The PRAXIS
®
Study Companion
Middle School
Mathematics
(5164)
www.ets.org/praxis
The Praxis
Middle School Mathematics Study Companion
2
Table of Contents
Middle School Mathematics (5164) ...................................................................................................... 3
Test at a Glance ..................................................................................................................................................... 3
About This Test ...................................................................................................................................................... 4
On-Screen Graphing Calculator ............................................................................................................................. 5
Using Your Calculator ............................................................................................................................................ 5
Content Topics ...................................................................................................................................................... 6
Discussion Questions ............................................................................................................................................. 6
Tasks of Teaching Mathematics .......................................................................................................................... 16
Middle School Mathematics (5164) Sample Test Questions .......................................................... 18
Information about Questions That Is Specific to the Middle School Mathematics Test ..................................... 18
Unit Conversions ................................................................................................................................................. 20
Formulas .............................................................................................................................................................. 20
Mathematics (5164) Sample Test Questions ...................................................................................................... 22
Answers ............................................................................................................................................................... 31
Understanding Question Types .......................................................................................................... 42
Understanding Selected-Response and Numeric-Entry Questions ..................................................................... 42
Understanding Constructed-Response Questions .............................................................................................. 43
General Assistance For The Test ........................................................................................................ 45
Praxi s
®
Interactive Practice Test .......................................................................................................................... 45
Doing Your Best ................................................................................................................................................... 45
Helpful Links ........................................................................................................................................................ 45
The Praxis
Middle School Mathematics Study Companion
3
Middle School Mathematics (5164)
Test at a Glance
The Praxis
®
Middle School Mathematics test is designed to measure knowledge and
competencies that are important for safe and effective beginning practice as a middle school
mathematics teacher. Test takers have typically completed a bachelor’s degree program with
appropriate coursework in mathematics and education.
Test Name
Middle School Mathematics
Test Code
5164
Time
180 minutes
Number of
Questions
66 selected-response and numeric-entry questions
Format
The test consists of a variety of selected-response questions, where you
select one or more answer choices; questions where you enter a
numeric answer in a box; and other types of questions. You can review
the possible question types in Understanding Question Types.
Calculator
An on-screen graphing calculator is provided.
Test Delivery
Computer Delivered
Content Categories
Approximate
Number of
Questions
Approximate
Percentage of
Examination
I. Numbers and Operations
16 23%
II. Algebra
15 23%
III. Functions
11 17%
IV. Geometry and Measurement
13 20%
V. Statistics and Probability
11 17%
All questions assess content from the above Middle School Mathematics domains.
Approximately 30% of questions assess content applied to a Task of Teaching Mathematics.
I.
II.
III.
IV.
V.
The Praxis
Middle School Mathematics Study Companion
4
About This Test
The Middle School Mathematics content topics span the middle school mathematics
curriculum, including content related to (I) Numbers and Operations, (II) Algebra, (III) Functions,
(IV) Geometry and Measurement, and (V) Statistics and Probability. A full list of the mathematics
topics covered is provided in Content Topics.
Test takers will find that approximately 30% of the questions call for application of mathematics
within a teaching scenario or an instructional task. Such questions—designed to measure
applications of mathematics knowledge and skills to the kinds of decisions and evaluations a
teacher must make during work with students, curriculum, and instruction—situate
mathematics content questions in tasks that are critical for teaching. A full list of the teaching
tasks covered, which have been identified based on research on mathematics instruction and
are a routine part of mathematics instruction, is provided in Tasks of Teaching Mathematics.
Test takers have access to an on-screen graphing calculator and a list of selected unit
conversions and formulas. This list is also provided in the Middle School Mathematics (5164)
Sample Test Questions section.
The assessment is designed and developed through work with practicing teachers and teacher
educators to reflect the mathematics curriculum as well as state and national standards for
mathematics, including the Standards for the Preparation of Middle Level Mathematics
Teachers (2020), by the National Council of Teachers of Mathematics (NCTM) and the Council
for the Accreditation of Educator Preparation (CAEP).
This test may contain some questions that will not count toward your score.
The Praxis
Middle School Mathematics Study Companion
5
On-Screen Graphing Calculator
An on-screen graphing calculator is
provided for the computer-delivered test.
Please consult the Praxis
®
Calculator Use
web page
(http://www.ets.org/praxis/test_day/policies
/calculators/) for further information and
for a link to download the calculator and
view tutorials on using the calculator.
You are expected to know how and when to
use the calculator since it will be helpful for
some questions. You are expected to
become familiar with its functionality before
taking the test. The calculator may be used
to perform calculations (e.g., division,
exponents, roots, finding the mean of a
data set), to graph and analyze functions, to
find numerical solutions to equations, and
to generate a table of values for a function.
Using Your Calculator
Take time to practice with the trial version
of the calculator. View the tutorials on the
website. Practice with the calculator so that
you are comfortable using it on the test.
There are only some questions on the test
for which a calculator is helpful or
necessary. First, decide how you will solve a
problem, then determine if you need a
calculator. For many questions, there is
more than one way to solve the problem.
Don’t use the calculator if you don’t need to;
you may waste time.
Sometimes answer choices are rounded, so
the answer that you get might not match
the answer choices in the question. Since
the answer choices are rounded,
substituting the choices into the question
might not produce an exact answer.
Don’t round any intermediate calculations.
For example, if the calculator produces a
result for the first step of a solution, keep
the result in the calculator and use it for the
second step. If you round the result from
the first step and the answer choices are
close to each other, you might choose the
incorrect answer.
Read the question carefully so that you
know what you are being asked to do.
Sometimes a result from the calculator is
NOT the final answer. If an answer you get
is not one of the choices in the question, it
may be that you didn’t answer the question
being asked. Read the question again. It
might also be that you rounded at an
intermediate step in solving the problem.
Think about how you are going to solve the
question before using the calculator. You
may only need the calculator in the final
step or two. Don’t use it more than
necessary.
Check the calculator modes (degree versus
radian, floating decimal versus scientific
notation) to see that these are correct for
the question being asked.
Make sure that you know how to perform
the basic arithmetic operations and
calculations (e.g., division, exponents,
roots). Your test may involve questions that
require you to do some of the following:
graph functions and analyze the graphs,
find zeros of functions, find points of
intersection of graphs of functions, find
minima/maxima of functions, find
numerical solutions to equations, and
generate a table of values for a function.
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Middle School Mathematics Study Companion
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Content Topics
This list details the topics that may be
included on the test. All test questions cover
one or more of these topics.
Note: The use of “e.g.” to start a list of
examples implies that only a few examples
are offered and the list is not exhaustive,
whereas the use of “i.e.” to start a list of
examples implies that the given list of
examples is complete.
Discussion Questions
In this section, discussion questions provide
examples of content that may be included
in the questions you receive on testing day.
They are open-ended questions or
statements intended to help test your
knowledge of fundamental concepts and
your ability to apply those concepts to
classroom or real-world situations. Answers
for the discussion questions are not
provided; however, thinking about the
answers will help improve your
understanding of fundamental concepts
and may help you answer a broad range of
questions on the test. Most of the questions
require you to combine several pieces of
knowledge to formulate an integrated
understanding and response. The questions
are intended to help you gain increased
understanding and facility with the test’s
subject matter. You may want to discuss
these questions with a teacher or mentor.
I. Numbers and Operations
A. Understands operations and
properties of the real number
system
1. Represents and solves word
problems involving addition,
subtraction, multiplication, and
division of real numbers
2. Represents and identifies the
effect that an operation has on a
given number (e.g., adding a
negative, adding the inverse,
dividing by a nonzero fraction)
3. Uses the order of operations to
simplify computations and solve
problems
4. Identifies and applies properties
of operations on a number
system (e.g., commutative,
associative, distributive, identity)
5. Compares and orders real
numbers, including absolute
values of real numbers
6. Classifies real numbers
(e.g., natural, whole, integer,
rational, irrational)
7. Identifies whether the sum or
product of rational and/or
irrational numbers must be
rational, must be irrational, or
can be rational or irrational
(e.g., the sum of two rational
numbers must be rational, the
product of two irrational
numbers can be rational or
irrational)
8. Performs operations involving
integer exponents
9. Approximates the value of a
radical expression
10. Uses scientific notation to
represent and compare
numbers and to perform
calculations
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Middle School Mathematics Study Companion
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B. Understands the relationships
among fractions, decimals, and
percents
1. Converts among fractions,
decimals, and percents
2. Represents repeating decimals
as fractions
3. Represents fractions, decimals,
and percents with models
(e.g., area models, base-10
bl
ocks, set models, colored rods)
C. Understands how to use ratios and
proportional relationships to solve
problems
1. Uses the language of ratio and
rate to describe relationships
between two quantities
2. Identifies and represents
proportional relationships an
d
u
ses them to solve problem
s
(
e.g., unit rates, scale factors,
constant of proportionality)
3. Solves percent problems
(e.g., expressing a percent as
a
rat
io per 100, discounts,
markups, taxes, tips, simple
interest, percent error)
D
. Understands how to reaso
n
qua
ntitatively and use units to
solve problems
1. Chooses and interprets unit
s
c
onsistently in formulas
2. Chooses and interprets the scale
in graphs and data displays
3. Solves problems involvin
g
d
imensional analysi
s
(
e.g., feet per second to miles
per hour, feet per second to
kilometers per hour)
E. Understands how to use basic
concepts of number theory
(e.g., divisibility, prime
factorization, multiples) to solve
problems
1. Uses the definitions of prime
and composite numbers to solve
problems
2. Solves problems involving
factors, multiples, and divisibility
Discussion Questions: Numbers and
Operations
Note that the use of “e.g.” to start a list of
examples implies that only a few examples
are offered and not an exhaustive list.
• Be able to convert repeating decimals
into fractions (e.g.,
=
7
0.583
12
).
• Be able to distinguish between a ratio
and a rate.
• Be able to calculate percent change,
percent (relative) error, and percents of
percents.
• Be able to determine the correct units in
an answer based on the units of the
initial measurements given in a
problem.
• Be able to identify a scale for a graph
that allows an entire set of data to be
represented on the graph.
• Be familiar with what unit conversions
are given on the math reference sheet.
Note that some other common unit
conversions (e.g., 1 yard = 3 feet,
1 minute = 60 seconds) are expected to
be known, and other unit conversions
(e.g., 1 mile = 1,760 yards,
1 gallon = 128 fluid ounces,
1 hour = 3,600 seconds) are expected to
be determined based on what is known
or what is given on the math reference
sheet.
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Middle School Mathematics Study Companion
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II. Algebra
A. Understands how to create,
evaluate, and manipulate
algebraic expressions, equations,
and formulas
1. Adds, subtracts, and multiplies
linear and quadratic
polynomials, including
polynomials with rational
coefficients
2. Evaluates, manipulates, and
compares algebraic expressions
involving rational exponents
(e.g., radicals, negative
exponents)
3. Uses variables to construct and
solve equations and inequalities
in real-world contexts
4. Translates verbal relationships
into algebraic equations or
expressions
5. Interprets parts of expressions
and equations in terms of a real-
world setting
6. Rewrites linear, quadratic, and
exponential expressions in
equivalent forms to reveal
properties of the quantity
represented by the expression
7. Determines the nature of the
solutions of a quadratic equation
(e.g., interprets the graph, finds
the discriminant, writes the
equation in factored form)
8. Rearranges formulas to solve for
a specified variable (e.g., solve
d rt=
for t)
B. Understands how to recognize and
represent linear relationships
algebraically
1. Determines the equation of a
line from information presented
in various forms (e.g., table,
graph, description)
2. Recognizes and is able to extract
information about a linear
equation when it is presented in
various forms
(e.g., slope-intercept, point-
slope, standard)
3. Converts among various forms
of linear equations
(e.g., slope-intercept, point-
slope, standard)
C. Understands how to solve
equations and inequalities
1. Solves one-variable linear
equations and inequalities
2. Solves one-variable nonlinear
equations and inequalities
(e.g., absolute value, quadratic)
3. Represents solutions to
equations and inequalities
(e.g., on a number line, in
the
-planexy
)
4. Justifies each step in solving
equations and inequalities
D. Understands how to solve systems
of equations and inequalities
1. Solves a system of two linear
equations or inequalities in two
variables algebraically and
graphically
2. Solves a system consisting of a
linear equation and a quadratic
equation in two variables
graphically
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Middle School Mathematics Study Companion
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3. Finds the solutions of
( ) ( )
=fx gx
approximately
(e.g., uses technology to graph
the functions); includes cases
wh
ere
( )
fx
and /or
( )
gx
are
linear, quadratic, or exponential
functions
4. Graphs the solution set to a
system of linear inequalities i
n
t
wo variables in the
-planexy
5. I
n a modeling context,
represents constraints by
systems of equations and/or
inequalities and interprets
solutions as viable or nonviable
options
Discussion Questions: Algebra
Note that the use of “e.g.” to start a list of
examples implies that only a few examples
are offered and not an exhaustive list.
• Be able to identify expressions that are
equivalent to expressions such
2
3
x
,
5
2
x
,
−4
x
,
( )
−1
3
x
, and
5
2
x
.
• Be able to write and solve equations,
inequalities, and systems of equations
or inequalities that represent real-worl
d
pro
blems.
• Be able to identify what parts of
expressions and equatio
ns
(
e.g., coefficients, terms, factors)
represent in the context of a real-worl
d
si
tuation.
• Be able to use the quadratic formula,
which is given on the math reference
sheet.
• Be able to determine the equation of a
line given two points on the line or one
point on the line and the slope of the
line.
• Be able to determine the slope of a line
or points on a line when an equation of
the line is given in standard form, slope-
intercept form, or point-slope form.
• Be able to solve one-variable linear
equations and inequalities that have
variables on both sides, involve
combining like terms, and involve usin
g
the distributive property.
• Be able to graph the solutions to linear
equations, linear inequalities, system
s
o
f linear equations, and systems of
linear inequalities in two variables in the
-planexy
.
• Be able to graph the solutions to one-
variable inequalities on the number line.
• Be able to identify the properties
(e.g., commutative property, distributive
property) that justify each step in a
given method for solving an equation or
inequality.
• Be able to solve systems of linear
equations graphically, by substitution,
or by elimination.
• Remember that the x-coordinates of the
points where the graphs of the
equations
( )
=y fx
and
( )
=y gx
i
ntersect are the solutions of the
equation
( ) ( )
=fx gx
.
III. Functions
A. Understands how to identify,
define, and evaluate functions
1. Determines whether a relation is
a function
2. Given a function (presented as
a
t
able of values, algebraically, o
r
g
raphically), determines if the
function is linear, quadratic, or
ex
ponential
3. Determines the value of a
function for a specified value i
n
it
s domain
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Middle School Mathematics Study Companion
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B. Knows how to determine and
interpret the domain and the
range of a function presented as a
table of values, algebraically, or
graphically
1. Determines the domain and
range of a function
2. Interprets domain and range in
real-world settings
C. Understands basic characteristics
of linear functions (e.g., intercepts,
slope)
1. Calculates the intercepts of a line
and interprets them in a
modeling context
2. Calculates the slope of a line
presented as a table of values,
algebraically, or graphically and
interprets it in a modeling
context
3. Interprets what a point
( )
,xy
on
the graph of a proportional
relationship means in terms of
the situation, with special
attention to the points
( )
0,0
and
( )
1, r
, where r is the unit rate
D. Understands the relationships
among functions, tables, and
graphs
1. Determines an equation to
represent a linear or quadratic
function presented graphically
2. Determines the type of equation
that best represents a given
graph
3. Sketches a graph, given an
equation of a function
(e.g., square root, absolute
value)
4. Compares properties
(e.g., intercepts, slope,
maximum) of two functions
presented as tables of values,
algebraically, graphically, or by
verbal descriptions
5. Identifies the symbolic
representation of a linear
function that is created when a
graph is translated horizontally
or vertically or reflected across
the x-axis
E. Knows how to analyze and
represent functions (i.e., linear,
quadratic, exponential) that model
given information
1. Interprets statements that use
function notation in terms of a
context
2. Interprets the parameters in a
linear or exponential function in
terms of a context
3. Calculates the rate of change of
a function over a given interval
and interprets it in a context
4. Determines and interprets the
x- and y-intercepts of quadratic
functions
5. Develops a function—
represented by a graph,
equation, or table—to model a
given set of conditions
6. Evaluates whether a particular
mathematical model (e.g., graph,
equation, table) can be used to
describe a given set of
conditions
7. Interprets a particular
mathematical model (e.g., graph,
equation, table) in a given
context
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Middle School Mathematics Study Companion
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F. Understands differences between
linear, quadratic, and exponential
models, including how their
equations are created and used to
solve problems
1. Identifies situations in which one
quantity changes at a constant
rate per unit interval relative to
another
2. Identifies situations in which a
quantity grows or decays by a
constant percent rate per unit
interval relative to another
3. Observes that a quantity
increasing exponentially
eventually exceeds a quantity
increasing linearly or
quadratically
G. Is familiar with how to represent
arithmetic sequences as functions
1. Writes arithmetic sequences
both recursively and with an
explicit formula and uses them
to model situations
Discussion Questions: Functions
Note that the use of “e.g.” to start a list of
examples implies that only a few examples
are offered and not an exhaustive list,
whereas the use of “i.e.” to start a list of
examples implies that the given list of
examples is complete.
• Remember that a function assigns
exactly one element of its range to each
element of its domain.
• Be able to identify features of a function
represented as a table, equation, or
graph that indicate whether the
function is linear, quadratic, or
exponential (e.g., the second differences
of a quadratic function are constant, a
quadratic function has degree 2, a
quadratic function has either a
maximum or a minimum).
• Be able to determine the domain
(x-values) and range (y-values) of a
function.
• Be able to determine the domain and
range of a function that is reasonable in
the context of a given real-world
situation (e.g., the domain in a certain
situation consists of positive integers,
the domain in a certain situation cannot
include values that would result in a
negative value for the height of an
object).
• Be able to determine the intercepts and
slope of a line represented as a table,
equation, or graph.
• Be able to interpret the meaning of the
intercepts and slope of a line in the
context of a real-world situation.
• Be able to interpret the meaning of a
point on the graph of a proportional
relationship in the context of a real-
world situation.
• Be able to determine whether a given
graph is best represented using a linear
equation, quadratic equation,
exponential equation, absolute value
equation, etc.
• Be able to write an equation of the
function that results after an existing
function is translated horizontally,
translated vertically, or reflected across
the x-axis.
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Middle School Mathematics Study Companion
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• Be able to interpret the meaning of m
and b in a function of the form
( )
= +f x mx b
in the context of a real-
world situation.
• Be able to interpret the meaning of a
and b in a function of the form
( )
= ⋅
x
f x ab
in the context of a real-
world situation.
• Be able to calculate the rate of change
of a function f on the interval
,ab
by
calculating
( ) ( )
−
−
fb fa
ba
, and be able to
interpret the meaning of the rate of
change in the context of the problem.
• Be able to interpret the meaning of the
intercepts of a quadratic function in the
context of a real-world situation.
• Given a linear, quadratic, or exponential
function represented as a table,
equation, graph, or description, be able
to determine a different representation
of the function (e.g., determine the
function that best matches a description
of a real-world situation).
• Be able to interpret the meaning of a
feature of a function (e.g., the maximum
on a graph, an ordered pair in a table of
values) in the context of a real-world
situation.
• Be able to identify real-world situations
that are best modeled by linear
functions or that are best modeled by
exponential functions.
• Be able to find the value of a term in an
arithmetic sequence.
• Be able to write an expression,
equation, or function that represents an
arithmetic sequence.
• Be familiar with the differences between
recursive and explicit rules for
arithmetic sequences.
• Consider becoming familiar with the
arithmetic sequence formula on the
math reference sheet.
IV. Geometry and Measurement
A. Knows the properties of types of
lines (e.g., parallel, perpendicular,
intersecting) and angles
1. Solves problems involving
parallel, perpendicular, and
intersecting lines
2. Applies angle relationships
(e.g., supplementary, vertical,
alternate interior) to solve
problems
B. Understands the properties of
triangles
1. Solves problems involving the
Pythagorean theorem in two
dimensions
2. Identifies characteristics of
special triangles (e.g., equilateral,
isosceles, right) and uses them
to solve problems
3. Determines whether given side
lengths or angle measures
would produce a triangle
(e.g., triangle inequality theorem)
and classifies triangles by their
sides or angles
4. Determines whether given
conditions would produce a
unique triangle, no triangle, or
more than one triangle
C. Knows the properties of
quadrilaterals and other polygons
1. Identifies the relationships
among various quadrilaterals
(e.g., parallelogram, rectangle,
rhombus)
2. Solves problems involving sides
and angles of polygons
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Middle School Mathematics Study Companion
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D. Knows the concepts of
transformations (i.e., translations,
reflections, rotations, dilations)
1. Applies properties of
translations, reflections, and
rotations (e.g., line segments are
taken to congruent line
segments, angles are taken to
congruent angles, parallel lines
are taken to parallel lines)
2. Applies properties of dilations
(e.g., angles are taken to
congruent angles, parallel lines
are taken to parallel lines)
3. Identifies a sequence of
transformations that maps a
preimage onto an image
4. Given a figure, describes the
transformations that map the
figure onto itself, including
reflection over a line of
symmetry
5. For a given transformation,
determines the coordinates of a
point on an image
E. Understands the concepts of
congruence and similarity
1. Determines whether two figures
are congruent or similar
2. Uses congruence and similarity
to solve problems with two-
dimensional and three-
dimensional figures
F. Understands the properties of
circles
1. Solves problems involving circles
(e.g., circumference, area)
G. Knows how to interpret
relationships between geometric
objects in the xy-plane
(e.g., distance, midpoint)
1. Uses coordinate geometry to
represent and identify the
properties of geometric shapes
and to solve problems
(e.g., Pythagorean theorem,
perimeter, area)
2. Determines the distance
between two points
3. Determines the midpoint of a
segment
H. Understands how to solve
problems involving perimeter and
area of polygons
1. Calculates and interprets
perimeter and area of polygons
that can be composed of
triangles and quadrilaterals,
including in real-world situations
2. Calculates changes in perimeter
and area as the dimensions of a
polygon change
I. Knows how to solve problems
involving solids
1. Calculates and interprets surface
area and volume of solids
(e.g., prisms, pyramids, cylinders,
spheres) and composite solids,
including in real-world situations
2. Calculates changes in surface
area and volume as the
dimensions of a solid change
3. Uses two-dimensional
representations (e.g., nets) of
three-dimensional objects to
visualize and solve problems
The Praxis
Middle School Mathematics Study Companion
14
J. Understands systems of
measurement (i.e., metric, United
States customary)
1. Solves measurement,
estimation, and conversion
problems involving time, length,
temperature, volume, and mass
in standard measurement
systems
2. Uses appropriate units of
measurement in a given context
Discussion Questions: Geometry and
Measurement
Note that the use of “e.g.” to start a list of
examples implies that only a few examples
are offered and not an exhaustive list,
whereas the use of “i.e.” to start a list of
examples implies that the given list of
examples is complete.
• Be familiar with the relationship
between the slopes of parallel lines and
the relationship between the slopes of
perpendicular lines.
• Be able to identify congruent and
supplementary angles given two parallel
lines and a transversal.
• Be able to distinguish among acute,
right, and obtuse triangles.
• Be able to identify and use special
characteristics of triangles
(e.g., equilateral, isosceles, right) to
solve problems involving lengths of
sides and measures of angles.
• Be able to distinguish among different
types of quadrilaterals.
• Be able to identify and use special
characteristics of squares, rectangles,
parallelograms, rhombuses, and
trapezoids to solve problems involving
lengths of sides and measures of
angles.
• Be able to find missing side lengths or
angle measures in polygons with more
than four sides.
• Be able to find the measures of interior
and exterior angles of regular polygons.
• Be familiar with the effects of
translations, reflections, rotations, and
dilations on figures.
• Be able to translate, reflect, rotate, and
dilate figures.
• Be able to distinguish between
congruent and similar figures and use
corresponding parts of congruent or
similar figures to solve problems.
• Be familiar with what geometric
formulas are given on the math
reference sheet, and be able to apply
these formulas to solve problems.
• Be able to solve problems that involve
the circumference or area of a circle and
the perimeter or area of a polygon
(e.g., finding the difference between the
area of a square and the area of a circle
inscribed in the square).
• Be able to find the distance between
any two points in the
-planexy
by using
a formula or the Pythagorean theorem.
• Be able to find the midpoint of a line
segment in the
-planexy
by using a
formula or another approach.
• Be familiar with the effect on the
perimeter, area, or volume of a figure as
the dimensions of the figure change by
different factors.
• Be able to use a net to find the surface
area and volume of a solid.
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Middle School Mathematics Study Companion
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• Be familiar with what unit conversions
are given on the math reference sheet.
Note that some other common unit
conversions (e.g., 1 yard = 3 feet,
1 minute = 60 seconds) are expected to
be known, and other unit conversions
(e.g., 1 mile = 1,760 yards,
1 gallon = 128 fluid ounces,
1 hour = 3,600 seconds) are expected to
be determined based on what is known
or what is given on the math reference
sheet.
• Be able to identify units that measure
length, area, volume, weight, etc.
V. Statistics and Probability
A. Understands statistical processes
and how to evaluate them
1. Recognizes a statistical question
as one that anticipates variability
in the data related to the
question and accounts for it in
the answers
2. Uses statistics to make
inferences about population
parameters based on a sample
from that population
3. Distinguishes between random
and biased sampling
B. Understands how to interpret,
analyze, and represent data
presented in a variety of displays
1. Represents and analyzes data in
various displays (e.g., bar graphs,
line graphs, circle graphs,
boxplots, histograms,
scatterplots, stem-and-leaf plots,
two-way tables)
2. Calculates relative frequencies
for rows or columns in two-way
tables and uses the calculations
to describe possible associations
between the two variables
3. Uses the equation of a linear
model to solve problems in the
context of bivariate
measurement data
(e.g., interpreting the slope and
intercept, interpolation)
4. Describes how two quantitative
variables are related (e.g., fit a
function to data, association,
correlation)
5. Chooses appropriate graphs
based on data type
(e.g., numerical, categorical)
C. Understands concepts associated
with measures of central tendency
and dispersion
1. Solves for the mean and
weighted average of given sets
of data
2. Determines and interprets
measures of center (e.g., mean,
median, mode) and spread
(e.g., range, interquartile range)
in a variety of problems
3. Summarizes a given numerical
data set in relation to its context
4. Describes the distribution of a
set of data by its center and
spread
5. Uses statistics appropriate to the
shape of the data distribution to
compare center and spread of
two or more different data sets
6. Interprets differences in center
and spread in the context of the
data sets, accounting for
possible effects of outliers
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Middle School Mathematics Study Companion
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D. Knows how to use and evaluate
probability models
1. Uses counting techniques
(e.g., the counting principle, tree
diagrams) to answer questions
involving a finite sample space
2. Solves probability problems
involving simple events
3. Solves probability problems
involving compound events
4. Interprets a probability model
and uses it to find probabilities
of events
5. Compares probabilities from a
model to observed frequencies
and identifies possible sources
of the discrepancy if the
agreement is not good
6. Interprets a uniform probability
model and uses it to determine
probabilities of events
Discussion Questions: Statistics and
Probability
Note that the use of “e.g.” to start a list of
examples implies that only a few examples
are offered and not an exhaustive list.
• Be able to make inferences about a
population based on a random sample
from the population (e.g., estimate the
number of people in a population for
which a certain characteristic is true).
• Be able to interpret a line of best fit
(trend line) and use it to solve problems.
• Be familiar with how to summarize
numerical data sets in relation to their
context (e.g., describe any overall
pattern and any notable differences
from the overall pattern, relate the
chosen measures of center and
variability to the shape of the data).
• Be able to express the difference
between the centers of two data sets as
a multiple of a measure of variability.
• Be able to solve counting problems by
using counting techniques or by
counting individual outcomes
(e.g., construct or interpret a tree
diagram that models a sample space).
• Be able to solve probability problems
involving independent events or
dependent events.
• Be able to solve problems involving a
probability model (which may not be
uniform) by observing frequencies in
data generated from a chance process.
• Be able to solve problems involving a
uniform probability model by assigning
equal probability to all outcomes.
Tasks of Teaching Mathematics
This list includes instructional tasks that
teachers engage in that are essential for
effective teaching of middle school
mathematics. Approximately 30% of test
questions will measure content knowledge
by assessing how that content knowledge is
applied in the context of one or more of
these tasks.
Mathematical explanations,
justifications, and definitions
1. Identifies valid explanations of
mathematical concepts (e.g., explaining
why a mathematical idea is considered
to be true), procedures,
representations, or models
2. Evaluates or compares explanations
and justifications for their validity,
generalizability, coherence, or precision,
including identifying flaws in
explanations and justifications
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Middle School Mathematics Study Companion
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3. Determines the changes that would
improve the validity, generalizability,
coherence, and/or precision of an
explanation or justification
4. Evaluates whether counterarguments
address a critique of a given justification
5. Evaluates definitions or other
mathematical language for validity,
generalizability, precision, usefulness in
a particular context, or support of key
ideas
Mathematical problems, tasks,
examples, and procedures
6. Identifies problems or tasks that fit a
particular structure, address the same
concept, demonstrate desired
characteristics, or elicit particular
student thinking
7. Identifies two or more problems that
systematically vary in difficulty or
complexity
8. Evaluates the usefulness of examples
for introducing a concept, illustrating an
idea, or demonstrating a strategy,
procedure, or practice
9. Identifies examples that support
particular strategies or address
particular student questions,
misconceptions, or challenges with
content
10. Identifies nonexamples or
counterexamples that highlight a
mathematical distinction or
demonstrate why a student conjecture
is incorrect or partially incorrect
11. Evaluates procedures for working with
mathematics content to identify special
cases in which the procedure might be
problematic or for validity,
appropriateness, or robustness
Mathematical representations, models,
manipulatives, and technology
12. Evaluates representations and models
(e.g., concrete, pictorial) in terms of
validity, generalizability, usefulness for
supporting students’ understanding, or
fit to the concept, calculation, etc. to be
represented
13. Evaluates how representations and
models (e.g., concrete, pictorial) have
been used to show particular ideas,
relationships between ideas, processes,
or strategies
14. Evaluates the use of technology
(e.g., graphing tools, software) for its
appropriateness or its support of key
ideas
Students’ mathematical reasoning
15. Identifies likely misconceptions about or
partial understanding of particular
mathematics content and practices
16. Identifies how new mathematics
content and practices can build on or
connect to students’ prior knowledge,
including misconceptions and errors
17. Evaluates or compares student work
(e.g., solutions, explanations,
justifications, representations) in terms
of validity, generalizability, coherence,
and/or precision
18. Evaluates student work to identify the
use of a particular concept, idea, or
strategy
19. Identifies how a student’s reasoning
would replicate across similar problems
20. Identifies different pieces of student
work that demonstrate the same
reasoning
21. Identifies situations in which student
work that seems valid might mask
incorrect thinking
The Praxis
Middle School Mathematics Study Companion
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Middle School Mathematics (5164) Sample Test Questions
Information about Questions That Is Specific to the Middle School
Mathematics Test
General
• All numbers used are real numbers.
• Unless otherwise stated, the domain of a given function f is the set of all real numbers x for
which
( )
fx
is a real number.
• Rectangular coordinate systems are used unless otherwise stated.
• Figures that accompany questions are intended to provide information that is useful in
answering questions.
o Figures are drawn to scale unless otherwise stated.
o Lines shown as straight are straight, and angle measures are positive. Positions of
points, angles, regions, etc., exist in the order shown.
Types of questions that may be on the test
• Selected-response questions—select one answer choice
o These are questions that ask you to select only one answer choice from a list of four
choices.
o Note that in most selected-response questions that ask for numerical values, the exact
answer should be found. If a selected-response question includes a word or phrase like
“approximately,” “best approximates,” or “is closest to,” it generally indicates that the
correct option will not be an exact value.
• Selected-response questions—select one or more answer choices
o These are questions that ask you to select one or more answer choices from a list of
choices. A question may or may not specify the number of choices to select. These
questions are marked with square boxes beside the answer choices, not circles or ovals.
See questions 13 and 16 in the Sample Test Questions.
o If a question of this type has exactly three answer choices, one, two, or three of the
choices may be correct.
o If a question of this type has more than three answer choices, the number of correct
choices will be at least 2 but fewer than the number of choices. For example, if a
question of this type has six answer choices, there will be two, three, four, or five correct
choices.
• Selected-response questions—select an area
o These are questions that ask you to select one or more locations on a picture or a figure
(e.g., the
-planexy
).
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Middle School Mathematics Study Companion
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• Numeric-entry questions
o Some of these questions ask you to enter your answer as an integer or a decimal in a
single answer box. Equivalent forms of the correct answer, such as 2.5 and 2.50, are all
correct. See questions 10 and 14 in the Sample Test Questions. Note that in these
questions, the exact answer should be entered unless the question asks you to round
your answer. Therefore, if one of these questions does not ask you to round your
answer, you should be able to enter the exact answer in the numeric-entry box. If you
are unable to do so, this may indicate that your answer is incorrect.
o Some of these questions ask you to enter your answer as a fraction in two separate
boxes—one for the numerator and one for the denominator. A negative sign can be
entered in either box. Equivalent forms of the correct answer, such as
1
2
and
6
12
, are all
correct, though there may be cases where you need to simplify your fraction so it fits in
the boxes. See question 9 in the Sample Test Questions.
• Drag-and-drop questions
o These questions ask you to pair up given phrases or expressions by dragging (with your
computer mouse) phrases from one location and matching them with given phrases or
expressions in another location. See question 8 in the Sample Test Questions.
• Table grid questions
o These questions refer to a table in which statements appear in the first column. For
each statement, select the correct properties by selecting the appropriate cell(s) in the
table. See question 2 in the Sample Test Questions.
• Text completion questions
o These questions ask you to select one or more answer choices to complete one or more
sentences. The choices may be located in drop-down menus in the sentences or in
columns at the end of the question. You will select one answer choice from each drop-
down menu or column.
The Praxis
Middle School Mathematics Study Companion
20
Unit Conversions
1 mile
=
5,280 feet 1 mile
≈
1.61 kilometers 1 inch
=
2.54 centimeters
1 pound
=
16 ounces 1 ton
=
2,000 pounds 1 kilogram
≈
2.2 pounds
1 cup
=
8 fluid ounces 1 quart
=
2 pints 1 gallon
≈
3.785 liters
1 pint
=
2 cups 1 gallon
=
4 quarts 1 liter
=
1,000 cubic centimeters
Formulas
Area
Rectangle with length
and width w: ................................................................
=Aw
Parallelogram with height h and base of length b: ...........................................
=A bh
Triangle with height h and base of length b: ......................................................
=
1
2
A bh
Trapezoid with height h and bases of length
1
b
and
2
b
:..................................
( )
= +
12
1
2
A b bh
Circle with radius r: ................................................................................................
π
=
2
Ar
Perimeter
Rectangle with length
and width w: .................................................................
= +22Pw
Circumference
Circle with radius r: ................................................................................................
π
= 2Cr
Volume
Right rectangular prism with length
, width w, and height h: .......................
=V wh
Right prism with height h and base of area B: ...................................................
=V Bh
Pyramid with height h and base of area B: ........................................................
=
1
3
V Bh
Right circular cylinder with height h and base of radius r: ...............................
π
=
2
V rh
Right circular cone with height h and base of radius r: ....................................
π
=
2
1
3
V rh
Sphere with radius r: ..............................................................................................
π
=
3
4
3
Vr
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Middle School Mathematics Study Companion
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Surface Area
Cube with side of length s: ....................................................................................
=
2
6As
Right rectangular prism with length
, width w, and height h: ......................
= ++2 22A w h wh
Right circular cylinder with height h and base of radius r: ...............................
ππ
= +
2
22A rh r
Sphere with radius r: ..............................................................................................
π
=
2
4Ar
Other Formulas
Quadratic formula: ................................................................................................
−± −
=
2
4
2
b b ac
x
a
Arithmetic sequence: ............................................................................................
( )
=+−
1
1
n
aa n d
Pythagorean theorem: ...........................................................................................
+=
222
abc
Sum of the measures of the interior angles of a polygon with n sides: ........
( )
= °−180 2Sn
The Praxis
Middle School Mathematics Study Companion
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Mathematics (5164) Sample Test Questions
1. A student used the same reasoning to evaluate four expressions. The four expressions and
the student’s answers are given as follows. The student incorrectly evaluated the first
two expressions but correctly evaluated the next two expressions.
1.
×−+=72635
2.
−+ ÷=9 5 16 8 2
3.
+ ÷ −=9 24 3 1 16
4.
×− ÷=7 2 18 6 11
If the student continues to use the same reasoning, which of the following expressions is
the student most likely to evaluate INCORRECTLY?
(A)
+− ÷8 7 12 3
(B)
−×+13 3 2 5
(C)
×÷ −10 6 15 3
(D)
×+ −4 5 10 12
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Middle School Mathematics Study Companion
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2. Ms. Kress asked her students to compare
1
3
and
7
8
. Four of her students correctly answered
that
7
8
is greater than
1
3
, but they gave different explanations when asked to describe their
strategies to the class.
Indicate whether each of the following student explanations provides evidence or does not
provide evidence of a mathematically valid strategy for comparing
1
3
and
7
8
.
Student Explanation
Provides
Evidence
Does Not
Provide
Evidence
When you look at the numbers, you see that 7 is bigger than 1, so
7
8
is the bigger fraction.
In the first fraction, 1 is less than half of 3, but in the second, 7 is
more than half of 8, so
7
8
is larger than
1
3
.
I multiplied 1 times 7 and 3 times 7, so
1
3
is the same as
7
21
. This
means that
7
8
is bigger than
1
3
because
1
8
is bigger than
1
21
.
I wanted to make a fraction equal to
1
3
with the same bottom
number as
7
8
, so I added 5 to 3 and got 8. Then I added 5 to 1 and
got 6, but 7 is greater than 6, so
7
8
is greater.
3. Ernesto bought 2 sport coats for $88.95 each. One of the coats needed alterations that cost
$15.50, and a 6% sales tax is applied to the cost of the coats but not to the alterations.
Which of the following values is closest to the total cost for the sport coats and the
alterations?
(A) $190
(B) $200
(C) $205
(D) $215
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Middle School Mathematics Study Companion
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4. The maximum speed at which a horse can run is 36 miles per hour.
What is the maximum speed of the horse in feet per second?
(A) 2.4
(B) 24.5
(C) 37.5
(D) 52.8
5.
A teacher wants to give an example in which the distributive property must be used to solve
a literal equation for a given variable.
Which of the following examples best serves the teacher’s purpose?
(A) Solve
( )
2
a bh
A
+
=
for b.
(B) Solve
( )
1A P rt= +
for r.
(C) Solve
22Pw= +
for w.
(D) Solve
2 22S w h wh= ++
for h.
6. A line in the
-planexy
passes through the point
( )
4,5
and is parallel to the graph
of
+=34xy
.
What is an equation of the line?
(A)
=−+3 17yx
(B)
=−+37yx
(C)
= −37yx
(D)
= −3 17yx
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Middle School Mathematics Study Companion
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7. Mr. Keller’s class is learning about algebraic equations. In his teacher’s edition of the
textbook, Mr. Keller finds a page that suggests he ask students to critique the following two
solutions to determine whether they are valid.
+=
=
=
4 2 66
6 66
66
11
x
x
x
= +
=
=
5 23
55
55
1
x
x
x
Which of the following is best addressed by the preceding task?
(A) Misunderstanding of the properties of operations
(B) Misunderstanding of the meaning of the equal sign
(C) Misunderstanding of how to identify and combine like terms
(D) Misunderstanding of how to use inverse operations to solve equations
8. The steps in a solution method for the equation
( )
+=
1
11 20 2
3
xx
follow.
Provide the justification for the result shown in each step in the solution method.
Addition/Subtraction Property of Equality
Multiplication/Division Property of Equality
Distributive Property
Step Justification
( )
+=
1
11 20 2
3
xx
Given
+=11 20 6xx
= −20 5x
−=4 x
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Middle School Mathematics Study Companion
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9. The graph of linear function f passes through the points
( )
−3,11
and
( )
−7, 4
.
What is the slope of the graph of f ?
Give your answer as a fraction.
10. The graph of the quadratic equation
= +
2
y ax c
is shown in the following
-planexy
.
If a and c are integers, what are the values of a and c ?
=
=
a
c
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Middle School Mathematics Study Companion
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11. A teacher wants to show the students in an Algebra I class two examples of functions that
are both linear and continuous. The teacher thinks of the following example.
The temperature in degrees Fahrenheit is a function of the temperature in
degrees Celsius.
Which of the following is also an example of a function that is both linear and continuous?
(A) The height in inches of a person is a function of the person’s age in years throughout
the person’s life.
(B) The number of calories consumed is a function of the volume, in ounces, of an energy
drink that is consumed.
(C) The total cost in dollars for purchasing hot dogs is a function of the number of hot dogs
purchased for $2.00 each.
(D) The total number of games in a tournament is a function of the number of teams in the
tournament when each team plays every other team once.
12. In the following figure, line
and line p are parallel, and
= 3yx
.
What is the value of x ?
(A) 75
(B) 60
(C) 45
(D) 30
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Middle School Mathematics Study Companion
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13. At the start of a lesson on finding the side length of a square given its area, Ms. Ruffin
reminded her students that a square has four sides of equal length. Then she asked them
to determine the side length of a square with an area of 36 square units. Several students
incorrectly answered that the side length is 9 units.
At the end of the lesson, Ms. Ruffin wants to give a similar problem to assess whether her
students are still making the same error. The students will write their final answers on slips
of paper and give them to Ms. Ruffin as they exit the class.
Which of the following area measurements would be useful for assessing student learning
in this situation?
Select ALL that apply.
(A) 16 square units
(B) 64 square units
(C) 100 square units
14. Reggie hiked 3,500 meters along a trail at a nearby park each day for the last 14 days.
How many kilometers did Reggie hike in the last 14 days?
kilometers
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Middle School Mathematics Study Companion
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15. An automobile company sold 6 different models in the United States in a certain year. The
following table shows the percent of total sales in the United States for each of the 6 models
that year.
Model
Percent of Sales
B
49.5%
C
24.7%
D
16.1%
E 5.0%
F 3.5%
G
1.2%
If a circle graph is constructed using the data in the table, which of the following values best
approximates the measure, in degrees, of the central angle for the sector representing the
sales of Model D ?
(A)
°4
(B)
°16
(C)
°29
(D)
°58
16. Each of the integers in list K (not shown) is greater than 75, and integers may appear more
than once in the list. List M consists of the integers in list K and 4 additional integers that are
each less than 75.
Which of the following statements about the centers or spreads of lists K and M must be
true?
Select ALL that apply.
(A) The mean of the integers in list M is less than the mean of the integers in list K.
(B) The median of the integers in list M is less than the median of the integers in list K.
(C) The mode of the integers in list M is less than the mode of the integers in list K.
(D) The range of the integers in list M is greater than the range of the integers in list K.
(E) The interquartile range of the integers in list M is greater than the interquartile range of
the integers in list K.
The Praxis
Middle School Mathematics Study Companion
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17. In a survey, 50 people were asked how many hours per day, h, they watched television. The
survey results are shown in the following table.
Hours of Television Watched per Day
Number of Hours Watched per Day
Number of People
< 1h
5
≤<12h
12
≤<23h
16
≤<34h
14
≥ 4h
3
If a person is selected at random from those surveyed, what is the probability that the
person selected will have watched at least 2 hours but less than 4 hours of television per
day?
(A)
3
10
(B)
8
25
(C)
1
2
(D)
3
5
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Middle School Mathematics Study Companion
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Answers
1. Option (B) is correct. A common misconception about the order of operations is that
multiplication always comes before division and that addition always comes before
subtraction. In the first expression, the student evaluated
(
)
72 63×− +
instead of
726 3
×−+
, and in the second expression, the student evaluated
( )
9 5 16 8−+ ÷
instead
of
9 5 16 8−+ ÷
. In the next two expressions, the misconception described does not
interfere with a student’s ability to correctly evaluate the expressions, and the student
obtained the correct answers, so it is likely that this misconception is the basis for the
student’s incorrect answers. In the expressions in (A), (C), and (D), this misconception
does not interfere with a student’s ability to correctly evaluate the expressions either,
but in the expression in (B), it is incorrect to add before subtracting, so the expression in
(B) is the one that the student is most likely to evaluate incorrectly.
Task of Teaching
Topic
Students’ mathematical reasoning
Task of Teaching
Subtopic
19. Identifies how a student’s reasoning would replicate
across similar problems
Category
I. Numbers and Operations
Topic
A. Understands operations and properties of the real
number system
Subtopic
3. Uses the order of operations to simplify computations
and solve problems
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Middle School Mathematics Study Companion
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2. The first and fourth explanations do not provide evidence of a mathematically valid
strategy for comparing
1
3
and
7
8
, but the second and third explanations do. In the first
explanation, the student compares only the numerators of the fractions, which is not a
valid strategy because it does not take into account the effect of the denominator on the
size of the pieces. In the second explanation, the student compares both fractions to the
benchmark fraction
1
2
, which is a valid strategy since
1
3
is less than
1
2
and
7
8
is greater
than
1
2
. In the third explanation, the student uses multiplicative reasoning to find a
common numerator, and then the student compares the fractions by reasoning about
the sizes of the unit fractions
1
8
and
1
21
. This is a valid strategy. In the fourth
explanation, the student uses additive reasoning to try to find a fraction equivalent to
1
3
that has a denominator of 8, but
6
8
is not equivalent to
1
3
, so this strategy is not valid.
Task of Teaching
Topic
Students’ mathematical reasoning
Task of Teaching
Subtopic
17. Evaluates or compares student work
(e.g., solutions, explanations, justifications,
representations) in terms of validity, generalizability,
coherence, and/or precision
Category
I. Numbers and Operations
Topic
A. Understands operations and properties of the real
number system
Subtopic
5. Compares and orders real numbers, including
absolute values of real numbers
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3. Option (C) is correct. Based on the information in the question, the total cost can be
calculated as
(
)
( )
( )
$88.95 2 1.06 $15.50 $204.07.+=
The choice that is closest to the total
cost is $205.
Category
I. Numbers and Operations
Topic
C. Understands how to use ratios and proportional
relationships to solve problems
Subtopic
3. Solves percent problems (e.g., expressing a
percent as a ratio per 100, discounts, markups, taxes,
tips, simple interest, percent error)
4. Option (D) is correct. To find the maximum speed of the horse in feet per second,
multiply the maximum speed of the horse in miles per hour by the appropriate
conversion factors. Note that certain conversion factors are provided on the test, such as
the conversion from miles to feet, but other conversion factors are not provided, such as
the conversions from hours to minutes and from minutes to seconds.
Since 1 mile
=
5,280 feet, 1 hour
=
60 minutes, and 1 minute
=
60 seconds, the
maximum speed of the horse in feet per second is equal to
, which is equal to 52.8 feet per second.
Remember that when calculations like this are performed, the units in the numerators
and denominators of the fractions need to be divided out so that only the required units
remain.
Category
I. Numbers and Operations
Topic
D. Understands how to reason quantitatively and use
units to solve problems
Subtopic
3. Solves problems involving dimensional analysis
(e.g., feet per second to miles per hour, feet per
second to kilometers per hour)
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5. Option (D) is correct. Remember that both
( )
+= +a b c ab ac
and
( )
+= +ab ac a b c
demonstrate the distributive property. To solve
2 22S w h wh= ++
for h, first subtract
2 w
from both sides of the equation to obtain
−=+2 22S w h wh
. Then use the
distributive property to factor the right-hand side of the equation to obtain
( )
−= +2 22S wh w
. Finally, divide both sides of the equation by
+22w
to obtain
−
=
+
2
22
Sw
h
w
. In this example, since h appears in two of the terms on the right-hand side
of the equation but does not appear in the third term, the distributive property must be
used to isolate h, which means (D) is the example that best serves the teacher’s purpose.
Each of the other examples can be solved for the given variable without using the
distributive property.
Task of Teaching Topic
Mathematical problems, tasks, examples, and
procedures
Task of Teaching
Subtopic
9. Identifies examples that support particular
strategies or address particular student questions,
misconceptions, or challenges with content
Category
II. Algebra
Topic
A. Understands how to create, evaluate, and
manipulate algebraic expressions, equations, and
formulas
Subtopic
8. Rearranges formulas to solve for a specified
variable (e.g., solve
d rt=
for t)
The Praxis
Middle School Mathematics Study Companion
35
6. Option (A) is correct. Since
+=34xy
is equivalent to
=−+34yx
, the slope of the graph
of
+=34xy
is
−3
. This means that the slope of the line that passes through the point
( )
4,5
is also
3−
since the line is parallel to the graph of
+=34xy
. Substituting the slope
of
−3
and the point
( )
4,5
into the point-slope form of the equation of a line yields
(
)
−=− −53 4yx
. Applying the distributive property yields
−=− +5 3 12yx
, and then
adding 5 to both sides of the equation yields
=−+3 17yx
, which is the equation in (A).
Category
II. Algebra
Topic
B. Understands how to recognize and represent linear
relationships algebraically
Subtopic
1. Determines the equation of a line from information
presented in various forms (e.g., table, graph,
description)
7. Option (C) is correct. In the first solution,
4x
and 2 are added to get
6x
, but the
4x
term
contains a variable, whereas the 2 is a constant term; it is incorrect to add
4x
and 2
because they are not like terms. Similarly, in the second solution,
2x
and 3 are added to
get
5x
, but
2x
and 3 are not like terms, so this strategy is not valid. Therefore, a
misunderstanding of how to identify and combine like terms is the option that is best
addressed by asking students to critique the two invalid strategies.
Task of Teaching Topic
Mathematical problems, tasks, examples, and
procedures
Task of Teaching
Subtopic
8. Evaluates the usefulness of examples for
introducing a concept, illustrating an idea, or
demonstrating a strategy, procedure, or practice
Category
II. Algebra
Topic
C. Understands how to solve equations and
inequalities
Subtopic
1. Solves one-variable linear equations and
inequalities
The Praxis
Middle School Mathematics Study Companion
36
8. The correct answer, from top to bottom, is the Multiplication/Division Property of
Equality, the Addition/Subtraction Property of Equality, and the Multiplication/Division
Property of Equality. The equation
( )
+=
1
11 20 2
3
xx
is multiplied by 3 on both sides to
obtain
+=11 20 6xx
, so this step is justified by the Multiplication/Division Property of
Equality. Then,
11x
is subtracted from both sides of the equation
+=11 20 6xx
to obtain
= −20 5x
, so this step is justified by the Addition/Subtraction Property of Equality. Finally,
both sides of the equation
= −20 5x
are divided by
−5
to obtain
−=4 x
, so this step is
justified by the Multiplication/Division Property of Equality.
Category
II. Algebra
Topic
C. Understands how to solve
equations and inequalities
Subtopic
4. Justifies each step in solving
equations and inequalities
9. The correct answer is
−
3
2
. The slope of a linear function can be found by substituting
into the formula
−
=
−
21
21
yy
m
xx
, where m is the slope and
( )
11
,xy
and
( )
22
,xy
are two
points on the linear function. Substituting the given points into the formula gives
( )
−− −
= = = −
−−
4 11 15 3
10 2
73
m
.
Category
III. Functions
Topic
C. Understands basic characteristics of linear
functions (e.g., intercepts, slope)
Subtopic
2. Calculates the slope of a line presented as a table
of values, algebraically, or graphically and interprets
it in a modeling context
The Praxis
Middle School Mathematics Study Companion
37
10. The correct answer is
= 2
a
and
= −5c
. Since the graph of the equation intersects the
y-axis at the point
(
)
−
0, 5
, the value of c must be
5−
. Then, one method to find the value
of a is to substitute the coordinates from another point on the graph into the equation
and solve for a. Using the point
(
)
2,3
and the fact that
= −5c
, it can be determined that
( )
= −
2
3 25a
, so
−=4 53a
. To solve this equation for a, add 5 to both sides of the
equation, and then divide both sides of the equation by 4, which leads to the answer
= 2a
.
Category
III. Functions
Topic
D. Understands the relationships among functions,
tables, and graphs
Subtopic
1. Determines an equation to represent a linear or
quadratic function presented graphically
11. Option (B) is correct. The function is linear because the number of calories consumed is
proportional to the number of ounces of the energy drink that are consumed, and the
function is continuous because the volume of the energy drink that is consumed can be
measured with any level of accuracy (e.g., to the nearest hundredth of an ounce, not
only the nearest ounce). The function in (A) is continuous but it is not linear because a
person does not grow linearly over time. The function in (C) is linear but it is not
continuous because it can be assumed that one can only buy a whole number of hot
dogs. The function in (D) is neither linear nor continuous because the function is
quadratic and there can only be a whole number of teams in the tournament.
Task of Teaching Topic
Mathematical problems, tasks, examples, and
procedures
Task of Teaching Subtopic
8. Evaluates the usefulness of examples for
introducing a concept, illustrating an idea, or
demonstrating a strategy, procedure, or practice
Category
III Functions
Topic
F. Understands differences between linear,
quadratic, and exponential models, including how
their equations are created and used to solve
problems
Subtopic
1. Identifies situations in which one quantity
changes at a constant rate per unit interval relative
to another
The Praxis
Middle School Mathematics Study Companion
38
12. Option (C) is correct. The properties of angles associated with parallel and transversal
lines can be used to show that the angle with measure x degrees and the angle with
measure y degrees are supplementary angles. The sum of the measures of
supplementary angles is
°
180
, so
+=180xy
. It is given that
= 3yx
. Substituting
3
x
for
y in the equation
+=180xy
yields
=4 180x
. Hence,
= 45x
.
Category
IV. Geometry and Measurement
Topic
A. Knows the properties of types of lines
(e.g., parallel, perpendicular, intersecting) and
angles
Subtopic
2. Applies angle relationships (e.g., supplementary,
vertical, alternate interior) to solve problems
The Praxis
Middle School Mathematics Study Companion
39
13. Options (B) and (C) are correct. At the start of the lesson, several students answered that
the side length of a square with area 36 square units is 9 units instead of giving the
correct side length of 6 units. Since the side length of a square with perimeter 36 units is
9 units, the students are probably confusing area and perimeter. Therefore, an area that
would NOT be useful for assessing student learning in this situation is one that would
allow a student to find the correct answer by dividing the number of square units by 4,
since the perimeter of a square is divided by 4 to find the side length of the square. In
both (B) and (C), the square root of the number of square units is not equal to the result
when the number of square units is divided by 4, so these areas would be useful for
assessing student learning in this situation. However, in (A),
=16 4
and
÷=16 4 4
.
Since the answers are the same, Ms. Ruffin would have no way of knowing whether
students were thinking about area or thinking about perimeter when finding the answer,
so the problem in (A) is not useful for assessing student learning in this situation.
Task of
Teaching
Topic
Mathematical problems, tasks, examples, and procedures
Task of
Teaching
Subtopic
6. Identifies problems or tasks that fit a particular structure, address
the same concept, demonstrate desired characteristics, or elicit
particular student thinking
Category
IV. Geometry and Measurement
Topic
H. Understands how to solve problems involving perimeter and area
of polygons
Subtopic
1. Calculates and interprets perimeter and area of polygons that can
be composed of triangles and quadrilaterals, including in real-world
situations
The Praxis
Middle School Mathematics Study Companion
40
14. The correct answer is 49 kilometers. Reggie hiked 3,500 meters along a trail each day for
14 days, so Reggie hiked
×=
3,500 14 49,000
meters during that time. Since there are
1,000 meters in 1 kilometer, dividing 49,000 by 1,000 gives the final answer: that Reggie
hiked 49 kilometers in the last 14 days.
Category IV. Geometry and Measurement
Topic
J. Understands systems of measurement (i.e., metric,
United States customary)
Subtopic
1. Solves measurement, estimation, and conversion
problems involving time, length, temperature,
volume, and mass in standard measurement
systems
15. Option (D) is correct. There are
360°
in a circle, and 16.1% of 360 is equal to
×=0.161 360 57.96
. So of the values given,
58
°
is the value that best approximates the
measure of the central angle of the sector representing the sales of Model D.
Category
V. Statistics and Probability
Topic
B. Understands how to interpret, analyze, and represent
data presented in a variety of displays
Subtopic
1. Represents and analyzes data in various displays
(e.g., bar graphs, line graphs, circle graphs, boxplots,
histograms, scatterplots, stem-and-leaf plots, two-way
tables)
The Praxis
Middle School Mathematics Study Companion
41
16. Options (A) and (D) are correct. Since the 4 additional integers that list M contains are
each less than each of the integers in list K, then each of the 4 additional integers that list
M contains must be less than the mean of the integers in list K, so the statement in (A)
must be true. Also, since the greatest integer in list M is equal to the greatest integer in
list K, but the least integer in list M is less than the least integer in list K, the statement in
(D) must be true.
However, the statements in (B), (C), and (E) may not be true. Remember that the
question asks which statements must be true (as opposed to asking which statements
could be true), so one example is sufficient to show that an option is not correct. Also,
the question states that integers may appear more than once in list K but does not state
how many integers are in list K. Suppose that list K consists of the integer 76 listed 15
times and that list M consists of the integers 71, 72, 73, 74, and the integer 76 listed 15
times. In this case, the median of each list is 76 and the mode of each list is 76, so the
statements in (B) and (C) are not true for this example. In addition, the first quartile of
each list is 76 and the third quartile of each list is 76, which means the interquartile
range of each list is 0, so the statement in (E) is not true for this example. Therefore, the
statements in (B), (C), and (E) may not be true.
Category
V. Statistics and Probability
Topic
C. Understands concepts associated with measures
of central tendency and dispersion
Subtopic
2. Determines and interprets measures of center
(e.g., mean, median, mode) and spread (e.g., range,
interquartile range) in a variety of problems
17. Option (D) is correct. Based on the data in the table, a total of
+=16 14 30
people
surveyed watched at least 2 hours of television but less than 4 hours of television per
day. If a person is selected at random from those surveyed, the probability that the
person selected will have watched at least 2 hours but less than 4 hours per day is
=
30 3
50 5
.
Category
V. Statistics and Probability
Topic
D. Knows how to use and evaluate probability models
Subtopic
2. Solves probability problems involving simple events
The Praxis
Middle School Mathematics Study Companion
42
Understanding Question Types
The Praxis® assessments include a variety of question types: constructed response (for which
you write a response of your own); selected response, for which you select one or more
answers from a list of choices or make another kind of selection (e.g., by selecting a sentence in
a text or by selecting part of a graphic); and numeric entry, for which you enter a numeric value
in an answer field. You may be familiar with these question formats from seeing them on other
standardized tests you have taken. If not, familiarize yourself with them so that you won't have
to spend time during the test figuring out how to answer them.
Understanding Selected-Response and Numeric-Entry Questions
For most questions you will respond by selecting an oval to choose a single answer from a list
of answer choices.
However, interactive question types may also ask you to respond by doing the following.
• Selecting more than one choice from a list of choices.
• Typing in a numeric-entry box. When the answer is a number, you may be asked to
enter a numerical answer. Some questions may have more than one entry box to enter
a response. Numeric-entry questions typically appear on mathematics-related tests.
• Selecting parts of a graphic. In some questions, you will select your answers by selecting
a location (or locations) on a graphic such as a map or chart, as opposed to choosing
your answer from a list.
• Selecting sentences. In questions with reading passages, you may be asked to choose
your answers by selecting a sentence (or sentences) within the reading passage.
• Dragging and dropping answer choices into targets on the screen. You may be asked to
select answers from a list of choices and to drag your answers to the appropriate
location in a table, paragraph of text, or graphic.
• Selecting answer choices from a drop-down menu. You may be asked to choose
answers by selecting choices from a drop-down menu (e.g., to complete a sentence).
Remember that with every question, you will get clear instructions.
The Praxis
Middle School Mathematics Study Companion
43
Understanding Constructed-Response Questions
Some tests include constructed-response questions, which require you to demonstrate your
knowledge in a subject area by writing your own response to topics. Essay questions and short-
answer questions are types of questions that call for a constructed response.
For example, an essay question might present you with a topic and ask you to discuss the
extent to which you agree or disagree with the opinion stated. For such questions, you must
support your position with specific reasons and examples from your own experience,
observations, or reading.
Following are a few sample essay topics to review:
• Brown v. Board of Education of Topeka
“We come then to the question presented: Does segregation of children in public
schools solely on the basis of race, even though the physical facilities and other
‘tangible’ factors may be equal, deprive the children of the minority group of equal
educational opportunities? We believe that it does.”
A. What legal doctrine or principle, established in Plessy v. Ferguson (1896), did the
Supreme Court reverse when it issued the 1954 ruling quoted above?
B. What was the rationale given by the justices for their 1954 ruling?
• In his self-analysis, Mr. Payton says that the better-performing students say small-group work
is boring and that they learn more working alone or only with students like themselves.
Assume that Mr. Payton wants to continue using cooperative learning groups because he
believes they have value for all students.
o Describe TWO strategies he could use to address the concerns of the students
who have complained.
o Explain how each strategy suggested could provide an opportunity to improve
the functioning of cooperative learning groups. Base your response on principles
of effective instructional strategies.
• “Minimum-wage jobs are a ticket to nowhere. They are boring and repetitive and teach
employees little or nothing of value. Minimum-wage employers take advantage of people who
need a job.”
o Discuss the extent to which you agree or disagree with this opinion. Support
your views with specific reasons and examples from your own experience,
observations, or reading.
The Praxis
Middle School Mathematics Study Companion
44
Keep the following things in mind when you respond to a constructed-response question.
1. Answer the question accurately. Analyze what each part of the question is asking you
to do. If the question asks you to describe or discuss, you should provide more than just
a list.
2. Answer the question completely. If a question asks you to do three distinct things in
your response, you should cover all three things for the best score. Otherwise, no
matter how well you write, you will not be awarded full credit.
3. Answer the question that is asked. Do not change the question or challenge the basis
of the question. You will receive no credit or a low score if you answer another question
or if you state, for example, that there is no possible answer.
4. Give a thorough and detailed response. You must demonstrate that you have a
thorough understanding of the subject matter. However, your response should be
straightforward and should not be filled with unnecessary information.
5. Take notes on scratch paper so that you don’t miss any details. Then you’ll be sure
to have all the information you need to answer the question.
6. Reread your response. Check that you have written what you intended to write. Do not
leave sentences unfinished or omit clarifying information.
The Praxis
Middle School Mathematics Study Companion
45
General Assistance For The Test
Praxis
®
Interactive Practice Test
This full-length Praxis
®
practice test lets you practice answering one set of authentic test
questions in an environment that simulates the computer-delivered test.
• Timed just like the real test
• Correct answers with detailed explanations
• Practice test results for each content category
ETS provides a free interactive practice test with each test registration. You can learn more
here
.
Doing Your Best
Strategy and Success Tips
Effective Praxis test preparation doesn’t just happen. You'll want to set clear goals and
deadlines for yourself along the way. Learn from the experts. Get practical tips to help you
navigate your Praxis test and make the best use of your time. Learn more at
Strategy and Tips
for Taking a Praxis Test.
Develop Your Study Plan
Planning your study time is important to help ensure that you review all content areas covered
on the test. View a sample plan and learn how to create your own. Learn more at
Develop a
Study Plan.
Helpful Links
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