Study Guide

Field 225: Middle Level/Intermediate Mathematics 
Sample Selected-Response Questions

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General Test Directions

This test contains two sections: (1) a section with selected-response questions and (2) a constructed-response section. The directions for the constructed-response assignment appear before that section.

Each question in the first section is a selected-response question with four answer choices. Read each question and answer choice carefully and choose the  start uppercase ONE end uppercase  best answer. Try to answer all questions. In general, if you have some knowledge about a question, it is better to try to answer it. You will  start uppercase NOT end uppercase  be penalized for guessing.

A calculator is available to you for this test. To access the calculator, click on the calculator icon located in the upper left corner of the screen. A pop-up window containing the calculator will appear. You can reposition the calculator by placing your cursor in the blue area above the calculator and dragging the window to the location of your choice.

Use the numbers on the keyboard and/or point and click with the mouse to enter your computations into the on-screen calculator. When you are finished, close the calculator by clicking the close calculator icon button in the upper right corner of the calculator.

Reference materials will also be available to you during this test. To access these reference materials, click on the reference materials icon located in the lower left corner of the screen.

You may work on and complete the selected-response questions and the constructed-response assignment in any order that you choose. Be sure to allocate your time carefully so that you are able to complete the entire test within the testing session.

Use of any other type of calculator or outside reference materials during the testing session is prohibited.

Sample Selected-Response Questions

Competency 0001 
Apply knowledge of the structure and property of the real number system.

1. Which of the following statements is true of the real number system?

  1. All integers are whole numbers
  2. Any real number must be either an integer or a rational number.
  3. Any integer is a rational number.
  4. All negative numbers are either integers or irrational numbers.

Correct Response: C. An integer is either a positive whole number, the opposite of a positive whole number, or 0. Any of these numbers may be converted into a rational number by writing the integer n as n divided by 1.

Competency 0002 
Apply knowledge of number operations and number theory.

2. A board that measures 33 and 3 eighths inches long is cut into as many 3 and 5 eighths-inch-long pieces as possible. If the blade removes 1 eighth of an inch of the original board with each cut, what is the length of the leftover scrap of wood?

  1. 3 and 1 fourth inches
  2. 3 and 5 sixteenths inches
  3. 3 and 3 eighths inches
  4. 3 and 1 half inches

Correct Response: C. The blade removes 1 eighth of an inch with each cut, so each piece of wood that is removed from the original board reduces its length by 3 and 5 eighths plus 1 eighth equals 3 and 6 eighths inches. Because 33 and 3 eighths divided by 3 and 6 eighths equals 267 eighths divided by 30 eighths equals 267 thirtieths equals 8 and 9 tenths, 8 whole cuts and a cut of 9 tenths of 3 and 6 eighths inches can be made from the board. This leaves a length of 9 tenths times the mixed number 3 and 6 eighths equals 9 tenths times 30 eighths equals 27 eighths equals 3 and 3 eighths remaining.

Competency 0003 
Apply the principles and properties of algebraic relations and functions.

3. The function f of x equals 572 minus 65 x relates the number of miles remaining in a road trip to the number of hours driven, x. What is the domain of this function in this context?

  1. 0 is less than or equal to x is less than or equal to 8.8
  2. 0 is less than or equal to x is less than or equal to 9
  3. 0 is less than or equal to x is less than or equal to 572
  4. 0 is less than or equal to x is less than infinity

Correct Response: A. The function f of x represents the number of miles remaining in the road trip. The end of the road trip occurs when f of x equals 0, and this takes place at x = 572 divided by 65 = 8.8 hours. The domain for the function in this context is 0 is less than or equal to x is less than or equal to 8.8.

Competency 0004 
Apply algebraic techniques.

4. A moving company charges 80 dollars for the first hour, and then 60 dollars for each hour after that. Which of the following expressions gives the total cost of the movers after x hours?

  1.  60 times the binomial x minus 1 plus 80 times x 
  2.  60 times x plus 80 times the binomial x minus 1 
  3.  80 plus 60 times the binomial x minus 1 
  4.  60 times x plus 80 

Correct Response: C. The movers are used for x hours. The first hour costs 80 dollars and each of the remaining x minus 1 hours costs 60 dollars. The total cost is 80 plus 60 times the binomial x minus 1.

Competency 0006 
Apply the properties of quadratic and higher-order polynomial functions and relations.

5. Consider the parabolas defined by f of x equals negative 2 x squared minus 4 x plus 3 and g of x equals x squared minus 6 x plus 12. Which of the following values represents the vertical distance between the y-coordinates for the vertices of the parabolas?

  1. 0
  2. 2
  3. 6
  4. 34

Correct Response: B. A parabola with an equation written in the standard form of a x squared plus b x plus c has a vertex with an x-coordinate located at minus b over 2 a. For the parabola described by the function f of x, this occurs at x equals the negative of the rationale expression with a numerator of negative 4 and a denominator of 2 times negative 2. This simplifies as negative 1. The corresponding y-coordinate can then be found by substitution: F of negative 1 equals negative 2 times open paren negative 1 close paren squared minus 4 times negative 1 plus 3 equals 5. The vertex for the function g of x is located at 3 comma 3 by the same reasoning. The vertical distance between these vertices is 5 minus 3 equals 2.

Competency 0007 
Analyze advanced functions and the conceptual foundations of calculus.

6. Which of the following expressions is equivalent to log of 20 x plus log of 10 y minus 1?

  1. log of x y
  2. log of the binomial 2 x plus y
  3. log of open paren 2 times x times y close paren
  4. log of open paren 20 times x times y minus 1 close paren

Correct Response: C. The expression can be simplified by applying the rules log of 10 equals 1 and log of the product a and b equals log a plus log b as follows:

log of 20 x plus log of 10 y minus 1
equals the sum of log 2 plus log 10 plus log x plus the sum of log 10 plus log y minus 1
equals the sum of log 2 plus 1 plus log x plus the sum of 1 plus log y minus 1
equals log of open paren 2 times x times y close paren plus 1

Competency 0008 
Apply principles and procedures related to measurement.

7.  start bold Use the diagram below to answer the question that follows. end bold 

A star-shaped polygon is shown with 5 points

A star-shaped polygon is shown with 5 points. The top point has a vertical orientation, and each point extends from a regular pentagon within the interior. The edges of the interior pentagon and the height of each of the triangles have a length of s.

A star shape is composed of a regular pentagon and five isosceles triangles, as shown in the diagram. Which of the following expressions represents the length of the perimeter of the star?

  1.  5 times the square root of 5-fourths times s 
  2.  5 times the square root of 5 times s 
  3.  510 times the square root of 2 times s 
  4.  10 times the square root of 3 times s 

Correct Response: B. The edges shown within the pentagon form the bases of 5 isosceles triangles. Each of the triangles can be decomposed into two adjoining right triangles with bases of 1 half s and height s. By the Pythagorean theorem, the length of each hypotenuse is the square root of the expression s squared plus open paren one-half s close paren squared equals the square root of the expression s squared plus one-fourth times s squared equals the square root of 5-fourths times s. The boundary of the star is composed of 10 of these hypotenuses, so its perimeter is 10 times the square root of 5-fourths times s equals 10-halves times the square root of 5 times s equals 5 times the square root of 5 times s.

Competency 0009 
Apply the principles and properties of Euclidean geometry in two and three dimensions.

8.  start bold Use the figure below to answer the question that follows. end bold 

A regular 12-sided polygon is shown

A regular 12-sided polygon is shown. The polygon is divided by a chord such that 4 edges of the polygon appear on the upper-left side of the chord and 8 edges of the polygon appear on the other side of the chord. The angle formed by the chord and an edge that intersects it from the upper-left side is labeled x.

The figure shown is a regular dodecagon. What is the measure of angle x?

  1. 30 degrees
  2. 36 degrees
  3. 45 degrees
  4. 55 degrees

Correct Response: C. Each of the interior angles of a regular dodecagon is 180 degrees times the fraction with numerator 12 minus 2 and denominator of 12 equals 150 degrees. The chord in the interior of the dodecagon splits it into two smaller polygons. The smaller polygon that contains the angle labeled x in the top left area is an irregular pentagon. The sum of its interior angles is 180 degrees times open paren 5 minus 2 close paren = 540 degrees, of which 450 degrees comes from the three 150 degrees angles opposite the chord. The remaining two angles sum to 90 degrees, and they must be congruent to each other because of the symmetry of the shape. It follows that x is 45 degrees.

Competency 0011 
Apply the principles, properties, and techniques of probability.

9. A student programs a game on their graphing calculator that asks the player to choose five unique numbers between 1 and 50, inclusive. Then, a random number generator creates a list of five random, unique numbers between 1 and 50. If exactly two of a player's numbers match the numbers chosen by the random number generator, they win. The order of the numbers does not affect the outcome. Which of the following expressions accurately represents the probability of winning a single turn of this game?

  1. A rational expression is shown with combinations of the form C open paren n comma k close paren, which is described here as n choose k. Its numerator is 5 choose 2 times 45 choose 3 and its denominator is 50 choose 5
  2. A rational expression is shown with a numerator of 45 times 5 times 4 and a denominator of 50 times 49 times 48
  3. A rational expression is shown with a numerator of 5 choose 2 times 45 choose 3 and a denominator of 50 times 49 times 48 times 47 times 46.
  4. A rational expression is shown with a numerator of 5 choose 2 a denominator of 50 choose 5.

Correct Response: A. The combination C open paren n comma k close paren represents the number of unique ways in which k objects can be chosen from a larger group of n objects without replacement and without regard to the order in which they are selected. Three such combinations occur in a winning round of the game: when the student draws 5 numbers from 50 in C open paren 50 comma 5 close paren ways, when the random number generator matches 2 of the student's 5 numbers in C open paren 5 comma 2 close paren ways, and when the random number generator selects 3 of the 45 non-matching numbers in C open paren 45 comma 3 close paren ways. The product of the last two combinations, C open paren 5 comma 2 close paren times C open paren 45 comma 3 close paren, describes the total number of ways that both of the corresponding events can occur (i.e., the number of ways the player can win a round of the game), and the fraction a rational expression is shown with combinations of the form C open paren n comma k close paren, which is described here as n choose k. Its numerator is 5 choose 2 times 45 choose 3 and its denominator is 50 choose 5 describes the probability of that outcome happening.

Competency 0012 
Apply the principles, properties, and techniques of statistics.

10. Students in a middle school are randomly selected and surveyed about how they get to school each morning. 54 students report that they ride a bus, and 22 students report that they are dropped off by parents/guardians. The remaining 4 students walk to school. If the middle school has 238 students, which of the following is the best estimate of the total number of students who ride the bus to school each morning?

  1. 153
  2. 161
  3. 169
  4. 184

Correct Response: B. Given that the students were randomly selected for the survey, it is reasonable to assume that the fraction of students who ride the bus to school in the morning sample, A rational expression with a numerator of 54 and a denominator of 54 plus 22 plus 4 equals the fraction 54 over 80, is equivalent to that of the students in the school's population: 54 over 80 equals x over 238. It follows that 80 x equals 12852 simplifies to x equals 160 point 65, which rounds to 161 students.