Mid Lle School Math

8/10/2019 Mid Lle School Math

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Middle School Mathematics (0069)

Test at a Glance

Test Name Middle School Mathematics

Test Code 0069

Time 2 hours

Number of Questions 40 multiple-choice (Part A)

3 short constructed response (Part B)

Format Multiple-choice questions and constructed-response questions, graphingcalculator allowed; calculators with QWERTY keyboards not allowed

Weighting Multiple-choice: 67% of total scoreShort constructed-response: 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. Problem-Solving 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 multiple-choice section and about 40 minutes on theconstructed-response section; the sections are not independently timed.


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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 real-world 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 four-function,

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 pen-input/stylus-drivendevices (e.g., Palm, PDAs, Casio Class Pad 300)

Devices with QWERTY keyboards (e.g., TI-92 PLUS,Voyage 200)

Cell-phone 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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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 real-world

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 two-dimensional and three-dimensional 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 thexy-plane (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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Demonstrate an intuitive understanding of a limit

Demonstrate an intuitive understanding of maximumand minimum

Estimate the area of a region in thexy-plane

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, absolute-value, 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 (x-values) and range (y-values) 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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V. Problem-Solving Exercises

Part B of the test contains three equally weightedconstructed-response questions that together

comprise 33 percent of the examinees score. Theprimary focus of the three constructed-response

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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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 they-axis 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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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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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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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 true-falsequestions 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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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 they-axis, 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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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 left-hand 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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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 constructed-response 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


Demonstrates a sufcient knowledge of subject matter,

concepts, theories, facts, procedures, or methodologies


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


Demonstrates a weak knowledge of subject matter,concepts, theories, facts, procedures, or methodologies


Demonstrates little understanding of signicant aspects of

any stimulus material presented

0

Blank, off-topic, 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 thex-axis and her monthly income on they-axis.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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(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.


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.


(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.


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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(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


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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(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


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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(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

.


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.


(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


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 3-8-10 triangle, but not in the 8-10-16 triangle. In the8-10-16 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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(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



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

3-8-10 triangle, the side of length 8 is longer than the side oflength 10. In the 8-10-16 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 2-letter 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 2-letter identication codescan be formed that begin with the letter A? How many

2-letter identication codes can be formed that beginwith the letter B? How many 2-letter identication codes

can be formed that begin with the letter C? How many2-letter 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.


(A) 4 identication codes can be used with a single letter.

3 2-letter codes beginning with A.

2 2-letter codes beginning with B.

1 2-letter codes beginning with C.

0 2-letter 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 aren-letters, there are n-possible 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.


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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(A) Single letter 4 (A, B, C, D)

2-Letter (Begin with A) 3 (AB, AC, AD)

2-Letter (B) 2 (BC, BD)

2-Letter (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


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

2-letter 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.


(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


Score of 1


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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87825-70428 PDF511


(A) 4

3

1 1

0

A, B, C, D, AB, BC, CD, AC, AD

(B) 331 codes

(C) 45


Score of 0


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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Mid Lle School Math

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