Factor quadratic polynomials.
3. Factor the difference of two squares.
4. Add algebraic fractions.
4. Subtract algebraic fractions.
4. Multiply algebraic fractions.
4. Divide algebraic fractions.
5. Check whether a given complex
number is a solution of a quadratic
equation by substituting it for the
variable and evaluating the expression,
using arithmetic with complex numbers.
6. Apply the properties of positive and
negative rational exponents to generate
equivalent algebraic expressions.
7. Justify steps in generating equivalent
expressions by identifying the properties
used.
7. Use substitution to check the equality
of expressions for some particular values
of the variables.
Introduction of
concept
Practice concept
Applications
Example
Algebra Standard 9.2.4 Represent real-world and mathematical situations using equations and inequalities
Benchmark
1. Represent relationships in various
contexts using quadratic equations and
inequalities
1. Solve quadratic equations and
inequalities by appropriate methods of
factoring, completing the square and
graphing.
1. Know how to use calculators,
graphing utilities or other technology to
solve quadratic equations and
inequalities.
2. Represent relationships in various
contexts using equations involving
exponential functions
2. Solve exponential equations either
graphically or numerically
2. Know how to use calculators,
graphing utilities or other technology to
solve exponential functions.
3. Recognize to solve equations, number
systems need to be extended from whole
numbers to integers to rational numbers
to real numbers to complex numbers
3. Recognize non-real complex numbers
are needed to solve some quadratic
equations with real coefficients
4. Represent relationships in various
contexts using systems of linear
inequalities.
4. Solve systems of linear inequalities
graphically
4. Appropriately indicate boundary lines
for systems of linear inequalities
5. Solve linear programming problems
in two variables using graphical methods
6. Represent relationships in various
contexts using absolute value
inequalities in two variables.
6. Solve absolute value inequalities in
two variables graphically.
7. Solve equations that contain radical
expressions.
Introduction of
concept
Practice concept
Applications
Example
7. Recognize extraneous solutions may
arise when using symbolic methods to
solve radical expressions.
8. Assess the reasonableness of a
solution in its given context.
8. Assess the reasonableness of a
solution compared to appropriate
graphical or numerical estimates.
8. Interpret a solution in the original
context.
Geometry Standard 9.3.1 Calculate measurements of plane and solid geometric; know that physical measurements depend on the
choice of unit and that they are approximations.
Benchmark
1. Determine the surface area of
pyramids.
1. Determine the surface area of cones.
1. Determine the surface area of spheres.
1. Determine the volume of pyramids.
1. Determine the volume of cones.
1. Determine the volume of spheres.
1. Use measuring devices as appropriate.
1. Use formulas as appropriate.
2. Compose and decompose two
dimensional figures.
2. Compose and decompose three
dimensional figures.
2. Use decomposition to determine
perimeter of various figures.
2. Use decomposition to determine area
of various figures.
2. Use decomposition to determine
volume of various figures.
3. Understand that quantities associated
with physical measurements must be
assigned units.
3. Apply units correctly in expressions
Introduction of
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Example
that involve measurements.
3. Apply units correctly in equations that
involve measurements.
3. Apply units correctly I problems
solutions that involve measurements.
3. Convert between measurement
systems.
4.Understand the fact that the effect of a
scale factor k on length is to multiply by
k.
4. Understand the fact that the effect of a
scale factor k on area is to multiply by
k2.
4. Understand the fact that the effect of a
scale factor k on volume is to multiply
by k3.
4.Apply the fact that the effect of a scale
factor k on length is to multiply by k.
4. Apply the fact that the effect of a
scale factor k on are is to multiply by k2.
4. Apply the fact that the effect of a
scale factor k on volume is to multiply
by k3.
5. Make reasonable estimates and
judgments about the accuracy of values
resulting from calculations involving
measurements.
Geometry Standard 9.3.2 Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other
results in geometry.
Benchmark
1. Understand the role of axioms in
logical arguments.
1. Understand the role of definitions in
logical arguments.
1. Understand the role of undefined terms
in logical arguments.
1. Understand the role of theorems in
logical arguments.
2. Accurately interpret and use words and
phrases in geometric proofs such as
“if…then,” “if and only if,” “all,” and
“not.”
2. Recongnize the logical relationship
between and “if…then”” statement and
its inverse.
2. Recongnize the logical relationship
between an “if...then” statement and its
converse.
3. Assess the validity of a logical
argument.
3. Give counterexamples to disprove a
statement.
4. Construct logical arguments and write
proofs of theorems and other results in
geometry, including proofs by
contradiction.
4. Express proofs in a form that clearly
justifies the reasoning, such as twocolumn proofs, paragraph proofs, flow
charts or illustrations.
5. Use technology tools to examine
theorems.
5. Use technology tools to examine test
conjectures.
5. Use technology tools to perform
constructions.
5. Use technology tools to develop
mathematical reasoning skills in multistep problems. The tools may include
compass and straight edge, dynamic
geometry software, design software or
Internet applets.
Introduction of
concept
Practice concept
Applications
Example
Geometry Standard 9.3.3 Know and apply properties of geometric figures to solve real-world and mathematical problems and to
logically justify results in geometry.
Benchmark
1. Know and apply properties of
parallel lines, including properties of
angles formed by a transversal, to solve
problems and logically justify results.
1. Know and apply properties of
perpendicular lines, including
properties of angles formed by a
transversal, to solve problems and
logically justify results.
2. Know and apply properties of angles,
including corresponding, exterior,
interior, vertical, complementary and
supplementary angles, to solve
problems and logically justify results.
3. Know and apply properties of
equilateral triangles to solve problems
and logically justify results.
3. Know and apply properties of
isosceles triangles to solve problems
and logically justify results.
3. Know and apply properties of
scalene triangles to solve problems and
logically justify results.
4. Apply the Pythagorean Theorem to
solve problems and logically justify
results.
4. Apply the converse of the
Pythagorean theorem to solve problems
and logically justify results.
5. Know the properties of right
triangles, including properties of 45-4590 and 30-60-90 triangles, to solve
problems and logically justify results.
5. Apply the properties of right
triangles, including properties of 45-4590 and 30-60-90 triangles, to solve
problems and logically justify results.
6. Knowing the properties of congruent
and similar figures to solve problems
and logically justify results.
6. Apply the properties of congruent
and similar figures to solve problems
and logically justify results.
Introduction of
concept
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Applications
Example
7. Use properties of polygons,
including quadrilaterals and regular
polygons, to define them.
7. Use properties of polygons,
including quadrilaterals and regular
polygons, to classify them.
7. Use properties of polygons,
including quadrilaterals and regular
polygons, to solve problems.
7. Use properties of polygons,
including quadrilaterals and regular
polygons, to logically justify results.
8. Know the properties of a circle to
solve problems and logically justify
results.
8. Apply the properties of a circle to
solve problems and logically justify
results.
Geometry Standard 9.3.4 Solve real-world and mathematical geometric problems using algebraic methods.
Benchmark
1. Understand how the properties of similar
right triangles allow the trigonometric ratios
to be defines.
1. Determine the sine of an acute angle in a
right triangle.
1. Determine the cosine of an acute angle in
a right triangle.
1. Determine the tangent of an acute angle
in a right triangle.
2. Apply the trigonometric ratios to solve
problems dealing with lengths in right
triangles.
2. Apply the trigonometric ratios to solve
problems dealing with areas in right
triangles.
2. Apply the trigonometric ratios to solve
problems dealing with lengths in figures
that can be decomposed into right triangles.
2. Apply the trigonometric ratios to solve
problems dealing with areas in figures that
can be decomposed into right triangles.
Introduction of
concept
Practice concept
Applications
Example
2. Know how to use calculators to evaluate
trigonometric ratios.
2. Know how to use tables to evaluate
trigonometric ratios.
2. Know how to use other technology to
evaluate trigonometric ratios.
3. Use calculators in connection with the
trig ratios to find angle measures in right
triangles in various contexts.
3. Use tables in connection with the trig
ratios to find angles measures in right
triangles in various contexts.
3. Use other technology in connection with
the trig ratios to find angle measures in right
triangles in various contexts.
4. Use coordinates geometry to represent
and analyze segments and polygons with
represent to determining lengths.
4. Use coordinates geometry to represent
and analyze segments and polygons with
respect to midpoints.
4. Use coordinates geometry to represent
and analyze segments and polygons with
respect to slopes of line segments.
5. Know the equation for the graph of a
circle with radius r and center.
5. Justify this equation using the
Pythagorean Theorem.
5. Justify this equation using properties of
translations.
6. Use numeric presentations of
transformations in two dimensions, such
as reflections, translations, scale changes
and rotations about the origin by
multiples on 90°, to solve problems
involving figures on a coordinate grid.
7.Use algebra to solve geometric
problems unrelated to coordinated
geometry, such as solving for an
unknown length in a figure involving
similar triangles, or using the
Pythagorean Theorem to obtain a
quadratic equation for a length in a
geometric figure.
Data Analysis and Probability Standard 9.4.1 Display and analyze data; use various measures associated with data to draw
conclusions, identify trends and describe relationships.
Benchmark
1. Describe a data set using data
displays, such as box-and whisker
plots.
1. Describe and compare data sets
using summary statistics, including
measures of center, location and
spread.
1. Measure of center and locations
include mean, median, quartile and
percentile.
1. Measure of spread include standards
deviation, range and interquartile range.
1. Know how to use calculators to
display data and calculate summary
statistics.
1. Know how to use spreadsheets to
display data and calculate summary
statistics.
1. Know how to use other technology
to display data and calculate summary
statistics.
2. Analyze the effects on summary
statistics of changes in data sets.
3.Use scatterplots to analyze patterns
3. Using technology, determine
regression lies (line of best fit).
3. Using technology, determine
correlation coefficients.
3. Use regression lines to make
predictions.
3. Use correlation coefficients to assess
the reliability of those predictions.
4.Use the mean and standard deviation
of a data set to fit it to a normal
distributions (bell shaped curve)
4. Use the mean and standard deviation
of a data set to estimate population
percentages.
Introduction of
concept
Practice concept
Applications
Example
4. Recognize that there are data sets for
which such a procedure is not
appropriate.
4. Use calculators to estimate areas
under the normal curve.
4. Use spreadsheets to estimate areas
under the normal curve.
4. Use tables to estimate areas under
the normal curve.
Data Analysis and Probability Standard 9.4.2 Explain the uses of data and statistical thinking to draw inferences make predictions
and justify conclusions.
Benchmark
1. Evaluate reports based on data
published in the media by identifying
the source of the data.
1. Evaluate reports based on data
published by the media by identifying
the design of the study.
1. Evaluate reports based on data
published by the media by identifying
the way the data are analyzed and
displayed.
1. Show how graphs and data can be
distorted to support different points of
view.
1. Know how to use spread sheet tables
and graphs to recognize and analyze
distortions in data displays.
1. Know how to use graphing
technology to recognize and analyze
distortions in data displays.
2. Identify misleading uses of data.
2. Explain misleading uses of data.
2. Recognize when arguments based on
data confuse correlation.
2. Recoginize when arguments based
on data confuse causation.
3. Explain the impact of sampling
methods during data collection.
Introduction of
concept
Practice concept
Applications
Example
3. Explain the impact of bias during
data collection.
3. Explain the impact of the phrasing of
questions during data collection.
Data Analysis and Probability Standard 9.4.3 Calculate probabilities and apply probability concepts to solve real-world and
mathematical problems.
Benchmark
1. Select and apply counting
procedures, such as the multiplication
and addition principles and tree
diagrams, to determine the size of a
sample space (the number of possible
outcomes).
1. Select and apply counting
procedures, such as the multiplication
and addition principles and tree
diagrams, to calculate probabilities.
2. Calculate experimental probabilities
by performing simulations or
experiments involving a probability
model.
2. Calculate experimental probabilities
by using relative frequencies of
outcomes.
3. Understand that the Law of Large
Numbers expresses a relationship
between the probabilities in a
probability model and the experimental
probabilities bound by performing
simulations or experiments involving
the model.
4. Use random numbers generated by a
calculator or a spreadsheet, or taken
from a table, to perform probability
simulations.
4. Use random numbers generated by a
calculator or a spreadsheet, or taken
from a table, to introduce fairness into
decision making.
5. Apply probability concepts such as
intersections, unions and complements
of events, and conditional probability
and independence, to calculated
probabilities and solve problems.
6. Describe the concepts of
intersections, unions and complements
using Venn diagrams. Understand the
Introduction of
concept
Practice concept
Applications
Example
relationships between these concepts
and the words AND, OR, NOT, as
used in computerized searches and
spreadsheets.
7. Understand simple probability
formulas involving intersections,
unions and complements of events.
7. Use simple probability formulas
involving intersections, unions and
complements of events.
8. Apply probability concepts to realworld situations to make informed
decisions.
9. Use the relationship between
conditional probabilities and relative
frequencies in contingency tables.