Study Guide

Elementary Mathematics Specialist
Sample Questions

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

1. Numbers that are expressed as repeating decimals are classified as:

  1. irrational.
  2. prime.
  3. rational.
  4. composite.
Enter to expand or collapse answer.Answer expanded
Correct Response: C. Rational numbers are numbers that can be expressed as the quotient of two integers, with the denominator not equal to zero. A rational number can be converted to its decimal representation in the base-10 system by dividing the denominator into the numerator of its fractional representation. If a non-zero part of the resulting decimal repeats periodically without ending, the number is classified as a repeating decimal.

Competency 0002
Analyze number operations and computational algorithms.

2. At the beginning of a 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 first?

  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
Enter to expand or collapse answer.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 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 and algebraic expressions.

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(x – 10) = 225x
  2. 25x + 225x – 10 = 500
  3. 500 + 25x = 225(x – 10)
  4. 225x – 10 – 25x = 500
Enter to expand or collapse answer.Answer expanded
Correct Response: C. Using the equation rate • time = distance, at time = x minutes the tortoise is (500 + 25x) meters from the hare's starting position. The hare starts ten minutes later, so his time is x – 10, and his distance from his starting position is 225(x – 10). In order to catch up to the tortoise, the hare's distance must equal the tortoise's distance, so 500 + 25x = 225(x – 10).

Competency 0004
Apply linear functions to model and solve problems.

4. Which of the following equations could be used to represent an equation that passes through the points (a, b) and (a + 2, b – 3)?







Enter to expand or collapse answer.Answer expanded
Correct Response: B. Use the point-slope form of a linear equation: y – y0 = m(x – x0), where m is the slope of the line and (x0, y0) are the coordinates of one of the points on the line. The slope of the line between two points is calculated as the difference in the y-coordinates of the points divided by the difference in the x-coordinates, in this case the slope m = . Substituting this value of m and the coordinates (a, b) in the point-slope equation gives .

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?







Enter to expand or collapse answer.Answer expanded
Correct Response: D. Use unit conversions and cancellation of units as follows:

. 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(0, 0), X(1, 3), Y(x, y), and Z(3, 2), which of the following coordinates of Y will make WXYZ a parallelogram?

  1. (4, 6)
  2. (2, 6)
  3. (4, 5)
  4. (2, 5)
Enter to expand or collapse answer.Answer expanded
Correct Response: C. Opposite sides of a parallelogram are parallel, so the slope of WX must be equal to the slope of ZY. The slope of WX = . Since slope is change in y over change in x, starting at Z(3, 2) increase y by 3 and x by 1, so Y = (4, 5).

Competency 0007
Analyze and interpret data.

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

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?







Enter to expand or collapse answer.Answer expanded
Correct Response: A. 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.

Type of Milk Gallons Sold in a Month
whole22
low-fat20
skim18
chocolate15

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?







Enter to expand or collapse answer.Answer expanded
Correct Response: A. 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(one not chocolate) = . Because the gallons are replaced as soon as they are sold, P(both not chocolate) = x , so P(at least one chocolate) = .

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.
Enter to expand or collapse answer.Answer expanded
Correct Response: D. 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 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 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
Enter to expand or collapse answer.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.