Study Guide

Field 082: Elementary Mathematics Specialist 
Sample Selected-Response Questions

Recommendation for individuals using a screenreader: please set your punctuation settings to "most."

Expand All | Collapse All

General Test Directions

This test consists of two sections: 1) a section with selected-response questions and 2) a constructed-response 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. Even if you are unsure of an answer, it is better to guess than not to answer a question at all. You will  start uppercase NOT end uppercase  be penalized for guessing.

The second section of this test consists of one constructed-response assignment. You will be asked to provide a written response to the assignment. Directions for completing your written response to the constructed-response assignment appear immediately before the assignment.

Reference materials will 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  start uppercase NOT end uppercase  use any type of calculator or outside reference materials during this testing session.

Sample Selected-Response Questions

Competency 0001 
Analyze the structure of number systems and the properties of the real number system.

1. Which of the following is the base-four representation of the base-ten number 39?

  1.  10013 sub four 
  2.  1113 sub four 
  3.  213 sub four 
  4.  20013 sub four  

Correct Response: C. This question requires the examinee to analyze the role of place value in a base system other than base 10. Place values in base four represent successive powers of four instead of powers of ten. 39 sub ten equals 2 times 4 squared plus 1 times 4 sup 1 plus 3 times 4 sup 0, so 39 sub ten equals 213 sub four.

Competency 0002 
Analyze number operations and computational algorithms.

2. At the beginning of the new school year, a second-grade teacher administers a pretest to determine how well students managed to retain basic arithmetic skills over the summer break. The teacher and the Elementary Mathematics Specialist are pleased to discover that many of the students were able to add and subtract numbers from 1 to 20 accurately and without using their fingers. In order to facilitate continued student growth in this area, which of the following instructional activities would it be most appropriate for the Elementary Mathematics Specialist to suggest  start italics first end italics ?

  1. guiding students to explore the computational algorithms they are employing
  2. having students use manipulatives to visually illustrate the problems they are solving
  3. asking students to explain the metacognitive processes they are using to solve the problems
  4. providing students with more complex problems that include numbers from 1 to 50

Correct Response: A. This question requires the examinee to use results of an assessment to plan strategies that enhance student mathematical understanding in relation to number operations and computational algorithms. Since the pretest shows that many students are able to add and subtract numbers from 1 to 20 accurately, asking them to explore the algorithms they are using will improve their depth of understanding of why the algorithms work and allow students to extend these algorithms to operations with larger numbers.

Competency 0003 
Analyze patterns, algebraic expressions, and functions.

3. A tortoise and a hare are 500 meters apart. The tortoise begins traveling away from the hare at a rate of 25 meters per minute. Ten minutes later, the hare starts running toward the tortoise 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 tortoise travels before the hare catches up to it?

  1.  25 left paren x minus 10 right paren equals 225 x 
  2.  25 x plus 225 x minus 10 equals 500 
  3.  500 plus 25 x equals 225 left paren x minus 10 right paren 
  4.  225 x minus 10 minus 25 x equals 500 

Correct Response: C. This question requires the examinee to analyze a real-world problem and translate it into an algebraic equation. Using the equation rate times time equals distance at time equals x minutes the tortoise is left paren 500 plus 25 x right paren meters from the hare’s starting position. The hare starts ten minutes later, so his time is x minus 10 and his distance from his starting position is 225 left paren x minus 10 right paren In order to catch up to the tortoise, the hare’s distance must equal the tortoise’s distance, so 500 plus 25 x equals 225 left paren x minus 10 right paren

Competency 0004 
Apply linear functions to model and solve problems.

4. A transformation applied to the linear function y equals 2 x minus 3 results in the linear function y equals 2 x plus 7. Which of the following statements describes the results of this transformation on the graph of y = 2 x minus 3?

  1. The graph shifts up 7 units.
  2. The graph shifts up 10 units.
  3. The graph shifts left 7 units.
  4. The graph shifts left 10 units.

Correct Response: B. This question requires the examinee to analyze the effect of a transformation on the graph of a linear function. Since left paren 2 x minus 3 right paren plus 10 equals 2 x plus 7 each y-value of y equals 2 x plus 7 s 10 units higher than the corresponding y-value of y equals 2 x minus 3 so the graph of y equals 2 x minus 3 has shifted up 10 units.

Competency 0005 
Apply concepts of measurement.

5. 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.  1,225 times 0.4 over 1,000 times 60 times 60 meters 
  2.  1,225 times 1,000 times 60 times 60 over 0.4 meters 
  3.  1,225 times 60 times 60 over 1,000 times 0.4 meters 
  4.  1,225 times 1,000 times 0.4 over 60 times 60 meters 

Correct Response: D. This question requires the examinee to use dimensional analysis to represent and solve problems. Use unit conversions and cancellation of units as follows: 1,225 kilometers over 1 hour times 1,000 meters over 1 kilometer times 1 hour over 60 minutes times 1 minute over 60 seconds times 0.4 seconds over 1. All units cancel except meters.

Competency 0006 
Apply concepts of Euclidean, transformational, and coordinate geometry.

6. Given a quadrilateral in a coordinate plane with three vertices represented by W left paren 0 comma 0 right paren , X left paren 1 comma 3 right paren , Y left paren x comma y right paren , and Z left paren 3 comma 2 right paren , which of the following coordinates of Y will make  W X Y Z  a parallelogram?

  1.  left paren 4 comma 6 right paren 
  2.  left paren 2 comma 6 right paren 
  3.  left paren 4 comma 5 right paren 
  4.  left paren 2 comma 5 right paren  

correct response: c. this question requires the examinee to analyze polygons in the coordinate plane in terms of slope and parallel lines. opposite sides of a parallelogram are parallel, so the slope of line segment w x must be equal to the slope of line segment z y. the slope of line segment w x equals 3 minus 0 over 1 minus 0 equals 3 over 1. since slope is change in y over change in x, starting at z left paren 3 comma 2 right paren increase y by 3 and x by 1, so y equals left paren 4 comma 5 right paren.

Competency 0007 
Analyze and interpret data.

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



   

The graph is a histogram. The vertical axis is labeled Frequency, with values marked from 0 to 8 in increments of 2. The horizontal axis is labeled Number of Miles Commuted to Work, and has labels for 0 to 5, 6 to 10, 11 to 15, 16 to 20, 21 to 25, 26 to 30, and 31 to 35. The data values are 6 for 0 to 5, 8 for 6 to 10, 7 for 11 to 15, 4 for 16 to 20, 4 for 21 to 25, 0 for 26 to 30, and 2 for 31 to 35.

Each of the workers at a company has reported the number of miles he or she commutes to work each day. The data are compiled in the graph above. Which of the following boxplots could represent the data?

Each response is a box and whisker plot labeled Miles Commuted. Number lines below the box and whiskers indicate the values for each point of the box and whiskers.

  1. The left end of the left whisker is at 1, the left end of the box is at 7, the line inside the box is at 12, the right end of the box is at 19, and the right end of the right whisker is at 32.
  2. The left end of the left whisker is at 2, the left end of the box is at 5, the line inside the box is at 13, the right end of the box is at 16, and the right end of the right whisker is at 33.
  3. The left end of the left whisker is at 1, the left end of the box is at 12, the line inside the box is at 15, the right end of the box is at 18, and the right end of the right whisker is at 33.
  4. The left end of the left whisker is at 2, the left end of the box is at 6, the line inside the box is at 14, the right end of the box is at 18, and the right end of the right whisker is at 29.

Correct Response: A. This question requires the examinee to analyze data presented in a variety of formats. According to the data in the graph, the range of the boxplot 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 the 16th value, which is between 11 and 15. 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.

Competency 0008 
Apply 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.


Milk Gallons sold by Type in a month
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 fifths
  3. 16 over 25
  4. 4 fifths

Correct Response: A. This question requires the examinee to apply concepts of probability to solve problems involving simple and compound events. The probability that at least one of the gallons sold is chocolate equals one minus the probability that both gallons sold are not chocolate. P left paren one not chocolate right paren equals 22 plus 20 plus 18 over 22 plus 20 plus 18 plus 15 equals 4 over 5 Because the gallons are replaced as soon as they are sold, P left paren both not chocolate right paren equals 4 over 5 times 4 over 5, so P left paren at least one chocolate right paren equals 1 minus 16 over 25 equals 9 over 25

Competency 0009 
Demonstrate knowledge of mathematics instruction and assessment.

9. A teacher poses the following problem to a fourth-grade class:

The population of a city is 83,427. Two different people each round the population of the city correctly, but they get different answers. How can this be?

The teacher is most likely asking this question to assess the students' ability to:

  1. write multidigit numbers in expanded form.
  2. interpret data.
  3. use estimation to judge the reasonableness of a solution.
  4. reason about place value.

Correct Response: D. This question requires the examinee to identify appropriate questions to assess students' mathematical understanding. Rounding results in reporting an amount in terms of a specified place value. The teacher is assessing the students' ability to recognize that the population could be reported as 80,000 if counting by ten-thousands, 83,000 if counting by thousands, etc.

Competency 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  start italics first end italics ?

  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 school mathematics curriculum 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

Correct Response: B. This question requires the examinee to recognize ways to establish a culture of collaboration in regard to the use of data to promote positive changes in the school mathematics program. 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.