# Study Guide

## Field 208: Mathematics

Sample Multiple-Choice Questions

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**

Understand mathematical practices.

1. Which of the following tasks can be completed by finding the determinant of a matrix?

- finding the probability of an event modeled by a normal curve
- modeling the exponential increase of a population over time
- finding the area of a polygon described on a coordinate plane
- calculating the greatest common factor of a set of integers

- Answer and Rationale
**Correct Response: C.**

Matrices can be used to find the area of a polygon whose vertices are given as coordinate pairs by dividing the polygon into non-overlapping triangles and summing the areas of the triangles. The area of each triangle is found by subtracting pairs of*x*-coordinates and pairs of*y*-coordinates to represent two of its sides as vectors: <*a*_{1},*b*_{1}> and <*a*_{2},*b*_{2}> and using the formula Area = times absolute value of .

**Objective 0002**

Understand how to select, integrate, and use appropriate technologies in the mathematics classroom.

2. Software that allows students to create regular tessellations of plane figures would be most appropriate for helping students learn about:

- dilations.
- inversions.
- isometries.
- projections.

- Answer and Rationale
**Correct Response: C.**

An isometry is a linear transformation that preserves length. Regular tessellations of a plane are created by reflections, rotations, and translations of plane figures. These transformations all preserve length, thus are isometries.

**Objective 0003**

Understand the process of reading, and apply knowledge of strategies for promoting students' reading development in the mathematics classroom.

3. A high school mathematics teacher regularly uses anticipation guides to promote students' disciplinary reading skills. Each anticipation guide consists of carefully constructed statements related to a mathematics topic students will be reading about in their textbook or in a supplemental text. Students must decide whether each statement is true or false based on their prior knowledge. Then, when reading the text, they must identify evidence in the text that either supports or refutes their initial choices. This instructional strategy supports the reading process primarily by:

- motivating students to read informational texts related to mathematics.
- helping students distinguish between fact and opinion in their reading.
- using writing to help deepen students' understanding of their reading.
- fostering in students the habit of applying reasoning during reading.

- Answer and Rationale
**Correct Response: D.**

Reading comprehension relies heavily on the ability to make connections not only between pieces of information in a text but between new information being read and the reader's prior knowledge. The instructional strategy described in the scenario requires students to read actively and purposefully to identify evidence in an assigned text that either supports or refutes their prior conceptions about a topic. The students must use metacognition to reflect on what they are reading and apply reasoning to make connections between new and prior knowledge.

**Objective 0004**

Understand the real number system and its operations.

4. Two containers each hold a number of cubes. The ratio of cubes in container X to those in container Y is 12:5. If one-third of the cubes in container X are moved to container Y, there will then be 135 cubes in container Y. What is the total number of cubes in both containers?

- 459
- 351
- 255
- 180

- Answer and Rationale
**Correct Response: C.**

One way to solve this is by drawing ratio diagrams. In the first diagram, a strip of 12 boxes compares to a strip of 5 boxes. After one-third of the cubes are moved, the strips have a ratio of 8 to 9.

Since 9 boxes represent 135 cubes, each box represents 15 cubes, and the total number of cubes in both boxes is 8(15) + 9(15) = 255.

Algebraically, let the number of cubes in box*X*= 12*a*and the number in box*Y*= 5*a*. Then(12 *a*) + 5*a*= 135, and*a*= 15,12(15) + 5(15) = 255. The diagram shows a graphic representation of this solution method. On the left are 2 strips of squares, representing the containers in the original state. The strip labeled x has 12 squares and the strip labeled y has 5 squares. Each square represents a number of cubes. Under the last 4 squares in strip y is a curly bracket and the number 1 third, representing the portion of items to be moved to the other container. On the right are 2 strips of squares, representing the containers after some cubes have been moved from container y to container x. The strip is labeled x now has 8 squares, which is the number remaining after 1 third of the original 12 has been removed. The strip labeled y now has 9 squares, which is the result of adding the 4 squares from strip x to the original 5 squares in strip y. Below strip y is a curly bracket and the number 135, indicating that these 9 squares represent 135 cubes.

**Objective 0005**

Understand the properties of complex numbers and linear algebra.

5. Let *z* = 2 + 2*i*. Which of the following expressions represents the polar form of *z*^{2}?

- Answer and Rationale
**Correct Response: D.**

Computing,*z*^{2}= (2 + 2*i*) (2 + 2*i*) = 4 + 8*i*− 4 = 8*i*. When graphed on a complex plane, the coordinates of*z*^{2}are (0, 8), the point is 8 units from the origin on the imaginary axis, and the angle between the positive real axis and the imaginary axis is 90° or in radians. On a polar graph, this point lies on a circle of radius 8 and its vector makes an angle of with the 0° axis. The polar form of a complex number is*z*=*r*(cos θ +*i*sin θ), where*r*is the radius and θ is the angle between the vector represented by*z*and the 0° axis, so*z*^{2}= . Alternatively, the polar form of*z*could be found and DeMoivre's Theorem applied to get the polar form of*z*^{2}.

**Objective 0005**

Understand the properties of complex numbers and linear algebra.

6. **Use the information below to answer the question that follows.**

**Activity: Complex Numbers**

Let *z* = 2 + *i* and *z* = 1 + 3*i*.

- Find
*z**z*. - Find the magnitudes of
*z*,*z*, and*z**z*. - Draw the vectors determined by
*z*,*z*, and*z**z*. - Use a protractor to measure the angles made by each of the three vectors with the positive real axis.
- Make conjectures about the magnitudes and angles of
*z*,*z*, and*z**z*.

As part of a unit on complex numbers, a teacher asks students to complete a series of examples based on the activity shown. This series of examples will enable students to develop an understanding of which of the following processes?

- multiplying complex numbers using their polar forms
- finding the dot product of vectors represented by complex numbers
- demonstrating that multiplication of complex numbers is commutative
- relating the product of the roots of quadratics to complex numbers

- Answer and Rationale
**Correct Response: A.**

Using the provided vectors, step 1 in the activity yields*z**z*= −1 + 7*i*; step 2 yields = , = , and = ; and steps 3 and 4 yield the graphs of vectors*z*,*z*, and*z**z*, which have angle measures of approximately θ = 26°, θ = 72°, and θ_{1,2}= 98°, as shown.

Inspection shows that 98° = 26° + 72° and = times ; that is, θ_{1,2}= θ + θ and = times . Recall that the magnitude of a vector is equal to the radius,*r*, of the circle the vector describes, so =*r*. That is, this activity enables students to develop an understanding that if_{j}*z*=*r*(cos θ + isin θ) and*z*=*r*(cos θ +*i*sin θ), then*z**z*=*r**r*[cos(θ + θ) +*i*sin(θ + θ)], which is multiplying the complex numbers*z*and*z*using their polar forms.The vectors are shown on a real versus imaginary graph. Vector z sub 1 extends from the origin to a point 2 units to the right and 1 unit up at an angle of 28°. Vector z sub 2 extends from the origin to a point 1 unit to the right and 3 units up at an angle of 72°. Vector z sub 1 z sub 2 extends from the origin to a point 1 unit to the left and 7 units up at an angle of 98° (from the positive real axis).

**Objective 0006**

Understand algebraic techniques.

7. **Use the table below to answer the question that follows.**

xf(x)−1 0.6667 −0.5 0.2667 0 0 .5 −0.2667 1 −0.6667 1.5 −1.714 2 ERROR 2.5 2.222 3 1.2 3.5 0.8485

The table shown could be a partial table of values for which of the following functions?

- Answer and Rationale
**Correct Response: C.**

A rational function is undefined for some value of*x*when the denominator of any fraction in the function equals 0. Using the table provided, the function is undefined when*x*= 2; therefore,*x*− 2 must be a factor in the denominator. The table shows that when*x*is −0.5 and −1,*f*(*x*) is positive. The function*f*(*x*) = can be rewritten as*f*(*x*) = , which has*x*− 2 as a factor in the denominator, and for*x*equal to −0.5 and −1,*f*(*x*) is positive. Alternatively, calculating*f*(−0.5) and*f*(−1) for*f*(*x*) = yields 0.2667 and 0.6667, respectively.

**Objective 0007**

Understand functions and the properties of linear relations and functions.

8. **Use the graph below to answer the question that follows.**

The function *f*(*x*) shown on the graph has intercepts (*a*, 0) and (0, *b*). Which of the following functions is an equation of its inverse, *f*^{−1}(*x*)?

- Answer and Rationale
**Correct Response: C.**

The inverse of a function maps values in the range of a function back to values in the domain. If (*a*, 0) and (0,*b*) are coordinates of points on the line represented by*f*(*x*), then (0,*a*) and (*b*, 0) are coordinates of points on the line represented by the inverse function. Calculate slope: = . Using the point-slope form with point (*b*, 0), .

**Objective 0008**

Understand quadratic functions and conic sections.

9. The graph of a parabola has an axis of symmetry at *x* = 1, an *x*-intercept at (−4, 0), and a *y*-intercept at (0, −12). Which of the following equations represents the graph of this parabola?

*y*= 0.5(*x*+ 4)(*x*− 6)*y*= 0.5(*x*− 4)(*x*+ 6)*y*= (*x*+ 4)(*x*− 3)*y*= (*x*− 4)(*x*+ 3)

- Answer and Rationale
**Correct Response: A.**

The intercept form of a parabola is*y*=*a*(*x*−*x*_{1})(*x*−*x*_{2}), where (*x*_{1}, 0) and (*x*_{2}, 0) are*x*-intercepts and the axis of symmetry is*x*= . Because the parabola has an*x*-intercept at (−4, 0),*x*_{1}= −4. Given that the axis of symmetry is*x*= 1, and that*x*_{1}= −4, 1 = , so*x*_{2}= 6. Therefore, the equation of the parabola is in the form*y*= a(*x*+ 4)(x − 6). To find*a*, substitute the values of the coordinate (0, −12) into the equation to obtain −12 =*a*(0 + 4)(0 − 6), so*a*= 0.5. The equation of the parabola is*y*= 0.5(*x*+ 4)(*x*− 6).

**Objective 0009**

Understand polynomial, absolute value, radical, and rational functions and inequalities.

10. Using dynamic graphing software, a teacher provides an activity in which students are given a function in the form of *y* = *x ^{p}* and students can use a sliding bar to change the value of

*p*. As a student slides the bar, the graph of the function changes. Which of the following statements represents a valid conclusion demonstrated by this activity?

- When
*p*is even, the graph of the function is in the first and second quadrants, and when*p*is odd, the graph of the function is in the first and third quadrants. - When
*p*is negative, the graph of the function translates*p*units to the right, and when*p*is positive, the graph of the function translates*p*units to the left. - When
*p*< 1, the graph of the function illustrates decay, and when*p*> 1, the graph of the function illustrates growth. - When
*p*> 1, the graph of the function stretches, and when 0 <*p*< 1, the graph of the function shrinks.

- Answer and Rationale
**Correct Response: A.**

The function*y*=*x*is a power function. For whole number values of^{p}*p*, when*p*is even,*x*is positive or zero, so the graph is in the first and second quadrants. When^{p}*p*is odd,*x*will be positive for^{p}*x*> 0, negative for*x*< 0 and 0 when*x*= 0. Thus, the graph will lie in the first and third quadrants.

**Objective 0009**

Understand polynomial, absolute value, radical, and rational functions and inequalities.

11. Given *f*(*x*) = 3 | *x* | − 2 and *g*(*x*) = | *x* − 3 |, over which of the following intervals on the *x*-axis is *f*(*x*) > *g*(*x*)?

- Answer and Rationale
**Correct Response: C.**

The vertex of the graph of*f*(*x*) is (0, −2), its right ray has a slope of 3, and its left ray has a slope of −3. The vertex of*g*(*x*) is (3, 0), its right ray has a slope of , and its left ray has a slope of −. A sketch of the graphs shows that they intersect once when 0 <*x*< 3 and again when*x*< 0. When 0 <*x*< 3,*x*is positive so| but*x*| =*x*, is negative, so*x*− 3| Solving the inequality for*x*− 3 | = − (x − 3).*x*yields*x*> 1. When and*x*< 0, |*x*| = −x| Solving for*x*− 3 | = − (x − 3).*x*yields , which is not among the response choices, so the interval asked for is (1, ∞ ).

**Objective 0010**

Understand exponential and logarithmic functions.

12. Atmospheric pressure and altitude are related by the formula ln , where *P* is pressure in kilopascals at altitude *h* in kilometers, P sub 0 is pressure at *h* = 0, and *k* is a constant. Which of the following equations expresses *P* as a function of *h*?

- Answer and Rationale
**Correct Response: C.**

To express*P*as a function of*h*, solve the equation In for*P*. Apply the exponential function to both sides of the equation: , so . Multiply both sides of the equation by P sub 0 to obtain .

**Objective 0011**

Understand trigonometric functions.

13. How many solutions are there for the system of equations *f*(*x*) = 3 sin 2*x* and *g*(*x*) = 2 on the interval [0, 2π]?

- 3
- 4
- 5
- 6

- Answer and Rationale
**Correct Response: B.**

The fundamental period of a sine function is 2π. The graph of*f*(*x*) = 3sin 2*x*has a period of = π, thus completing two cycles in the interval [0, 2π]. The amplitude of*f*(*x*) is 3, so the highest value of the function is*y*= 3, which will occur twice on this interval. The graph of*g*(*x*) = 2 is a horizontal line. Since 2 < 3,*g*(*x*) intersects*f*(*x*) four times.

**Objective 0012**

Understand principles of differential calculus.

14. The radius of a spherical balloon is decreasing at the rate of 5 cm/s. At what rate is the volume of the balloon decreasing when the diameter of the balloon is 20 cm?

- 10,667π cm
^{3}/s - 4,500π cm
^{3}/s - 2,667π cm
^{3}/s - 2,000π cm
^{3}/s

- Answer and Rationale
**Correct Response: D.**

For a sphere,*V*= πr^{3}. The rate at which the volume is changing is expressed as the derivative of the volume function with respect to time: = 4πr^{2}. At the time when the*diameter*= 20 cm and = −5, = 4π(10^{2})(−5) = −2000π.

**Objective 0013**

Understand principles of integral calculus.

15. A rocket is launched vertically from the ground with an initial acceleration of 250 m/s^{2}. Its velocity, *v*, in meters per second at time *t* is given by *v*(*t*) = 0.15*t*^{2} + 250*t*. What is the height above the ground of the rocket at *t* = 6 s?

- 1,505.4 m
- 4,510.8 m
- 6,027.0 m
- 9,032.4 m

- Answer and Rationale
**Correct Response: B.**

The position function (height) is the antiderivative of the velocity function. Thus,*h*(*t*) = 0.05*t*^{3}+ 125*t*^{2}+*C*. Since the rocket was launched from the ground,*h*= 0 when*t*= 0, making*C*= 0 as well. Then*h*(6) = 0.05(216) + 125(36) = 4510.8 meters.

**Objective 0014**

Understand principles and applications of measurement.

16. The volume of a cube is 512 cubic inches. What is the ratio of the volume of the sphere that can be inscribed in the cube to the volume of the sphere that can be circumscribed about the cube?

- Answer and Rationale
**Correct Response: A.**

The largest sphere that can be inscribed in a cube is tangent to each face of the cube, so its diameter is the length of one side of the cube. Since the volume of the cube is 512 cubic inches, its edge length is or 8 inches. Its radius is 4 and its volume is π(4^{3}). The smallest sphere that can be circumscribed about the cube passes through the vertices of the cube, so its diameter is the length of a diagonal of the cube. Since the cube’s edge length is 8, its diagonal length is = , the radius of the circumscribed sphere is , and the volume of the sphere is . The ratio of the volumes reduces to = .

**Objective 0015**

Understand Euclidean geometry.

17. A teacher displays a large sheet of paper in the shape of a sector of a circle and then joins the straight edges with tape. This demonstration would be most appropriate for which of the following purposes?

- introducing a unit on platonic solids
- illustrating the lateral surface area of a cone
- showing that oblique and right solids have the same volume
- indicating the vertices and edges of polyhedra

- Answer and Rationale
**Correct Response: B.**

A sector of a circle is formed by a central angle and its subtended arc. When the sector is cut out of the circle and the straight edges are joined, the result is a cone. The area of the sector of the circle is the lateral area of the cone.

**Objective 0016**

Understand coordinate and transformational geometry.

18. **Use the diagram below to answer the question that follows.**

The diagram shows triangle A B C and triangle A prime B prime C prime plotted on an x, y graph. In triangle A B C, the coordinates of point A are 2, 4. In triangle A prime B prime C prime, the coordinates of point A prime are 6, 12. Both points lie on the line y = 2x. Points B, C, B prime, and C prime all lie on the line y = x over 2.

triangle*ABC* is enlarged to triangle*A'B'C'* by a point projection from the origin of the coordinate plane as shown in the diagram. What is the ratio of the area of triangle*ABC* to the area of triangle*A'B'C'*?

- 1:4
- 1:9
- 2:3
- 4:9

- Answer and Rationale
**Correct Response: B.**

A point projection, also called a dilation, increases or decreases measurements proportionally. Since each coordinate in (2, 4) is multiplied by 3 to get (6, 12), the ratio of linear measurements in the two triangles is 1:3. The ratio of the areas is the square of the ratio of the linear measures, or 1:9.

**Objective 0017**

Understand the process of collecting, organizing, and representing data.

19. **Use the normal model below to answer the question that follows.**

The model is a normal distribution graph of N mu sigma from mu minus 3 sigma to mu + 3 sigma. From mu minus sigma to mu + sigma is dimensioned as 68%. From mu minus 2 sigma to mu + 2 sigma is dimensioned as 95%. From mu minus 3 sigma to mu + 3 sigma is dimensioned as 99.7%.

A certain type of dry cat food is packaged by weight in 3-pound (48-ounce) bags. The machine used to fill the bags is set so that, on average, a bag contains 48.5 ounces. The company has determined that the weights of all bags are normally distributed with a standard deviation of 0.5 ounce. If the company ships 200 randomly selected bags to a pet store, about how many bags can be expected to contain less than 3 pounds of cat food?

- 16
- 27
- 32
- 45

- Answer and Rationale
**Correct Response: C.**

Weights are normally distributed with a mean of 48.5 and standard deviation 0.5, so 50% of all weights fall below 48.5 and 34% of the weights fall within one standard deviation below the mean, which is between 48 and 48.5. This leaves 16% of weights below 48. Of 200 randomly selected bags, about 16% of them or 0.16(200) = 32 can be expected to weigh less than 48 ounces.

**Objective 0018**

Understand probability and discrete mathematics.

20. If two numbers are selected at random and without replacement from the set {3, 5, 7, 11, 13}, what is the probability that their product is divisible by 7?

- Answer and Rationale
**Correct Response: C.**

If two numbers are selected without replacement, then there are 5 choices for the first number and 4 choices for the second number, which makes 20 pairs; however, each pair of numbers is counted twice, since choosing 5 first and 11 second, for example, is the same as choosing 11 first and 5 second. Thus, there are really 10 pairs to consider. Since 3, 5, 11, and 13 are not divisible by 7, the only products divisible by 7 are those which have 7 as a factor. There are 4 of these: 3 × 7, 5 × 7, 11 × 7, and 13 × 7. Thus, the probability that the product is divisible by 7 is = .