Middle School Mathematics (0069) Test at a ... - · PDF fileThe Praxis Middle School Mathematics test is designed to certify examinees as teachers of middle school mathematics

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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, graphing calculator allowed; calculators with QWERTY keyboards not allowed

Weighting Multiple-choice: 67% of total score Short constructed-response: 33% of total score

IV

VI

II

III

Content CategoriesApproximate Number of Questions

Approximate Percentage of Examination

I. Arithmetic and Basic Algebra 12 20%

II. Geometry and Measurement 10 17%

III. Functions and Their Graphs 8 13%

IV. Data, Probability, and Statistical Concepts; Discrete Mathematics 10 17%

V. Problem-Solving Exercises

3(constructed

response)

33%

Process Categories

Mathematical Problem SolvingMathematical Reasoning and ProofMathematical Connections Distributed Across ContentMathematical Representation CategoriesUse of Technology

Pacing and Special Tips

In allocating time on this assessment, you should plan to spend about 80 minutes on the multiple-choice section and about 40 minutes on the constructed-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 bachelor’s program

with an emphasis in mathematics education, mathematics, or

education. 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 particular

school 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,

scientifi c, 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-driven devices (e.g., Palm, PDA’s, 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 your

calculator’s 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

examinee’s 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 numbers expressed in various forms; apply the order of operations; 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 is most appropriate in the context of a problem

• Order any fi nite set of real numbers and recognize equivalent forms of a number; classify a number as rational, irrational, real, or complex; estimate values of expressions involving decimals, exponents, and radicals; fi nd powers and roots

• Given newly defi ned operations on a number system, determine whether the closure, commutative, associative, or distributive properties hold

• Demonstrate an understanding of concepts associated with counting numbers (e.g., prime or composite, 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 the reasonableness of results

• Work with algebraic expressions, formulas, and equations; add, subtract, and multiply polynomials; divide polynomials; add, subtract, multiply, and divide algebraic fractions; perform standard algebraic operations involving complex numbers, radicals, and exponents, including fractional and negative exponents

• Determine the equations of lines, given suffi cient information; recognize and use the basic forms of the equation for a straight line

• Solve and graph linear equations and inequalities in one or two variables; solve and graph systems of linear equations and inequalities in two variables; solve and graph nonlinear algebraic equations; solve equations 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 are combinations of these fi gures; compute the surface area and volume of right prisms, cones, cylinders, spheres, and solids that are combinations of these fi 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 fi 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; describe relationships among sets of special quadrilaterals; solve problems involving angles, diagonals, and vertices of polygons with more than four sides

• Solve problems that involve using the properties of circles, including problems involving inscribed angles, central angles, radii, tangents, arcs, and sectors

• Solve problems involving refl ections, rotations, and translations of points, lines, or polygons in the plane

• Solve problems that can be represented on the xy-plane (e.g., fi nding the distance between two points or fi nding the coordinates of the midpoint of a line segment)

• Estimate absolute and relative error in the numerical answer to a problem by analyzing the effects of round-off and truncation errors introduced in the course of solving a problem

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• Demonstrate an intuitive understanding of a limit

• Demonstrate an intuitive understanding of maximum and minimum

• Estimate the area of a region in the xy-plane

III. Functions and Their Graphs

• Understand function notation for functions of one variable and be able to work with the algebraic defi nition of a function (e.g., for every x there is one y)

• Identify whether a graph in the plane is the graph of a function; given a set of conditions, decide if they determine a function

• Given a graph (for example, a line, a parabola, a step, absolute value, or simple exponential), select an equation that best represents the graph; given an equation, show an understanding of the relationship between the equation and its graph

• Determine the graphical properties and sketch a graph of 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 a model 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 two seemingly different situations. For example, given two different word problems, determine whether a particular equation can represent the relationship between the variables in the problems

• Find the domain (x-values) and range (y-values) of a function without necessarily knowing the defi nitions; 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 and interpret geometric representations of numeric and algebraic concepts

IV.a. Data, Probability, and Statistical Concepts

• Organize data into a presentation that is appropriate for 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, circle, double line, double bar, scatterplot, stem plot, line plot, box plot); draw conclusions from data

• Solve probability problems involving fi nite sample spaces by actually counting outcomes; solve probability problems by using counting techniques; solve probability problems involving independent and dependent events; solve problems by using geometric probability

• Solve problems involving average, including arithmetic mean and weighted average; fi 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; fi nd and interpret 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, if—then) as well as logical quantifi ers (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 Pascal’s triangle to solve problems

• Solve problems that involve simple sequences or number patterns (e.g., triangular numbers or other geometric numbers); fi nd rules for number patterns

• Use and interpret matrices as tools for displaying data

• Draw conclusions from information contained in simple diagrams, fl owcharts, paths, circuits, networks, or algorithms

• Explore patterns in order to make conjectures, predictions, or generalizations

5




V. Problem-Solving Exercises

Part B of the test contains three equally weighted constructed-response questions that together comprise 33 percent of the examinee’s score. The primary focus of the three constructed-response questions will be distributed across the four previously described content categories. Also, the examinee will be expected to integrate knowledge from different areas of mathematics.

Mathematical Process CategoriesIn addition to having knowledge of the mathematics content explicitly described in the Content Categories section, entry-level middle school mathematics teachers must also be able to think mathematically; moreover, they must have an understanding of the ways in which mathematical content knowledge is acquired and used. Answering questions on this assessment may involve one or more of the processes described in the process categories below, and all of the processes may be applied to any of the content topics.

Mathematical Problem Solving

• Solve problems that arise in mathematics and those involving mathematics in other contexts

• Build new mathematical knowledge through problem solving

• Apply and adapt a variety of appropriate strategies to solve 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 arguments and proofs

Mathematical Connections

• Recognize and use connections among mathematical ideas

• Apply mathematics in context outside of mathematics

• Demonstrate an understanding of how mathematical ideas interconnect and build on one another

Mathematical Representation

• Select, apply, and translate among mathematical representations 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 was 25 percent less than the original price. If is the cost of the car to the dealer and is the selling price, which of the following represents in terms of ?

(A)

(B)

(C)

(D)

5. Which fi gure below results if right triangle ABC above is fl ipped (refl ected) across the y-axis and then turned (rotated) clockwise about point C' by 90 degrees?

Sample Test QuestionsThe sample questions that follow illustrate the kinds of questions in the test. They are not, however, representative of the entire scope of the test in either content or diffi culty. 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 table above?

(A) As x decreases, y increases.

(B) As x increases, y does not change.

(C) As x increases, y decreases.

(D) As x increases, y increases.

2. The average number of passengers who use a certain airport each year is 350 thousand. A newspaper reported the 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 number of people at the show?

(A) 102

(B) 80

(C) 36

(D) 30

y

x

C

B

A

7




8. A square is inscribed in each of the circles above. The radius of circle A is 1, and the radius of circle B is 2. What is the ratio of the area of the square inscribed in circle A to the area of the square inscribed in circle B?

(A)

(B)

(C)

(D)

9. Which of the following defi nes y as a function of x ?

(A)

(B)

(C)

(D)

6. The large rectangular block pictured above was made by stacking smaller blocks, all of which are the same size. What are the dimensions in centimeters of each of the smaller blocks?

(A)

(B)

(C)

(D)

7. In the fi gure above, line and line are parallel and What is the value of x ?

(A) 30

(B) 45

(C) 60

(D) 75

8




10. A taxi ride costs $2.50 for the fi 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 dog and 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 county’s trash. If approximately 60 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%

9




Questions 13–14 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 were in 1950?

(A) None

(B) One

(C) Two

(D) Three

14. The number of students in medical schools increased by approximately 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 certain woodland, a biologist captured and marked 84 snails that were then released back into the woodland. Fifteen days 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 were captured, what should the biologist estimate the snail population to be, based on the probabilities suggested by this experiment?

(A) 630

(B) 1,010

(C) 1,040

(D) 1,080

16. If a student takes a test consisting of 20 true-false questions and randomly guesses at all of the answers, what is the probability that all 20 guesses will be correct?

(A) 0

(B)

(C)

(D)

ROBIN’S TEST SCORES

88, 86, 98, 92, 90, 86

17. In an ordered set of numbers, the median is the middle number if there is a middle number; otherwise, the median is the average of the two middle numbers. If Robin 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. As x moves from – 4 to 0 (that is, from left to right on the number line), its value increases.

Similarly, the value of y increases from –2 to 0. Thus, it can be seen that as x increases, y increases. 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. 350,000,000 � 350,000 = 1,000, so the correct answer is (C).

3. If a represents the number of adults, then 5a represents the number of children and 6a represents the total number of people at the show. Since 6a represents 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 triangle ABC is refl ected across the y-axis, the fi gure formed is located in quadrant I and is the mirror image of the given fi 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 = 4 centimeters long. Similarly, the width of a small block is 8 � 2 = 4 centimeters and the height of a small block is 9 � 3 = 3 centimeters. 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 measured x degrees and angle measured y degrees are supplementary angles. Recall that the sum of the measures of supplementary angles is 180°. That is, It is given that Substituting for y, 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 fi 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 fi gure. If you like to compute, here is what you might do. First consider circle A. The radius of circle A is 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 square A is . So the area of square A is

Likewise, the area of square B is Thus the ratio of the area of square A to the area of square B is 2 : 8, which is 1 : 4. The correct answer is (D).

Alternatively, you may recall that when you are comparing two similar fi gures whose corresponding linear dimensions have a ratio of 1 to 2, as in this problem, the ratio of the areas of the fi gures is

the ratio of the square of the linear dimensions; that is, to , which is 1 to 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 should recall that if y is a function of x, then each value of x (in the domain of the function) results in only one value of y. In choices (A) and (B), most values of x have two different corresponding values of y. You can see this by solving the equations in (A) and (B) for y. In (A),

or Similarly, in (B), So neither (A) nor (B) defi nes y as a function of x. In choice (D), for each value of x, there is more than one value of y that satisfi esthe inequality. So (D) does not defi ne y as a function of x. However, in (C), for each value of x, there is only one value of y that corresponds to that value of x. Thus, the correct answer choiceis (C).

11




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 fi rst step and the correct interval between

steps. Since the cost for the fi rst mile or less is $2.50, the cost

for the fi rst step (the value on the vertical axis) should be 2.5 over

the horizontal interval from 0 to mile, with a solid dot at mile.

(There should be no cost at a distance of 0 miles, since there is no charge 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 content in percents; the question asks for the weight of 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 the trash consist of paper; the graph shows that this amount equals 40% of the total, so

60 = 0.4 � (total weight)

and the total weight is 600 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

tons of plasticstons of paper

= =840

15

%%

ortons of plastics

6015

=

tons of plastics= 605

= 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 fi 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 fi 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).

12




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 fi nd the percent increase, divide this number by the base number; that is, the number of students before the increase, or 40,000.

30 00040 000

34

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, about 1290

, or 215

, of the population had been marked.

Thus, the original 84 snails marked represented approximately

215

of the entire population and the biologist should estimate the

snail population to be 84 152

, 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, the guesses are independent events. Thus, the probability of guessing

all 20 answers correctly is 12

20

, and the correct answer is (B).

17. The problem gives a set of test scores and the defi nition of the median. The fi rst part of the defi nition 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 two middle numbers in the set, 88 and 90, and the average of the two middle numbers is

88 902

1782

89+ = =

Thus, the median of Robin’s scores is 89 and the correct answer is (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 questions and sample responses along with the standards used in scoring the responses. When you read these sample responses, keep in mind that they will be less polished than if they had been developed at home, edited, and carefully presented. Examinees do not know what questions will be asked and must decide, on the spot, how to respond.

Readers take these circumstances into account when scoring the responses. Readers will assign scores based on the following scoring 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 signifi cant aspects of any stimulus material presented

2

• Responds appropriately to most parts of the question• Where required, provides an explanation that is suffi ciently

supported by relevant evidence• Demonstrates a suffi cient knowledge of subject matter,


• Demonstrates a basic understanding of the most signifi cant aspects of any stimulus material presented

1

• Responds appropriately to some parts of the question• Where required, provides a weak explanation that is not well

supported by relevant evidence• Demonstrates a weak knowledge of subject matter,


• Demonstrates little understanding of signifi cant 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

14




Sample Response That Received a Score of 3

(A) ,� �

y x x

y x

� � �

� � �

0 2,000 10 000

120

10 000 2,000

( , )if

see note at end of response

�

� (if x > 10,000)

(B)

0 300

900

1,800

2,400

3,000

600

1,500 1,200

2,100

2,700

3,600 3,900 4,200 4,500

3,300

0 E

mily

's I

ncom

e(E

ach

Uni

t = $

300)

Emily’s 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 Emily’s income would be $1,000. If she made $8,000 in monthly sales, her income would be $2,000 since her sales did not exceed $10,000. If she

made $30,000 in monthly sales, her income would be

y = − + =120 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 for calculation though not as concrete) would be

y x� �120

1,500 .

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 was correctly 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

Emily’s 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. (For example, if Emily’s monthly sales are $15,000, she receives a

commission of 5 percent of $5,000.)

(A) Write an equation that gives Emily’s monthly income, y, in terms of her monthly sales, x, if her monthly sales are less than or equal to $10,000 x 10 000, . Write a second equation that gives Emily’s monthly income, y, in terms of her monthly sales, x, if her monthly sales are greater than $10,000 x >10,000 .

(B) Using the equations you wrote in part (a) of this question, draw a graph of Emily’s monthly income, y, as a function of her monthly sales, x, for monthly sales of $0 to $60,000 0 60 000x , . Graph her monthly sales on the x-axis and her monthly income on the y-axis. Label each axis and show the units and scales used.

(C) Use the graph you drew in part (b) of this question to estimate the increase in Emily’s 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

xeer 10,000

y x

y x

� ��

2 000 10 000

2 000 05 10 000

, ,

, . ,

(b)

8 7 6 5 4 3 2 1 0

10 0 20 30

Monthly Sales$ Thousands

MonthlyIncome

$Thousands

EMILY’S MONTHLY INCOME

40 50 60 x

y

(c) Increase if sales were to go from $8,000 to $30,000

At $8,000 sales she gets no extra since 8 000 10 000, , . So only gets $2,000 base salary.

At $30,000 she gets basic $2,000 plus .05 times 20,000 since that’s the amount over $10,000. So it goes from $2,000 on the graph to $3,500 at 30,000 monthly sales.


Score of 2

This response received a score of 2 because it responds appropriately to most parts of the question and demonstrates a suffi cient knowledge of the concepts relevant to the question. In particular, both of the equations requested in part (a) are correct. The graph in part (b) is partially correct. The values plotted for x > 10,000 refl ect calculating Emily’s income without subtracting $10,000 when evaluating and graphing the second equation. It should not have the discontinuity shown at monthly sales x = $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 the examinee missed an opportunity to identify the mistake in part (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

10,000020,000

30,000

Income

40,00050,000

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 of being 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, the discrete values that are graphed in part (b) are correct. In part (c), the general description of the increase in income is correct for values of sales (x) greater than $30,000; however, the estimation of the income associated with sales of $8,000 as different from $2,000 is further evidence of misunderstanding of this piecewise linear function. There was also an arithmetic error in fi nding the difference between $3,000 and $1,800. The response received some credit because portions of parts (b) and (c) are correct.

16





(A) y x. ( ,05 10 000

y x2 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 10–1–2–3–4–5 20 30

1

–1–2

–3

–4

–5

2

3

4

5

x

y

(C) 8,000


Score of 0

This response received a score of 0 because it demonstrates extremely limited understanding of the topic. The equations in part (a) are not correct. The graph in part (b) does not receive any credit because it is inaccurately drawn and does not refl ect the piecewise linear characteristics of the situation. Most of the discrete values were graphed incorrectly, even though, based on the table, they appear to have been computed correctly. Part (c) is either incomplete or totally incorrect. The evidence of correct thinking provided by this response does not reach the threshold required to receive a score above 0.

Sample Question 2

In triangle EFG, side EF has length 8 and side FG has length 10.

(A) Two of the possible lengths of side EG are 3 and 16. Draw triangle EFG with side EG of length 3. Draw a second triangle EFG with side EG of length 16. For each of the triangles, label all of the vertices and show the lengths of all of the sides. [Note: Your drawings are not expected to be exact but should reasonably represent the relative lengths of the sides.]

(B) The length of side EG could not be 1 or 20. Explain why not. Draw fi gures to support your explanation.

(C) If triangle EFG is a right triangle, what are the two possible lengths of side EG? Draw the two right triangles. For each triangle, label all of the vertices and show the lengths of all of the sides. Indicate the right angle.

17





(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 triangle EFG, if the length of side EG was 1, the sum of the lengths of sides EG and EF would be 9. This value is less than the length of side FG, which is 10. Also if the length of side EG was 20, the sum of the lengths of sides EF and FG would 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 EFG were a right , the length of side EG could be either 6 or approximately 12.81.

8

10

F E

G

12.81

6

8

E G

F

10


Score of 3

This response received a score of 3 because it responds appropriately to all parts of the question and demonstrates strong 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 F is acute when side EG has length 3 and obtuse when side EG has length 16. Part (b) both describes and illustrates the triangle inequality property correctly. Part (c) correctly shows the two possible right triangles with sides of 8 and 10. (The one minor error in the response is that the right angle is not labeled in the second drawing. However, angle E appears to be a right angle and is consistent with the opposite side being the hypotenuse; thus, the error is not considered signifi cant enough to warrant a lower score.)

8

16

10

E G

F

18





(A)

F

E

G

E G

F

(B) From Pythagorean theorem a b c2 2 2 .

FG is hypotenuse C

EF EG = FG2 2 2

2 2 28 10b =

++

b could 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 demonstrates a suffi cient 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 E is acute when side FG has length 3 and obtuse when side EG has length 16. Part (b) receives no credit because it assumes incorrectly that triangle EFG must be a right triangle. Part (c) correctly shows the two possible right triangles 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

19




(B) There are no ways to construct triangles with measured lengths of the 3 sides to be 1, 8, 10 due to the fact that the 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

1 8

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 the EG can be 6 according to Pyth. theorem.

FE

G

6

8

10

FE

G

10

8

√164 or

2√41


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), both triangles are drawn (incorrectly) as right triangles. The relative lengths of the sides are acceptable in the 3-8-10 triangle, but not in the 8-10-16 triangle. In the 8-10-16 triangle, the side of length 10 is shown longer than the side of length 16 and the largest angle (which should be obtuse) is not opposite the longest side. The response to part (b) is incorrect. Credit is given for the correct possible right triangles shown in part (c).

20





(A)

8 10

E

F

G3

8 10

E

F

G16

(B)

8 10

E

F

G1

8

10

E

F

G20

For EG to have a length of 1 or 20 would not be possible because 1 would create a triangle too small to the relative dimensions of the other lengths while 20 would create a triangle which has one length too big.

(C) F

E G? = 6EG = 6

8

10

F

E

G10

8

12


Score of 0

This response received a score of 0 because it demonstrates extremely limited understanding of the topic. Both triangles in part (a) are drawn as right triangles. Neither of the triangles drawn shows reasonable relative lengths of the sides. In the 3-8-10 triangle, the side of length 8 is longer than the side of length 10. In the 8-10-16 triangle, the side of length 8 is shown as the longest side and is opposite the greatest angle. None of the angles in the second triangle is shown as obtuse. No credit is given for part (b) since the two fi gures are 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 not suffi ciently explain the triangle inequality property. Part (c) contains one of the two possible confi gurations of right triangle EFG. Although a portion of part (c) is correct, the evidence of knowledge about triangle geometry provided by this response does not reach the threshold required to receive a score above 0.

21




Sample Question 3

In a certain experiment, a researcher plans to label each sample with an identifi cation code consisting of either a single 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 identifi cation codes that can be formed: A, B, C, AB, BC, and AC. The 2-letter combinations BA, CA, and CB would not be identifi cation codes because the letters are not in alphabetical order.

(A) If the researcher uses the 4 letters A, B, C, and D, how many identifi cation codes can be formed that consist of a single letter? How many 2-letter identifi cation codes can be formed that begin with the letter A? How many 2-letter identifi cation codes can be formed that begin with the letter B? How many 2-letter identifi cation codes can be formed that begin with the letter C? How many 2-letter identifi cation codes can be formed that begin with the letter D? List all of the identifi cation codes that can be formed using the letters A, B, C, and D.

(B) Recall the formula 1 2 3 12

1n n n .

Explain how this formula can be applied to answer the following question: If the researcher uses all 26 letters, what is the maximum possible number of identifi cation codes that can be formed?

(C) How many different letters did the researcher use if a maximum of 45 possible identifi cation codes could have been formed? Show your work.


(A) 4 identifi cation 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 3 12

1n n n

This formula can be applied to this problem because to fi nd the number of possible labels when there are n-letters, there are n-possible single letter labels, n – 1 possible labels beginning with the fi rst letter, n – 2 possible ways beginning with the second letter, and so on all the way down to one possible way for the second to last letter and zero ways for the last letter. When adding these numbers to fi nd the total possible labels, we get the equation n n n n1 2 3

1 0. This is the sum of all the numbers from 0 to

n which is what the expression 12

1n n gives us.

Therefore, there are 12

26 27 351 codes that can

be made using all 26 letters.

(C) 12

1

12

4512

n n

n n 1 90

n n2 90

n n2 90 0

n n10 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 responds appropriately 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 possible identifi cation 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 using all 26 letters and provides an appropriate explanation of how the formula for fi nding the sum of the fi rst n integers can be applied to this question. Part (c) provides a correct algebraic solution that shows that a maximum of 45 possible identifi cation codes can be formed if 9 letters are used.

22





(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, AD 1

24

BC, BD 2 4 1 10 CD

(B) The formula is for combinations and can be used to fi nd how many different combinations are created using 26 letters.

12

26 26 1 13 27 351. It is the same as 26!

(factorial)

(C) 45 12

1n n

90 1n n

90 2n n

n n2 90 0

n n10 9

n n10 0 9 0

n n10 9

cannot be negative


Score of 2

This response received a score of 2 because it demonstrates a suffi cient knowledge of the concepts relevant to the question. Part (a) shows a systematic identifi cation and counting of the possible codes that can be formed with the letters A, B, C, and D. The response omits the number of 2-letter codes that can be formed beginning with the letter C but correctly identifi es the one code that begins with the letter C as part of the complete list of the 10 possible codes that can be formed with these 4 letters. The calculations to the right in the response to part (a) appear to be an application of the formula given in part (b) to predict or confi rm the total number of codes in part (a). These calculations are considered irrelevant in evaluating the response 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 totally incorrect. Part (c) provides a correct algebraic solution that shows that a maximum of 45 possible identifi cation codes can 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

12

1n n

12

26 26 1 351

351 combinations

(C) 45 12

1n n

45 1

212

2n n


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 identifi cation code, it does present a systematic and correct identifi cation of the 10 possible codes that can be formed using the letters A, 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.

23




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

This response received a score of 0 because it demonstrates extremely limited understanding of the topic. The response to part (a) shows a partially complete list of the possible codes that can be formed with the letters A, B, C, and D. The response 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 or explanation of how the formula given can be applied to this question. The response to part (c) is either just recording the information given or is an incorrect response with no work shown. The evidence of correct thinking provided by this response does not reach the threshold required to receive a score above 0.

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Middle School Mathematics (0069) Test at a ... - · PDF fileThe Praxis Middle School Mathematics test is designed to certify examinees as teachers of middle school mathematics