Download VERTICAL ALIGNMENT DOCUMENT – MATHEMATICS GRADE 8

VERTICAL ALIGNMENT DOCUMENT
MATHEMATICS: GRADE 8, ALGEBRA 1, and GEOMETRY
8.1
8.1A
8.1B
8.1C
EIGHTH GRADE
Number, operation, and quantitative
reasoning. The student understands that
different forms of numbers are appropriate for
different situations. The student is expected to:
Compare and order rational numbers in various
forms including integers, percents, and positive
and negative fractions and decimals.
Including numbers represented as fractions and
decimals.
Select and use appropriate forms of rational
numbers to solve real-life problems.
8.1
8.1B
Gray shading: Knowledge Statement
Gold Shading: Student Expectations on TAKS
Yellow Shading: SE’s that align at 2 grade levels
Blue Shading: SE’s that are tested at the next grade level
ALGEBRA I and NINTH GRADE TAKS
Number, operation, and quantitative reasoning.
The student understands that different forms of
numbers are appropriate for different situations.
The student is expected to:
GEOMETRY and TENTH GRADE TAKS
Select and use appropriate forms of rational
numbers to solve real-life problems.
Including those involving proportional
relationships.
Approximate (mentally and with calculators)
the value of irrational numbers as they arise
from problem situations (such asπ, √2).
Including using geometric problems using the
square root of a number
Including the square root of any number less
than 14.
8.1D
Express numbers in scientific notation,
including negative exponents, in appropriate
problem situation.
Including:
•Converting numbers back to standard form
1
8.2
8.2A
8.2B
8.2C
8.2D
•Scientific notation using positive or negative
exponents
Number, operation, and quantitative
reasoning. The student selects and uses
appropriate operations to solve problems and
justify solutions. The student is expected to:
Select appropriate operations to solve problems
involving rational numbers and justify the
selections.
Including formulating equations with
appropriate order of operations (addition,
subtraction, multiplication, division, square,
and square root) with positive and negative
integers, fractions, decimals, and percents.
Use appropriate operations to solve problems
involving rational numbers in problem
situations.
Including problems with multi-operations
(addition, subtraction, multiplication, division,
square and square root) and mixed forms of
rational numbers (positive and negative
integers, fractions, decimals, and percents).
Evaluate a solution for reasonableness.
Including application problems for money,
measurement, and percent.
Use multiplication by a constant factor (unit
rate) to represent proportional relationships.
Including:
•Using multiple forms of fractions, decimals,
percents, positive and negative integers within a
single problem. (Example: 1 gallon = 4 quarts
(g=4q))
2
•Referring to the measurement side of the
TAKS chart
8.3
8.5
Including percents, fractions, and decimals.
Patterns, relationships, and algebraic
thinking. The student identifies proportional
or non-proportional linear relationships in
problem situations and solves problems. The
student is expected to:
Patterns, relationships, and algebraic
thinking. The student uses graphs, tables, and
algebraic representations to make predictions
and solve problems. The student is expected
to:
A.1
8.3
Foundations for functions. The student
understands that a function represents a
dependence of one quantity on another and can
be described in a variety of ways. The student is
expected to:
Patterns, relationships, and algebraic thinking.
The student identifies proportional or nonproportional linear relationships in problem
situations and solves problems. The student is
expected to:
A.1
8.3
8.5
Foundations for functions. The student
understands that a function represents a
dependence of one quantity on another and can
be described in a variety of ways. The student is
expected to:
Patterns, relationships, and algebraic thinking.
The student identifies proportional or nonproportional linear relationships in problem
situations and solves problems. The student is
expected to:
Geometric patterns. The student uses a variety of
representations to describe geometric
relationships and solve problems. The student is
expected to:
A.1A
A.1B
Describe independent and dependent quantities in
functional relationships.
Including:
•Linear and quadratic functions
•Explaining a functional relationship by using one
variable to describe another variable.
•Selecting the independent or dependent quantity
in an equation or verbal description and justifying
the selection
Gather and record data and use data sets to
determine functional relationships between
quantities.
Including:
•Students collecting data that models linear and
quadratic functions
•Writing equations from a table of data
A.1A
A.1B
Describe independent and dependent quantities in
functional relationships.
Including:
•Linear and quadratic functions •Explaining a
functional relationship by using one variable to
describe another variable.
•Selecting the independent or dependent quantity
in an equation or verbal description and justifying
the selection
Gather and record data and use data sets to
determine functional relationships between
quantities.
Including:
•Students collecting data that models linear and
quadratic functions.
•Writing equations from a table of data
3
8.5B
Find and evaluate an algebraic expression to
determine any term in an arithmetic sequence
(with a constant rate of change)
A.1C
Including:
 Expressions in which the constant rate of
change is expressed as a fraction or a
decimal
 nth term table
 Finding the nth term
 Using the nth term to find a specific term
 The formula for the arithmetic sequence
(answers should be in distributive format)
 The first term + common difference (n – 1)
 Vocabulary (i.e. substitute, algebraic
expression, rule, expression, nth term,
prediction, pattern, correlation, term,
sequence)
 Number’s position in a sequence
8.3B
Estimate and find solutions to application
problems involving percents and other
proportional relationships such as similarity and
rates.
Including:
•Ratios that may not be in lowest terms
represented in a table, graph, equation, verbal
description and geometric representations.
•Setting up a proportion problem from a verbal
description
•Using data in a table
•Generating a list of data from a functional
relationship
•Using a graphing calculator (specifically using the
table function in the calculator). An option would
be to teach linear regression using the calculator.
Describe functional relationships for given
problem situations and write equations or
inequalities to answer questions arising from the
situations.
Including:
 Areas of circles and squares
 Perimeters of squares, equilateral triangles, and
circumference
 Constant rate of change (i.e. slope)
 Literal equations (a = lw solve for l)
A.1D
8.3B
Represent relationships among quantities using
concrete models, tables, graphs, diagrams, verbal
descriptions, equations, and inequalities.
Including quadratic relationships and linear
relationships (constant rate of change) with and
without a graphing calculator.
Estimate and find solutions to application
problems involving percents and other
proportional relationships such as similarity and
rates.
•Generating a list of data from a functional
relationship
•Using a graphing calculator.
G.5A
A.1C
A.1D
8.3B
Use numeric and geometric patterns to develop
algebraic expressions representing geometric
properties
Including describing functional relationships in
writing equations or inequalities as they pertain to:
 Areas of circles and polygons
 Perimeters of polygons and circumference
of circles
Describe functional relationships for given
problem situations and write equations or
inequalities to answer questions arising from the
situations.
Including:
 Areas of circles and squares
 Perimeters of squares, equilateral triangles,
and circumference
 Constant rate of change
Represent relationships among quantities using
concrete models, tables, graphs, diagrams, verbal
descriptions, equations, and inequalities.
Including quadratic relationships (areas of circles
and squares) and linear relationships (perimeters of
squares, equilateral triangles, circumference, and
constant rate of change) with and without a
graphing calculator.
Estimate and find solutions to application
problems involving percents and other
proportional relationships such as similarity and
4
•Dilations (Enlargements and reductions) of
geometric figures
•Measurements using standard and metric units
•Unit conversions
8.5A
Including linear relationships (constant rate of
change, and similar figures) with and without a
graphing calculator.
rates.
Including linear relationships (perimeters of
squares, equilateral triangles, circumference,
constant rate of change, and similar figures) with
and without a graphing calculator.
Predict, find, and justify solutions to application
problems using appropriate tables, graphs, and
algebraic equations.
Including:
•Multiple representations of a table, graph,
equation, sequence or verbal description within
a single context of a problem
•Present and future incremental predictions
•Vocabulary: (i.e. Interval, scale, nth term,
coordinate plane, position, sequence, trend,
correlation, relationships, variables, positive,
negative, algebraic equations, evaluate, rule
prediction, between, pattern, exceed, arithmetic
sequence, term)
•Positive, negative and no correlation or trend
•Answer choices in the form of an
inclusive/exclusive relationship (Example:
Between 5 and 12) (Example: >, <, ≤, ≥)
Graphs to include:
•Line Graph
•Bar Graph
•Multiple Bar Graph
•Pie Chart
•Histogram
•Scatter plot
•Box and Whiskers
•Pictograph
•Circle Graph
5
•Line Plots
•Stem and leaf
A.1E
Interpret and make decisions, predictions, and
critical judgments from functional relationships.
A.1E
Including linear relationships (constant rate of
change) quadratic relationships communicated
with concrete models, tables, graphs, diagrams,
verbal descriptions, equations, and inequalities.
8.3
8.4
8.12
8.3A
Patterns, relationships, and algebraic
thinking. The student identifies proportional
or non-proportional linear relationships in
problem situations and solves problems. The
student is expected to:
Patterns, relationships, and algebraic
thinking. The student makes connections
among various representations of a numerical
relationship. The student is expected to:
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
Compare and contrast proportional and nonproportional linear relationships.
Including
•Ratios that may not be in lowest terms
represented in a table, graph, equation, verbal
description and geometric representations.
•Setting up a proportion problem from a verbal
description
•Using data in a table
•Dilations (Enlargements and reductions) of
geometric figures
A.2
Foundations for functions. The student uses the
properties and attributes of functions. The
G.4
student is expected to:
A.2
Interpret and make decisions, predictions, and
critical judgments from functional relationships.
Including linear relationships (perimeters of
squares and equilateral triangles, circumference,
constant rate of change, and similar figures) and
quadratic relationships (area of circle and square)
communicated with concrete models, tables,
graphs, diagrams, verbal descriptions, equations,
and inequalities.
Geometric structure. The student uses a variety of
representations to describe geometric
relationships and solve problems. The student is
expected to:
Foundations for functions. The student uses the
properties and attributes of functions. The
student is expected to:
A.2A
Identify and sketch the general forms of linear (y =
x) and quadratic (y = x2) parent functions.
Including :
 Investigations with and without a graphing
calculator
 Specifically using the terminology “parent
functions”
 Including parent functions that have been
altered (for example a parabola turned
upside down still belongs to the parent
function y=x2)
A.2A
Identify and sketch the general forms of linear (y =
x) and quadratic (y = x2) parent functions.
Including investigations with and without a
graphing calculator. This SE was not tested in
2003 or 2004 at this grade.
6
•Measurements using standard and metric units
•Unit conversions
A.2B
8.4A
The student is expected to generate a different
representation of data given another
representation of data (such as table, graph,
equation, or verbal description).
Including:
•Multiple representations of a table, graph,
equation, sequence or verbal description within
a single context of a problem
•Present and future incremental predictions
•Vocabulary: (i.e. Interval, scale, nth term,
coordinate plane, position, sequence, trend,
correlation, relationships, variables, positive,
and negative
A.2C
Identify mathematical domains and ranges and
determine reasonable domain and range values for
given situations, both continuous and discrete.
Including:
•Values displayed in a table
•Values displayed by an equation
•Values displayed in a graph.
•Values displayed by an inequality.
•Values from a verbal description of everyday
experiences such as temperature, money, height,
etc.
A.2B
Interpret situations in terms of given graphs or
creates situations that fit given graphs.
G.4A
Including interpreting real-world situations in
terms of graphs and also describing a real-world
situation that fits a graph.
A.2C
Identify mathematical domains and ranges and
determine reasonable domain and range values for
given situations, both continuous and discrete.
Including:
•A range of values displayed in a table
•A range of values displayed in a graph
•A range of values displayed by an
inequality
•A range of values from a verbal description of
everyday experiences such as temperature, money,
height, etc.
The student is expected to select an appropriate
representation (concrete, pictorial, graphical,
verbal, or symbolic) in order to solve problems.
Including:
•Interpreting real-world geometric situations in
terms of graphs, tables, and literal equations
•Describing real-world geometric situations that
fit appropriate representations
Interpret situations in terms of given graphs or
creates situations that fit given graphs.
Including interpreting real-world situations in
terms of graphs and also describing a real-world
situation that fits a graph.
Graphs to include:
•Line Graph
•Bar Graph
•Multiple Bar Graph
•Histogram
•Scatter plot
•Pictograph
7
8.12B
•Circle Graph
•Line Plots
•Stem and leaf
•Venn Diagram
Draw conclusions and make predictions by
analyzing trends in scatter plots.
A.2D
Including:
Describe the scatterplot in words
(increasing/decreasing)
•Scatter plots that show no trend
•Positive, negative and no correlations or trends
A.3
A.3A
A.3B
A.4
Collect and organize data, make and interpret
scatter plots (including recognizing positive,
negative, or no correlation for data approximating
linear situations) and model, predict, and make
decisions and critical judgments in problem
situations.
Including organizing data that demonstrates a
positive linear correlation, a negative linear
correlation, and no correlation with and without a
graphing calculator
Foundations for functions. The student
understands how algebra can be used to express
generalizations and recognizes and uses the
power of symbols to represent situations. The
student is expected to:
Use symbols to represent unknowns and variables.
Including similarity, constant rate of change, area,
perimeter, circumference, and proportions. Write
an expression to represent the solution to a
problem.
Look for patterns and represent generalizations
algebraically.
Including expressions in the form of, but not
limited to:

an, an±b, a/n, n/a, a/n ± b, n/a ± b, a ±n, n – a

geometric sequence

arithmetic sequence

common ratios and differences
Foundations for functions. The student
A.2D
A.3
A.3A
Collect and organize data, make and interpret
scatter plots including recognizing positive,
negative, or no correlation for data approximating
linear situations) and model, predict, and make
decisions and critical judgments in problem
situations.
Including organizing data that demonstrates a
positive linear correlation, a negative linear
correlation, and no correlation with and without a
graphing calculator
Foundations for functions. The student
understands how algebra can be used to express
generalizations and recognizes and uses the
power of symbols to represent situations. The
student is expected to:
Use symbols to represent unknowns and variables.
Including similarity, constant rate of change, area,
perimeter, circumference, and proportions.
A.3B
A.4
Look for patterns and represent generalizations
algebraically.
Including expressions in the form of, but not
limited to:

an, an±b, a/n, n/a, a/n ± b, n/a ± b, a ±n, n – a

geometric sequence

arithmetic sequence

common ratios and differences
Foundations for functions. The student
8
A.4A
A.4B
A.4C
A.5
A.5A
understands the importance of the skills required
to manipulate symbols in order to solve problems
and uses the necessary algebraic skills required
to simplify algebraic expressions and solve
equations and inequalities in problem situations.
The student is expected to:
Find specific function values, simplify polynomial
expressions, transform and solve equations, and
factor as necessary in problem situations.
Including:
•Areas of rectangles and squares.
•Factoring binomials and trinomials.
•Apply the commutative, associative, and
distributive properties to solve equations.
•Substitute a value for a variable.
•Use a graphing calculator to find specific function
values (e.g. zeros of quadratic functions)
Use the commutative, associative, and distributive
properties to simplify algebraic expressions.
Connect equation notation with function notation,
such as y = x + 1 and f(x) = x + 1.
Including examples of functions such as linear and
quadratic relationships, and non-examples such as
y2 = x.
Linear functions. The student understands that
linear functions can be represented in different
ways and translates among their various
representations. The student is expected to:
Determine whether or not given situations can be
represented by linear functions.
Including:
•Verbal descriptions that describe a constant rate
of change and a rate of change that is not constant
A.4A
understands the importance of the skills required
to manipulate symbols in order to solve problems
and uses the necessary algebraic skills required
to simplify algebraic expressions and solve
equations and inequalities in problem situations.
The student is expected to:
Find specific function values, simplify polynomial
expressions, transform and solve equations, and
factor as necessary in problem situations.
Including:
•Areas of rectangles and squares
•Factoring binomials and trinomials
•Apply the commutative, associative, and
distributive properties to solve equations
•Substitute a value for a variable
•Using a graphing calculator
A.4B
Use the commutative, associative, and distributive
properties to simplify algebraic expressions.
A.5
Linear functions. The student understands that
linear functions can be represented in different
ways and translates among their various
representations. The student is expected to:
Determine whether or not given situations can be
represented by linear functions.
A.5A
Including:
 Verbal descriptions that describe a constant
rate of change and a rate of change that is not
9
•A table of values with a constant rate of change
and a table of values in which the rate of change is
not constant.
A.5B
A.5C
A.6
A.6A

constant
A table of values with a constant rate of change
and a table of values in which the rate of
change is not constant.
Determine the domain and range for linear
functions in given situations.
Including:
•Earning a salary and/or commission
•Speed
•Temperature, etc…
Use, translate, and make connections among
algebraic, tabular, graphical, or verbal descriptions
of linear functions.
Including:
•Real-world verbal descriptions of a constant rate
of change such as earning an hourly wage or a
constant speed.
•Connecting the graph of a line to a description of
a real-world experience.
•Connecting an algebraic expression to a
description of a real-world experience.
•Using a graphing calculator.
Linear functions. The student understands the
meaning of the slope and intercepts of the graphs
of linear functions and zeros of linear functions
and interprets and describes the effects of
changes in parameters of linear functions in
real-world and mathematical situations. The
student is expected to:
Develop the concept of slope as rate of change and
determine slopes from graphs, tables, and
algebraic representations.
Including algebraic equations in which the
A.5C
A.6
A.6A
Use, translate, and make connections among
algebraic, tabular, graphical, or verbal descriptions
of linear functions.
Including:
•Real-world verbal descriptions of a constant rate
of change such as earning an hourly wage or a
constant speed
•Connecting the graph of a line to a description of
a real-world experience
•Connecting an algebraic expression to a
description of a real-world experience
•Using a graphing calculator
Linear functions. The student understands the
meaning of the slope and intercepts of the graphs
of linear functions and zeros of linear functions
and interprets and describes the effects of
changes in parameters of linear functions in
real-world and mathematical situations. The
student is expected to:
Develop the concept of slope as rate of change and
determine slopes from graphs, tables, and
algebraic representations.
Including algebraic expressions in which the
10
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
A.6B
Such as:
 Formulas with a linear relationship (i.e. d =
r t)
 Slope formula
 Sketch of a line on a coordinate plane
(given a table)
Interpret the meaning of slope and intercepts in
situations using data, symbolic representations, or
graphs.
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
A.6B
Including algebraic equations in slope-intercept
form, point-slope form, and standard form with
and without a graphing calculator.
A.6C
Such as:
•Symbolic representations including use of tables
and real world applications
•Representation of slope as integers, fractions,
decimals and mixed numbers
Investigate, describe, and predict the effects of
changes in m and b on the graph of y = mx + b;
Including algebraic equations in which the
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
Interpret the meaning of slope and intercepts in
situations using data, symbolic representations, or
graphs.
Including algebraic expressions in which the
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
A.6C
Investigate, describe, and predict the effects of
changes in m and b on the graph of y = mx + b;
Including algebraic equations in which the
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
Such as:
 Transformation
 Changing Slope and/or y intercept
 Doubling/halving slope
 Parallel and perpendicular slope
11
A.6D
relationships
Graph and write equations of lines given
characteristics such as two points, a point and a
slope, or a slope and y-intercept.
A.6D
Including algebraic equations in slope-intercept
form, point-slope form, and standard form with
and without a graphing calculator.
A.6E
Determine the intercepts of the graphs of linear
functions and zeros of linear functions from
graphs, tables, and algebraic representations.
A.6E
Including algebraic equations in slope-intercept
form, point-slope form, and standard form with
and without a graphing calculator.
A.6F
Interpret and predict the effects of changing slope
and y-intercept in applied situations.
A.6F

A.6G
Including Real-world situations tht model a
constant change such as a salary, commission,
a ride in a taxi, renting a car, speed, buying
gasoline, etc.
 Algebraic equations in slope-intercept form,
point-slope form, and standard form
Relate direct variation to linear functions and solve
problems involving proportional change.
Including:
 Real-world situations that model a constant
change such as a salary, commission, a ride in
a taxi, renting a car, speed, buying gasoline,
etc.
 Algebraic equations in slope-intercept form,
point-slope form, and stand form
Graph and write equations of lines given
characteristics such as two points, a point and a
slope, or a slope and y-intercept.
Including algebraic expressions in which the
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
Determine the intercepts of the graphs of linear
functions and zeros of linear functions from
graphs, tables, and algebraic representations.
Including algebraic expressions in which the
equation is in slope-intercept form, point-slope
form, and standard form with and without a
graphing calculator.
Interpret and predict the effects of changing slope
and y-intercept in applied situations.

A.6G
Including Real-world situations that model a
constant change such as a salary, commission,
a ride in a taxi, renting a car, speed, buying
gasoline, etc.
 Algebraic equations in slope-intercept form,
point-slope form, and standard form
Relate direct variation to linear functions and solve
problems involving proportional change.
Including:
 Real-world situations that model a constant
change such as a salary, commission, a ride in
a taxi, renting a car, speed, buying gasoline,
etc.
 Algebraic equations in slope-intercept form,
point-slope form, and stand form
12

A.7
A.7A
Using a graphing calculator
Linear functions. The student formulates
equations and inequalities based on linear
functions, uses a variety of methods to solve
them, and analyzes the solutions in terms of the
situation. The student is expected to:
Analyze situations involving linear functions and
formulate linear equations or inequalities to solve
problems.

A.7
A.7A
Including:
•Real-world problems involving a constant rate of
change where the value of the y-intercept is zero or
not zero.
•Algebraic equations in slope-intercept form,
point-slope form, and standard form.
A.7B
A.7C
Investigate methods for solving linear equations
and inequalities using concrete models,
graphs, and the properties of equality, select
a method, and solve the equations and
inequalities.
A.7B
Using a graphing calculator
Linear functions. The student formulates
equations and inequalities based on linear
functions, uses a variety of methods to solve
them, and analyzes the solutions in terms of the
situation. The student is expected to:
Analyze situations involving linear functions and
formulate linear equations or inequalities to solve
problems.
Including:
•Real-world problems involving a constant rate of
change with a constant and a constant rate of
change without a constant
•Algebraic expressions in which the equation is in
slope-intercept form, point-slope form, and
standard form.
Investigate methods for solving linear equations
and inequalities using concrete models,
graphs, and the properties of equality, select
a method, and solve the equations and
inequalities.
Including:
Including:


Using information from concrete models to
write linear equations and inequalities, plot
graphs, and solve equations and inequalities
 Use graphs to solve linear equations and
inequalities
 Algebraic equations and inequalities in slopeintercept form, point-slope form, and standard
form
 Using a graphing calculator
Interpret and determine the reasonableness of
solutions to linear equations and inequalities.
A.7C
Using information from concrete models to
write linear equations and inequalities, plot graphs,
and solve equations and inequalities
 Use graphs to solve linear equations and
inequalities
 Algebraic equations and inequalities in slopeintercept form, point-slope form, and
standard form
 Using a graphing calculator
Interpret and determine the reasonableness of
solutions to linear equations and inequalities.
13
Including:
•Linear relationships in tables, equations,
inequalities, and verbal descriptions
•Algebraic equations and inequalities in slopeintercept form, point-slope form, and standard
form
•Using a graphing calculator
A.8
Linear functions. The student formulates systems
of linear equations from problem situations, uses
a variety of methods to solve them, and analyzes
the solutions in terms of the situation. The
student is expected to:
G.7
A.8
A.8A
Analyze situations and formulate systems of linear
equations in two unknowns to solve problems.
A.8A
Including setting up a system given a real world
situation.
A.8B
Solve systems of linear equations using concrete
models, graphs, tables, and algebraic methods.
Including:
• Using the addition method (aka elimination
G.7B
A.8B
Including:
•Linear relationships in tables, equations,
inequalities, and verbal descriptions
•Equations in the form of y = mx and y = mx+b
•Algebraic expressions in which the equation is in
slope-intercept form, point-slope form, and
standard form
•Using a graphing calculator
Dimensionality and the geometry of location. The
student understands that coordinate systems
provide convenient and efficient ways of
representing geometric figures and uses them
accordingly. The student is expected to:
Linear functions. The student formulates systems
of linear equations from problem situations, uses
a variety of methods to solve them, and analyzes
the solutions in terms of the situation. The
student is expected to:
Analyze situations and formulate systems of linear
equations in two unknowns to solve problem.
Including:
•Algebraic expressions in which the equation is in
slope-intercept form, point-slope form, and
standard form
• Using the addition method to solve a system in
which there is no solution, one solution, and
infinite solutions
• Using the substitution method to solve a system
in which there is no solution, one solution, and
infinite solutions.
Use slopes and equations of lines to investigate
geometric relationships, including parallel lines,
perpendicular lines, and special segments of
triangles and other polygons.
Solve systems of linear equations using concrete
14
method or combinations method) to solve a system
in which there is no solution, one solution, and
infinite solutions
• Using the substitution method to solve a system
in which there is no solution, one solution, and
infinite solutions
• Using a graphing calculator to find the
intersection of the system (i.e. the solution)
A.8C
A.9
A.9A
Interpret and determine the reasonableness of
solutions to systems of linear equations.
Including:
•Algebraic equations in slope-intercept form,
pointslope form, and standard form.
• Using the addition method to solve a system in
which there is no solution, one solution, and
infinite solutions.
• Using the substitution method to solve a system
in which there is no solution, one solution, and
infinite solutions.
• Using graphing calculators
Quadratic and other nonlinear functions. The
student understands that the graphs of quadratic
functions are affected by the parameters of the
function and can interpret and describe the
effects of changes in the parameters of quadratic
functions. The student is expected to:
Determine the domain and range for quadratic
functions in given situations:
models, graphs, tables, and algebraic methods.
A.8C
A.9
Including:
•Algebraic expressions in which the equation is in
slope-intercept form, point-slope form, and
standard form.
• Using the addition method to solve a system in
which there is no solution, one solution, and
infinite solutions.
• Using the substitution method to solve a system
in which there is no solution, one solution, and
infinite solutions.
• Using a graphing calculator
Interpret and determine the reasonableness of
solutions to systems of linear equations.
Including:
•Algebraic expressions in which the equation is in
slope-intercept form, point-slope form, and
standard form
• Using the addition method to solve a system in
which there is no solution, one solution, and
infinite solutions.
• Using the substitution method to solve a system
in which there is no solution, one solution, and
infinite solutions
• Using graphing calculators
Quadratic and other nonlinear functions. The
student understands that the graphs of quadratic
functions are affected by the parameters of the
function and can interpret and describe the
effects of changes in the parameters of quadratic
functions. The student is expected to:
15
A.9B
A.9C
A.9D
A.10
A.10A
Including graphs, tables, verbal descriptions, and
equations.
Investigate, describe, and predict the effects of
changes in a on the graph of y = ax2 + c.
Including:
•Equations in which a is a number less than 0 and
greater than 0.
•Using a graphing calculator.
Investigate, describe, and predict the effects of
changes in c on the graph of y = ax2 + c.
Including:
•Equations in which c is a number less than 0
•Equations in which c is a number greater than 0
•Using a graphing calculator
Analyze graphs of quadratic functions and draw
conclusions.
Including:
•Naming the vertex
•Naming the zeros (roots, solutions, and xintercepts)
•Determine whether ‘a’ is positive or negative
•Finding the domain and range
•Applying quadratics to real world applications
Quadratic and other nonlinear functions. The
student understands there is more than one way
to solve a quadratic equation and solves them
using appropriate methods. The student is
expected to:
Solve quadratic equations using concrete models,
tables, graphs, and algebraic methods.
Including:
•Factoring
A.9B
A.9C
A.9D
Investigate, describe, and predict the effects of
changes in a on the graph of y = ax2 + c.
Including:
•Equations in which a is a number less than 0 and
greater than 0.
•Using a graphing calculator.
Investigate, describe, and predict the effects of
changes in c on the graph of y = ax2 + c.
Including:
•Equations in which c is a number less than 0 and
greater than 0
•Using a graphing calculator
Analyze graphs of quadratic functions and draw
conclusions.
Including:
•Naming the vertex
•Naming the zeros •Estimate a in y = ax2 + c.
•Finding the domain and range
A.10
A.10A
Quadratic and other nonlinear functions. The
student understands there is more than one way
to solve a quadratic equation and solves them
using appropriate methods. The student is
expected to:
Solve quadratic equations using concrete models,
tables, graphs, and algebraic methods.
Including:
•Algebra tiles.
16
A.10B
A.11
A.11A
A.11B
A.11C
8.6
Geometry and spatial reasoning. The student
uses transformational geometry to develop
spatial sense. The student is expected to:
8.6
•Graphing calculators to find zeros (roots,
solutions, and x-intercepts)
•Other methods such as algebra tiles
Make connections among the solutions (roots) of
quadratic equations, the zeros of their related
functions, and the horizontal intercepts (xintercepts) of the graph of the function.
Including:
•Using a graphing calculator
•Factoring
•Real world problems such as area of a rectangle
•Other methods such as algebra tiles
•Use terminology (i.e. solutions, roots, zeros, and
x-intercepts)
Quadratic and other nonlinear functions. The
student understands there are situations modeled
by functions that are neither linear nor quadratic
and models the situations. The student is
expected to:
Use patterns to generate the laws of exponents and
apply them in problem-solving situations.
Analyze data and represent situations involving
inverse variation using concrete models, tables,
graphs, or algebraic methods.
Analyze data and represent situations involving
exponential growth and decay using concrete
models, tables, graphs, or algebraic methods.
Geometry and spatial reasoning. The student
uses transformational geometry to develop spatial
sense. The student is expected to:
•Factoring
•Graphing calculators
A.10B
Make connections among the solutions (roots) of
quadratic equations, the zeros of their related
functions, and the horizontal intercepts (xintercepts) of the graph of the function.
Including:
•Using a graphing calculator
•Factoring
•Algebra tiles
•Real world problems such as area of a rectangle
A.11
A.11A
G.5
G.11
Quadratic and other nonlinear functions. The
student understands there are situations modeled
by functions that are neither linear nor quadratic
and models the situations. The student is
expected to:
Use patterns to generate the laws of exponents and
apply them in problem-solving situations.
Geometric patterns. The student uses a variety of
representations to describe geometric
relationships and solve problems. The student is
expected to:
Similarity and the geometry of shape. The student
applies the concepts of similarity to justify
properties of figures and solve problems. The
student is expected to:
17
8.6A
Generate similar figures using dilations
including enlargements and reductions.
8.6A
Including:
•Figures graphed on a coordinate grid
•Figures with dimensions labeled in the
diagram
•Vocabulary: (i.e. similar, dilation,
enlargement, reduction, coordinate plane,
vertex, dimension, proportional, corresponding
side, scale factor)

Multiply to solve for dilations by using the
scale factor

Enlargements – scale factor greater than 1

Reductions – scale factor less than 1
Generate similar figures using dilations including
enlargements and reductions.
G.5A
Including:
•Figures graphed on a coordinate grid
•Figures with dimensions labeled in the diagram.
•Problems in which vertices are given and require
the student to plot the figure.
G.11B
8.6A
8.6B
Graph dilations, reflections, and translations on
a coordinate plane.
Including:
•All four quadrants
•Reflections across the x or y axes
•Dilations include enlargements or reductions
•Vocabulary: (i.e. similar, dilation,
enlargement, reduction, coordinate plane,
vertex, dimension, translation, reflection
8.6B
Graph dilations, reflections, and translations on a
coordinate plane.
8.6B
Use numeric and geometric patterns to develop
algebraic expressions representing geometric
properties.
Including:
•Finding the sum of the interior angles of polygons
•Deriving volume formulas
•Discovering the area formulas for a regular
polygon
•Discovering the relationship among the sides of
45-45-90 and 30-60-90 triangles
Use ratios to solve problems involving similar
figures.
Including:
•Comparing the areas, perimeters and volumes of
similar polygons and solids
•Dilations
Generate similar figures using dilations including
enlargements and reductions.
Including:
•Figures graphed on a coordinate grid •Figures
with dimensions labeled in the diagram
•Problems in which vertices are given and require
the student to plot the figure
Graph dilations, reflections, and translations on a
coordinate plane.
Including terminology:
•“mapped”
•x ’ , y ’ , z ’
18
8.7
proportional, corresponding side, scale factor)
Geometry and spatial reasoning. The student
uses geometry to model and describe the
physical world. The student is expected to:
8.7
Geometry and spatial reasoning. The student
uses geometry to model and describe the physical
world. The student is expected to:
G.6
G.7
8.7A
Draw three-dimensional figures from different
perspectives.
8.7A
Include:
•Drawing three dimensional figures when
given a specified view
•Drawing two dimensional views when a three
dimensional figure is given
Draw three-dimensional figures from different
perspectives.
G.6C
Include:
•nets
•review of classification of polygons and
polyhedrons
Including the use of unit blocks to explore
concrete models.
8.7A
8.7B
Use geometric concepts and properties to solve
problems in fields such as art and architecture.
8.7B
Include:
 Using the given data to solve for perimeter,
circumference, area, volume or a dimension
 Various representation of limits of measures
8.7C
Use pictures or models to demonstrate the
Pythagorean Theorem.
Including:
•When inscribed in a circle or polygon and/or
real life pictorial examples (see sample
Use geometric concepts and properties to solve
problems in fields such as art and architecture.
Dimensionality and the geometry of location. The
student analyzes the relationship between threedimensional geometric figures and related twodimensional representations and uses these
representations to solve problems. The student is
expected to:
Dimensionality and the geometry of location. The
student understands that coordinate systems
provide convenient and efficient ways of
representing geometric figures and uses them
accordingly. The student is expected to:
Use orthographic and isometric views of threedimensional geometric figures to represent and
construct three-dimensional geometric figures and
solve problems.
8.7B
Draw three-dimensional figures from different
perspectives.
Use geometric concepts and properties to solve
problems in fields such as art and architecture
Include:
 Scale factors and measurement conversion
 Area and perimeter
8.7C
Use pictures or models to demonstrate the
Pythagorean Theorem.
8.7C
Use pictures or models to demonstrate the
Pythagorean Theorem.
Include:
•The introduction in the use of TAKS formula
chart
19
8.7D
8.8
8.8A
questions)
•Vocabulary: (i.e. hypotenuse, leg, radius,
diameter)
Locate and name points on a coordinate plane
using ordered pairs of rational numbers.
Including:
•Using all four quadrants
•Vocabulary: (i.e. x-axis, y-axis, x-coordinate,
y- coordinate, quadrants, origin)
Measurement. The student uses procedures to
determine measures of three-dimensional
figures. The student is expected to:
Find lateral and total surface area of prisms,
pyramids, and cylinders using concrete models
and nets (2 dimensional models).
•Teaching how to find the square roots on the
calculator
8.7D
8.8B
G.7A
8.7D
8.8
Measurement. The student uses procedures to
determine measures of three-dimensional figures.
The student is expected to:
G.8
8.8A
Find lateral and total surface area of prisms,
pyramids, and cylinders using concrete models and
nets (2 dimensional models).
G.8A
No spheres, No cones
Including:
 Unit conversions in two and three
dimensions
 Using formula chart rulers and formulas
 Various representations of limits of
measures of edges
 Vocabulary (i.e. surface area, prism,
rectangular prism, triangular prism,
cylinder, pyramid, lateral surface area, edge,
face vertex, height, base, total surface area,
net)
 Measurements in metric and standard units
 Recognizing symbol ≈ means
approximately equal to
Connect models of prisms, cylinders, pyramids,
Locate and name points on a coordinate plane
using ordered pairs of rational numbers.
Including:
 Using the original TAKS formula chart rulers
consistently when measuring
 Reviewing how to read and use a ruler
 Measurements in metric and standard units
8.8B
Connect models of prisms, cylinders, pyramids,
G.8D
8.8A
G.8D
Use one- and two-dimensional coordinate systems
to represent points, lines, rays, line segments, and
figures.
Including triangles and quadrilaterals.
Locate and name points on a coordinate plane
using ordered pairs of rational numbers.
Congruence and the geometry of size. The
student uses tools to determine measurements of
geometric figures and extends measurement
concepts to find perimeter, area, and volume in
problem situations. The student is expected to:
Find area of regular polygons, circles, and
composite figures.
Find surface areas and volumes of prisms,
pyramids, spheres, cones, cylinders, and
composites of these figures in problem situations.
Find lateral and total surface area of prisms,
pyramids, and cylinders using concrete models and
nets (two-dimensional models).
Find surface areas and volumes of prisms,
20
spheres, and cones to formulas for volume of
these objects.
spheres, and cones to formulas for volume of these
objects.
Including:
•Matching nets and models to appropriate
formulas to problem solve
•Real-life models (i.e. sphere-basketball)
Include:
•Reviewing how to read EXIT level formula chart
for Volume
•Reviewing how to find the Volume of solids
•Reviewing how to read and use a ruler
•Using the original TAKS formula chart rulers
consistently when measuring; instead of handheld
rulers
Estimate measurements and use formulas to solve
application problems involving lateral and total
surface area and volume.
8.8B
Including measurements in metric and standard
units.
8.8C
Estimate measurements and use formulas to
solve application problems involving lateral and
total surface area and volume.
8.8C
Including:
 Measurement in standard and metric units
for cubes, cylinders, cones, spheres, and
prisms.
 Rounding all dimensions to whole numbers.
 Using “3” for pi symbol.
 The capital B on the formula chart is the
area of the base.
 Vocabulary (i.e. surface area , prism,
rectangular prism, triangular prism,
cylinder, pyramid, lateral surface area, edge,
face, vertex, height, base, total surface area,
net, volume).
 Real-life models (i.e. rectangular prism = a
present or a shoe box)
8.9
Including measurements in metric and standard
units for cubes, cylinders, cone, spheres, and
prisms.
Measurement. The student uses procedures to
8.8C
Connect models of prisms, cylinders, pyramids,
spheres, and cones to formulas for volume of these
objects.
pyramids, spheres, cones, cylinders, and
composites of these figures in problem situations.
Connect models of prisms, cylinders, pyramids,
spheres, and cones to formulas for volume of these
objects.
Estimate measurements and use formulas to solve
application problems involving lateral and total
surface area and volume.
Connect models of prisms, cylinders, pyramids,
spheres, and cones to formulas for volume of these
objects.
Include:
 Reviewing the concepts of estimation,
rounding and place value
 Reviewing how to read and use a ruler
 Using the TAKS formula chart ruler
consistently, instead of a handheld ruler
8.9
Measurement. The student uses procedures to
G.5
Geometric patterns. The student uses a variety of
21
determine measure of three-dimensional
figures. The student is expected to:
determine measure of three-dimensional figures.
The student is expected to:
G.8
G.11
8.9A
Use the Pythagorean Theorem to solve real-life
problems
8.9A
Including:
 When inscribed in a circle or polygon
and/or real life pictorial examples
 Vocabulary: (i.e. hypotenuse, leg, radius,
diameter)
Use the Pythagorean Theorem to solve real-life
problems
Include:
 Using TAKS formula chart
 Teaching how to find square roots on the
calculator
Examples of Pythagorean triples are (3, 4, 5),
(6, 8, 10), (5, 12, 13), (15, 8, 17),(12, 16, 20),
(7, 24, 25) etc.
G.5D
G.8C
G.11C
8.9A
8.9B
Use proportional relationships in similar twodimensional figures or similar three-
8.9B
Use proportional relationships in similar twodimensional figures or similar three-dimensional
8.9B
representations to describe geometric
relationships and solve problems. The student is
expected to:
Congruence and the geometry of size. The
student uses tools to determine measurements of
geometric figures and extends measurements
concepts to find perimeter, area, and volume in
problem situations. The student is expected to:
Similarity and the geometry of shape. The
student applies the concepts of similarity to justify
properties of figures and solve problems. The
student is expected to:
Identify and apply patterns from right triangles to
solve meaningful problems including special right
triangles (45-45-90 and30-60-90) and triangles
whose sides are Pythagorean triples.
Including trig ratios sine, cosine, tangent
Derive, extend, and use the Pythagorean Theorem
Including:
 Distance formula
 Unknown lengths in polygons and circles
Develop, apply, and justify triangle similarity
relationships, such as right triangle ratios, trig
ratios, and Pythagorean triples using a variety of
methods.
Including:
 Triangle Prop Theorem
 Angle Bisector Proportionality
 Sine, Cosine, & Tangent
Use the Pythagorean Theorem to solve real-life
problems
Use proportional relationships in similar twodimensional figures or similar three-dimensional
22
dimensional figures to find missing
measurements.
8.10
8.10A
Including:
 Setting up proportions or using a scale
factor
 Identifying the corresponding sides of
similar figures when the figure is rotated
and/or not rotated
 Vocabulary (i.e. similar, corresponding,
scale factor, dimensions, rotation,
proportional and two-and three-dimensional
figures)
Measurement: The student describes how
changes in dimensions affect linear, area, and
volume measures. The student is expected to:
Describe the resulting effects on perimeter and
area when dimensions of a shape are changed
proportionally.
Including:
 Using a scale factor and/or dilations with
whole numbers or fractions • Rectangles
 Finding missing dimensions or area or
perimeter •Squares
 Using the same scale factor proportionately
in a figure the effects • Circles
 Vocabulary: (i.e. perimeter, area, scale
factors, dilation/dilated, •A review of the
scale factor concepts enlargement,
reduction, similar, dimension, proportional)
 Generalizing the effects on perimeter, area
and volume if the length, Examples include:
width, and height are changed by the same
scale factor
figures to find missing measurements.
figures to find missing measurements.
8.10
Measurement: The student describes how
changes in dimensions affect linear, area, and
volume measures. The student is expected to:
G.11
8.10A
Describe the resulting effects on perimeter and
area when dimensions of a shape are changed
proportionally.
G.11D
Including:
 Rectangles
 Squares
 Circles
 A review of the scale factor concepts
8.10A
Similarity and the geometry of shape. The
student applies the concepts of similarity to justify
properties of figures and solve problems. The
student is expected to:
Describe the effects on perimeter, area, and
volume when one or more dimensions of a figure
are changed and apply this idea in solving
problems.
Describe the resulting effect on perimeter and area
when dimensions of a shape are changed
proportionally.
23
8.10B
8.11
8.11A
8.11B
8.11C
Describe the resulting effects on volume when
dimensions of a solid are changed
proportionally
Probability and statistics. The student applies
concepts of theoretical and experimental
probability to make predictions. The student is
expected to:
Find the probabilities of dependent and
independent events.
Including:
 Displaying the results as a fraction or a
decimal or percent
 Working the problem from a verbal
description
 Analyzing data from a table or graph
 Using experimental results and comparing
those results with the theoretical results
Use theoretical probabilities and experimental
results to make predictions and decisions
Including:
 Displaying the results as a fraction or a
decimal or percent
 Working the problem from a verbal
description
 Analyzing data from a table or graph
 Using experimental results and comparing
those results with the theoretical results
Select and use different models to simulate an
event.
8.10B
8.11
8.11A
Describe the resulting effects on volume when
dimensions of a solid are changed proportionally.
Including:
 Rectangular prisms
 Cylinders
Probability and statistics. The student applies
concepts of theoretical and experimental
probability to make predictions. The student is
expected to:
Find the probabilities of dependent and
independent events.
G.11D
8.10B
8.11
8.11A
Describe the effects on perimeter, area, and
volume when one or more dimensions of a figure
are changed and apply this idea in solving
problems.
Describe the resulting effect on volume when
dimensions of a solid are changed proportionally.
Probability and statistics. The student applies
concepts of theoretical and experimental
probability to make predictions. The student is
expected to:
Find the probabilities of dependent and
independent events.
Including:
 Using the terminology dependent and
independent events
 Reviewing fraction, decimal, and %
conversions
 Teaching calculator concepts (i.e. decimal to
fraction)
8.11B
Use theoretical probabilities and experimental
results to make predictions and decisions
8.11B
Use theoretical probabilities and experimental
results to make predictions and decisions
Including:
 Teaching difference between theoretical and
experimental probability’
 Reviewing fraction, decimal, and %
conversions
 Calculator use
Including:
24

8.12
8.12A
8.12B
Displaying the results as a fraction or a
decimal or percent
 Using experimental results from
independent and dependent events and
comparing those results with
the theoretical results
(Such as using spinners, dice, and/or
marbles
in a sack in a probability event)
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
Select the appropriate measure of central
tendency or range to describe a set of data and
justify the choice for a particular situation.
Including:
 Finding mean, median, mode and range to
justify an answer
 The effects of changing data on mean,
median, mode and range
See SE Algebra I (2D)
8.12
8.12A
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
Select the appropriate measure of central tendency
or range to describe a set of data and justify the
choice for a particular situation.
8.12
8.12A
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
Select the appropriate measure of central tendency
or range to describe a set of data and justify the
choice for a particular situation.
Including:
 Mean
 Median
 Mode
8.12B
See SE Algebra I (2D
8.12C
Select and use an appropriate representation for
presenting and displaying relationships among
collected data, including line plots, line graphs,
stem and leaf plots, circle graphs, bar graphs, box
Draw conclusions and make predictions by
analyzing trends in scatterplots.
8.12C
Including:
 Scatter plots that show no trend
 Positive, negative and no correlations or
trends
 Describe the scatter plot in words
(increasing/decreasing)
Select and use an appropriate representation for
presenting and displaying relationships among
collected data, including line plots, line graphs,
stem and leaf plots, circle graphs, bar graphs,
8.12C
Select and use an appropriate representation for
presenting and displaying relationships among
collected data, including line plots, line graphs,
stem and leaf plots, circle graphs, bar graphs, box
25
8.13
8.13A
box and whisker plots, histograms, and Venn
diagrams, with and without the use of
technology.
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
Evaluate methods of sampling to determine
validity of an inference made from a set of data.
and whisker plots, histograms, and Venn diagrams,
with and without the use of technology.
and whisker plots, histograms, and Venn diagrams,
with and without the use of technology.
8.13
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
8.13
Probability and statistics. The student uses
statistical procedures to describe data. The
student is expected to:
8.13B
Recognize misuses of graphical or numerical
information and evaluates predictions and
conclusions based on data analysis
8.13B
Recognize misuses of graphical or numerical
information and evaluates predictions and
conclusions based on data analysis
Including biased sampling due to method of
collecting the data.
8.13B
8.14
8.14A
8.14B
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Recognize misuses of graphical or numerical
information and evaluates predictions and
conclusions based on data analysis
Including analyzing all parts of a bar graph
(title, vertical and horizontal scale) and table of
values for possible misleading information.
Underlying processes and math tools. The
student applies grade 8 math to solve problems
connected to everyday experiences,
investigations in other disciplines, and
activities in and outside of school. The
student is expected to:
Identify and apply mathematics to everyday
experiences, to activities in and outside of
school, with other disciplines, and with other
mathematical topics.
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Use a problem-solving model that incorporates
8.14
Underlying processes and math tools. The
student applies grade 8 math to solve problems
connected to everyday experiences, investigations
in other disciplines, and activities in and outside
of school. The student is expected to:
8.14
Underlying processes and math tools. The
student applies grade 8 math to solve problems
connected to everyday experiences, investigations
in other disciplines, and activities in and outside
of school. The student is expected to:
8.14A
Identify and apply mathematics to everyday
experiences, to activities in and outside of school,
with other disciplines, and with other mathematical
topics.
8.14A
Identify and apply mathematics to everyday
experiences, to activities in and outside of school,
with other disciplines, and with other mathematical
topics.
8.14B
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
Use a problem-solving model that incorporates
8.14B
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
Use a problem-solving model that incorporates
26
8.14C
understanding the problem, making a plan,
carrying out the plan, and evaluating the
solution for reasonableness.
understanding the problem, making a plan,
carrying out the plan, and evaluating the solution
for reasonableness.
understanding the problem, making a plan,
carrying out the plan, and evaluating the solution
for reasonableness.
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
Select or develop an appropriate problemsolving strategy from a variety of different
types, including drawing a picture, looking for a
pattern, systematic guessing and checking,
acting it out, making a table, working a simpler
problem, or working backwards to solve a
problem.
8.14D
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Select tools such as real objects, manipulatives,
paper/pencil, and technology or techniques such
as mental math, estimation, and number sense
to solve problems.
8.15
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Underlying processes and math tools. The
student communicates about Grade 8 math
through informal and mathematical language,
representations, and models. The student is
8.14C
Including:
 Review of key vocabulary words (i.e. “per’,
“each”, and “of” means to multiply)
 Review of problem solving strategies (i.e. draw
a picture or a table)
Select or develop an appropriate problem-solving
strategy from a variety of different types, including
drawing a picture, looking for a pattern, systematic
guessing and checking, acting it out, making a
table, working a simpler problem, or working
backwards to solve a problem.
8.14C
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
8.15
Underlying processes and math tools. The
student communicates about Grade 8 math
through informal and mathematical language,
representations, and models. The student is
Select or develop an appropriate problem-solving
strategy from a variety of different types, including
drawing a picture, looking for a pattern, systematic
guessing and checking, acting it out, making a
table, working a simpler problem, or working
backwards to solve a problem.
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
8.15
Underlying processes and math tools. The
student communicates about Grade 8 math
through informal and mathematical language,
representations, and models. The student is
27
expected to:
8.15A
8.15B
8.16
8.16A
8.16B
Communicate mathematical ideas using
language, efficient tools, appropriate units and
graphical, numerical, physical, or algebraic
mathematical models.
expected to:
8.15A
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Evaluate the effectiveness of different
representations to communicate ideas.
This student expectation can be tested in every
strand including one or more than one TEKS at
a time.
Underlying processes and math tools. The
student uses logical reasoning to make
conjectures and verify conclusions. The
student is expected to:
Make conjectures from patterns or sets of
examples and non-examples.
Including:
 Defining a concept introduced in a higher
grade
 Showing a pattern, examples, and/or nonexamples
 Expecting students to choose a correct
response by analyzing the pattern,
examples, or non-examples
Validate his/her conclusions using
mathematical properties and relationships.
This student expectation can be tested in every
strand including one or more than one TEKS at
Communicate mathematical ideas using language,
efficient tools, appropriate units and graphical,
numerical, physical, or algebraic mathematical
models.
expected to:
8.15A
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
8.16
8.16A
8.16B
Underlying processes and math tools. The
student uses logical reasoning to make
conjectures and verify conclusions. The student
is expected to:
Make conjectures from patterns or sets of
examples and non-examples.
Including:
 Defining a concept introduced in a higher
grade
 Showing a pattern, examples, and/or nonexamples
 Expecting students to choose a correct
response by analyzing the pattern, examples, or
non-examples
Validate his/her conclusions using mathematical
properties and relationships.
This student expectation can be tested in every
strand including one or more than one TEKS at a
Communicate mathematical ideas using language,
efficient tools, appropriate units and graphical,
numerical, physical, or algebraic mathematical
models.
This student expectation can be tested in every
strand including one or more than one TEKS at a
time.
8.16
8.16A
8.16B
Underlying processes and math tools. The
student uses logical reasoning to make
conjectures and verify conclusions. The student
is expected to:
Make conjectures from patterns or sets of
examples and non-examples.
Including:
 Defining a concept introduced in a higher
grade
 Showing a pattern, examples, and/or nonexamples
 Expecting students to choose a correct
response by analyzing the pattern, examples, or
non-examples
Validate his/her conclusions using mathematical
properties and relationships.
This student expectation can be tested in every
strand including one or more than one TEKS at a
28
a time.
time.
G.1
G.1A
G.1B
G.1C
time.
The student understands the structure of, and
relationships within, an axiomatic system. The
student is expected to:
Develop an awareness of the structure of a
mathematical system, connecting definitions,
postulates, logical reasoning, and theorems.
Including the use of direct proofs, manipulatives
and technology to draw conclusions and discover
relationships about geometric shapes and their
properties.
Recognize the historical development of geometric
systems and know mathematics is developed for a
variety of purposes.
Including:
 The discovery of Pi and it’s applications
 A historical discussion of Euclid’s
elements and how they are used in the
development of modern geometry
 A time line of geometry’s developments
Compare and contrast the structures and
implications of Euclidean and non-Euclidean
geometries.
Including parallelism as exhibited in Euclid’s 5th
postulate.
G.2
Non-Euclidean geometries include:
 Spherical to show parallel lines do not exist
as defined in Euclidean geometry
 Cylindrical to show parallel lines do exist
as defined in Euclidean geometry
Geometric Structure. The student analyzes
geometric relationships in order to make and
29
G.2A
G.2B
G.3
G.3A
verify conjectures. The student is expected to:
Use constructions to explore attributes of
geometric figures and to make conjectures about
geometric relationships.
Including:
 The use of manipulatives and technology
 The construction of angle bisectors,
perpendicular bisectors, parallel lines,
congruent angles, congruent segments,
perpendicular lines at a point on a line,
perpendicular lines from a point to a line and
segment bisectors
Make conjectures about angles, lines, polygons,
circles, and three-dimensional figures and
determine the validity of the conjectures, choosing
from a variety of approaches such as coordinate,
transformational, or axiomatic.
Including:
 Reflections
 Translations
 Rotations
 The use of direct proofs, manipulatives and
technology to draw conclusions and discover
relationships about geometric shapes and their
properties.
Geometric structure. The student applies logical
reasoning to justify and prove mathematical
statements. The student is expected to:
Determine the validity of a conditional
statement, its converse, inverse, and
contrapositive.
Including consistent usage as it applies to
30
geometric figures and relationships.
G.3B
Construct and justify statements about geometric
figures and their properties;
Including:
G.3C

The formulation of conclusions in the form of
a conditional statement

The use of manipulatives and technology to
draw conclusions about geometric figures
Use logical reasoning to prove statements are true
and find counter examples to disprove statements
that are false.
Examples include:
The statement “All right angles are congruent” is
true. Is the converse also true? If not, provide a
counterexample that disproves the statement.
G.3D
Use inductive reasoning to formulate a conjecture.
Including:

G.3E
The student discovery of the sum of the
interior angles of a polygon
 Finding the volume of cones and pyramids
 The student discovery of relationships among
similar polygons and solids
Use deductive reasoning to prove a statement.
Including:
 Triangle congruence statements (angle-side31
G.5
angle, side-side-side, angle-angle-side, sideangle-side and hypotenuse-leg)
 The relationships among the angles of parallel
lines (i.e. alternate interior angles, same side
interior angles, corresponding angles
Geometric patterns. The student uses a variety
of representations to describe geometric
relationships and solve problems. The student
is expected to:
G.5B
Use numeric and geometric patterns to make
generalizations about geometric properties,
including properties of polygons, ratios in
similar figures and solids, and angle
relationships in polygons and circles.
G.5C
Use properties of transformations and their
compositions to make connections between
mathematics and the real world, such as
tessellations.
G.6
Dimensionality and the geometry of location.
The student analyzes the relationship between
three-dimensional geometric figures and
related two-dimensional representations and
uses these representations to solve problems.
The student is expected to:
G.6A
Describe and draw the intersection of a given
plane with various three-dimensional geometric
figures.
G.6B
G.7
Including conics and other cross-sectional views of
geometric solids.
Use nets to represent and construct threedimensional geometric figures
Dimensionality and the geometry of location.
The student understands that coordinate
32
systems provide convenient and efficient ways
of representing geometric figures and uses
them accordingly. The student is expected to:
G.7C
G.8
G.8B
Derive and use formulas involving length,
slope, and midpoint
Including:
The relationship between Pythagorean theorem
and the distance formula
The application of the formulas to prove properties
of figures such as rhombi, squares, rectangles, etc.
Congruency and the geometry of size. The
student uses tools to determine measurements
of geometric figures and extends measurement
concepts to find perimeter, area, and volume
in problem situations. The student is expected
to:
Find areas of sectors and arc lengths of circles
using proportional reasoning.
Including:
ArcLength
=
Circumference
G.9
G.9A
Central angle
360°
Area of sector
=
Central angle
Area of circle
360°
Congruence and the geometry of size. The
student analyzes properties and describes
relationships in geometric figures.
The
student is expected to:
Formulate and test conjectures about the
properties of parallel and perpendicular lines
based on explorations and concrete models.
Including:
 Finding the slopes of lines to determine their
33
G.9B
G.9C
G.9D
relationship (parallel, perpendicular or
intersecting)
 Student discovery of Mid-segment theorem.,
Dual Parallels theorem, Dual Perpendiculars
theorem and Triangle Proportionality theorem
Formulate and test conjectures about the
properties and attributes of polygons and their
component parts based on explorations and
concrete models.
Including:
 Recognizing polygons (through decagons)
 Properties of regular polygons
 Properties of quadrilaterals, triangles, and
special polygons (e.g. hexagons)
Formulate and test conjectures about the
properties and attributes of circles and the lines
that intersect them based on explorations and
concrete models.
Including:
 Identifying tangents, secants, chords,
diameters, radii, inscribed angles, central
angles
 Student exploration of the properties of
intersecting chords, secants and tangents.
 Exploration of the relationships among angles
in circles
 Application of central angles to the reading of
circle graphs
Analyze the characteristics of polyhedra and
other three-dimensional figures and their
component parts based on explorations and
concrete models.
Including:
 Prisms (with regular polygon bases to 10 sides)
 Pyramids
34



G.10
G.10A
G.10B
Cones
Cylinders
Spheres
Congruence and the geometry of size. The
student applies the concept of congruence to
justify properties of figures and solve problems.
The student is expected to:
Use congruence transformations to make
conjectures and justify properties of geometric
figures including figures represented on a
coordinate plane.
Including rotations, reflections, translations, and
combinations of these.
Justify and apply triangle congruence
relationships.
Including:


G.11
G.11A
SAS, SSS, ASA, AAS, HL
The use of triangle congruence to prove
corresponding parts of triangles are congruent
Similarity and the geometry of shape. The
student applies the concepts of similarity to justify
properties of figures and solve problems. The
student is expected to:
Use and extend similarity properties and
transformations to explore and justify conjectures
about geometric figures.
Including:
 Dilations
 Rotations
 Reflections
 Translations
35