Mississippi College and Career Readiness Standards for
Mathematics Scaffolding Document
Grade 8
September 2016 Page 1 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers
8.NS.1
Know that numbers that
are not rational are called
irrational. Understand
informally that every
number has a decimal
expansion; for rational
numbers show that the
decimal expansion repeats
eventually, and convert a
decimal expansion which
repeats eventually into a
rational number.
Desired Student Performance
A student should know
• Real numbers are the set of rational
numbers together with the set of irrational
numbers.
• A rational number is a number expressed
in the form a/b or -a/b for some fraction
a/b. The rational numbers include the
integers.
• An irrational number is a number that
cannot be expressed as the ratio a/b,
where a and b are integers and b ≠ 0.
• The decimal form of a fraction is called a
repeating or terminating decimal.
• A repeating decimal is the decimal form
of a rational number. Repeating decimals
can be represented using bar notation
where a bar is drawn only over the
digit(s) that repeat. For example,
0.333333…. = 0. 3
.
• A decimal is called terminating if its
repeating digit is 0. For example, 0.250
is
typically written 0.25.
A student should understand
• Real numbers are either rational or
irrational.
• That the set of real numbers can be
represented with a Venn diagram.
A student should be able to do
• Write a fraction or mixed number as
a repeating decimal by showing,
filling in, or otherwise producing the
steps of long division.
• Write a repeating decimal as a
fraction or mixed number in simplest
form.
• Name all sets of numbers to which a
given real number belongs.
• Convert a repeating decimal into a
rational number.
September 2016 Page 2 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers
8.NS.2
Use rational approximations
of irrational numbers to
compare the size of irrational
numbers, locate them
approximately on a number
line diagram, and estimate the
value of expressions (e.g., π
2
).
For example, by truncating the
decimal expansion of
,
show that
is between 1
and 2, then between 1.4 and
1.5, and explain how to
continue on to get better
approximations.
Desired Student Performance
A student should know
• The square root of a number is one of
its two equal factors. If a
2
= b, then a =
±
.
• A perfect square is a rational number
whose square root is a whole number.
For example, 36 is a perfect square
because its square root is 6.
• The cube root of a number is one of
three equal factors of a number. If a
3
=
b, then a =
.
• Real numbers is the set of rational
numbers together with the set of
irrational numbers.
• A rational number is a number
expressed in the form a/b or -a/b for
some fraction a/b. The rational
numbers include the integers.
• An irrational number is a number that
cannot be expressed as the ratio a/b,
where a and b are integers and b ≠ 0.
• The decimal form of a fraction is called
a repeating or terminating decimal.
A student should understand
• Every positive number has both a
positive and negative square root. In
real-world situations, only the
positive or principal square root is
considered.
• How to compare and order rational
and irrational numbers.
• The value of a square root can be
approximated between integers.
• The square root of a non-perfect
square is irrational.
• Square roots may be negative and
written as -
24.
• How to plot irrational numbers on a
number line.
A student should be able to do
• Find the square and cube roots of
numbers.
• Estimate square roots and cube
roots to the nearest integer using
perfect squares and perfect cubes.
• Estimate square roots and cube
roots to an appropriate
approximation by truncating, or
dropping, the digits after the first
decimal place, then after the second
decimal place and so on.
• Compare and order rational and
irrational numbers using a number
line.
• Use the estimated value of an
irrational number to evaluate an
expression.
September 2016 Page 3 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Work with radicals and integer exponents
8.EE.1
Know and apply the
properties of integer
exponents to generate
equivalent numerical
expressions.
For example,
3
2
× 3
–5
= 3
–3
= 1/3
3
= 1/27.
Desired Student Performance
A student should know
• A power is a product of repeated factors
using an exponent and a base.
• The base in a power is the number that is
the common factor.
• The exponent in a power is the number
of times the base is used as a factor.
• A monomial is a number, a variable, or a
product of a number and one or more
variables.
• How to multiply powers with the same
base, add their exponents (product of
powers)
×
=
• How to divide powers with the same
base, subtract their exponents. (quotient
of powers)
=
• How to find the power of a power,
multiply the exponents. (power of a
power) (
)
=
×
• How to find the power of a product, find
the power of each factor, and multiply.
(power of a product) (ab)
m
= a
m
b
m
• Any nonzero number to the zero power is
1.
= 1, x 0.
• Any nonzero number to the negative n
power is the multiplicative inverse of its
nth power,
=
, x
0.
A student should understand
• All operations involving the
properties of addition and the
distributive property of
multiplication over addition can be
used to simplify expressions.
• Variables can be used to represent
quantities in a real-world or
mathematical problem.
• Expressions are powerful tools for
exploring, reasoning about, and
representing situations.
• Two or more expressions may be
equivalent even when their
symbolic forms differ.
A student should be able to do
• Write an expression using
exponents.
• Evaluate an expression containing
exponents.
• Simplify expressions involving one,
two, or three properties using the
laws of exponents.
• Write an expression using a
positive exponent.
• Write a fraction as an expression
using a negative exponent other
than -1.
• Multiply and divide with negative
exponents.
• Classify expressions by their
equivalence to a given expression.
September 2016 Page 4 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Work with radicals and integer exponents
8.EE.2
Use square root and cube
root symbols to represent
solutions to equations of the
form x
2
= p and x
3
= p, where
p is a positive rational
number. Evaluate square
roots of small perfect squares
and cube roots of small
perfect cubes. Know that √2 is
irrational.
Desired Student Performance
A student should know
• The radical sign is the symbol
placed
before a number or quantity to indicate
the extraction of a root, which will be the
square root. The value of a higher root is
indicated by a raised digit in front of the
symbol, as in
,
, …
, where n is
an integer.
• The square root of a number is one of its
two equal factors. If a
2
= b, then a = ±
.
• A perfect square is a rational number
whose square root is a whole number.
For example, 36 is a perfect square
because its square root is 6.
• The cube root of a number is one of three
equal factors of a number. If a
3
= b, then
a =
.
• A perfect cube is a rational number
whose cube root is a whole number. For
example, 64 is a perfect cube because its
cube root is 4.
• A rational number is a number expressed
in the form a/b or –a/b for some fraction
a/b. The rational numbers include the
integers.
• An irrational number is a number that
cannot be expressed as the ratio a/b,
where a and b are integers and b ≠ 0.
A student should understand
• How to recognize perfect squares.
• How to recognize perfect cubes.
• That non-perfect squares and non-
perfect cubes are irrational
numbers.
• Squaring a number and taking the
square root (
) of a number are
inverse operations.
• Cubing a number and taking the
cube root (
) of a number are
inverse operations.
• When solving x
2
= 36, there are two
solutions, ±6 since 6 x 6 = 36
and -6 x -6 = 36.
A student should be able to do
• Find square roots of numbers.
• Find cube roots of numbers.
• Estimate square roots and cube
roots to the nearest integer.
• Order and compare real numbers.
• Find the distance between two
points using the distance formula.
• Find parts of a right triangle using
the Pythagorean Theorem.
• Find the edge length of a cubical
object with a given volume.
September 2016 Page 5 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Work with radicals and integer exponents
8.EE.3
Use numbers expressed in
the form of a single digit
times an integer power of 10
to estimate very large or very
small quantities, and to
express how many times as
much one is than the other.
For example, estimate the
population of the United States
as 3 times 10
8
and the
population of the world as 7
times 10
9
, and determine that
the world population is more
than 20 times larger.
Desired Student Performance
A student should know
• Scientific notation is when you
express a number as the product of
two factors. The first factor must be
greater than or equal to one but less
than 10 and the second factor is a
power of 10. For example, × 10
,
1 < 10 and n is an integer
• How to write a number in scientific
notation from standard form.
• How to write a number in standard
form from scientific notation.
• Exponential and standard forms of
powers of 10. For example, 0.1 is
10
-1
.
A student should understand
• Scientific notation is used to
express very large or very small
numbers.
• When looking at a number in
scientific notation, if the exponent
increases by one, the value
increases 10 times.
• When looking at a number in
scientific notation, if the exponent
decreases by one, the value
decreases 10 times.
A student should be able to do
• Compare and interpret scientific
notation quantities in the context of
the situation.
• Evaluate expressions involving
addition, subtraction, multiplication,
or division and express the answer
in scientific notation.
September 2016 Page 6 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Work with radicals and integer exponents
8.EE.4
Perform operations with
numbers expressed in
scientific notation, including
problems where both decimal
and scientific notation are
used. Use scientific notation
and choose units of
appropriate size for
measurements of very large
or very small quantities (e.g.,
use millimeters per year for
seafloor spreading). Interpret
scientific notation that has
been generated by
technology.
Desired Student Performance
A student should know
• Scientific notation is when you
express a number as the product of
two factors. The first factor must be
greater than or equal to one but less
than ten and the second factor is a
power of ten. For example, ×
10
, 1 < 10 and n is an
integer
• How to write a number in scientific
notation from standard form.
• How to write a number in standard
form from scientific notation.
• Exponential and standard forms of
powers of 10. For example, 0.1 is
10
-1
.
• How to convert a number from
standard form to scientific notation
with and without the use of
technology.
A student should understand
• Scientific notation is used to
express very large or very small
numbers.
• How to compare and interpret
scientific notation quantities in the
context of the situation with or
without a scientific calculator.
• When looking at a number in
scientific notation, if the exponent
increases by one, the value
increases 10 times.
• When looking at a number in
scientific notation, if the exponent
decreases by one, the value
decreases 10 times.
• How to read a number that is
written in scientific notation using
technology. For example, 3.7E-2 is
3.7 x 10
-2
.
A student should be able to do
• Perform operations with numbers
expressed in both decimal and
scientific notation and express the
answer in scientific notation without
a scientific calculator.
• Compare and order numbers
expressed as decimals and
scientific notation without a
calculator.
• Choose a meaningful unit of
measure in the context of the
situation with and without a
scientific calculator.
• Interpret scientific notation that has
been generated by a scientific
calculator.
September 2016 Page 7 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations
8.EE.5
Graph proportional
relationships, interpreting the
unit rate as the slope of the
graph. Compare two different
proportional relationships
represented in different ways.
For example, compare a
distance-time graph to a
distance-time equation to
determine which of two moving
objects has greater speed.
Desired Student Performance
A student should know
• Students build on their work with
ratios, unit rates, and proportional
relationships from 6
th
and 7
th
grade.
• A rate is a ratio that compares two
quantities with different kinds of
units.
• A unit rate is a rate that has a
denominator of one unit.
• A proportional relationship exists
when the rate is constant.
• Constant rate of change is when the
rate of change between any two
points is the same.
• How to identify the constant of
proportionality (unit rate) in tables,
graphs, equations, diagrams, and
verbal descriptions of proportional
relationships.
• Constant of proportionality (unit rate)
is the constant ratio in a proportional
linear relationship.
A student should understand
• A linear relationship has a constant
rate of change and a straight line
graph.
• Slope is the rate of change
between any two points on a line.
The ratio of the rise, or vertical
change, to the run, or horizontal
change.
• The rise is the vertical change
between any two points on a line.
• The run is the horizontal change
between any two points on a line.
• Slope =
• A linear relationship is a direct
variation when the ratio of y to x is
a constant, m. We say y varies
directly with x.
=
= , where m is
the constant of variation and m
0
• In a direct variation equation y =
mx, m represents the constant of
variation, the constant of
proportionality, the slope, and the
unit rate.
A student should be able to do
• Graph real-world proportional
relationships such as earnings per
hour.
• Determine whether the relationship
between two quantities is linear.
• Find the constant rate of change in
a linear relationship.
• Compare the proportional
relationship between two different
quantities represented in different
forms.
• Find the slope of a line using a
table, graph, equation, diagram,
and verbal description.
• Find the slope of a line that passes
through two given points.
• Given an equation of a proportional
relationship, graph the relationship
and recognize that the unit rate is
the coefficient of x.
Sep
tember 2016 Page 8 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations
8.EE.6
Use similar triangles to
explain why the slope m is
the same between any two
distinct points on a non-
vertical line in the coordinate
plane; derive the equation y =
mx for a line through the
origin and the equation y =
mx + b for a line intercepting
the vertical axis at b.
Desired Student Performance
A student should know
• Similar triangles have the same
shape.
• The ratio of the rise to the run of two
slope triangles formed by a line is
equal to the slope of the line.
• The slope m of a line passing
through points (x
1,
y
1
) and
(x
2
, y
2
) is the ratio of the difference in
the y-coordinates to the
corresponding difference in the x-
coordinates.
=
x
x
1
, where x
2
x
1
A student should understand
• Since the ratios of the rise to the run
of two similar triangles are the same
as the slope of the line, the slope m
of a line is the same between any
two distinct points on a non-vertical
line in the coordinate plane.
• The ratio of the vertical leg to the
horizontal leg of given similar slope
triangles formed by a line is
equivalent to the absolute value of
the slope of the line.
• How to use the slope formula, point
(x,y) and the origin (0,0) to derive
the equation y = mx.
• How to use the slope formula, point
(x,y), and point (0,b) to derive y =
mx + b.
A student should be able to do
• Graph two triangles given the
vertices of both and determine if
they are similar.
• Graph a pair of similar triangles,
write a proportion comparing the
rise to the run for each of the
similar slope triangles, and find the
numeric value.
• Given the hypotenuse of a right
triangle in a coordinate plane,
choose two pairs of points and
record the rise, run, and slope
relative to each pair and verify that
they are the same.
September 2016 Page 9 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.7a
Solve linear equations in one
variable.
a. Give examples of linear
equations in one variable with
one solution, infinitely many
solutions, or no solutions.
Show which of these
possibilities is the case by
successively transforming
the given equation into
simpler forms, until an
equivalent equation of the
form x = a, a = a, or a = b
results (where a and b are
different numbers).
Desired Student Performance
A student should know
• The product of a number and its
multiplicative inverse is 1.
×
= 1,
where a 0 and b 0.
• The coefficient is the numerical
factor of a term that contains a
variable.
• An equation is a sentence stating
that two quantities are equal.
• The solution of an equation is the
value of a variable that makes the
equation true.
• Addition property of equality.
• Subtraction property of equality
• Multiplication property of equality.
• Division property of equality.
• A two-step equation contains two
operations.
• How to solve simple one-step
equations.
A student should understand
• How to find the multiplicative inverse of
a number.
• To solve an equation in which the
coefficient is not 1, you must multiply or
divide each side by the coefficient of
the variable. For example, in the
equation -3x = 12, you must divide both
sides by -3. A common error in
problems of this type is for students to
divide both sides by 3.
• Some equations have variables on
each side of the equals sign. To solve,
use the properties of equality to write
an equivalent equation with the
variables on one side of the equal sign
and then solve the equation.
• Some equations have no solution.
When this occurs, the solution is the
null set or empty set and is shown by
the symbol or { }. After solving the
equation, the solution will look like a =
b, where a and b are different numbers.
• Other equations may have every
number as their solution. An equation
that is true for every value of the
variable is called an identity. After
solving the equation, the solution will
look like a = a.
A student should be able to do
.
• Solve an equation using the
multiplicative inverse.
• Solve an equation using the
addition, subtraction, multiplication,
or division properties of equality to
justify the steps to the solution.
• Solve multistep equations in which
coefficients and constants may be
any rational number.
• Translate a word phrase or real-
world problem into an equation.
• Solve equations with variables on
both sides of the equal sign.
• Determine if an equation has no
solution.
• Determine if an equation is an
identity with infinitely many
solutions.
• Create equations that have one
solution, infinitely many solutions,
or no solution.
• Classify equations by the number
of solutions.
September 2016 Page 10 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.7b
Solve linear equations in one
variable.
b. Solve linear equations and
inequalities with rational
number coefficients,
including those whose
solutions require expanding
expressions using the
distributive property and
collecting like terms.
Desired Student Performance
A student should know
• The product of a number and its
multiplicative inverse is 1.
×
= 1,
where a 0 and b 0
• The coefficient is the numerical
factor of a term that contains a
variable.
• An equation is a sentence stating
that two quantities are equal.
• An inequality is a statement that two
quantities are not equal. The
symbols >, <, >, and < are used to
express inequalities.
• The solution of an equation or
inequality is the value of a variable
that makes the equation or inequality
true.
• Addition property of equality.
• Subtraction property of equality
• Multiplication property of equality.
• Division property of equality.
• Properties of inequality.
A student should understand
• How to find the multiplicative inverse of
a number.
• To solve an equation or inequality in
which the coefficient is not 1, you must
multiply or divide each side by the
coefficient of the variable. For example,
in the equation -3x = 12, both sides
must be divided by -3.
• Multiplying or dividing an inequality by a
negative number coefficient results in a
reversed inequality symbol. If > and
< 0, then × < × . If > and
< 0, then ÷ < ÷
• Some equations have variables on
each side of the equal sign. To solve,
use the properties of equality to write
an equivalent equation with the
variables on one side of the equal sign
and then solve the equation.
• Some inequalities have variables on
each of the inequality symbol. To solve,
use the properties of inequality to write
an equivalent inequality with variables
on one side of the inequality symbol
and then solve the inequality.
• How to use the distributive property.
For example, 3(x+2) is equivalent to
3x+6.
A student should be able to do
• Solve an equation or inequality
using the multiplicative inverse.
• Solve an equation using the
addition, subtraction, multiplication,
or division properties of equality to
justify the steps to the solution.
• Solve an inequality using the
properties of inequality to justify the
steps to the solution.
• Solve multistep equations and
inequalities with rational number
coefficients and constants.
• Create equivalent expressions by
combining like terms and using the
distributive property.
• Translate a word phrase or real-
world problem into an equation or
inequality.
• Solve equations and inequalities
with variables on both sides of the
equal sign or inequality symbol.
• Solve equations and inequalities
containing grouping symbols.
September 2016 Page 11 of 34
College- and Career-Readiness Standards for Mathematics
• How to solve simple one-step
equations and inequalities.
•
How to combine like terms. For
example, 2r+r+5r = 8r.
• Determine if an equation has no
solution or is an identity with
infinitely many solutions.
September 2016 Page 12 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.8a
Analyze and solve pairs of
simultaneous linear
equations.
a. Understand that solutions
to a system of two linear
equations in two variables
correspond to points of
intersection of their graphs,
because points of
intersection satisfy both
equations simultaneously.
Desired Student Performance
A student should know
• A line represents the infinite number
of solutions to a linear equation with
two variables.
• Linear equations graph a straight
line.
• Solutions of an equation are the
values of the variables that make the
equation true.
• A system of linear equations is two
or more linear equations that
represent constraints on the
variables used.
• The point of intersection is the point
where two lines intersect.
• How to reason abstractly and
quantitatively.
A student should understand
• The points (x, y) on a nonvertical
line are the solutions of the
equation y = mx + b.
• The relationship between
equivalent forms of linear
equations.
• Three solutions to systems of two
linear equations: one solution, no
solution, and infinitely many
solutions.
• When there is no solution, the lines
are parallel, the slopes are the
same, and the y-intercepts are
different.
• When there are an infinite number
of solutions, the lines are the same,
and both the slopes and y-
intercepts are the same.
• When there is only one solution,
the lines intersect, and both the
slopes and y-intercepts are
different.
A student should be able to do
• Graph lines in a plane.
• Use graphs and tables and relate
them to equations.
• Interpret a point as an ordered pair
(x, y).
• Identify the point of intersection of
two lines as the solution to the
system.
• Verify by computation that a point
of intersection is a solution to each
equation in the system.
• Determine the number of solutions
using the slope and y-intercepts.
• Write a second equation to create a
specified solution.
September 2016 Page 13 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.8b
Analyze and solve pairs of
simultaneous linear
equations.
b. Solve systems of two linear
equations in two variables
algebraically, and estimate
solutions by graphing the
equations. Solve simple
cases by inspection. For
example, 3x + 2y = 5 and 3x +
2y = 6 have no solution because
3x + 2y cannot simultaneously
be 5 and 6.
Desired Student Performance
A student should know
• A system of linear equations is two
or more linear equations that
represent constraints on the
variables used.
• Solutions of an equation are the
values of the variables that make the
equation true.
• Expressions in different forms can
be equivalent.
• Coordinates are ordered pairs of
numbers used to locate a point on a
coordinate grid.
• How to solve linear equations with
one variable.
• How to look for and make use of
structure.
A student should understand
• Pairs of lines in a plane intersect,
are parallel, or are the same line.
• The relationship between linear
equations in two variables and
lines in a plane.
• The relationship between
equivalent forms of linear
equations.
• Three solutions to systems of two
linear equations: one solution, no
solution, and infinitely many
solutions.
• Point-slope form:
y - y
1
= m (x - x
1
).
• Standard form: Ax+By = C.
• Slope-intercept form: y = mx + b.
• Substitution is an algebraic model
that can be used to find the exact
solution of a system of equations.
A student should be able to do
• Decide whether two quantities are
in a proportional relationship, and
identify the constant of
proportionality.
• Use algebraic and mathematical
reasoning.
• Solve pairs of simultaneous linear
equations using various methods
such as substitution.
• Use properties of equality.
• Use technology to graph two linear
equations to estimate the solution
of the system.
• Perform operations with a zero
coefficient, and with non-zero
rational coefficients.
September 2016 Page 14 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.8c
Analyze and solve pairs of
simultaneous linear
equations.
c. Solve real-world and
mathematical problems
leading to two linear
equations in two variables.
For example, given coordinates
for two pairs of points,
determine whether the line
through the first pair of points
intersects the line through the
second pair.
Desired Student Performance
A student should know
• A line represents the infinite number
of solutions to a linear equation with
two variables.
• The x-intercept is the point where
the graph crosses the x-axis.
• The y-intercept is the point where
the graph crosses the y-axis.
• Coordinates are ordered pairs of
numbers used to locate a point on a
coordinate grid.
• The slope is the ratio of the vertical
change to the horizontal change
between any two points on a line.
• The slope formula: m = (y
2
– y
1
) / (x
2
– x
1
).
• How to look for and make use of
structure.
A student should understand
• Algebraic expressions and
equations are used to model real-
world problems and represent
quantitative relationships.
• In the equation y = mx + b, m is the
slope of the line as well as the unit
rate of a proportional relationship
(in this case, b = 0).
• The slope of a line is a constant
rate of change.
• The relationship between the slope
formula and point-slope form of a
linear equation.
A student should be able to do
• Analyze the relationship between
the dependent and independent
variables.
• Use variables to represent two
quantities in a real-world problem.
• Write an equation to express one
quantity in terms of the other
quantity.
• Represent proportional
relationships by equations.
• Explain what a point on the graph
of a proportional relationship means
in terms of the situation.
• Interpret solutions in the context of
the problem.
• Graph two linear equations on the
coordinate grid and find their
intersection point.
September 2016 Page 15 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Functions
Define, evaluate, and compare functions
8.F.1
Understand that a function is
a rule that assigns to each
input exactly one output. The
graph of a function is the set
of ordered pairs consisting of
an input and the
corresponding output.
Desired Student Performance
A student should know
• Input is the number or piece of data
that is put into a function.
• Output is the number or piece of
data that is the result of an input of a
function.
• A rule is a summary of a predictable
relationship that tells how to find the
value of a variable.
• This standard extends the
understanding of constant rate.
• How to reason abstractly and
quantitatively.
• The parts of the coordinate plane.
A student should understand
• Functions are useful in making
sense of patterns and making
predictions.
• Functions describe situations
where one quantity determines
another.
• A function represents a relationship
between an input and an output,
where the output depends on the
input; therefore, there can be only
one output for each input.
• How to graph ordered pairs.
• How to name ordered pairs from a
graph.
A student should be able to do
• Determine functions from non-
numerical data.
• Graph inputs and outputs as
ordered pairs in the coordinate
plane.
• Graph functions in the coordinate
plane.
• Read inputs and outputs from the
graph of a function in the
coordinate plane.
• Tell whether a set of points in the
plane represent a function.
September 2016 Page 16 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Functions
Define, evaluate, and compare functions
8.F.2
Compare properties of two
functions each represented in
a different way (algebraically,
graphically, numerically in
tables, or by verbal
descriptions.) For example,
given a linear function
represented by a table of values
and a linear function
represented by an algebraic
expression, determine which
function has the greater rate of
change.
Desired Student Performance
A student should know
• Rate of change is the amount of
change in the dependent variable
produced by a given change in the
independent variable.
• A function is a rule that assigns to
each input exactly one output.
• A linear function is a function whose
graph is a line.
• The y-intercept is the point where
the graph crosses the y-axis.
• How to reason abstractly and
quantitatively.
A student should understand
• The slope (m) of a line is a
constant rate of change.
• Functions can be represented in
a table, as a rule, as a formula or
equation, as a graph, or as a
verbal description.
• Functions describe situations
where one quantity determines
another.
• How to determine the rate of
change (slope) from an equation,
a graph, a table, and a verbal
description.
• How to find the y-intercept.
A student should be able to do
• Translate among representations
and partial representations of
functions.
• Determine the properties of a
function from a verbal description,
table, graph, or algebraic form.
• Make comparisons between the
properties of two functions
represented differently.
September 2016 Page 17 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Functions
Define, evaluate, and compare functions
8.F.3
Interpret the equation y = mx
+ b as defining a linear
function, whose graph is a
straight line; give examples of
functions that are not linear.
For example, the function A = s
2
giving the area of a square as a
function of its side length is not
linear because its graph
contains the points (1,1), (2,4),
and (3,9), which are not on a
straight line.
Desired Student Performance
A student should know
• A linear function graphs a straight
line.
• In the equation y = mx + b, m is the
slope of the line, and b is the y-
intercept of the line.
• Functions that are not linear will not
graph straight lines.
• Interpretation means to
communicate symbolically,
numerically, abstractly, and/or with a
model.
• How to look for and make use of
structure.
A student should understand
• Functions are described in terms
of their inputs and outputs.
• Linear functions may not always
be in the form y = mx + b.
• The slope and y-intercept in
relation to the function
represented by the equation y =
mx + b.
• Constant rates and proportional
relationships can be described
by a function.
• Non-linear functions do not have
a constant rate of change.
• A function machine may use y as
an input and x as an output or
vice versa.
A student should be able to do
• Identify the rate of change between
input and output values.
• Provide examples of relationships
that are non-linear functions.
• Create a table of values that can be
defined as non-linear functions.
• Analyze rates of change to
determine linear and non-linear
functions.
• Determine rate of change from
equations in forms other than the
slope-intercept form.
September 2016 Page 18 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Functions
Use functions to model relationships between quantities
8.F.4
Construct a function to model
a linear relationship between
two quantities. Determine the
rate of change and initial
value of the function from a
description of a relationship
or from two (x, y) values,
including reading these from
a table or from a graph.
Interpret the rate of change
and initial value of a linear
function in terms of the
situation it models, and in
terms of its graph or a table of
values.
Desired Student Performance
A student should know
• The equation y = mx + b defines a
linear function whose graph is a line.
• Rate of change is the amount of
change in the dependent variable
produced by a given change in the
independent variable.
• The y-intercept is the point where
the graph crosses the y-axis.
• The initial value of a linear function is
the value of the y-variable when the
x value is zero.
• How to reason abstractly and
quantitatively.
• How to model with mathematics.
A student should understand
• Functions describe situations
where one quantity determines
another.
• Functions are useful in solving
problems involving quantitative
relationships.
• In the linear equation y = mx + b,
the slope m represents the rate
of change and the y-intercept b
represents the initial value.
• Linear functions can have
discrete rates and continuous
rates.
A student should be able to do
• Use variables to represent
quantities in a real-world or
mathematical problem.
• Analyze a variety of function
representations such as verbal
description, table, two (x,y) values,
graph, and equation.
• Write a linear function modeling a
situation.
• Find the initial value of the function
in relation to the situation.
• Find the rate of change in relation
to the situation.
• Find the y-intercept in relation to
the situation.
• Explain constraints on the domain
in relation to the situation.
September 2016 Page 19 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Functions
Use functions to model relationships between quantities
8.F.5
Describe qualitatively the
functional relationship
between two quantities by
analyzing a graph (e.g., where
the function is increasing or
decreasing, linear or
nonlinear). Sketch a graph
that exhibits the qualitative
features of a function that has
been described verbally.
Desired Student Performance
A student should know
• Qualitative graphs are graphs used
to represent situations that may not
have numerical values or graphs in
which numerical values are not
included.
• A positive rate of change indicates
that a linear function is increasing.
• A negative rate of change indicates
that a linear function is decreasing.
• A linear function graphs a straight
line.
• A non-linear function does not graph
a straight line.
• How to reason abstractly and
quantitatively.
• How to look for and make use of
structure.
A student should understand
• Functions describe situations
where one quantity determines
another.
• The slope of a line can provide
useful information about the
functional relationship between
the two types of quantities.
• The graph of a function can be
used to help describe the
relationship between two
quantities.
• The information represented on
the axes of the graph.
• How to interpret the y-axis.
A student should be able to do
• Match the graph of a function to a
given situation. For example, the
speed of a school bus on its route
to school.
• Create a graph of a function that
describes the relationship between
two variables.
• Write a verbal description of the
functional relationship between two
variables depicted on a graph.
September 2016 Page 20 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.1a
Verify experimentally the
properties of rotations,
reflections, and translations:
a. Lines are taken to lines,
and line segments to line
segments of the same length.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• Symmetry transformations produce
images that are identical in size and
shape to the original figure.
• How to verify means to demonstrate
something is true, accurate, or
justified.
• How to look for and express
regularity in repeated reasoning.
A student should understand
• Ideas about how distance
behaves under transformations
are used to describe and analyze
two-dimensional figures.
• Translations do not change the
orientation.
• Reflections reverse the
orientation.
• Rotations change the orientation.
• Geometric attributes of lines
provide descriptive information
about an object’s properties and
position in space.
• Reflections, rotations, and
translations are symmetry
transformations.
A student should be able to do
• Identify lines and line segments in
two-dimensional figures.
• Measure and compare lengths of a
figure and its image.
• Verify that after a figure has been
translated, reflected, or rotated,
corresponding lines and line
segments remain the same length.
• Determine the change in orientation
to isolate the transformations used.
September 2016 Page 21 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.1b
Verify experimentally the
properties of rotations,
reflections, and translations:
b. Angles are taken to angles
of the same measure.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• Symmetry transformations produce
images that are identical in size and
shape to the original figure.
• How to verify means to demonstrate
something is true, accurate or
justified.
• An angle is a figure formed by two
rays or line segments that have a
common vertex.
• How to look for and express
regularity in repeated reasoning.
A student should understand
• Ideas about how angles behave
under transformations are used
to describe and analyze two-
dimensional figures.
• Geometric attributes of angles
provide descriptive information
about an object’s properties and
position in space.
• Reflections, rotations, and
translations are symmetry
transformations.
A student should be able to do
• Identify angles in two-dimensional
figures.
• Measure and compare angle
measures of a figure and its image.
• Verify that after a figure has been
translated, reflected, or rotated,
corresponding angles have the
same measure.
September 2016 Page 22 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.1c
Verify experimentally the
properties of rotations,
reflections, and translations:
c. Parallel lines are taken to
parallel lines.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• Symmetry transformations produce
images that are identical in size and
shape to the original figure.
• How to verify means to demonstrate
something is true, accurate or
justified.
• Parallel lines are lines in a plane that
never meet.
• How to look for and express
regularity in repeated reasoning.
A student should understand
• Ideas about how distance
behaves under transformations
are used to describe and analyze
two-dimensional figures.
• Geometric attributes of lines
provide descriptive information
about an object’s properties and
position in space.
• Reflections, rotations, and
translations are symmetry
transformations.
A student should be able to do
• Identify parallel lines in two-
dimensional figures.
• Measure and compare parallelism
of a figure and its image.
• Verify that after a figure has been
translated, reflected, or rotated,
corresponding parallel lines remain
parallel.
September 2016 Page 23 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.2
Understand that a two-
dimensional figure is
congruent to another if the
second can be obtained from
the first by a sequence of
rotations, reflections, and
translations; given two
congruent figures, describe a
sequence that exhibits the
congruence between them.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• A rigid motion is a sequence of one
or more rotations, reflections, and/or
translations.
• Understand means to know how
something works or happens.
• How to identify corresponding sides
and angles from congruency
statement and/or figures.
• How to look for and make use of
structure.
A student should understand
• Transformations can be used to
prove that two figures are
congruent.
• Geometric attributes of figures
provide descriptive information
about an object’s position in
space.
• The connection between
congruence and transformations.
• Ideas about congruence can be
used to describe and analyze
two-dimensional figures and to
solve problems.
• Two plane figures are congruent
if one can be obtained from the
other by rigid motion.
• Matching tick marks and arcs
may be used to show
congruency of sides and angles.
A student should be able to do
• Perform a series of transformations
to prove or disprove that two given
figures are congruent.
• Describe a sequence of
transformations that exhibit
congruence of two figures.
September 2016 Page 24 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.3
Describe the effect of
dilations, translations,
rotations, and reflections on
two-dimensional figures
using coordinates.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• Symmetry transformations produce
images that are identical in size and
shape to the original figure.
• Dilations move each point along the
ray through the point emanating from
a fixed center and multiply distances
from the center by a common scale
factor.
• A similarity transformation is a rigid
motion followed by a dilation.
• Coordinates are ordered pairs of
numbers used to locate points on a
coordinate grid.
• How to reason abstractly and
quantitatively.
A student should understand
• The relationship between x- and
y- coordinates and the x- and y-
axes.
• In a dilation, each coordinate of
the original image is multiplied by
the scale factor.
• In a translation, the x and y
coordinates of the original image
change by the value of the
horizontal and vertical changes.
• In a rotation, each point of the
original figure and its new image
are the same distance from the
center of rotation.
• In a reflection, each point of the
original image and its new image
are the same distance from the
line of reflection.
A student should be able to do
• Name an ordered pair as the
coordinates of a point in a
coordinate plane.
• Graph coordinates in a coordinate
plane.
• Describe the changes occurring to
coordinates of a figure after
transformations and dilations.
• Determine the new coordinates of
an image given the original
coordinates and a series of
transformations and/or dilations to
be applied.
September 2016 Page 25 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.4
Understand that a two-
dimensional figure is similar
to another if the second can
be obtained from the first by a
sequence of rotations,
reflections, translations, and
dilations; given two similar
two-dimensional figures,
describe a sequence that
exhibits the similarity
between them.
Desired Student Performance
A student should know
• A transformation is a geometric
operation that relates each point of a
figure to an image point.
• A rigid motion is a sequence of one
or more rotations, reflections, and/or
translations.
• Understand means to know how
something works or happens.
• Two polygons are similar when their
corresponding angles are congruent
and the measures of their
corresponding sides are
proportional.
• Dilations move each point along the
ray through the point emanating from
a fixed center and multiply distances
from the center by a common scale
factor.
• A similarity transformation is a rigid
motion followed by a dilation.
• How to look for and make use of
structure.
A student should understand
• Transformations and dilations
can be used to prove that two
figures are similar.
• Geometric attributes of figures
provide descriptive information
about an object’s position in
space.
• Dilations create similar figures.
• Ideas about similarity can be
used to describe and analyze
two-dimensional figures and to
solve problems.
• Similarity transformation is a rigid
motion followed by a dilation.
A student should be able to do
• Perform a series of transformations
and dilations to prove or disprove
that two given figures are similar.
• Describe a sequence of
transformations and dilations that
exhibit similarity of two figures.
September 2016 Page 26 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.5
Use informal arguments to
establish facts about the
angle sum and exterior angle
of triangles, about the angles
created when parallel lines
are cut by a transversal, and
the angle-angle criterion for
similarity of triangles. For
example, arrange three copies
of the same triangle so that the
sum of the three angles appears
to form a line, and give an
argument in terms of
transversals why this is so.
Desired Student Performance
A student should know
• An exterior angle is an angle at a
vertex of a polygon where the sides
of the angle are one side of the
polygon and the extension of the
other side meeting at the vertex.
• An interior angle is the angle inside a
polygon formed by two adjacent
sides of the polygon.
• Parallel lines are lines in a plane that
never meet.
• A transversal is a line that intersects
two or more lines.
A student should understand
• The angle-angle criterion for
similarity of triangles states that
two triangles with two pairs of
equal angles are similar.
• The sum of any triangle’s interior
angles will have the same
measure as a straight angle.
• The measure of an exterior angle
of a triangle is equal to the sum
of the measures of its two
remote interior angles.
• The relationships and
measurements of the angles
created when two parallel lines
are cut by a transversal.
A student should be able to do
• Construct triangles from three
measures of angles.
• Construct viable arguments.
• Make conjectures regarding
relationships and measurements of
the angles created when two
parallel lines are cut by a
transversal.
• Apply proven relationships to
establish properties to justify
similarity.
• Show that the sum of the angles in
a triangle is the angle formed by a
straight line.
September 2016 Page 27 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand and apply the Pythagorean Theorem
8.G.6
Explain a proof of the
Pythagorean Theorem and its
converse.
Desired Student Performance
A student should know
• Legs are the sides of a right
triangle that are adjacent to the
right angle.
• The hypotenuse is the side of a
right triangle that is opposite the
right angle.
• The hypotenuse is the longest
side of a right triangle.
• The Pythagorean Theorem
states that if a and b are the
lengths of the legs of a right
triangle and c is the length of the
hypotenuse, then a
2
+ b
2
= c
2
.
• The converse of the
Pythagorean Theorem states
that if side lengths of a triangle a,
b, c satisfy a
2
+ b
2
= c
2
, then the
triangle is a right triangle.
A student should understand
• Visual models can be used to
demonstrate the relationship of
the three side lengths of any
right triangle.
• The converse of the
Pythagorean Theorem can be
used to determine if a given
triangle is a right triangle.
• There are various proofs of the
Pythagorean Theorem.
• The Pythagorean Theorem and
its converse can be used to
solve problems.
A student should be able to do
• Use algebraic reasoning to relate a
visual model to the Pythagorean
Theorem.
• Explain why the Pythagorean
Theorem holds.
September 2016 Page 28 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand and apply the Pythagorean Theorem
8.G.7
Apply the Pythagorean
Theorem to determine
unknown side lengths in right
triangles in real-world and
mathematical problems in two
and three dimensions.
Desired Student Performance
A student should know
• Legs are the sides of a right triangle
that are adjacent to the right angle.
• The hypotenuse is the side of a right
triangle that is opposite the right
angle.
• The hypotenuse is the longest side
of a right triangle.
• Irrational numbers cannot be written
as a quotient of two integers where
the denominator is not 0.
• How to decompose polygons into
triangles.
• The Pythagorean Theorem states
that if a and b are the lengths of the
legs of a right triangle and c is the
length of the hypotenuse, then a
2
+
b
2
= c
2
.
A student should understand
• The Pythagorean Theorem
relates to work in irrational
numbers.
• The Pythagorean Theorem is
useful in practical problems.
A student should be able to do
• Apply the Pythagorean Theorem to
find an unknown side length of a
right triangle.
• Use the Pythagorean Theorem in a
diagram to solve real-world
problems involving right triangles.
• Find right triangles in a three-
dimensional figure.
• Use the Pythagorean Theorem to
calculate various dimensions of
right triangles found in a three-
dimensional figure.
• Provide answers as whole numbers
and irrational numbers
approximated to three decimal
places with the use of a calculator.
September 2016 Page 29 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Understand and apply the Pythagorean Theorem
8.G.8
Apply the Pythagorean
Theorem to find the distance
between two points in a
coordinate system.
Desired Student Performance
A student should know
• Legs are the sides of a right triangle
that are adjacent to the right angle.
• The hypotenuse is the side of a right
triangle that is opposite the right
angle.
• The Pythagorean Theorem states
that if a and b are the lengths of the
legs of a right triangle and c is the
length of the hypotenuse, then a
2
+
b
2
= c
2
.
A student should understand
• Geometric attributes of figures
provide descriptive information
that support visualization.
• Applying the Pythagorean
Theorem to find the distance
between two points is related to
finding lengths and analyzing
polygons.
A student should be able to do
• Connect any two points on a
coordinate grid to a third point so
that the three points form a right
triangle.
• Use a right triangle built from two
original points connecting a third
point in a coordinate grid and the
Pythagorean Theorem to find the
distance between the two original
points.
September 2016 Page 30 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Geometry
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
8.G.9
Know the formulas for the
volumes of cones, cylinders,
and spheres and use them to
solve real-world and
mathematical problems.
Desired Student Performance
A student should know
• Volume is the capacity of a three-
dimensional shape.
• Recognize three-dimensional
shapes—cone, cylinder, and sphere.
• The formulas used to find the
volumes of cones, cylinders, and
spheres.
• This is the culminating standard of
acquiring a well-developed set of
geometric measurement skills.
A student should understand
• The volume is the number of unit
cubes that will fit into a three-
dimensional figure.
• The similarity between finding
the volume of a cylinder and the
volume of a right prism.
• The relationship between the
volume of a cylinder and the
volume of a cone with the same
base.
• The relationship between the
volume of a sphere and the
volume of a circumscribed
cylinder.
A student should be able to do
• Use the formulas to find the
volumes of cylinders, cones, and
spheres.
• Solve real-world problems involving
the volumes of cylinders, cones,
and spheres.
September 2016 Page 31 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Statistics and Probability
Investigate patterns of association in bivariate data
8.SP.1
Construct and interpret
scatter plots for bivariate
measurement data to
investigate patterns of
association between two
quantities. Describe patterns
such as clustering, outliers,
positive or negative
association, linear
association, and nonlinear
association.
Desired Student Performance
A student should know
• A Scatter plot is a graph in the
coordinate plane representing a set
of bivariate data.
• Bivariate data are pairs of linked
numerical observations.
• A positive linear association is one
that would be modeled using a line
with a positive slope.
• A negative linear association is one
that would be modeled using a line
with a negative slope.
• A cluster is a group of numerical
data values that are close to one
another.
• An outlier is a value that does not
seem to fit the general pattern in a
scatter plot.
A student should understand
• A pattern in a scatter plot
suggests that there may be a
relationship between the two
variables used to construct the
scatter plot.
• A scatter plot may show a linear
association, a nonlinear
association, or no association.
• The variable not changed by
other variables or the
independent variable is
represented on the horizontal
axis.
• The variable to be predicted by
the independent variable or the
dependent variable is
represented on the vertical axis.
A student should be able to do
• Plot ordered pairs on a coordinate
grid representing the relationship
between two data sets.
• Describe patterns in the context of
the measurement data.
• Interpret patterns of association in
the context of the data sample.
September 2016 Page 32 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Statistics and Probability
Investigate patterns of association in bivariate data
8.SP.2
Know that straight lines are
widely used to model
relationships between two
quantitative variables. For
scatter plots that suggest a
linear association, informally
fit a straight line, and
informally assess the model
fit by judging the closeness of
the data points to the line.
Desired Student Performance
A student should know
• A scatter plot is a graph in the
coordinate plane representing a set
of bivariate data.
• A line can be used to represent the
trend in a scatter plot.
• Linear association is when the data
on a scatter plot show an upward or
downward trend.
• Whether or not data plotted on a
scatter plot have a linear
association.
A student should understand
• A good line for prediction is one
that goes through the middle of
the points in a scatter plot for
which the points tend to fall close
to the line.
• A trend line on a scatter plot
shows the association more
clearly.
• A line of best fit is the most
accurate trend line on a scatter
plot showing the relationship
between two sets of data.
A student should be able to do
• Draw a straight trend line to
approximate the linear relationship
between the plotted points of two
data sets.
• Make inferences regarding the
reliability of the trend line by noting
the closeness of the data points to
the line.
September 2016 Page 33 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Statistics and Probability
Investigate patterns of association in bivariate data
8.SP.3
Use the equation of a linear
model to solve problems in
the context of bivariate
measurement data,
interpreting the slope and
intercept. For example, in a
linear model for a biology
experiment, interpret a slope of
1.5 cm/hr as meaning that an
additional hour of sunlight each
day is associated with an
additional 1.5 cm in mature
plant height.
Desired Student Performance
A student should know
• Bivariate data are pairs of linked
numerical observations.
• The y-intercept is the point where
the graph crosses the y-axis.
• Slope of a line is a constant rate of
change between the two variables.
• The initial value of a linear function is
the value of the y-variable when the
x value is zero.
• How to model with mathematics.
• How to reason abstractly and
quantitatively.
A student should understand
• A trend line on a scatter plot
shows the association more
clearly.
• A line of best fit is the most
accurate trend line on a scatter
plot showing the relationship
between two sets of data.
• The equation of the trend line
can be used to summarize the
given data and make predictions
regarding additional data points.
• The slope and y-intercept in
relation to the function
represented by the equation y =
mx + b.
A student should be able to do
• Determine the equation of the trend
line that approximates the linear
relationship between the plotted
points of two data sets.
• Use a linear equation to describe
the association between two
quantities in bivariate data.
• Interpret the slope of the equation
in the context of the collected data.
• Interpret the y-intercept of the
equation in the context of the
collected data.
September 2016 Page 34 of 34
College- and Career-Readiness Standards for Mathematics
GRADE 8
Statistics and Probability
Investigate patterns of association in bivariate data
8.SP.4
Understand that patterns of
association can also be seen
in bivariate categorical data
by displaying frequencies and
relative frequencies in a two-
way table. Construct and
interpret a two-way table
summarizing data on two
categorical variables
collected from the same
subjects. Use relative
frequencies calculated for
rows or columns to describe
possible association between
the two variables. For example,
collect data from students in your
class on whether or not they have a
curfew on school nights and
whether or not they have assigned
chores at home.
Is there evidence that those who
have a curfew also tend to have
chores?
Desired Student Performance
A student should know
• Categorical data are non-numerical
data sets.
• Bivariate data are pairs of linked
numerical observations.
• A two-way table organizes data
about two categorical variables into
rows and columns.
• Frequency is the number of times a
given data value occurs in a data
set.
• Relative frequency is the ratio of the
number of desired results to the total
number of trials.
A student should understand
• Rules of probability can lead to
more valid and reliable
predictions about the likelihood
of an event occurring.
• Categorical data can have
patterns of association.
• Venn diagrams can also be used
to display data from a two-way
table.
A student should be able to do
• Create a two-way table to record
the frequencies of bivariate
categorical values.
• Compute marginal sums or
marginal percentages.
• Determine the relative frequencies
for rows and/or columns of a two-
way table.
• Use the relative frequencies and
context of the problem to describe
possible associations between the
two sets of data.
