Study Guide

Elementary Education (Grades 1–6)
Sample Multiple-Choice Questions

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Field 198: Mathematics (Grades 1–6)

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 college algebra and statistics.

1. Which of the following graphs could represent the polynomial equation y = (x + 3)(x + 5)(x – 2)?







Answer and Rationale
Correct Response: B.
The equation y = (x + 3)(x + 5)(x – 2) is the factored form of a third-degree polynomial function. The roots of a polynomial equation are the values of x that will make the value of y equal zero and show up on the graph as the
x-intercepts. One method of solving for the roots of a polynomial equation is to factor the equation and then set each factor equal to zero. In the given question, the equation is already factored, and setting the individual factors equal to zero results in x + 3 = 0, so x = –3; x + 5 = 0, so x = –5; and x – 2 = 0, so x = 2. The three roots are x = –3, –5, or 2, so the graph will cross the x-axis at the points (–3, 0), (–5, 0), and (2, 0). While each graph represents a third-degree polynomial function, the only graph that has the correct x-intercepts is the one shown in response B.

Objective 0001
Apply knowledge of college algebra and statistics.

2. A community health center conducted a survey to determine how long clerical staff had been in their current positions. The results reported a mean of 28 months, a median of 23 months, and an interquartile range of 14 months. Which of the following box plots could represent these data?







Answer and Rationale
Correct Response: C.
A box plot displays the median, upper quartile, lower quartile, minimum, and maximum as parts of the graph. The median, given to be 23 months, is true for either of the box plots shown in response A or C, as indicated by the vertical line displayed inside the box of each plot corresponding to the value of 23. To find the interquartile range, find the difference between the upper quartile and the lower quartile. The plot in response C has an upper quartile of 30, as indicated by the vertical line displayed on the right side of the box corresponding to the value of 30, and a lower quartile of 16, displayed as a vertical line on the left side of the box corresponding to the value of 16. The difference between these two values is 30 – 16, or 14. The box plot that could represent this data set is shown in response C.

Objective 0002
Apply knowledge of the mathematics curriculum and strategies for teaching counting and cardinality, numbers, and operations in base ten.

3. Use the problem and student work below to answer the question that follows.

Problem:
A school is planning an assembly for the last day of the school year. There are 100 fourth graders in the school. The principal has arranged the chairs in the auditorium into rows of eight. How many rows does the principal need to save to seat all the fourth graders?

Student work:
100 ÷ 8 = _______
12 x 8 = 96
100 - 96 = 4
The principal has to save 12 rows.

Which of the following statements is the best evaluation of the student's work?

  1. The student's answer is incorrect and the work shows a misunderstanding of how to interpret the remainder in a real-world problem.
  2. The student's answer is incorrect and the work shows a misunderstanding of how to perform the common division algorithm.
  3. The student's answer is correct and the work shows an understanding of the relationship between division and multiplication.
  4. The student's answer is correct and the work shows an understanding of how to model division problems using repeated subtraction.
Answer and Rationale
Correct Response: A.
This problem requires the candidate to divide the total number of fourth graders by the number of chairs per row. All the fourth graders need to be seated so any remainder will require a new row of chairs. The student chose the correct operation to solve the problem by dividing 100 by 8, and correctly checked that 12 8, or 96, is the closest multiple of 8 to 100. However, the student's answer of 12 rows is incorrect because the remainder of 4 represents 4 fourth graders who will need chairs. The principal needs to have 12 rows of 8, plus 4 more chairs, or 13 rows in all. The student's response indicates that he or she has a misunderstanding of the meaning of the remainder in the context of this problem.

Objective 0002
Apply knowledge of the mathematics curriculum and strategies for teaching counting and cardinality, numbers, and operations in base ten.

4. Use the problem and student work below to answer the question that follows.

Problem:
A third grader wants to read 95 pages by the end of the week. The student has already read 37 pages. How many pages does the student have left to read?

Student work:
37 + 3 = 40
40 + 50 = 90
90 + 5 = 95
So the student has 58 pages left to read.

A third-grade teacher asks students to solve story problems and explain the solutions. One student's solution method is displayed. The teacher wants to ensure that all students, including English language learners, understand the language of the problem, and plans to present and explain a visual model that best represents the strategy employed by this student. Which of the following visual models would be most appropriate for the teacher to use?







Answer and Rationale
Correct Response: A.
The student is using a counting on method to find the difference between 95 and 37. First the student counts up to the nearest multiple of ten from 37 to get to 40, then counts up a group of tens to get to 90. Lastly, the student counts up from 90 to 95. The number line model shows the same procedure, starting at 37 and jumping up to 40, then jumping forward by tens to 90, and last jumping up to 95. The arrows above the number line indicate how much each jump is worth and summing the arrow values gives the difference between 95 and 37, which is the number of pages the student has left to read. The number line shown in response A best models this strategy.

Objective 0003
Apply knowledge of properties of numbers and operations involving fractions and strategies for teaching these concepts to students.

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

A teacher uses a tape diagram during a lesson on fractions. If the labeled section on the diagram represents 1 whole, which of the following number bonds (addition facts) best matches the tape diagram?







Answer and Rationale
Correct Response: C.
In the tape diagram shown, the whole is labeled as four sections of the tape, so each section is one-fourth. The entire tape diagram is made up of 6 of those sections, and resulting in the improper fraction . A bond diagram is a representation of the part-part-whole relationship of a number showing how smaller numbers make up a larger number. The bond diagram in response C models the fraction as the sum of and , which matches the tape diagram.

Objective 0003
Apply knowledge of properties of numbers and operations involving fractions and strategies for teaching these concepts to students.

6. Use the student work below to answer the question that follows.

A student multiplies two decimals as shown. Which of the following approaches would most likely help the student gain understanding of where to correctly position the decimal point when finding the product of two decimals?

  1. writing out step-by-step instructions for multiplying decimals
  2. converting the decimals to their fraction equivalents and then multiplying the fractions
  3. creating a word problem to model decimal multiplication
  4. using a calculator to multiply pairs of decimals and recording all the results in a table
Answer and Rationale
Correct Response: B.
The student incorrectly placed the decimal point so that it lined up with the decimal points in the numbers used in the multiplication. Converting the decimals to their fraction equivalents would allow the student to get the correct result using prior knowledge of how to multiply fractions. The fraction product of allows the student to understand why the result should be the decimal value of 75 hundredths, or 0.75.

Objective 0004
Apply knowledge of operations and algebraic thinking and strategies for teaching these concepts to students.

7. A student needs to find the greatest common factor of 12 and 18 to solve a word problem. Which of the following models of student work shows a valid method for finding this greatest common factor?







Answer and Rationale
Correct Response: C.
The greatest common factor of a pair of numbers is the largest whole number value that is a factor of both numbers. One method of finding the greatest common factor is to do the prime factorization of both numbers and find the product of the prime factors that the two numbers have in common. The prime factorization of 12 is 2 2 3 and of 18 is 2 3 3. The prime factors that 12 and 18 have in common are 2 and 3 so the greatest common factor is 6.

Objective 0005
Apply knowledge of measurement and data and strategies for teaching these concepts to students.

8. Use the stem-and-leaf plot below to answer the question that follows.

What is the median for the set of test scores shown?

  1. 65
  2. 68
  3. 73
  4. 78
Answer and Rationale
Correct Response: C.
To find the median of a data set, locate the value in the middle of the data set when all the values are arranged in numerical order. This stem and leaf plot has tens digits as the stems in the left hand column and ones digits as the leaves in the right hand column, as indicated by the key at the bottom. In a stem and leaf plot the values are already ordered. There are 13 test scores shown in this stem and leaf plot, so the middle value will be in the seventh position. The seventh value in the list is 73.

Objective 0005
Apply knowledge of measurement and data and strategies for teaching these concepts to students.

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

A spinner is divided into thirds and each third is shaded a different color, as shown in the diagram. If the radius of the spinner is 6 inches, what is the perimeter of the green section of the spinner?

  1. 16π
  2. 22π
  3. 4π + 12
  4. 12π + 12
Answer and Rationale
Correct Response: C.
The green section of the spinner has three edges, the curved edge that is one third the circumference of the circle and two straight edges that are each equal to the radius of the circle. To find the circumference of a circle, multiply the diameter of the circle by π. This spinner has diameter of 12 inches so the circumference of this spinner is 12π. One third of that is . The radius of the spinner is 6 inches so the perimeter of the green section is 4π + 6 + 6 = 4π + 12.

Objective 0006
Apply knowledge of geometry and strategies for teaching geometry concepts to students.

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

The diagram shows a sketch of the relative positions of four lines on a plane. If angle FBC is congruent to angle DCG , then which of the following statements must be true?

  1. Line BC is parallel to line FG.
  2. Line BF is perpendicular to line CB.
  3. Line CG is parallel to line BF.
  4. Line FG is perpendicular to line CG.
Answer and Rationale
Correct Response: C.
Corresponding angles are formed when a transversal passes through two lines. The angles that are in the same position in terms of the transversal are called corresponding angles. Two lines are parallel if the corresponding angles formed by the transversal passing through them are congruent. Angles FBC and DCG are corresponding angles for the two lines CG and BF with transversal AD. Line CG is parallel to line BF.