Study Guide

Field 035: Mathematics
Sample Multiple-Choice Questions

Expand All | Collapse All

Objective 0001
Number and Quantity (Standard 1)

1. A company manufactures beads, some of which are cubes whose edge length needs to be 5 mm. If there is an error of 2% in edge length, what will be the approximate percent error in the volume of a cube-shaped bead?

  1. 2%
  2. 4%
  3. 6%
  4. 8%
Answer
Correct Response: C. This question requires the examinee to demonstrate knowledge of attending to accuracy and precision with different representations of numbers. The length of a bead edge may deviate by as much as 2% from 5 mm. This means that a bead edge may be as long as 1.02 x 5 = 5.1 mm, and the bead's volume could be as great as (5.1)3 = 132.651 cu. mm. A bead with an edge length of 53 mm will have a volume of 125 cu. mm. The error is calculated as , or 6%.

Objective 0001
Number and Quantity (Standard 1)

2. Use the problem below to answer the question that follows.

Finding the solution to this problem is based on an application of:

  1. congruence classes.
  2. greatest common factors.
  3. linear combinations.
  4. least common multiples.
Answer
Correct Response: D. This question requires the examinee to demonstrate knowledge of the application and use of factors. The least common multiple of 18, 30, and 42 is the smallest number of which all three given numbers are factors. The simplest way to find the LCM in this case is to find the prime factorizations of 18, 30, and 42, i.e., 2 • 32, 2 • 3 • 5, and 2 • 3 • 7. The prime factorization of the least common multiple is the product of each prime factor, the maximum number of times it occurs in any one of the three numbers. In this case, 2 • 32 • 5 • 7 = 630.

Objective 0002
Algebra (Standard 2)

Use the system of inequalities below to answer the question that follows.

2xy ≤ 4
x + y ≤ 5
x – 2y ≤ 8

3. Which of the following graphs represents the solution of this system of inequalities?







Answer
Correct Response: D. This question requires the examinee to demonstrate knowledge of creating and solving equations and inequalities. Convert each of the three linear inequalities to the slope intercept form:
y ≥ 2x – 4, y ≤ 5 – x, and . Use the x- and y- intercepts to identify the line associated with each inequality and interpret the y-value in each case as being above or below the line, as appropriate. The correct response can also be checked by substituting the values of various coordinate pairs from the shaded portion of response D and confirming that each pair satisfies all three of the original inequalities.

Objective 0002
Algebra (Standard 2)

4. Given a set R together with two binary operations, addition and multiplication, which of the following is a necessary but not sufficient condition for R to be a ring?

  1. R contains a multiplicative identity.
  2. Multiplication is associative.
  3. Each element in R has a multiplicative inverse.
  4. Multiplication is commutative.
Answer
Correct Response: B. This question requires the examinee to demonstrate knowledge of the concepts of abstract algebra involving both real and complex numbers. A ring is an algebraic structure consisting of a set of elements with two binary operations, addition and multiplication. By definition a ring must be associative (but not necessarily commutative) under multiplication, and it need not have a multiplicative identity or multiplicative inverses.

Objective 0003
Functions (Standard 3)

5. A child kicks a ball that is resting on the ground and it follows a trajectory modeled by the function y = –0.2x2 + 3x – 3, where x is the horizontal distance the ball travels in feet. Approximately how far from the child does the ball hit the ground?

  1. 12.85 ft.
  2. 13.92 ft.
  3. 15.00 ft.
  4. 16.39 ft.
Answer
Correct Response: A. This question requires the examinee to demonstrate knowledge of modeling problems with nonlinear functions and their representations. This problem can be solved by using the quadratic formula to find the roots of the equation y = -0.2x2 + 3x – 3, where a = –0.2, b = 3, and c = –3. Thus Note that in this case the horizontal distance traveled by the ball equals the distance between the two roots along the x-axis, i.e., 13.925 – 1.075 or approximately 12.85 feet.

Objective 0003
Functions (Standard 3)

6. A student is using the function y = 3 cos(4x + Π) – 2 to model a set of periodic data. What is the period of the data?





Answer
Correct Response: B. This question requires the examinee to demonstrate knowledge of modeling periodic phenomena with trigonometric functions. The period, p, of a periodic function is the change along the x-axis corresponding to one complete cycle of the function. If the general sinusoidal equation is written as y = A cos B(xC) + D, then the period is calculated as In this case, the particular equation is Thus,

Objective 0004
Calculus (Standard 6)

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


A and B are positive numbers that represent the areas of the semicircle and the triangle that make the graph shown of function f(x). Which of the following expressions can be used to evaluate ?

  1. B + 3A
  2. BA
  3. B + A
  4. B – 3A
Answer
Correct Response: C. This question requires the examinee to demonstrate knowledge of the interpretation and properties of the definite integral. Given the properties of definite integrals, . Also, . Therefore .

Objective 0004
Calculus (Standard 6)

8. What is the estimated value of , using rectangular approximation with left endpoints and four subintervals?

Answer
Correct Response: A. This question requires the examinee to demonstrate knowledge of numerical approximations of definite integrals. The shaded portion of the graph represents the area asked for in the question:
This can be estimated as the sum of the areas of the four shaded rectangles:

Objective 0005
Measurement and Geometry (Standard 4)

9. Tuktut Nogait National Park of Canada, covering 16,340 km2, is the calving ground of the Bluenose caribou. If 1 m = 3.28 ft., what is the size of this park in square miles?

  1. 63 mi.2
  2. 1,015 mi.2
  3. 6,306 mi.2
  4. 10,150 mi.2
Answer
Correct Response: C. This question requires the examinee to demonstrate knowledge of principles, procedures, and applications of measurement. Use dimensional analysis to solve this conversion problem:

Objective 0005
Measurement and Geometry (Standard 4)

10. Parallelogram ABCD is drawn on the coordinate plane with vertices A(3, 2), B(6, 2), C(3, 0), and D at the origin. If ABCD is rotated 90° counterclockwise about point A to form A'B'C'D', what translation must be applied to A'B'C'D' so that D' returns to the origin?

  1. T(x, y) → (x – 5, y + 1)
  2. T(x, y) → (x – 1, y – 5)
  3. T(x, y) → (x + 5, y – 1)
  4. T(x, y) → (x + 1, y + 5)
Answer
Correct Response: A. This question requires the examinee to demonstrate knowledge of the use of translations and rotations in relation to similarity, congruence, and symmetries. The coordinates of ABCD after the described transformations are A'(3, 2), B'(3, 5), C'(5, 2), and D'(5, –1), so point D must be translated 5 units in the negative x direction and 1 unit in the positive y direction to return it to the origin.

Objective 0006
Statistics and Probability (Standard 5)

11. Let x be a normally distributed random variable with mean 5.2 and standard deviation 1.8. What is the approximate probability that x ≥ 3.4?

  1. 16%
  2. 34%
  3. 66%
  4. 84%
Answer
Correct Response: D. This question requires the examinee to demonstrate knowledge of solving problems with normal distributions. Since this a normal distribution, 50% of the x values will be above the mean, 5.2, and approximately 68% of the values will be within one standard deviation of the mean. Thus the approximate probability that 3.4 ≤ x ≤ 5.2 is 34%, and the probability that x ≥ 3.4 is calculated as 34% + 50% = 84%.

Objective 0006
Statistics and Probability (Standard 5)

12. Use the Venn diagram below to answer the question that follows.


The Venn diagram shows the results of a survey of elementary school children that asked what flavors of ice cream they liked. Given that a particular child is known to like either vanilla or chocolate, what is the approximate probability that the child likes strawberry?

  1. 12%
  2. 16%
  3. 28%
  4. 38%
Answer
Correct Response: D. This question requires the examinee to demonstrate knowledge of probabilities of simple and compound events. The particular child selected must be one of the 33 + 17 + 5 + 11 + 6 + 8 = 80 who liked either vanilla or chocolate. Of these, 17 + 5 + 8 = 30 like strawberry, so the desired probability is calculated as .

Objective 0007
Discrete Mathematics (Standard 7)

13. Use the incomplete truth table below to answer the question that follows.


Which of the following truth tables represents [(p → q)∧q] → p?

Answer
Correct Response: A. This question requires the examinee to demonstrate knowledge of symbolic logic. Use this logic table to analyze the truth of the expression [(p → q)∧q] → p.

In each response, the four possibilities for the values of p and q are presented, in order, from top down: both p and q are true, p is true and q is false, p is false and q is true, and both p and q are false. For example, in response A, if p is false and q is true (the third row), then according to the logic table above, [(p → q)∧q] → p evaluates as [T∧T] → F, which gives T → F. This statement is not true, so there is an F in the table of response A.

Objective 0008
Mathematics Instruction and Assessment (Standard 8)

14. According to the Indiana Academic Standards for Mathematics, which of the following learning objectives would be the most advanced topic?

  1. Solve quadratic equations in the complex number system.
  2. Represent complex numbers in polar form.
  3. Use the inverse relationship between squares and square roots.
  4. Solve systems of two linear equations by substitution.
Answer
Correct Response: B. This question requires the examinee to demonstrate knowledge of the Indiana Academic Standards for Mathematics. The representation of complex numbers on the complex plane in rectangular and polar form is a topic appropriate for pre-calculus (PC.PCN.2), according to the standards.

Objective 0008
Mathematics Instruction and Assessment (Standard 8)

15. High school mathematics students have been exploring similarity in triangles and have discovered that for right triangles, the ratios of the side lengths are properties of the angles in the triangle. Building on this discovery, which of the following concepts should the teacher introduce?

  1. the Pythagorean theorem
  2. trigonometric ratios
  3. the law of sines
  4. side-angle-side congruence
Answer
Correct Response: B. This question requires the examinee to demonstrate knowledge of instructional strategies for promoting student understanding of concepts related to mathematics. The study of the relative lengths of sides in relation to the measures of angles in triangles is the basis of trigonometry. The first step will be to extend what the students know about the ratios observed in right triangles to all triangles.