Test Information Guide
Field 65: Middle School Mathematics
Sample Multiple-Choice Questions
The following reference material will be available to you during the test:
Formulas
Calculators will NOT be provided for the Middle School Mathematics test.
Number System and Quantity
Objective 0001
Apply the structure and properties of number systems.
1. A panel contains 108 switches, each of which can be turned on and off. All the switches start in the "on" position. Starting at the first switch, person A flips every fourth switch on the panel. Next, also starting at the first switch, person B flips every sixth switch. How many switches are in the "on" position when both people are done flipping switches?
- 54
- 63
- 72
- 81
- Answer
- Correct Response: D.
Correct Response: D.
Objective 0001
Apply the structure and properties of number systems.
2. Use the equations below to answer the question that follows.
A = 1 times 10 to the negative first power plus 6 times 10 to the negative second power
B = 2 x 10 to the power of zero plus 7 times ten to the negative second power
Two numbers, A and B, are shown above in expanded notation. What is the sum of A and B?
- 0 point 2 2 3
- 0 point 2 8 6
- 2 point 2 3
- 2 point 8 6
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0001
Apply the structure and properties of number systems.
3. Which of the following statements is true for all integers?
- the absolute value of the quantity m plus n is greater than or equal to the absolute value of m plus the absolute value of n
- the absolute value of the quantity m plus n is less than or equal to the absolute value of m plus the absolute value of n
- the absolute value of the quantity m plus n is less than the absolute value of m plus the absolute value of n
- the absolute value of the quantity m plus n is equal to the absolute value of m plus the absolute value of n
- Answer
- Correct Response: B.
Correct Response: B.
Objective 0002
Use rational numbers, ratios, and proportional relationships.
4. Map software suggests that it will take 1 hour and 20 minutes to travel 65 miles on a stretch of roadway if a driver's current speed remains constant. If a driver travels, on average, about 5 miles per hour slower than this speed, how far will the driver have traveled after 30 minutes?
- 21 and seven eighths miles
- 24 and three eighths miles
- 24 and seven twelfths miles
- 26 and seven eighths miles
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0002
Use rational numbers, ratios, and proportional relationships.
5. A coffee company recommends that consumers use 4and 1 half tablespoons of coffee grounds to make 6 cups of coffee. How many tablespoons of grounds should consumers use to make 10 cups of coffee?
- 7and 1 sixth tablespoons
- 7and 1 half tablespoons
- 8and 1 half tablespoons
- 13and 1 third tablespoons
- Answer
- Correct Response: B.
Correct Response: B.
Objective 0002
Use rational numbers, ratios, and proportional relationships.
6. Use the information below to answer the question that follows.
A business sells sedans and S U Veez from two different brands. The distribution of vehicles in the business's inventory is partially described in the incomplete two-way frequency table below.
| left intentionally blank | S U Vs | Sedans | Total |
|---|---|---|---|
| Brand A | left intentionally blank | left intentionally blank | left intentionally blank |
| Brand B | 135 | left intentionally blank | left intentionally blank |
| Total | left intentionally blank | 80 | left intentionally blank |
If 75 percent of the vehicles from Brand B are S U Veez and 60 percent of all of the S U Veez in the inventory are from Brand B, what fraction of all of the vehicles in the inventory are S U Veez from Brand A?
- 2 fifths
- 5 ninths
- 18 sixty-firsts
- 45 sixty-firsts
- Answer
- Correct Response: C.
Correct Response: C.
Algebra, Functions, and Modeling
Objective 0003
Use patterns to model and solve problems.
7. Use the diagram below to answer the question that follows.
A diagram showing three patterns of linked polygons. Each pattern consists of a single polygon, followed by two of the same polygon joined along a side, followed by three of the same polygon consecutively joined along sides. The first pattern is labeled 3 gons, and the polygons used are equilateral triangles. The second pattern is labeled 4 gons, and the polygons used are squares. The third pattern is labeled 5 gons, and the polygons used are regular pentagons.
A student arranges regular n-gons, each with a side length measuring 1 unit, as shown in the diagram. The student then calculates the perimeter, P, for each iteration, k, of the given pattern. For any regular n-gon, which of the following equations can be used to calculate the perimeter of the k eth iteration?
- P equals open paren N minus 2 close paren K plus 2
- P equals open paren 2 K plus 2 close paren N
- P equals open paren N minus 1 close paren K plus 1
- P equals open paren K plus 1 close paren N
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0003
Use patterns to model and solve problems.
8. Use the diagram below to answer the question that follows.
A pattern begins with a center square with congruent squares connected to each side for a total of 5 squares. In step 2, there is a 2 by 2 array of squares with 2 squares bordering each side of the center arrangement. In step 3, there is a 3 by 3 array bordered by 3 squares on each side.
The diagram shows the first three steps of a pattern made with square blocks. Assuming that the pattern continues, which expression represents the number of blocks in the enth step?
- n squared plus 4 n
- 3 n squared minus 4
- the quantity n plus 2 squared
- 3 n squared minus 4
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0004
Apply algebraic techniques to expressions and equations.
9. A farmer has 100 feet of fencing. The farmer will use some of the fencing to enclose a circular area that has a radius of r feet and the rest of the fencing to enclose a square area. Which of the following inequalities most precisely models the constraint on r?
- 0 is less than r is less than 25
- 0 is less than r is less than 50 over pi
- 0 is less than r is less than 50
- 0 is less than r is less than 10 over the square root of pi
- Answer
- Correct Response: B.
Correct Response: B.
Objective 0004
Apply algebraic techniques to expressions and equations.
10. A sports stadium can accommodate a maximum of 6200 spectators. For any given event, at least 2500 seats must be sold to cover the stadium's operating costs. If 800 spectators are already inside the stadium and additional spectators enter at a constant rate of 26 people per minute, which of the following inequalities represents m, the number of minutes the gates should remain open to admit new spectators?
- 26 m is less than or equal to 6200 minus 2500 plus 800
- 26 m plus 800 is less than or equal to 6200 minus 2500
- 2500 is less than or equal to 26 m plus 800 which is less than or equal to 6200
- 2500 plus 800 is less than or equal to 26 m which is less than or equal to 6200
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0005
Demonstrate knowledge of relations and functions.
11. The function h of x can be defined by h of x = f of g of x where f of x = x to the power of negative 1 half and g of x = open paren 2 minus x close paren squared. What is the domain of h of x?
- open bracket 0 comma infinity close paren
- open interval negative infinity comma infinity
- open interval negative infinity comma 2 union open interval 2 comma infinity
- open bracket 0 comma 2 close paren union open interval 2 comma infinity
- Answer
- Correct Response: D.
Correct Response: D.
Objective 0005
Demonstrate knowledge of relations and functions.
12. Use the graph below to answer the question that follows.
The graph of the function f of x equals the square root of the quantity x plus 3 is shown.
The graph of the function f of x = the square root of the quantity x plus 3 is shown. Which of the following functions, g of x, represents a translation of the graph of f of x four units to the left on the x-axis?
- g of x = the square root of the quantity x plus 7
- g of x = the square root of the quantity x minus 1
- g of x = the square root of the quantity x plus 3 plus 4
- g of x = the square root of the quantity x plus 3 minus 4
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0006
Apply the properties of linear relations and functions.
13. Use the equations below to answer the question that follows.
a times x + b times y + c =0 d times x + e times y + f = 0
The coefficients of the terms in the equations shown satisfy the proportion a colon b = d colon e. Which of the following relationships best describes the graphs of the equations?
- coincident
- intersecting
- parallel
- perpendicular
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0006
Apply the properties of linear relations and functions.
14. A landscaping company charges a fixed price per cubic yard for mulch plus an additional delivery charge. For a delivery of 8 cubic yards, the total cost is 328 dollars, and for 11 cubic yards, the total cost is 424 dollars. Assuming that the delivery charge is constant, what is the y-intercept of the graph that relates x, the number of cubic yards of mulch delivered, to y, cost?
- 12
- 32
- 72
- 96
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0007
Apply the principles and properties of nonlinear relations and functions.
15. Each year, the profits at a company increase by 7 percent. If the profits at the end of the first year are 4000 dollars, which of the following expressions represents the profits at the end of the tenth year?
- 4000 times 9 times 0 point 7
- 4000 times 10 times 1 point 0 7
- 4000 plus 10 times 1 point 0 7 to the tenth power
- 4000 times 1 point 0 7 to the ninth power
- Answer
- Correct Response: D.
Correct Response: D.
Objective 0008
Apply principles, concepts, and procedures related to measurement.
16. Points A and B are on a circle with center O and radius 12. If arc A B has length 9pi, what is the measure of angle A O B?
- 108 degrees
- 135 degrees
- 216 degrees
- 270 degrees
- Answer
- Correct Response: B.
Correct Response: B.
Geometry and Measurement
Objective 0009
Apply the principles of Euclidean geometry and proof.
17. For which of the following statements are both the original statement and its converse true in Euclidean space?
- Vertical angles are congruent.
- Linear pairs of angles are supplementary.
- Nonintersecting planes are parallel.
- Nonintersecting lines are parallel.
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0009
Apply the principles of Euclidean geometry and proof.
18. Which of the following statements is true in Euclidean space?
- If segment A M is congruent to segment M B, then M is the midpoint of segment A B.
- If the measure of angle A plus the measure of angle B = 180 degrees, then angle A and angle B are a linear pair.
- Only one line exists that is parallel to a given line through a point not on the line.
- If the angles formed by two pairs of rays are congruent, then the corresponding rays are parallel.
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0010
Apply properties of two- and three-dimensional figures.
19. Use the diagram below to answer the question that follows.
A diagram showing a composite figure consisting of five congruent isosceles triangles that share a single apex and are joined consecutively along their sides, with a regular pentagon sharing a side with the base of the fifth triangle. The diagram is labeled with lowercase letters as follows: a is the distance from the center of the pentagon to a vertex; b is the distance from the center of the pentagon to the middle of a side; c is the length of each of the two congruent sides of every triangle; d is the length of the base of each triangle; e is the length of the altitude of each triangle.
The diagram shows a net of a right solid with a regular base. The height of the solid formed from the net can be computed by evaluating which of the following expressions?
- the root of the quantity c squared minus a squared
- the root of the quantity c squared minus b squared
- the root of the quantity e squared minus a squared
- the root of the quantity e squared minus d squared
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0010
Apply properties of two- and three-dimensional figures.
20. Use the information below to answer the question that follows.
Quadrilaterals are placed in the appropriate sections of a Venn diagram based on the properties of their diagonals, as shown below.
A Venn diagram is shown with three overlapping circles. One circle is labeled "diagonals are congruent", another circle is labeled "Diagonals bisect each other", and the third circle is labeled, "diagonals are perpendicular". The overlapping region that represents the intersection between "Diagonals bisect each other" and "diagonals are perpendicular" but not "diagonals are congruent" is shaded.
Which of the following quadrilaterals is included in the shaded section?
- kite
- rectangle
- rhombus
- parallelogram
- Answer
- Correct Response: C.
Correct Response: C.
Objective 0011
Apply the principles and properties of coordinate and transformational geometries.
21. Point (a, b) is on a circle with center (h, k). What is the slope of the line tangent to the circle at (a, b)?
- a minus h all over k minus b
- b minus k all over a minus h
- a minus h all over b minus k
- k minus b all over h minus a
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0011
Apply the principles and properties of coordinate and transformational geometries.
22. Trapezoid A B C D is graphed on a coordinate plane with vertices A2, 1, B3, 2, C4, 2, and D5, 1. The trapezoid is reflected over the line y = x. A dilation centered at the origin with scale factor 3 halves is applied to the reflected image. What are the coordinates of c prime, the final image of vertex C?
- 3, 3
- 3, 6
- 6, 3
- 6, 6
- Answer
- Correct Response: B.
Correct Response: B.
Statistics and Probability
Objective 0012
Understand the principles, techniques, and applications of statistics.
23. Medical personnel often use blood tests to determine whether patients have acceptable levels of different substances in their bodies and then use these tests to establish appropriate treatment. For one particular substance, doctors expect to see 4.8 mmol/L with a standard deviation of 0.6 mmol/L of the substance in patients' bodies. A doctor discovers that one patient's level of this substance is 1.2 standard deviations below the mean. How much of the substance is in the patient?
- 3.24 mmol/L
- 4.08 mmol/L
- 5.52 mmol/L
- 6.36 mmol/L
- Answer
- Correct Response: B.
Correct Response: B.
Objective 0012
Understand the principles, techniques, and applications of statistics.
24. A two-way frequency table would be most appropriate for displaying which of the following data sets?
- the numbers of gas-powered, hybrid, or electric cars owned by people in the age groups 16to30, 31to45, 46to65, and over 65
- the percent of movies released last year in each of six rating categories established by the Motion Picture Association of America
- the number of oil spills associated with each of seven categories of causes of these spills over the past 25 years
- the proportion of students entering U S colleges next year who are from the eastern, western, and middle regions of the country and from outside the country
- Answer
- Correct Response: A.
Correct Response: A.
Objective 0013
Understand the principles of probability.
25. Use the Venn diagram below to answer the question that follows.
A Venn diagram is shown with two partially overlapping circles. One circle is labeled "Tennis" and the other circle is labeled "Softball". The area inside both circles is shaded, except for the overlapping region.
The Venn diagram represents students participating in two spring sports, tennis and softball. The shaded region of the Venn diagram can be used to answer which of the following questions?
- What is the probability that a randomly selected student is participating in tennis but not softball?
- What is the probability that a randomly selected student is participating in tennis or softball, but not both?
- What is the probability that a randomly selected student is participating in tennis or softball?
- What is the probability that a randomly selected student is participating in neither tennis nor softball?
- Answer
- Correct Response: B.
Correct Response: B.