Test Information Guide

Field 47: Mathematics (Middle School)
Sample Multiple-Choice Questions

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The following reference material will be available to you during the test:

Formulas PDF opens in new window.

Calculators will NOT be provided for the Mathematics (Middle School) test.

Number Sense and Operations

Objective 0002
Understand principles and operations related to integers, fractions, decimals, percents, ratios, and proportions.

1. Use the sequence of steps below to answer the question that follows.

The sequence of steps above could be used to answer which of the following questions?

  1. How can a decimal be converted to a fraction reduced to lowest terms?
  2. Where should the decimal point be placed in the product of two decimals?
  3. Why is it necessary to invert and multiply when solving problems involving fractions?
  4. How is scientific notation used to multiply decimals?
Answer
Correct Response: B.

Correct Response: B.


Objective 0003
Understand and solve problems involving integers, fractions, decimals, percents, ratios, and proportions.

2. Which of the following situations represents 2and two fifths ÷ one fourth?

  1. A 2and two fifths-acre lot needs to be plowed. If four workers split the plowing evenly, how many acres will each person plow?
  2. One side of a one-fourth-square-foot rectangle is 2and two fifths feet. How long is the other side?
  3. Alix sawed off one fourth of a 2and two fifths-yard log. How many yards did Alix saw off?
  4. Terry picked 2and two fifths pounds of berries and put them into one-quarter-pound containers. How many containers of berries did Terry fill?
Answer
Correct Response: D.

Correct Response: D.


Patterns, Relations, and Algebra

Objective 0007
Understand properties of functions and relations.

3. Use the diagram below to answer the question that follows.

Which of the following graphs represents the function described in the above diagram?

  1. Graph with a rather narrow parabola that has its vertex at open parenthesis zero comma negative two close parenthesis and is opened upward.


  2. Graph with a rather narrow parabola that has its vertex at open parenthesis zero comma two close parenthesis and is opened downward.


  3. Graph with a parabola that has its vertex at open parenthesis two comma zero close parenthesis and is opened upward.


  4. Graph with a parabola that has its vertex at open parenthesis two comma zero close parenthesis and is opened downward.
Answer
Correct Response: B.

Correct Response: B.


Objective 0008
Understand properties and applications of linear relations and functions.

4. Use the graph below to answer the question that follows.

The graph shown above represents the relationship between the length of a geyser's eruption, ℓ, and the time until the next eruption, t. Which of the following linear equations best models the data?

  1. t + 12ℓ = 33
  2. 2t 3ℓ = 66
  3. 12ℓ t = 33
  4. 3ℓ + 2t = 66
Answer
Correct Response: C.

Correct Response: C.


Objective 0009
Understand properties and applications of quadratic relations and functions.

5. In solving the quadratic equation x2 + 14x 4 = 30 by completing the square, the first step is to add 4 to both sides of the equation. The second step is to:

  1. add 49 to both sides of the equation.
  2. factor x from the binomial x2 + 14x.
  3. factor the number 2 from 14, 4, and 30.
  4. take the square root of both sides of the equation.
Answer
Correct Response: A.

Correct Response: A.


Geometry and Measurement

Objective 0012
Understand the principles of Euclidean geometry and use them to prove theorems.

6. Use the geometric proof below to answer the question that follows.

Which of the following statements and reasons would be most appropriate in step 5 of this proof?

  1. Statement, Angle QPR is congruent to angle QPS.	Reason, Line PQ is perpendicular to line RS.

  2. Statement, P is the center of the circle. Reason, Definition of a circle.

  3. Statement, Angle PSQ is congruent to angle PRQ.	Reason, Properties of isosceles triangle.

  4. Statement, Line PQ is congruent to line PQ. Reason, Reflexive property of congruence.
Answer
Correct Response: D.

Correct Response: D.


Objective 0014
Solve problems involving three-dimensional shapes.

7. Use the diagram below to answer the question that follows.

Rectangular piece of cardboard that is fifteen units by twenty five units has squares cut from the corners that are five units on each side. The cardboard is then folded to make a box.

Square corners, 5 units on a side, are removed from a 15-unit-by-25-unit rectangular piece of cardboard. The sides of the cardboard are then folded to form an open box. Which of the following is the volume, in cubic units, of the box?

  1. 375
  2. 625
  3. 1000
  4. 1125
Answer
Correct Response: A.

Correct Response: A.


Data Analysis, Statistics, and Probability

Objective 0016
Understand descriptive statistics and the methods used in collecting, organizing, reporting, and analyzing data.

8. Which of the following situations best represents a random sampling?

  1. Ask every tenth person coming out of a health spa how many times a week they exercise to determine how often people in the town exercise.
  2. Survey students in advanced biology classes to determine the average amount of time students in a certain school study each week.
  3. Find the heights of all boys in a senior gym class to determine the average height of all boys in the school.
  4. Count the number of chocolate chips in every fifth cookie to determine the average number of chocolate chips per cookie in a bag of cookies.
Answer
Correct Response: D.

Correct Response: D.


Objective 0017
Understand the fundamental principles of probability.

9. Use the diagram below to answer the question that follows.

A point is randomly selected within the rectangle shown in the diagram above. What is the probability that the y-coordinate of the point is less than or equal to 2?

  1. one eighth
  2. two ninths
  3. one fourth
  4. one half
Answer
Correct Response: C.

Correct Response: C.


Trigonometry, Calculus, and Discrete Mathematics

Objective 0019
Understand the conceptual basis of calculus.

10. The graph of the function f(x) = 2x2 6x + 4 has its minimum value at the point (three halves, one half). Which of the following statements must be true?

  1. The derivative of f(x) is zero when x is equal to three halves.
  2. The value of f(x) approaches a limit of zero as x approaches three halves.
  3. The graph of f(x) shifts from a concave downward shape to a concave upward shape.
  4. The slope of the line tangent to f(x) at x = three halves is equal to one half.
Answer
Correct Response: A.

Correct Response: A.