# TExES Mathematics 7-12: Ultimate Guide and Practice Test

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### TExES Mathematics 7-12 Quick Facts

The TExES Mathematics 7-12 exam is designed to assess whether a beginning teacher in the state of Texas has the necessary knowledge and capability to teach mathematics in the seventh through twelfth grades.

**Cost: **

$116

**Scoring: **

The score range is from 100 to 300. A score of 240 is needed to pass.

**Pass rate: **

80%

**Study time: **

In order to pass, you should break the test topics down into your strengths and weaknesses. Give yourself enough time to work on each topic until you feel confident with the majority of the subjects. Create a study plan detailing the time you need to study each topic and the days you plan to do that.

**What test takers wish they would’ve known: **

- An onscreen definitions and formulas sheet is provided.
- You must bring your own graphing calculator.
- You should guess if you don’t know the answer.

Information and screenshots obtained from the National Evaluation Series website: http://www.tx.nesinc.com/PageView.aspx?f=GEN_Tests.html

## Exam Content

### Overview

The exam has six domains:

- Number Concepts (14%)
- Patterns and Algebra (33%)
- Geometry and Measurement (19%)
- Probability and Statistics (14%)
- Mathematical Processes and Perspectives (10%)
- Mathematics Learning, Instruction, and Assessment (10%)

So, let’s talk about Number Concepts first.

### Number Concepts

The Number Concepts domain has about 14 questions. These questions account for 14% of the entire exam.

This domain can be neatly divided into 3 competencies:

- The Real Number System
- The Complex Number System
- Number Theory Concepts and Principles

Let’s talk about a concept that you will more than likely see on the test.

**Subsets of the Real Number System**

### Patterns and Algebra

The Patterns and Algebra domain has about 33 questions. These questions account for 33% of the entire exam.

This domain can be neatly divided into 7 competencies:

- Patterns
- Functions, Relations, and Their Graphs
- Linear and Quadratic Functions
- Polynomial, Rational, Radical, Absolute Value, and Piecewise Functions
- Exponential and Logarithmic Functions
- Trigonometric and Circular Functions
- Calculus

Here are some concepts you will see on the test.

**Compound Interest**

To calculate continuous compound interest, use the formula below:

After *t* years, the balance *A* in the account with principal *P* and annual interest rate *r* is given by the equation:

The principal *P* is the amount in the account at the beginning of the period. The rate *r* should be written as a decimal.

For example, find the accumulated value of an investment of $4,000 for 8 years at an interest rate of 3.4% compounded continuously.

**Finding Inverse Functions**

**Solving Quadratic Equations**

Quadratic equations are equations where the variable has degree 2.

Let’s solve the quadratic equation:

5x² + x = 2

Some quadratic equations can be solved by factoring; however, this equation can only be solved algebraically by using the quadratic formula or by graphing. We will solve using the quadratic formula.

To use this method, first put the equation into the form ax² + bx + c = 0. To do this, subtract 2 on both sides of the equation:

5x² + x – 2 = 0

Now we can see that *a* = 5, *b* = 1, and *c* = -2. We will plug these coefficients into the quadratic formula:

### Geometry and Measurement

The Geometry and Measurement domain has about 19 questions. These questions account for 19% of the entire exam.

This domain can be neatly divided into 4 competencies:

- Measurement
- Euclidean Geometry as Axiomatic Systems
- Euclidean Geometry Results, Uses, and Applications
- Coordinate, Transformational, and Vector Geometry

Take a look at these concepts.

**Finding Arc Length**

To find the length of an arc of a circle, we need to determine the ratio of the entire circle that it encompasses. The formula for the circumference of a circle is 2𝜋*r*, where *r* is the circumference of the circle; therefore, the length of an arc is a proportion of that.

Arc length = 2𝜋*r*C360°, where *C* is the measure of the central angle in degrees.

Look at the circle below. What is the arc length of arc *BC*?

**Angle Relationships**

Angle relationships help us to find missing angles when we are given a diagram, often involving parallel and intersecting lines.

Some important angle relationships are supplementary, complementary, vertical angles, alternate interior angles, and alternate exterior angles.

Supplementary angles are angles whose measures add to 180 degrees. If the angles are next to each other, they will form a straight line.

Complementary angles are angles whose measures add to 90 degrees. If the angles are next to each other, they will form a right angle.

For example, if ∡a=48° and ∡b=42°, then angle *a* and angle *b* are complementary since they sum to 90°.

Vertical angles are opposite each other on other sides of two intersecting lines. Vertical angles are always congruent (have equal measure).

Also, angle *a* and angle *d* are supplementary, as are angle *a* and angle *b*. Other supplementary pairs are:

Angle *b* and angle *c*

Angle *c* and angle *d*

A transversal is a line that cuts through at least two other lines.

When we have transversals that cut through more than one line, they form alternate angles. Alternate interior angles are on opposite sides of the transversal but inside the two lines being cut by the transversal.

*NOTE: When the two lines cut by the transversal are parallel to each other, then alternate interior angles are always congruent.*

Likewise, alternate exterior angles are congruent if the two lines cut by a transversal are parallel. These are angles on the outside of the two lines, but opposite each other.

Angle *e* and angle *h* are therefore congruent since they are alternate exterior angles. Angle *g* and angle *f* are also alternate exterior angles and therefore congruent.

**Pythagorean Theorem**

The Pythagorean Theorem is used to find the missing side of a right triangle.

For the right triangle, where *a* and *b* are the shorter sides of the triangle, and *c* is the hypotenuse, then *a² **+ **b**² **= **c**²**.*

Example 1: Suppose we know one side of a right triangle is 10 cm, and the hypotenuse of the triangle is 26 cm. Find the remaining side.

Let *a* = 10 cm and *c* = 26 cm. Then we will use the Pythagorean Theorem to find *b*:

*a² **+ **b**² **= **c**²*

*10² **+ **b**² **= **26**² *

*b² **= **26**² **– **10**² **= *576

*b = √**576 **= *24

Therefore, the remaining side of the triangle is 24 cm.

Example 2: Suppose two people start walking from the same point. After 1 hour, Person A has walked 2.8 miles due north. Person B has walked 3.2 miles due east. Find the distance between them at this time.

Since the two people are walking at right angles to each other, we can apply the Pythagorean Theorem. In this case, the distance between Person A and Person B is the hypotenuse of a right triangle.

Let *a* = 2.8 and *b* = 3.2. Then *a² **+ **b² **= **c**² *gives:

2.8*² *+ 3.2*² *= c*²*

18.08 = c*²*

*c* ≈ 4.25

Therefore, Person A and Person B are approximately 4.25 miles apart after 1 hour.

### Probability and Statistics

The Probability and Statistics domain has about 14 questions. These questions account for 14% of the entire exam.

This domain can be neatly divided into 3 competencies:

- Analyzing and Representing Data
- Probability
- Probability, Sampling, and Statistics

Let’s talk about some concepts that you will more than likely see on the test.

**Measures of Central Tendency**

Measures of central tendency help us to determine how data is distributed. We will consider several measures of central tendency below.

The mean of a data set is the average value of a data set. This can be found by adding together all of the values and dividing by the total number of values.

The mode of a data set is the value that occurs most frequently in the data set. It is possible to have more than one mode in a data set if several values occur the most.

The median of a data set is the middle value in the set. If there is an even number of data points, there will not be an exact middle. In this case, the median is found by taking the average of the two data points closest to the middle.

For example, suppose that the ages for a group of ten students were collected and are listed below:

9, 11, 13, 11, 8, 7, 13, 9, 9, 12

The mean of this data set can be found by adding all of these ages together and dividing by 10, since there are 10 students:

To find the mode and median of a data set, it is helpful to reorder the set from lowest to highest.

7, 8, 9, 9, 9, 11, 11, 12, 13, 13

Now we can see that the mode of the data set is 9, since 9 occurs 3 times, which is more than any other data point.

The middle of the data set is between the two 11’s in the middle of the set. Therefore, the median of the set is 11.

An outlier is a data point that is far outside of the normal range of the data set. It is far away from the rest of the data points. For example, suppose that some daily high temperatures in the month of May for a particular area are given below in degrees Fahrenheit:

66, 56, 61, 45, 48, 52, 23, 66, 53, 58, 59

Reordering gives 23, 45, 48, 52, 53, 56, 58, 59, 61, 66, 66

The mean of this data is (66 + 56 + 61 + 45 + 48 + 52 + 23 + 66 + 53 + 58 + 59) / 11 = 53.364

The mode of the data is 66 since this value occurs twice.

The median is 56, since that is the middle value.

However, the temperature of 23 degrees is an outlier, because there is a 22-degree difference between it and any other temperature. Therefore, we can remove this outlier from the data and calculate the mean, mode, and median again to get a better description of the central tendencies of this data set.

45, 48, 52, 53, 56, 58, 59, 61, 66, 66

After the temperature of 23 is discarded, the new mean is 56.4. The mode of the data is still the same in this case since 66 is the only temperature that occurred more than once. The median is now the average of 56 and 58, which is 57 degrees since now there are only ten data points.

**Scatterplots**

A scatterplot is a graph made up of points in the *x*–*y* plane, that show a relationship between two variables *x* and *y*.

If the points go up as *x* increases, then there is a positive correlation between the two variables. For example, there is a positive correlation between the temperature outside and ice cream sales, since as it gets hotter, ice cream sales increase.

If the points go down as *x* increases, then there is a negative correlation between the two variables. For example, there is usually a negative correlation between the number of absences a student has and their grade, since as the absences increase their grade usually decreases.

If the points on the scatterplot don’t follow any pattern as *x* increases, then there is no correlation between the variables. Here are some examples of scatterplots below:

By finding a line of best fit on a scatter plot, predictions can be made about future data points.

For example, a line of best fit is shown for the scatterplot below:

Using this line, we can estimate that when the temperature is 21 degrees Celsius, ice cream sales are around $460.

### Mathematical Processes and Perspectives

The Mathematical Processes and Perspectives domain has about 10 questions. These questions account for 10% of the entire exam.

This domain can be neatly divided into 2 competencies:

- Mathematical Reasoning and Problem Solving
- Connections and Communication

Take a look at these concepts that are likely to appear on the test.

**Problem Solving Process**

The problem-solving process can be broken into steps in various ways, but the basic idea is the same no matter how you state it.

- Re-read the problem. Determine what the question is asking for.
- Determine what information you already know by finding the key details in the problem.
- Choose a strategy to solve the problem.
- Solve the problem.
- Check your answer and make sure that it makes sense in context.

Some problem-solving strategies for mathematics include working backward, looking for a pattern, making an estimate, guess and check, and drawing a picture.

Let’s look at a problem together.

Penny bought some cookies at a bake sale. She ate 3 of them and gave 2 of them to her sister. After dividing the rest equally with her brother, she had 8 cookies remaining. How many cookies did she start with?

This problem can be solved by working backward from the final number of cookies she has at the end of this process. She ends with 8 cookies, which was half of what she had before she shared with her brother. At that point, she had 16 cookies. She gave 2 cookies to her sister. So, she had 18 cookies before doing that. She ate 3 cookies first; therefore, she started with 21 cookies.

Guess and check works well with finding the roots of polynomial functions. In order to determine if a value is a root of the function, you must make a reasonable guess, and then apply synthetic division to determine if the remainder of the division is 0. If so, then you have found a root.

**Formal versus Informal Reasoning**

Formal reasoning, also called deductive reasoning, starts with premises that are known to be true and continues towards a logical conclusion.

For example, humans breathe air. David is a human; therefore, David breathes air.

Informal reasoning, also called inductive reasoning, starts with specific observations and combines them to make broad generalizations.

For example, you notice that Julia shows up late to work. Julia is a teenager; therefore, you conclude that teenagers are irresponsible.

Conclusions reached using inductive reasoning can be wrong sometimes, but this type of reasoning is still important. Inductive reasoning is most often used to form hypotheses, which can then be tested using the scientific method.

### Mathematics Learning, Instruction, and Assessment

The Mathematics Learning, Instruction, and Assessment domain has about 10 questions. These questions account for 10% of the entire exam.

This domain can be neatly divided into 2 competencies:

- Mathematics Learning and Instruction
- Mathematics Assessment

Let’s talk about some concepts.

**Inquiry-Based Learning**

Inquiry-based learning is about sparking a student’s curiosity through exploration and discovery. This kind of learning is not instructor-led; instead, the teacher helps facilitate learning by careful scaffolding.

In mathematics, this might look like working in small groups to explore a problem and share ideas with one another, or it might involve the teacher posing a problem and the entire class brainstorming together to come up with a solution.

For example, the teacher might present the diagram below and ask the students to make their own observations and pose their own questions about the figures.

Students may ask questions like:

*“What fraction of each of the shapes is shaded?”*

*“How can you find the area of each of the shaded regions?”*

*“Is the triangle equilateral?”*

The teacher can use scaffolding strategies and give students hints (if needed) to help answer these questions and others.

**Formative versus Summative Assessments**

Formative assessment is given regularly throughout the school year to provide ongoing feedback and help teachers determine concepts that need to be covered in more detail. Formative assessments also help students identify where their own understanding can be improved. Formative assessment is usually lower stakes and can include very informal tasks such as games, projects, and group work, as well as writing a summary about the main point of the lesson that day or submitting an outline of a paper before writing it.

Summative assessments are usually given at the end of a chapter, unit, or course to determine how much the student has learned and retained. These types of assessments are usually higher stakes and include tests, quizzes, final papers, and cumulative projects.

And that’s some basic info about the exam.

## Exam Content Practice Test

#### Question 1

What is the sum of the first 100 terms in the sequence of perfect square integers (1 + 4 + 9 + 25 +… +a₁₀₀)?

- 189,000
- 450,000
- 500,000
- 338,350

Correct answer: **4. **The sum of a series of perfect square integers can be found by using the following formula:

Sum_(perfect squares) = [n(n + 1)(2n + 1)] / 6

= [(100(101)(201)]/6

= 338,350

#### Question 2

If $1,000 is deposited into an annuity account each year for 10 years, about how much will the annuity be worth at the end of the ten-year period if the interest rate is 8%?

- $15,000
- $15,500
- $16,000
- $1,500

Correct answer: **2. **Use the annuity formula:

Total Amount = {A[(1+r) ⁽ᴺ ⁺ ¹⁾ – 1] ÷ r} – A

Substitute the values: A = $1,000, r = 8% = 0.08, t = 10 years

Total Amount = {1,000[(1+.08) ⁽¹⁰ ⁺ ¹⁾ – 1] ÷ .08} – 1,000 ≈ 15,645

Of the choices presented, $15,500 is the closest to $15,645 and is the correct answer.

#### Question 3

To which subset(s) does -4 belong?

- Negative integers
- Negative integers, real numbers
- Negative integers, integers, rational numbers, real numbers
- Negative integers, rational numbers

Correct answer: **3. **If a number is a negative integer, that number is automatically a member of all preceding subsets. This means that -4 is a negative integer, an integer, a rational number, and a real number.

#### Question 4

What is the equation of the graph pictured below?

- y = ½(x + 1)² + 4
- y = 2(x + 1)² + 4
- y = ½(x – 1)² + 4
- y = (x – 1)² + 4

Correct answer: **1. **y = (x – 1)² + 4 is incorrect. This equation is in vertex form and has a vertex of (1,4). The graph pictured has a vertex of (-1,4).

y = 2(x + 1)² + 4 is incorrect. Although this parabola has the correct vertex, the point (-3,6), which is a point labeled on the graph pictured, is not a solution to the quadratic 6 ≠ 2(-3 + 1)² + 4; 6 ≠ 8

y = ½(x – 1)² + 4 is incorrect, because it has a vertex at (1,4).

#### Question 5

If logₓ 5 = 0.864 and logₓ 2 = 0.214, what is the value of logₓ 20?

- 1.070
- 1.292
- 3.456
- 1.942

Correct answer: **2. **Since logₓ 20= logₓ (2∙2∙5), this means that:

logₓ 20 = logₓ 2 + logₓ 2 + logₓ 5

logₓ 20 = 0.214 + 0.214 + 0.864

= 1.292

#### Question 6

Which of the following equations might occur in the process of solving the equation: (5²ˣ⁻³)⁻² = 25⁻³ˣ ⁺ ⁸?

- -4x + 30 = -6x + 80
- 4x – 3 = -3x + 8
- 6 – 4x = 16 – 6x
- -4x – 5 = -6x +10

Correct answer: **3. **(5²ˣ⁻³)⁻² = 25⁻³ˣ ⁺ ⁸ can be simplified to 5⁻⁴ˣ ⁺ ⁶ = 5⁻⁶ˣ ⁺ ¹⁶ → -4x + 6 = -6x + 16.

#### Question 7

What is the horizontal asymptote for the following function? f(x) = 2ˣ⁻³ + 5?

- y = -3
- y = 5
- y = -5
- y = 3

Correct answer: **2. **The constant term, 5, tells us that the horizontal asymptote is y = 5. As x approaches -∞, f(x) approaches but never reaches 5, because 2ˣ⁻³ approaches but never reaches 0.

#### Question 8

Which of the following models an exponential decay?

- f(x) = 0.5(1.2)ˣ
- f(x) = 0.5(2)ˣ
- f(x) = 2(0.5)ˣ
- f(x) = 2(1.5)ˣ

Correct answer: **3. ** f(x) = 2(0.5)ˣ is the correct answer, because the base of the exponential term, (0.5), is less than 1.

#### Question 9

The function f(x) = x³ is scaled by -2. Which of the following is a valid description of the resulting graph?

- This transformation causes the graph of f(x) to shift horizontally by 2 units to the left
- This transformation causes the graph of f(x) to compress by a factor of 2 in the x direction
- This transformation causes a reflection of f(x) over the y-axis and x-axis and a compression by a factor of 2 in the x-direction
- This transformation causes a reflection of f(x) over the x-axis and compression by a factor of 2 in the x-direction

Correct answer: **4. **When a factor is applied to the x-term of a function, the graph will compress or expand in the x-direction; it will become “thinner” or “wider.” If the factor k is 0 < k < 1, then the function will expand. If k > 1, the function will compress. A negative factor will cause the graph to reflect over the x-axis.

#### Question 10

Look at the graphs and their functions above.

Which function is neither even nor odd?

- I
- II
- III
- IV

Correct answer: **3. **III is neither even nor odd. Although this graph does have a line of symmetry, that line is not the y-axis. And so, the left plane and the right plane are not mirror images of each other.

#### Question 11

Use the functions below to answer the question:

f(x)= x + 8

g(x) = x² + 1

h(x) = 2x –1

j(x) = x³

Find the product: (h•g•j)(x).

- 2x⁶ – x⁵ + 2x⁴ – x³
- 2x⁹ – x⁶ + 2x⁴ – x³
- 2x⁶ – x³
- x⁵ + 2x⁴

Correct answer: **1. **

(h•g•j)(x)

= h(x)•g(x)•j(x)

= (2x – 1)(x² + 1)(x³)

= (2x³ + 2x – x² –1)(x³)

= 2x⁶ – x⁵ + 2x⁴ – x³

#### Question 12

Which of the following statements about functions and their inverses is true?

- The graph of the inverse of the function f(x) is the reflection of f(x) over the x-axis
- All functions have inverses
- The composite of f(x) and its inverse is x
- The product of a function and its inverse is 1

Correct answer: **3. **The graph of the inverse of a function f(x) is the reflection of f(x) over the line y=x. Only functions whose graphs pass both the horizontal and vertical line tests have inverses. The product of a function and its reciprocal is 1. Reciprocals and inverses are not the same.

#### Question 13

What is the derivative of f(x) = √(4x+5) in simplest form?

- 2/√(4x + 5)
- (½)[1/√(4x + 5)] = [1/√(4x + 5)](½)
- 4/√(4x + 5)
- ½(4)(4x + 5)⁻¹/² = 2(4x + 5)⁻¹/²

Correct answer: **1. **Use the chain rule to find the derivative of this function. Let u = (4x + 5). Then f(x) = u¹/² and f’x =

½ u⁻¹/²du

= ½ (4x + 5)⁻¹/²(4)

= 2/√(4x + 5)

#### Question 14

If f'(x) = 9x² – 4x + 5, what is f(x)?

- f(x) = 3x² – 2x + 5 + C
- f(x) = 18x – 4 + C
- f(x) = 3x³ – 2x² + 5x + C
- f(x) = 9x³ – 4x² + 5x + C

Correct answer: **3. **If f(x) = 18x – 4, then f’(x) = 18. This is not what f’ was given to be.

If f(x) = 9x³ – 4x² + 5x, then f’(x) = 27x² – 8x + 5. This is not what was given for f’(x).

If f(x) = 3x² – 2x + 5, then f’(x) = 6x – 2; this is not the value for f’(x) that was given.

#### Question 15

At what point will the tangent to the curve, y = x² – x, be parallel to the line y = x?

- (0, 0)
- (0, 1)
- (1, 0)
- (1, 1)

Correct answer: **3. **The first derivative, f’(x) = 2x – 1, gives the slope of the tangent line as 2. We want to know when the slope of the tangent line will be 1 (the slope of the line y = x is 1 and all lines parallel to the line y = x must have the same slope). So, when will 2x – 1= 1? → 2x – 1 = 1, 2x = 2, x = 1. So, return to the original equation and find the value of y when x = 1: y=x² – x = (1)² – 1 = 0: y=0. Therefore, the line tangent to the curve y = x² – x will be parallel to y + x at the point (1,0).

#### Question 16

What is the measure of arc AFD?

- 170°
- 190°
- 135°
- It cannot be determined

Correct answer: **2. **m∠AXD = mBC + mAFD2. Substitute known values: 5° = 80 + mAFD2; mAFD = 190°.

#### Question 17

If an interior angle of a regular polygon measures 135˚, how many sides does the polygon have?

- 7 sides
- 8 sides
- 6 sides
- 9 sides

Correct answer: **2. **If an interior angle measures 135˚, the exterior angle that is its linear pair is equal to 45˚. Since the sum of the exterior angles of a polygon, one at each vertex, is 360˚, then 360˚/45˚ = 8. There are 8 sides in this polygon.

#### Question 18

What is the area of the sector formed by a 40˚ central angle in a circle with a radius of 3 cm? (Round your answer to the nearest tenth.)

- 3.1 cm²
- 5.3 cm²
- 4.8 cm²
- 2.4 cm²

Correct answer: **1. **The area of this circle is π(3)² = 9π. This sector represents a portion of the total area: (40/360)(9π) = π. Since the approximate value of pi is 3.14, 3.1 cm² is the correct answer.

#### Question 19

The measure of angle 1 is:

- 80°
- 45°
- 20°
- 40°

Correct answer: **3. **Angle 1 is an inscribed angle. If m∠1 = 40, then its inscribed angle would measure 80°. However, in this figure, arc AB is 40°. If m∠1 = 80, then its intercepted arc would have a measure of 160°. This contradicts the fact that arc AB has a measure of 40°. If m∠1 = 45, then its intercepted arc would have a measure of 90°. This contradicts the fact that arc AB has a measure of 40°.

#### Question 20

For what value of x would a and b be parallel?

- 30
- 20
- 10
- 40

Correct answer: **4. **If a and b are parallel, then (x + 15) and 125° would be supplementary angles, because the angles adjacent to and below the 125° angle correspond to the (x + 15)° angle. This means that 125 + (x + 15) = 180 → x + 15 = 55 → x = 40°: This choice is the correct answer.

#### Question 21

Two hikers start from the same campsite for a morning hike. Hiker A walks at a rate of 2.5 miles per hour due north, while hiker B walks at a rate of 4 miles per hour due east. After two hours of hiking, how far apart are the two hikers?

- About 9.5 miles
- About 9 miles
- About 10 miles
- About 13 miles

Correct answer: **1. **Draw a picture and label it. The distance that we are trying to find is the distance directly from A to B, the “hypotenuse.” 5² + 8² = (AB)² → 25 + 64 = 89 = (AB)² → AB = √89 ≈ 9.43 → about 9.5 miles.

#### Question 22

When asked to find the slope of the line through the points (5, -2) and (-3, -1), a student determined that the slope was equal to 8/(-1) = -8. Which of the following best describes this student’s response?

- The student’s answer is correct
- The student added integers incorrectly
- The student subtracted integers incorrectly
- The student has confused the x and y values in determining the slope

Correct answer: **4. **This student found the slope by doing the following work: (5 – (-3))/(-2 – (-1)) = 8/1. The student confused the formula for finding the slope: (y₂ – y₁)/(x₂ – x₁) = (-1 – (-2))/(-3 – 5) = 1/-8.

#### Question 23

75, 82, 68, 95, 74, 72, 91, 60, 72, 80

What is the mean, median, mode, and range for the data above?

- Mean: 76.9; Median: 74.5; Mode: 72; Range: 31
- Mean: 76.9; Median: 74.5; Mode: 72; Range: 35
- Mean: 77; Median: 73.5; Mode: 72; Range: 31
- Mean: 80; Median: 73; Mode: 72; Range: 35

Correct answer: **2. **Begin by rearranging the data in order from least to greatest: 60, 68, 72, 72, 74, 75, 80, 82, 91, 95. The median is the average of the two middle terms since there is an even number of data: 74 + 75 = 149; 149/2 = 74.5. The range should be 35 (95 – 60).

#### Question 24

Based on the box-and-whisker plot above that represents scores on a recent trigonometry test, which of the following statements is true?

- This data has a positive skew
- As many students scored between 65 and 97 as those who scored between 97 and 100
- Half of the students scored below 70
- Only one student had a perfect score (100) on this test

Correct answer: **3. **In a box plot, the vertical lines represent the medians of the data and each quartile, so the median of all of the data is 70, which means half of the students scored above 70 and a half scored below 70. This choice is the correct answer.

#### Question 25

James and Mary are tossing a coin, rolling a pair of dice, and summing the faces shown. What is the probability that Mary will roll a 12 and toss heads?

- 1/2 • 1/36 = 1/72
- 1/2 • 1/11 = 1/22
- 1/2 • 1/18 = 1/36
- 1/2 • 1/12 = 1/24

Correct answer: **1. **1/2 • 1/11 = 1/22 is incorrect because not all sums are equally likely. This choice assumes the probability of rolling a 12 is 1/11 since 11 unique sums are possible. But the probability of rolling any one of the unique sums varies.

1/2 • 1/12 = 1/24 is incorrect because the probability of rolling a sum of 12 is not 1/12.

1/2 • 1/18 = 1/36 is incorrect because the probability of rolling a sum of 12 is not 1/18.

#### Question 26

Identify the correct sequence of prerequisite skills required for solving AND graphing the problem: y ≥ -x² + 9.

- Plotting points on a Cartesian plane, solving equations
- Solving quadratics, graphing quadratics, graphing inequalities on a Cartesian plane
- Solving inequalities, solving systems of equations, using a graphing calculator
- Factoring and finding square roots, graphing radical expressions, graphing on a Cartesian plane with and without a calculator

Correct answer: **2. **This choice is correct. BEFORE graphing this inequality, students need to have these prerequisite skills.

#### Question 27

A student, Isaac, has made the following conjecture:

If p is an odd integer and p = a + b where a and b are also integers, then a and b are also odd.

Which of the following is a counterexample that another student could use to show Isaac that his conjecture is incorrect?

- 17 = 5.5 + 11.5
- 9 = 24 + 6.6
- 6 = 1 + 5
- 21 = 12 + 9

Correct answer: **4. **17 = 5.5 + 11.5 is incorrect, because the addends are not integers.

9 = 24 + 6.6 is incorrect because only one of the addends is an integer.

6 = 1 + 5 is incorrect, because the sum is an even integer.

#### Question 28

An eighth-grade teacher is beginning a unit on multiplication of fractions. Which of the following is the least appropriate way to model this process?

- Model the process with arrays
- Model the process with pattern blocks
- Model the process with base-10 blocks
- Model the process by drawing a picture

Correct answer: **3. **Using arrays is an excellent way to model multiplication of fractions. Pattern blocks are an excellent manipulative to use for modeling the multiplication of fractions. Drawing pictures is another excellent way to model the multiplication of fractions.

#### Question 29

Which of the following activities would best support an algebra student’s development of a linear relationship?

- Weigh an empty beaker and record the weight. Then add 5 ml of water, weigh again, and record the amount of water in the beaker and the weight. Continue with this process until 10 measures and weights have been taken and recorded. Graph the results.
- Start with one penny, record trial # 1 … 1 penny. Then, add two pennies to your total and record trial #2 … (1 + 2) = 3 pennies; add four pennies, record trial #3 … (1 + 2 + 4) = 7 pennies, the total amount of money you have. Then add 8 pennies, record trial #4 … (1 + 2 + 4 + 8) = 15. At each trial, the number of pennies added is two times more than the number added on the previous trial. Complete at least 8 trials. Graph the results.
- If y = 1/x, choose values for x and find the corresponding value for y. In a two column chart, record the values you have chosen for x and what you have found to be the value for y. Choose 10 values for x and find the corresponding values for y. Graph your results.
- All of the above would be appropriate linear activities.

“If y = 1/x…” is incorrect. This is an inverse proportion and its graph is that of a rational function in quadrants I and III.

Correct answer: **1. **“Start with one penny…” is incorrect. This situation describes an exponential growth, not a linear one.

#### Question 30

Maria has recently moved from Mexico City to the U.S. She is an 8th-grade student who speaks little English but who came from Mexico City with excellent grades. What would be an appropriate accommodation for Maria’s math teacher to use with Maria?

- Repeat the instructions that are given to the rest of the class more slowly and privately to Maria
- Make sure that Maria has all the materials she needs to complete the tasks assigned
- Pair another student who speaks Spanish with Maria to clarify instructions in Spanish as needed
- Allow Maria to be an observer in math class for a few days until she feels a bit more at ease

Correct answer: **3. **This choice is correct. If possible, pairing Maria with a student who speaks Spanish would be an excellent strategy.