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Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
Middle School Mathematics (0069)
Test at a Glance
Test Name Middle School Mathematics
Test Code 0069
Time 2 hours
Number of Questions 40 multiplechoice (Part A)
3 short constructed response (Part B)
Format Multiplechoice questions and constructedresponse questions, graphingcalculator allowed; calculators with QWERTY keyboards not allowed
Weighting Multiplechoice: 67% of total scoreShort constructedresponse: 33% of total score
IV
V
I
II
III
Content CategoriesApproximate
Number ofQuestions
ApproximatePercentage ofExamination
I. Arithmetic and Basic Algebra 12 20%II. Geometry and Measurement 10 17%
III. Functions and Their Graphs 8 13%
IV. Data, Probability, and StatisticalConcepts; Discrete Mathematics 10 17%
V. ProblemSolving Exercises
3(constructed
response)
33%
Process Categories
Mathematical Problem Solving
Mathematical Reasoning and ProofMathematical Connections Distributed Across ContentMathematical Representation CategoriesUse of Technology
Pacing andSpecial Tips
In allocating time on this assessment, you should plan to spend about80 minutes on the multiplechoice section and about 40 minutes on theconstructedresponse section; the sections are not independently timed.

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
About This Test
The Praxis Middle School Mathematics test is designed to
certify examinees as teachers of middle school mathematics.
Examinees have typically completed a bachelors program
with an emphasis in mathematics education, mathematics, oreducation. Course work will have included many of the
following topics: theory of arithmetic, foundations of
mathematics, geometry for elementary and middle school
teachers, algebra for elementary and middle school teachers,
the big ideas of calculus, data and their uses, elementary
discrete mathematics, elementary probability and statistics,
history of mathematics, mathematics appreciation, and the
use of technology in mathematics education.
The examinee will be required to understand and work with
mathematical concepts, to reason mathematically, to make
conjectures, to see patterns, to justify statements using
informal logical arguments, and to construct simple proofs.
Examinees will be expected to know common mathematical
formulae, but any infrequently used formulae will be provided
within the relevant question. Additionally, the examinee will be
expected to solve problems by integrating knowledge from
different areas of mathematics, to use various representations
of concepts, to solve problems that have several solution
paths, and to develop mathematical models and use them to
solve realworld problems.
The test is not designed to be aligned with any particularschool mathematics curriculum, but it is intended to be
consistent with the recommendations of national studies
on mathematics education such as the National Council of
Teachers of Mathematics (NCTM) Principles and Standards
for School Mathematics (2000) and the National Council
for Accreditation of Teacher Education (NCATE) Program
Standards for Initial Preparation of Mathematics Teachers (2003).
This test may contain some questions that will not count
toward your score.
Calculators
The examinee will be allowed to use a fourfunction,
scientic, or graphing calculator during the examination.
However, computers, calculators with QWERTY (typewriter)
keyboards, and electronic writing pads are NOT allowed.Unacceptable machines include the following:
Powerbooks and portable/handheld computers
Pocket organizers
Electronic writing pads or peninput/stylusdrivendevices (e.g., Palm, PDAs, Casio Class Pad 300)
Devices with QWERTY keyboards (e.g., TI92 PLUS,Voyage 200)
Cellphone calculators
More information on the calculator use policy for Praxis tests
can be found at www.ets.org/praxis/prxcalc.html.
PRAXIS
Graphing Calculator Policy
Test administration staff will clear the memory of all
graphing calculators both before and after test
administration.
We recommend that you
back up any important information in yourcalculators memory, including applications,
before arriving at the test site
know how to clear the memory on the approved
calculator that you plan to use during the test
Note: Instructions on how to back up and clear the
memory of calculators can be found on various
calculator Web sites.

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
Topics Covered
Content Categories
In each of the content categories, the test will assess an
examinees ability to use appropriate mathematical language
and representations of mathematical concepts, to connect
mathematical concepts to one another and to realworld
situations, and to integrate mathematical concepts to solve
problems. Because the assessments were designed to
measure the ability to integrate knowledge of mathematics,
answering any question may involve more than one
competency and may involve competencies from more than
one content category. Representative descriptions of topics
covered in each category are provided below.
I. Arithmetic and Basic Algebra
Add, subtract, multiply, and divide rational numbersexpressed in various forms; apply the order ofoperations; identify the properties of the basic
operations on the standard number systems (e.g.,closure, commutativity, associativity, distributivity);identify an inverse and the additive and multiplicative
inverses of a number; use numbers in a way that ismost appropriate in the context of a problem
Order any nite set of real numbers and recognizeequivalent forms of a number; classify a number as
rational, irrational, real, or complex; estimate values ofexpressions involving decimals, exponents, and
radicals; nd powers and roots
Given newly dened operations on a number system,
determine whether the closure, commutative,associative, or distributive properties hold
Demonstrate an understanding of conceptsassociated with counting numbers (e.g., prime orcomposite, even or odd, factors, multiples, divisibility)
Interpret and apply the concepts of ratio, proportion,
and percent in appropriate situations
Recognize the reasonableness of results within the
context of a given problem; using estimation, test thereasonableness of results
Work with algebraic expressions, formulas, andequations; add, subtract, and multiply polynomials;divide polynomials; add, subtract, multiply, and divide
algebraic fractions; perform standard algebraicoperations involving complex numbers, radicals, and
exponents, including fractional and negative exponents
Determine the equations of lines, given sufcient
information; recognize and use the basic forms of theequation for a straight line
Solve and graph linear equations and inequalities in
one or two variables; solve and graph systems oflinear equations and inequalities in two variables;
solve and graph nonlinear algebraic equations; solveequations and inequalities involving absolute values
Solve problems that involve quadratic equations,using a variety of methods (e.g., graphing, formula,calculator)
II. Geometry and Measurement
Solve problems that involve measurement in both
metric and traditional systems
Compute perimeter and area of triangles,
quadrilaterals, circles, and regions that arecombinations of these gures; compute the surface
area and volume of right prisms, cones, cylinders,
spheres, and solids that are combinations ofthese gures
Apply the Pythagorean theorem to solve problems;solve problems involving special triangles, such as
isosceles and equilateral
Use relationships such as congruency and similarity to
solve problems involving twodimensional and threedimensional gures; solve problems involving parallel
and perpendicular lines
Solve problems using the relationships among the
parts of triangles, such as sides, angles, medians,
midpoints, and altitudes Solve problems using the properties of special
quadrilaterals, such as the square, rectangle,
parallelogram, rhombus, and trapezoid; describerelationships among sets of special quadrilaterals;
solve problems involving angles, diagonals, andvertices of polygons with more than four sides
Solve problems that involve using the properties ofcircles, including problems involving inscribed angles,central angles, radii, tangents, arcs, and sectors
Solve problems involving reections, rotations, andtranslations of points, lines, or polygons in the plane
Solve problems that can be represented on thexyplane (e.g., nding the distance between twopoints or nding the coordinates of the midpointof a line segment)
Estimate absolute and relative error in the numericalanswer to a problem by analyzing the effects of round
off and truncation errors introduced in the course ofsolving a problem

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
Demonstrate an intuitive understanding of a limit
Demonstrate an intuitive understanding of maximumand minimum
Estimate the area of a region in thexyplane
III. Functions and Their Graphs
Understand function notation for functions of onevariable and be able to work with the algebraicdenition of a function (e.g., for everyxthere is oney)
Identify whether a graph in the plane is the graph of a
function; given a set of conditions, decide if theydetermine a function
Given a graph (for example, a line, a parabola, a step,absolute value, or simple exponential), select anequation that best represents the graph; given an
equation, show an understanding of the relationshipbetween the equation and its graph
Determine the graphical properties and sketch a graphof a linear, step, absolutevalue, quadratic, or
exponential function
Demonstrate an understanding of a physical situation
or a verbal description of a situation and develop amodel of it, such as a chart, graph, equation, story,
or table
Determine whether a particular mathematical model,
such as an equation, can be used to describe twoseemingly different situations. For example, given two
different word problems, determine whether aparticular equation can represent the relationshipbetween the variables in the problems
Find the domain (xvalues) and range (yvalues) of afunction without necessarily knowing the denitions;
recognize certain properties of graphs (e.g., slope,intercepts, intervals of increase or decrease, axis of
symmetry)
Translate verbal expressions and relationships into
algebraic expressions or equations; provide andinterpret geometric representations of numeric and
algebraic concepts
IV.a. Data, Probability, and Statistical Concepts
Organize data into a presentation that is appropriatefor solving a problem (e.g., construct a histogram and
use it in the estimation of probabilities)
Read and analyze data presented in various forms(e.g., tables, charts, graphs, line, bar, histogram, circledouble line, double bar, scatterplot, stem plot, line
plot, box plot); draw conclusions from data
Solve probability problems involving nite sample
spaces by actually counting outcomes; solveprobability problems by using counting techniques;solve probability problems involving independent and
dependent events; solve problems by using geometricprobability
Solve problems involving average, including arithmeticmean and weighted average; nd and interpret
common measures of central tendency (e.g., mean,sample mean, median, mode) and know which is the
most meaningful to use in a given situation; nd andinterpret common measures of dispersion (e.g., range,
spread of data, standard deviation, outliers)
IV.b. Discrete Mathematics
Use and interpret statements that contain logical
connectives (and, or, ifthen) as well as logicalquantiers (some, all, none)
Solve problems involving the union and intersection of
sets, subsets, and disjoint sets Solve basic counting problems involving permutations
and combinations without necessarily knowing
formulas; use Pascals triangle to solve problems
Solve problems that involve simple sequences or
number patterns (e.g., triangular numbers or othergeometric numbers); nd rules for number patterns
Use and interpret matrices as tools for displaying data
Draw conclusions from information contained in
simple diagrams, owcharts, paths, circuits, networks,or algorithms
Explore patterns in order to make conjectures,predictions, or generalizations

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
V. ProblemSolving Exercises
Part B of the test contains three equally weightedconstructedresponse questions that together
comprise 33 percent of the examinees score. Theprimary focus of the three constructedresponse
questions will be distributed across the four previouslydescribed content categories. Also, the examinee willbe expected to integrate knowledge from different
areas of mathematics.
Mathematical Process CategoriesIn addition to having knowledge of the mathematics contentexplicitly described in the Content Categories section, entry
level middle school mathematics teachers must also be ableto think mathematically; moreover, they must have an
understanding of the ways in which mathematical contentknowledge is acquired and used. Answering questions on this
assessment may involve one or more of the processesdescribed in the process categories below, and all of theprocesses may be applied to any of the content topics.
Mathematical Problem Solving
Solve problems that arise in mathematics and thoseinvolving mathematics in other contexts
Build new mathematical knowledge throughproblem solving
Apply and adapt a variety of appropriate strategies tosolve problems
Mathematical Reasoning and Proof
Select and use various types of reasoning and
methods of proof
Make and investigate mathematical conjectures
Develop and evaluate mathematical argumentsand proofs
Mathematical Connections
Recognize and use connections among mathematical
ideas
Apply mathematics in context outside of mathematics
Demonstrate an understanding of how mathematicalideas interconnect and build on one another
Mathematical Representation
Select, apply, and translate among mathematicalrepresentations to solve problems
Use representations to model and interpret physical,
social, and mathematical phenomena Create and use representations to organize, record,
and communicate mathematical ideas
Use of Technology
Use technology appropriately as a tool for
problem solving
Use technology as an aid to understanding
mathematical ideas

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
4. The original price of a certain car was 25 percent greater
than its cost to the dealer. The actual selling price was25 percent less than the original price. If is the cost ofthe car to the dealer and is the selling price, which of
the following represents in terms of ?
(A)
(B)
(C)
(D)
5. Which gure below results if right triangleABCabove isipped (reected) across theyaxis and then turned
(rotated) clockwise about point C'by 90 degrees?
Sample Test QuestionsThe sample questions that follow illustrate the kinds ofquestions in the test. They are not, however, representative ofthe entire scope of the test in either content or difficulty.Answers with explanations follow the questions.
Directions: Each of the questions or statements below is
followed by four suggested answers or completions. Select
the one that is best in each case.
1. Which of the following is true about the data in the tableabove?
(A) Asxdecreases,yincreases.
(B) Asxincreases,ydoes not change.
(C) Asxincreases,ydecreases.
(D) Asxincreases,yincreases.
2. The average number of passengers who use a certain
airport each year is 350 thousand. A newspaper reportedthe number as 350 million. The number reported in the
newspaper was how many times the actual number?
(A) 10
(B) 100
(C) 1,000
(D) 10,000
3. If there are exactly 5 times as many children as adults at
a show, which of the following CANNOT be the numberof people at the show?
(A) 102
(B) 80
(C) 36
(D) 30
y
x
C
B
A

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
8. A square is inscribed in each of the circles above. Theradius of circleAis 1, and the radius of circleBis 2.What is the ratio of the area of the square inscribed incircleAto the area of the square inscribed in circleB?
(A)
(B)
(C)
(D)
9. Which of the following denesyas a function ofx?
(A)
(B)
(C)
(D)
6. The large rectangular block pictured above was made bystacking smaller blocks, all of which are the same size.
What are the dimensions in centimeters of each of thesmaller blocks?
(A)
(B)
(C)
(D)
7. In the gure above, line and line are parallel and
What is the value ofx?
(A) 30
(B) 45
(C) 60
(D) 75

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
10. A taxi ride costs $2.50 for the rst mile or fraction
thereof plus $0.50 for each additional mile or
fraction thereof. Which of the following graphs
represents the total cost of a ride as a function of
distance traveled?
(A)
(B)
(C)
(D)
11. In a class of 29 children, each of 20 children has a dogand each of 15 has a cat. How many of the children have
both a dog and a cat?
(A) None of the children necessarily has both.
(B) Exactly 5
(C) Exactly 6
(D) At least 6 and at most 15
12. The graph above shows the distribution of the content,
by weight, of a countys trash. If approximately60 tons of the trash consists of paper, approximately
how many tons of the trash consist of plastics?
(A) 24
(B) 20
(C) 15
(D) 12
Paper40%
Other36%
Glass
Plastics
Metals
7%
8%
9%

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
Questions 1314 refer to the following graph.
13. In how many of the years shown were there more than
twice as many students in medical schools as there werein 1950?
(A) None
(B) One
(C) Two
(D) Three
14. The number of students in medical schools increased byapproximately what percent from 1970 to 1980?
(A) 75%
(B) 60%
(C) 50%
(D) 45%
15. In order to estimate the population of snails in a certainwoodland, a biologist captured and marked 84 snails
that were then released back into the woodland. Fifteendays later the biologist captured 90 snails from the
woodland, 12 of which bore the markings of the
previously captured snails.
If all of the marked snails were still active in the
woodland when the second group of snails werecaptured, what should the biologist estimate the snail
population to be, based on the probabilities suggestedby this experiment?
(A) 630
(B) 1,010
(C) 1,040
(D) 1,080
16. If a student takes a test consisting of 20 truefalsequestions and randomly guesses at all of theanswers, what is the probability that all 20 guesses
will be correct?
(A) 0
(B)
(C)
(D)
ROBINS TEST SCORES
88, 86, 98, 92, 90, 86
17. In an ordered set of numbers, the median is the middlenumber if there is a middle number; otherwise, the
median is the average of the two middle numbers. IfRobin had the test scores given in the table above, what
was her median score?
(A) 89
(B) 90
(C) 92
(D) 95

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
Answers 1. Asxmoves from 4 to 0 (that is, from left to right on the
number line), its value increases.
Similarly, the value ofyincreases from 2 to 0. Thus, it can beseen that asxincreases,yincreases. The correct answer is (D).
2. The number of passengers who use the airport each year,
350 thousand, can be written as 350,000; 350 million can be
written as 350,000,000. 350,000,000350,000 = 1,000, so the
correct answer is (C).
3. If arepresents the number of adults, then 5arepresents thenumber of children and 6arepresents the total number of people
at the show. Since 6arepresents a whole number that is a multiple
of 6, there cannot be 80 people at the show, for 80 is not a
multiple of 6. The correct answer is (B).
4. This question asks you to apply your knowledge of percent
increase or decrease to determine a selling price based on cost
of a car to the dealer, c. Since the original price of the car was
25 percent greater than the cost to the dealer, the original price
was c+ 0.25c= 1.25c. Since the selling price was 25 percent
less than this amount, only 75 percent of this amount will be paid,
so the selling price of the car was 0.75(1.25c). Thus, the correct
answer choice is (D).
5. When triangleABCis reected across theyaxis, the gure
formed is located in quadrant I and is the mirror image of the given
gure. Rotating the triangle 90 degrees clockwise about vertex C'
yields choice (A).
6. The length of the large block, 12 centimeters, is 3 times the
length of a small block, so each small block is 12 3 = 4centimeters long. Similarly, the width of a small block is 82 = 4centimeters and the height of a small block is 93 = 3centimeters. Thus, the correct answer is (D).
7. This question asks you to apply your understanding of angles
in a plane and, in particular, properties of angles associated with
parallel and transversal lines. You should be able to show, using
pairs of alternate interior angles and corresponding angles, that
angle measuredxdegrees and angle measuredydegrees are
supplementary angles. Recall that the sum of the measures of
supplementary angles is 180. That is, It is given
that Substituting fory, you get Hence,
Therefore, the correct answer is (B).
8. This question asks you to apply your knowledge of circles,
squares, and proportional reasoning to nd the ratio of the areas
of two squares. There are many ways to approach this problem.One approach is to use the information given and many things that
you know about circles, squares, and triangles and do lots of
computation. Another is to use your knowledge of what happens
to area when you scale up corresponding linear dimensions in a
gure. If you like to compute, here is what you might do. First
consider circleA.The radius of circleAis 1, and the diameter is 2.This diameter is also the diagonal of the inscribed square and the
hypotenuse of a right triangle with side a. By the Pythagorean
theorem, ; , , and a, thus
the length of a side of squareAis .So the area of squareAis
Likewise, the area of squareBis Thus
the ratio of the area of squareAto the area of squareBis 2 : 8,which is 1 : 4.The correct answer is (D).
Alternatively, you may recall that when you are comparing two
similar gures whose corresponding linear dimensions have a ratio
of 1to2,as in this problem, the ratio of the areas of the gures isthe ratio of the square of the linear dimensions; that is, to ,which is 1to 4. Hence, the correct choice is (D).
9. This question asks you to identify a function by applying your
understanding of functions to different mathematical statements.
To answer questions, such as this, that ask which of the
following, you need to consider only the choices given. There are
usually other correct answers to the question, as in this case, that
you are not asked to consider. To answer this question, you shouldrecall that ifyis a function ofx, then each value ofx(in the domain
of the function) results in only one value ofy. In choices (A) and
(B), most values ofxhave two different corresponding values ofy.
You can see this by solving the equations in (A) and (B) fory.In (A),
or Similarly, in (B),
So neither (A) nor (B) denesyas a function ofx.In choice (D), foreach value ofx, there is more than one value ofythat satises
the inequality. So (D) does not dene y as a function ofx. However,in (C), for each value ofx, there is only one value ofythat
corresponds to that value ofx.Thus, the correct answer choiceis (C).

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
10. This question asks you to apply your knowledge of graphing
data in a coordinate plane to a situation involving graduated rate.
You should notice that each of the choices given is the graph of a
step function. You will need to identify the graph that includes the
correct cost for the rst step and the correct interval between
steps. Since the cost for the rst mile or less is $2.50, the cost
for the rst step (the value on the vertical axis) should be 2.5 over
the horizontal interval from0to mile, with a solid dot at mile.
(There should be no cost at a distance of 0miles, since there is nocharge if there is no ride.) In each of the subsequent horizontal
intervals of mile, the cost value on the vertical axis should show
an increment of $0.50, with a solid dot at the right endpoint of
each interval. Only choice (A) illustrates this correctly. Choice (C)
has the correct cost values for each step but does not represent
the endpoints of each interval correctly. The correct answer
choice, therefore, is (A).
11. Since the 29 children have a total of 35 dogs and cats, at
least 6 children must have both a dog and a cat. If there are
exactly 6 children with both a cat and a dog, then 14 children have
only a dog and 9 children have only a cat. On the other hand, all
15 cat owners could also own a dog; then 5 children have only a
dog and 9 children have neither a dog nor a cat. Thus, the correct
answer is (D).
12. The circle graph shows the distribution of the trash contentin
percents; the question asks for the weightof the plastics content
in tons. From the graph we see that plastics account for 8% of the
total weight of the trash. The problem states that 60 tons of thetrash consist of paper; the graph shows that this amount equals
40% of the total, so
60 = 0.4(total weight)
and the total weight is60
0 4.= 150 tons.
The weight of plastics equals 8%of 150 tons, or (0.08)(150) =12 tons.
There is another, slightly faster, way to solve this problem. We use
the fact that the ratio of plastics to paper in the trash is the same,
whether the two amounts are given as percents or in tons. This
gives us the proportion
tonsof plastics
tons of paper= =
8
40
1
5
%
%
or
tons of plastics
60
1
5=
tonsof plastics=60
5= 12
The correct answer is (D).
13. The bar graph presents information for eight different years.
The vertical scale goes from 0 to 80,000. The zeros are left off the
scale because the title tells you to read the numbers as
thousands. To nd the number of students in any one year, read
the height of the corresponding bar from the lefthand scale and
multiply that height by 1,000.
The bar for 1950 has a height of about 27, so the number of
students in 1950 was about 27,000. You have to nd the number
of years in which there were more than twice as many; that is,
more than 54,000 students. To do this, count the number of bars
that are higher than 54. These are the bars for 1975, 1980, and
1985. Thus, there were three years in which there were more than
twice as many students as in 1950. The correct answer is (D).

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Middle School Mathematics (0069)
Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING., PRAXIS I, PRAXIS II, and PRAXIS IIIare registered trademarks of Educational Testing Service (ETS) in the United States and other countries. PRAXIS and THE PRAXIS SERIES are trademarks of ETS. 8601
14. To compute a percent increase, you need the increase in
the number of students and the number of students before the
increase.
The graph shows that the number of students in 1970 was
40,000 and the number of students in 1980 was 70,000, an
increase of 30,000 students. To nd the percent increase, divide
this number by the base number; that is, the number of students
before the increase, or 40,000.
30 000
40 000
3
4 0 75 75
,
,. %= = =
The correct answer is (A).
15. Given the conditions of the experiment, it is reasonable to
assume that the 90 snails captured by the biologist, 15 days after
the markings were made, represent a random sample of the snail
population.
Thus, about12
90,or
2
15, of the population had been marked.
Thus, the original 84 snails marked represented approximately
2
15of the entire population and the biologist should estimate the
snail population to be 8415
2,or 630.
The correct answer is (A).
16. The probability that the student guesses any one answer
correctly is 1/2, and, since the student is randomly guessing, theguesses are independent events. Thus, the probability of guessing
all 20 answers correctly is1
2
20
,and the correct answer is (B).
17. The problem gives a set of test scores and the denition of
the median. The rst part of the denition tells you to order the
scores; that is, to arrange them in order from smallest to largest.
Here are the numbers ordered from smallest to largest:
86, 86, 88, 90, 92, 98
Because there is an even number of scores (6), there are twomiddle numbers in the set, 88and 90, and the average of thetwo middle numbers is
88 90
2
178
2 89
+
= =
Thus, the median of Robins scores is 89and the correct answeris (A). (Notice that the median of a set of numbers need not be one
of the numbers in the set.)

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Sample Test QuestionsThis section presents sample constructedresponse questionsand sample responses along with the standards used in scoringthe responses. When you read these sample responses, keepin mind that they will be less polished than if they had beendeveloped at home, edited, and carefully presented. Examinees
do not know what questions will be asked and must decide, onthe spot, how to respond.
Readers take these circumstances into account when scoringthe responses. Readers will assign scores based on the followingscoring guide.
SCORING GUIDES
3
Responds appropriately to all parts of the question
Where required, provides a strong explanation that is well
supported by relevant evidence
Demonstrates a strong knowledge of subject matter,concepts, theories, facts, procedures, or methodologies
relevant to the question
Demonstrates a thorough understanding of the most
signicant aspects of any stimulus material presented
2
Responds appropriately to most parts of the question
Where required, provides an explanation that is sufciently
supported by relevant evidence
Demonstrates a sufcient knowledge of subject matter,
concepts, theories, facts, procedures, or methodologies
relevant to the question
Demonstrates a basic understanding of the mostsignicant aspects of any stimulus material presented
1
Responds appropriately to some parts of the question
Where required, provides a weak explanation that is not wel
supported by relevant evidence
Demonstrates a weak knowledge of subject matter,concepts, theories, facts, procedures, or methodologies
relevant to the question
Demonstrates little understanding of signicant aspects of
any stimulus material presented
0
Blank, offtopic, or totally incorrect response
Does nothing more than restate the question or some
phrases from the question
Demonstrates extremely limited understanding or a
misunderstanding of the topic

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Sample Response That Received a Score of 3
(A),
y x x
y x
0 2,000 10 000
1
20
10 000 2,000
( , )if
see note at enof response
(if x> 10,000)
(B)
0300
900
1,800
2,400
3,000
600
1,500
1,200
2,100
2,700
3,6003,900
4,2004,500
3,300
0
Emily'sIncome
(EachUnit=$300)
Emilys Sales in Thousands of Dollars
10 15 20 25 30 35 40 45 50 55 60 655
$1,000
(8,000, 2,000)
(10,000, 2,000)
(C) The increase in Emilys income would be $1,000. If shemade $8,000in monthly sales, her income would be$2,000since her sales did not exceed $10,000. If she
made $30,000in monthly sales, her income would be
y = + =1
20 30 000 10 000 2 000 3 000( , , ) , , , which
can also be read from the graph. Therefore the difference
in income would be $3,000 $2,000 = $1,000.
Note: An alternative to this equation (easier forcalculation though not as concrete) would be
y x 1
201,500 .
Commentary on Sample Response That Earned a
Score of 3
This response received a score of 3 because it responded
appropriately to all parts of the question: the response to
part (a) included both equations requested; the graph wascorrectly drawn in part (b); and the response to part (c) was
obtained from the graph, as requested. Two additional
strengths of this response are 1) the observation that the
algebraic solution to part (c) is consistent with the graphical
solution and 2) the footnote giving an alternative form of the
second equation in part (a), with the explanation that,
although it would be easier to use for computation, it is not
as concrete.
Sample Question 1
Emilys monthly income consists of a monthly salary of
$2,000, plus a commission if her monthly sales exceed
$10,000. The commission is equal to 5 percent of the
amount by which her monthly sales exceed $10,000. (Forexample, if Emilys monthly sales are $15,000, she receives a
commission of 5 percent of $5,000.)
(A) Write an equation that gives Emilys monthly income,y,in terms of her monthly sales,x, if her monthly sales areless than or equal to $10,000 x 10 000, .Write asecond equation that gives Emilys monthly income,y, interms of her monthly sales,x, if her monthly sales aregreater than $10,000 x>10,000 .
(B) Using the equations you wrote in part (a) of this
question, draw a graph of Emilys monthly income,y,as a function of her monthly sales,x, for monthly salesof $0 to $60,000 0 60 000 x , . Graph her monthlysales on thexaxis and her monthly income on theyaxis.Label each axis and show the units and scales used.
(C) Use the graph you drew in part (b) of this question toestimate the increase in Emilys monthly income if her
monthly sales were to increase from $8,000 to $30,000.Show your work.

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Sample Response That Received a Score of 2
(a) Monthly income = Monthly salary + .05
any sales ov
x
eer 10,000
y x
y x
2 000 10 0002 000 05 10 000, ,, . ,
(b)
876543
210
100 20 30
Monthly Sales$ Thousands
MonthlyIncome
$
Thous
ands
EMILYS MONTHLY INCOME
40 50 60x
y
(c) Increase if sales were to go from $8,000to$30,000
At $8,000sales she gets no extra since8 000 10 000, , . So only gets $2,000 base salary.
At $30,000she gets basic $2,000plus .05times 20,000since thats the amount over $10,000. So it goes from$2,000on the graph to $3,500at 30,000monthly sales.
Commentary on Sample Response That Earned a
Score of 2
This response received a score of 2 because it responds
appropriately to most parts of the question anddemonstrates a sufcient knowledge of the conceptsrelevant to the question. In particular, both of the equations
requested in part (a) are correct. The graph in part (b) ispartially correct. The values plotted forx> 10,000 reect
calculating Emilys income without subtracting $10,000 whenevaluating and graphing the second equation. It should not
have the discontinuity shown at monthly salesx= $10,000.The response to part (c) is correct based on the graph in part
(b). The correct process is described in part (c), but theexaminee missed an opportunity to identify the mistake inpart (b) by not carrying out this process and then comparing
the algebraic solution to the graphical solution.
Sample Response That Received a Score of 1
(a) 2,000 10 000
2,000 05 10 000
y x
y x
,
. ,
(b)
5,000
4,500
4,000
3,500
3,000
2,500
2,000
10,0000
20,000
30,000
Income
40,000
50,000
60,000
x
y
(c) $2,200
The graph moves up $500 for every 10,000 sales.
At 8,000 sales, the income would be $1,800
At 30,000 sales, the income would be $3,000 judging by the dots.
Commentary on Sample Response That Earned a
Score of 1
This response received a score of 1 because it demonstrates
a weak knowledge of the concepts relevant to the question.In part (a), the equations (actually inequalities as written) are
not correct. The response does not provide evidence ofbeing able to translate a verbal representation into algebraic
equations. Because part (a) is incorrect, the graph in part (b)is not the graph of a piecewise linear function. However, thediscrete values that are graphed in part (b) are correct. In
part (c), the general description of the increase in income iscorrect for values of sales (x) greater than $30,000; however,the estimation of the income associated with sales of $8,000as different from $2,000 is further evidence of misunderstanding
of this piecewise linear function. There was also an arithmeticerror in nding the difference between $3,000 and $1,800.The response received some credit because portions of
parts (b) and (c) are correct.

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Sample Response That Received a Score of 0
(A) y x . ( ,05 10 000
y x 2 000 05 10 000, . ( ) ,
(B) x y
15,000 2,250
20,000 2,500
25,000 2,750
30,000 3,000
35,000 3,250
40 50
Monthly Salesin (1,000)
Monthly Incomein (1,000)
O 1012345 20 30
1
1
2
3
4
5
2
3
4
5
x
y
(C) 8,000
Commentary on Sample Response That Earned aScore of 0
This response received a score of 0 because it demonstrates
extremely limited understanding of the topic. The equationsin part (a) are not correct. The graph in part (b) does notreceive any credit because it is inaccurately drawn and does
not reect the piecewise linear characteristics of thesituation. Most of the discrete values were graphed
incorrectly, even though, based on the table, they appear tohave been computed correctly. Part (c) is either incomplete
or totally incorrect. The evidence of correct thinking providedby this response does not reach the threshold required toreceive a score above 0.
Sample Question 2
In triangleEFG, sideEF has length 8 and sideFG haslength 10.
(A) Two of the possible lengths of sideEG
are 3 and 16.Draw triangleEFGwith sideEGof length 3. Draw asecond triangleEFGwith sideEGof length 16. Foreach of the triangles, label all of the vertices and show
the lengths of all of the sides. [Note: Your drawings arenot expected to be exact but should reasonablyrepresent the relative lengths of the sides.]
(B) The length of sideEGcould not be 1 or 20. Explain whynot. Draw gures to support your explanation.
(C) If triangleEFGis a right triangle, what are the twopossible lengths of sideEG? Draw the two right
triangles. For each triangle, label all of the vertices andshow the lengths of all of the sides. Indicate the right
angle.

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Sample Response That Received a Score of 3
(A)
8
3
10
F
E G
(B) The sum of the lengths of two sides of a triangle must be
greater than the length of the other side. In triangleEFG,if the length of sideEGwas 1, the sum of the lengths ofsidesEGandEFwould be 9. This value is less than thelength of side FG, which is 10. Also if the length of sideEGwas 20, the sum of the lengths of sidesEFand FGwould be 18. This value is less than the length of side
EG, which is 20.
8
20
10
F
F
E G
8
10
1
F FG
E
(C) If EFGwere a right , the length of sideEGcould beeither 6 or approximately 12.81.
8
10
FE
G
12.81
6
8
EG
F
10
Commentary on Sample Response That Earned aScore of 3
This response received a score of 3 because it responds
appropriately to all parts of the question and demonstratesstrong knowledge of the concepts relevant to the question.Part (a) includes two drawings that show reasonable relative
lengths of the sides and that show clearly that angle Fisacute when sideEGhas length 3 and obtuse when sideEGhas length 16. Part (b) both describes and illustrates thetriangle inequality property correctly. Part (c) correctly shows
the two possible right triangles with sides of 8 and 10. (Theone minor error in the response is that the right angle is notlabeled in the second drawing. However, angleEappears tobe a right angle and is consistent with the opposite sidebeing the hypotenuse; thus, the error is not considered
signicant enough to warrant a lower score.)
8
16
10
E G
F

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Sample Response That Received a Score of 2
(A)
F
E
G
E G
F
(B) From Pythagorean theorem a b c2 2 2
.
FGis hypotenuse C
EF EG = FG2 2 2
2 2 28 10b =
+
+
bcould not be true by being 1 or 20.
(C) c b a
a
2 2 2
100 64 36
6
E a
F
G
c= 10b= 8
c
c
c
c
2 2 2
2
10 8
100 64
164
12 8
.
Commentary on Sample Response That Earned a
Score of 2
This response received a score of 2 because it demonstratesa sufcient knowledge of the concepts relevant to thequestion. Part (a) includes two drawings that show
reasonable relative lengths of the sides and that show clearlythat angleEis acute when side FGhas length 3 and obtusewhen sideEGhas length 16. Part (b) receives no creditbecause it assumes incorrectly that triangleEFGmust be aright triangle. Part (c) correctly shows the two possible righttriangles with sides of 8 and 10.
Sample Response That Received a Score of 1
(A)
FE
G
3
8
10
FE
G
16
10
8
F
E
G
c
a= 10
b= 8

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(B) There are no ways to construct triangles with measured
lengths of the 3 sides to be 1, 8, 10 due to the fact thatthe angles have to add up to 180. Using trigonometry
sin, cos,and tan, they will not add up.
There is no way that the angles will add up to 180.
G
E
F10
18
Once again, it just proves that there is no way the angles
are going to add up to 180.
20
8
10 ?
(C) One possible solution is theEG can be 6 according toPyth. theorem.
FE
G
6
8
10
FE
G
10
8
164 or
241
Commentary on Sample Response That Earned aScore of 1
This response received a score of 1 because it demonstratesa weak knowledge of the concepts relevant to the question.
In part (a), both triangles are drawn (incorrectly) as righttriangles. The relative lengths of the sides are acceptable
in the 3810 triangle, but not in the 81016 triangle. In the81016 triangle, the side of length 10 is shown longer than
the side of length 16 and the largest angle (which should beobtuse) is not opposite the longest side. The response to
part (b) is incorrect. Credit is given for the correct possibleright triangles shown in part (c).

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Sample Response That Received a Score of 0
(A)
8 10
E
F
G3
810
E
F
G16
(B)
8 10
E
F
G1
8
10
E
F
G20
ForEG to have a length of 1 or 20 would not be possiblebecause 1 would create a triangle too small to therelative dimensions of the other lengths while 20 would
create a triangle which has one length too big.
(C) F
E G? = 6
EG= 6
8
10
F
E
G10
8
12
Commentary on Sample Response That Earned aScore of 0
This response received a score of 0 because it demonstrates
extremely limited understanding of the topic. Both trianglesin part (a) are drawn as right triangles. Neither of the trianglesdrawn shows reasonable relative lengths of the sides. In the
3810 triangle, the side of length 8 is longer than the side oflength 10. In the 81016 triangle, the side of length 8 is
shown as the longest side and is opposite the greatestangle. None of the angles in the second triangle is shown as
obtuse. No credit is given for part (b) since the two guresare drawn showing triangles with sides of 1, 8, and 10 and 8,10, and 20, respectively, but the explanation indicates that
such triangles are not possible. The response does notsufciently explain the triangle inequality property. Part (c)
contains one of the two possible congurations of righttriangle EFG. Although a portion of part (c) is correct, the
evidence of knowledge about triangle geometry provided bythis response does not reach the threshold required to
receive a score above 0.

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Sample Question 3
In a certain experiment, a researcher plans to label each
sample with an identication code consisting of either asingle letter or 2 different letters in alphabetical order. For
example, if the researcher uses the 3 letters A, B, and C,
then there are 6 possible identication codes that can beformed: A, B, C, AB, BC, and AC. The 2letter combinationsBA, CA, and CB would not be identication codes becausethe letters are not in alphabetical order.
(A) If the researcher uses the 4 letters A, B, C, and D, how
many identication codes can be formed that consist ofa single letter? How many 2letter identication codescan be formed that begin with the letter A? How many
2letter identication codes can be formed that beginwith the letter B? How many 2letter identication codes
can be formed that begin with the letter C? How many2letter identication codes can be formed that begin
with the letter D? List all of the identication codes thatcan be formed using the letters A, B, C, and D.
(B) Recall the formula 1 2 31
21 n n n .
Explain how this formula can be applied to answer thefollowing question: If the researcher uses all 26 letters,
what is the maximum possible number of identicationcodes that can be formed?
(C) How many different letters did the researcher use if amaximum of 45 possible identication codes could have
been formed? Show your work.
Sample Response That Received a Score of 3
(A) 4 identication codes can be used with a single letter.
3 2letter codes beginning with A.
2 2letter codes beginning with B.
1 2letter codes beginning with C.
0 2letter codes beginning with D.
A, B, C, D, AB, AC, AD, BC, BD, CD
(B) 1 2 31
21 n n n
This formula can be applied to this problem because
to nd the number of possible labels when there arenletters, there are npossible single letter labels, n 1possible labels beginning with the rst letter, n 2possible ways beginning with the second letter, and so
on all the way down to one possible way for the secondto last letter and zero ways for the last letter. When
adding these numbers to nd the total possible labels,we get the equation n n n n 1 2 3
1 0 . This is the sum of all the numbers from 0 to
nwhich is what the expression 1
21n n gives us.
Therefore, there are1
226 27 351 codes that can
be made using all 26 letters.
(C)
1
2 11
2
451
2
n n
n n 1 90
n n
2 90
n n2 90 0
n n 10 9 0
n n
n n
10 0 9 0
10 9
9 letters were used.
Commentary on Sample Response That Earned a
Score of 3
This response received a score of 3 because it respondsappropriately to all parts of the question and demonstrates
strong knowledge of the concepts relevant to the question.Part (a) presents the correct number for each possibleidentication code and a complete and correct list of all the
possible codes that could be formed with the letters A, B, C,and D. The response demonstrates a systematic approach to
counting and identifying the possible codes. The response to
part (b) correctly shows that 351 codes can be formed usingall 26 letters and provides an appropriate explanation of howthe formula for nding the sum of the rst nintegers can beapplied to this question. Part (c) provides a correct algebraicsolution that shows that a maximum of 45 possibleidentication codes can be formed if 9 letters are used.

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Sample Response That Received a Score of 2
(A) Single letter 4 (A, B, C, D)
2Letter (Begin with A) 3 (AB, AC, AD)
2Letter (B) 2 (BC, BD)
2Letter (B) 0 A, B, C, D
AB, AC, AD1
24
BC, BD 2 4 1 10
CD
(B) The formula is for combinations and can be used to ndhow many different combinations are created using 26
letters.
1
226 26 1 13 27 351 . It is the same as 26!
(factorial)
(C) 451
21 n n
90 1 n n
902
n n
n n2 90 0
n n 10 9
n n 10 0 9 0
n n 10 9
cannot be negative
Commentary on Sample Response That Earned a
Score of 2
This response received a score of 2 because it demonstratesa sufcient knowledge of the concepts relevant to thequestion. Part (a) shows a systematic identication and
counting of the possible codes that can be formed with theletters A, B, C, and D. The response omits the number of
2letter codes that can be formed beginning with the letter Cbut correctly identies the one code that begins with the
letter C as part of the complete list of the 10 possible codesthat can be formed with these 4 letters. The calculations tothe right in the response to part (a) appear to be an
application of the formula given in part (b) to predict orconrm the total number of codes in part (a). These
calculations are considered irrelevant in evaluating theresponse to part (a). In part (b), the given formula is used
correctly to calculate the number of possible codes that
could be formed with 26 letters, but the explanation of how
this formula can be applied to this question is totallyincorrect. Part (c) provides a correct algebraic solution that
shows that a maximum of 45 possible identication codescan be formed if 9 letters are used.
Sample Response That Received a Score of 1
(A) Single letter A, B, C, D
2 Letter (A) AB, AC, AD
(B) BC, BD
(C) CD
(D) none
(B) Each letter will only be paired up once, so the formula
shows that for 26 letters, there would be
1
21n n
1
226 26 1 351
351 combinations
(C) 451
21 n n
451
2
1
22
n n
Commentary on Sample Response That Earned a
Score of 1
This response received a score of 1 because it demonstrates
a weak knowledge of the concepts relevant to the question.Although the response to part (a) does not explicitly answer
the questions about the numbers of each type of identicationcode, it does present a systematic and correct identication
of the 10 possible codes that can be formed using the lettersA, B, C, and D. The evaluation of the formula given in part (b)for 26 letters is correct, but the explanation of how this
formula can be applied to this question is totally incorrect.The response to part (c) is incomplete and does not provide
evidence of the ability to solve equations such as these.

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8782570428 PDF511
Sample Response That Received a Score of 0
(A) 4
3
1 1
0
A, B, C, D, AB, BC, CD, AC, AD
(B) 331 codes
(C) 45
Commentary on Sample Response That Earned a
Score of 0
This response received a score of 0 because it demonstrates
extremely limited understanding of the topic. The responseto part (a) shows a partially complete list of the possiblecodes that can be formed with the letters A, B, C, and D. Theresponse does not demonstrate an understanding of how to
identify and count the possible codes systematically,beginning with each of the letters. Part (b) provides an
incorrect numerical answer with no accompanying work orexplanation of how the formula given can be applied to this
question. The response to part (c) is either just recording theinformation given or is an incorrect response with no workshown. The evidence of correct thinking provided by this
response does not reach the threshold required to receive ascore above 0.

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