Study Guide

Field 314: Elementary Mathematics Specialist 
Sample Multiple-Choice Questions

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

Formulas PDF

Each multiple-choice question has four answer choices. Read each question and its answer choices carefully and choose the ONE best answer.

During the test you should try to answer all questions. Even if you are unsure of an answer, it is better to guess than not to answer a question at all. You will NOT be penalized for choosing an incorrect response.

Objective 0001 
Apply knowledge of number sense, number systems, and the properties of the real number system.

1. A sixth-grade teacher introduces which of the following vocabulary terms to describe any number that can be written as a fraction?

  1. integers
  2. natural numbers
  3. rational numbers
  4. whole numbers
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: C.
A rational number is any number that can be written as the fraction  a over b , where a and b are both integers and b  is not equal to 0.

Objective 0002 
Analyze number operations and computational algorithms.

2. A third-grade teacher is introducing their students to the inverse relationship between multiplication and division. Which of the following pairs of equations would best demonstrate this relationship?

  1. 3  times 4 = 12 and 4  times 3 = 12
  2. 12  divided by 3 = 4 and 12  divided by 4 = 3
  3. 3  times 4 = 12 and 12  divided by 2 = 6
  4. 12  divided by 3 = 4 and 4  times 3 = 12
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: D.
The inverse relationship between multiplication and division requires that if the product of numbers a and b is a number c, then the quotient of c divided by b is a and the quotient of c divided by a is b. In the correct response, a = 4, b = 3, and c = 12 are used to illustrate this relationship in algebraic notation.

Objective 0003 
Apply knowledge of patterns, algebraic and proportional reasoning, expressions, and equations.

3. A turtle and a rabbit are 500 meters apart. The turtle begins traveling away from the rabbit at a rate of 25 meters per minute. Ten minutes later, the rabbit starts running toward the turtle at a rate of 225 meters per minute. Which of the following equations could be used to find x, the number of minutes that the turtle travels before the rabbit catches up to it?

  1.  25 times the difference x minus 10 equals 225 x
  2.  25 x plus 225 x minus 10 equals 500
  3.  500 plus 25 x equals 225 times the difference x minus 10
  4.  225 x minus 10 minus 25 x equals 500
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: C.
Using the equation rate  times  time = distance, at time = x minutes, the tortoise is  the sum 500 plus 25 x  meters from the rabbit's starting position. The rabbit starts ten minutes later, so its time is x  minus  10, and its distance from the starting position is  225 times the difference x minus 10 . In order to catch up to the turtle, the rabbit's distance must equal the turtle's distance, so  500 plus 25 x equals 225 times the difference x minus 10 . The left side of the equation is their distance apart, the rate of the turtle using the given 25 meters, and the unknown. The right side of the equation is the rate of the rabbit times the unknown and ten subtracted from it. By setting the distance and rate of each animal opposite to each other, solving for x will determine the number of minutes the turtle will travel before the rabbit catches up to it.

Objective 0004 
Apply knowledge and concepts of linear functions to model and solve problems.

4. A person collecting signatures for a ballot initiative begins the day with 67 signatures and collects 12 signatures an hour. Which of the following equations is a linear function that can be used to find the total number of signatures the person has collected (y) after a certain number of hours (x)?

  1.  x equals 12 y plus 67
  2.  y equals 12 x
  3.  y equals 12 x plus 67
  4.  y equals 67 x plus 12
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: C.
The number of hours spent collecting signatures is represented by x. The term 12x represents the number of signatures collected per hour, multiplied by the number of hours spent collecting signatures. The term 67 represents the signatures that have already been collected at the start of the day. Adding these two terms equals the total number of signatures, y.

Objective 0005 
Apply knowledge and concepts of Euclidean and coordinate geometry.

5. Which of the following nets will be a square-based pyramid when folded into a three-dimensional shape?

  1.  A figure composed of equilateral triangles arranged around a square such that two equilateral triangles share a side with the square along consecutive sides of the square. Each of the first two equilateral triangles share a side with a second equilateral triangle, oriented along the same corresponding sides of the first two equilateral triangles. 
  2.  A figure composed of equilateral triangles arranged around a square such that two equilateral triangles share a side with the square along consecutive sides of the square. One of the first two equilateral triangles shares a side with a second equilateral triangle, which shares a side with a third equilateral triangle, such that the combined three triangles create a trapezoidal shape for which one of the nonparallel sides forms the edge of the center square. 
  3.  A figure composed of equilateral triangles arranged around a square such that three equilateral triangles share a side with the square along consecutive sides of the square. The equilateral triangle in the middle side of the square shares a side with a second equilateral triangle. 
  4.  A figure composed of equilateral triangles arranged around a square such that two equilateral triangles share a side with the square along consecutive sides of the square. Each of the first two equilateral triangles share a side with a second equilateral triangle, such that each of the 4 triangles share a vertex with the same corner of the square. 
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: B.
A net is a pattern that can be cut and folded to make a model of a solid shape. If each of the triangles were folded along the lines connecting them to other triangles or the square, they would form the edges of a solid. By manipulating the square and triangles in response B, the result would be a square-based pyramid.

Objective 0006 
Apply knowledge and concepts of measurement.

6. An observer at a baseball game sees a batter hit the ball and then hears the sound of the hit 0.4 seconds later. If sound travels at 1,225 kilometers per hour, which of the following expressions could be used to estimate the distance in meters between the batter and the observer?

  1.  a ratio in meters of one thousand two hundred twenty-five times four tenths over one thousand times sixty times sixty 
  2.  a ratio in meters of one thousand two hundred twenty-five times one thousand times sixty times sixty over four-tenths 
  3.  a ratio in meters of one thousand two hundred twenty-five times sixty times sixty over one thousand times four-tenths 
  4.  a ratio in meters of one thousand two hundred twenty-five times one thousand times four-tenths over the expression sixty times sixty 
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: D.
Dimensional analysis is used to represent and solve problems involving unit conversions. Multiplying the given 1,225 kilometers per hour by the time in hours that the sound took to reach the observer estimates the distance from the batter to the observer. To do this, the 0.4 s travel time can be converted into hours by multiplying it by the conversion factors  1 hour over 60 minutes  and  1 minute over 60 seconds . The result is a distance in kilometers, which can be multiplied by  1000 meters over 1 kilometer  to convert it to a distance in meters.

Objective 0007 
Analyze and interpret data.

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

SHORT ALT TEXT

A bar graph depicting the frequency of miles commuted to work by individuals. The horizontal axis (x-axis) is labeled the number of miles commuted to work, divided into intervals: 0-5, 6-10, 11-15, 16-20, 21-25, 26-30, and 31-35. The vertical axis (y-axis) indicates the frequency and is labeled 0, 2, 4, 6, 8.

The bars show the following frequencies for each mileage range:
0 to 5 miles: 6 individuals
6 to 10 miles: 8 individuals
11 to 15 miles: 7 individuals
16 to 20 miles: 4 individuals
21 to 25 miles: 4 individuals
26 to 30 miles: 0 individuals
31 to 35 miles: 2 individuals

Each of the workers at a company has reported the number of miles they commute to work each day. The data are compiled in the graph shown. Which of the following box plots could represent the data?

  1.  A box and whisker plot over a number line from 1 to 32. The box starts at 7, has a line at 12, and ends at 19. 
  2.  A box and whisker plot over a number line from 2 to 33. The box starts at 5, has a line at 13, and ends at 16. 
  3.  A box and whisker plot over a number line from 1 to 33. The box starts at 12, has a line at 15, and ends at 18. 
  4.  A box and whisker plot over a number line from 2 to 29. The box starts at 6, has a line at 14, and ends at 18. 
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: A.
According to the data in the graph, the range of the box plot needs to include a lowest number between 0 and 5 and a highest number between 31 and 35, so choice D is not possible. When the 31 data values from the graph are listed in increasing order, the median is between the 15th and 16th value, and can be found by taking the average of the 15th (11) and 16th (15) data values. The first quartile value is the 8th data value, which is between 6 and 10. Choice A is the only plot for which both of these statements are true.

Objective 0008 
Apply knowledge and concepts of probability.

8. A small store sells one-gallon containers of different kinds of milk. The store keeps track of the number of gallons sold for each kind of milk over the course of a month. The data are shown in the table below.

Table depicting the types of milk a small store sells per month, in gallons
Type of Milk Gallons Sold in a Month
whole 22
low-fat 20
skim 18
chocolate 15

On Monday, the store sold 2 gallons of milk. If a stockperson immediately replaces each gallon as it is sold, what is the probability that at least one of the gallons sold was chocolate?

  1.  9 over 25 
  2.  3 over 5 
  3.  16 over 25 
  4.  4 over 5 
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: A.
The probability that at least one of the gallons sold is chocolate equals 1 minus the probability that both gallons sold are not chocolate. When selecting one gallon of milk at random, the probability of selecting a gallon that is not chocolate is  the sum of 22 plus 20 plus 18 over the sum of 22 plus 20 plus 18 plus 15 equals four-fifths . As a result, the probability that two gallons that are not chocolate are selected in a row is  four-fifths multiplied by four fifths equals sixteen twenty-fifths . Subtracting this from 1 yields the correct response.

Objective 0009 
Demonstrate knowledge of mathematics instruction and assessment.

9. After analyzing the data from formative assessments completed at the beginning of a new school year, a second-grade teacher and the Elementary Mathematics Specialist are pleased to discover that many of the students were able to add and subtract numbers within 20 accurately. In order to facilitate continued student growth in this area, which of the following instructional activities would be most appropriate for the Elementary Mathematics Specialist to suggest first?

  1. encouraging students to justify the strategies they are employing
  2. having students use manipulatives to model the problems they are solving
  3. supplying students with a timed online program to practice basic facts
  4. providing students with more complex problems that include numbers from 1 to 50
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: A.
Since the pretest shows that many students are able to add and subtract numbers from 1 to 20 accurately, asking them to explore the strategies they are using will improve their depth of understanding of why their methods work and allow students to extend these strategies to operations with larger numbers.

Objective 0010 
Demonstrate knowledge of instructional leadership in mathematics.

10. An Elementary Mathematics Specialist is reviewing recent standardized assessment data that show that the overall performance of students in the school is below expectations in several key areas of mathematics. The Elementary Mathematics Specialist can best promote positive changes in the school's mathematics curriculum and instruction by taking which of the following steps first?

  1. developing a comprehensive written plan of action to address the data and presenting it to the Board of Education for approval
  2. communicating the data to teachers and engaging them in a dialogue to develop a clear shared purpose
  3. redesigning the available materials and lesson plans in light of the data and working with the teachers to implement it
  4. researching professional development options and recommending particular professional development opportunities for specific teachers
Answer and Rationale. Enter to expand or collapse. Answer expanded
Correct Response: B.
To best promote changes, it is necessary that the teachers recognize and support the purpose of the changes. Communicating the data to the teachers allows them to recognize needs and engaging them in a dialogue allows them to share in evaluating and improving instruction, thereby gaining their support.