# TExES Math 4-8 (115) Ultimate Guide and Practice Test

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### TExES Math 4-8 Quick Facts

This test is for individuals planning to teach mathematics to upper elementary or middle school students. It assesses your understanding of mathematical concepts at an intermediate level.

**Cost:**

$116

**Scoring:**

The score received will range from 100-300 with a passing score of 240.

**Study time:**

This test is comprehensive in evaluating your understanding of mathematical concepts. In order to be successful on this exam, you must demonstrate a knowledge of the concepts and how to apply them. For example, a teacher must know how to calculate the area of a shape and how to teach students to use this knowledge to identify the area of an irregular object. When studying, be sure you understand the concepts as well as how to do the calculations. Spend time on vocabulary and practice calculations.

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

- You will receive a formula sheet for this exam. Be sure you know how to use these formulas and which problems call for a specific formula.
- There is an on-screen calculator for the test. You will not be able to bring your own calculator.
- You will receive a pencil and scratch paper from the testing center. You may not bring your own.
- After working through the test, you will see a screen with each question number and the answer you selected. This page also shows any questions you did not answer. Look through this carefully before submitting your exam.
- You are able to flag a challenging question and review it at the end. There is a button to click that will highlight a question on the screen at the end of the test.
- You can take snack or restroom breaks during the test, but your time does not stop. Plan accordingly.
- Many testing centers are cold so it’s best to dress in layers.

Information and screenshots obtained from Pearson.

## Domain I: Number Concepts

### Overview

**The Real Numbers**

Real numbers are all the numbers in the world that can be represented on a number line. Numbers not considered real numbers are called imaginary and result from seemingly impossible operations such as the square root of a negative number.

Real numbers are categorized as irrational or rational. Irrational numbers have a never-ending, non-repeating decimal point. Examples of irrational numbers include √2 and π. Rational numbers have a finite decimal or a repeating decimal.

Rational numbers are divided into subcategories. The first category is natural numbers. These are also called counting numbers as they are positive, whole numbers and the first numbers children learn to say aloud. Zero is not a natural number. The next category is whole numbers – all natural numbers and zero. The next group is integers – all whole numbers and all negative numbers without a decimal. Numbers with decimals or fractions are considered rational but do not fall into one of the other subcategories.

**Properties of Exponents**

Exponents are numbers that represent the power to which a number is raised and are expressed through multiplication. In the expression 83, the base number is 8 and the exponent is 3. The exponent is always written above and to the right of the base number. The example expression,83, is equal to 888 or 512. Because exponents represent multiplication, there are properties that allow expressions to be simplified.

- Product of Powers Property: If two identical bases are multiplied, the exponents can be added together.

3⁵ × 3³ = 3⁵⁺³ = 3⁸

- Product to a Power Property: If two numbers are multiplied together and raised to a power, they can separately be raised to the power and then multiplied.

(3 × 5)² = 3² × 5² because (3 × 5)² = 15² = 225 and 3² = 9 and 5² = 25 and 9 × 25 = 225

- Quotient of Powers Property: If two identical bases are divided, the exponents can be subtracted.

3⁵ ÷ 3³ = 3⁵⁻³ = 3²

- Quotient to a Power Property: If two numbers are divided and raised to a power, they can separately be raised to the power and then divided.

(12 ÷ 3)² = 12² ÷ 3² because (12 ÷ 3)² = 4² = 16 and 12²= 144 and 3²= 9 and 144 ÷ 9 = 16

- Power to a Power Property: If an exponent is raised to an exponent, they can be multiplied together.

(3³)⁴ = 3³ˣ⁴ = 3¹²

- Negative Exponents Property: If a base has a negative exponent, it represents the reciprocal, or inverse fraction, of the base raised to the same power.

3⁻⁵ = ⅓⁵

- Zero Power Property: Any base raised to the zero power equals 1.

3⁰ = 1

**Combinations and Permutations**

Combinations and permutations are used to solve problems involving lists or groups of numbers. Formulas for both provide a simplified counting technique that can be used when a selection is made from a larger group – for example, selecting pieces of candy from a bowl. A combination does not consider order, while a permutation does. A permutation would be to select three favorite candies in order, from eight in a bowl. A combination would be to select three pieces of candy from a bowl of eight candies.

Both formulas use the factorial function, represented by the symbol !:

5! = 5 x 4 x 3 x 2 x 1 = 120

In permutations, the formula is

where *n* is the number of available choices and *r* is the number of choices made. In the example above, *n* is 8 because there are 8 pieces of candy available and *r* is 3 because 3 pieces are selected. The numerator of the fraction represents all the choices. The denominator is the number of choices minus the number of pieces selected. Once the first piece of candy is chosen, there are only 7 pieces remaining for the second selection. After the second selection is made, there are only 6 pieces left.

Let’s look at the example and simplify it before solving.

Substitute numbers into the formula:

Complete the subtraction within the denominator parenthesis:

Write the expanded form:

Simplify:

8 x 7 x 6

Solve:

336

In combinations, order does not matter. In the example, picking 3 candies such as chocolate, strawberry, and caramel is the same as picking caramel, strawberry and chocolate. It does not matter what order is selected so the combination should only be counted once rather than twice.

The formula for solving a combination is

where *n* is the number of available choices and *r* is the number of choices made. In the example above, *n* is 8 because there are 8 pieces of candy to choose from and *r* is 3 because 3 pieces are selected. The numerator of the fraction represents all the choices. The denominator is the number of choices less the number of pieces selected, multiplied by the number of pieces because order does not matter.

Let’s look at the example to solve.

Substitute numbers into the formula:

Complete the subtraction within the denominator parenthesis:

Write the expanded form:

Simplify:

Simplify:

Solve:

56

## Domain II: Patterns and Algebra

### Overview

**Solving Systems of Linear Equations**

A linear equation is an equation written with the variables x and y which represent points on a line. Linear equations may have additional variables which increase the dimensions. A system of linear equations is two or more equations that work together. The equations must share at least one variable to be considered a system. The lines generally intersect at a point representing the answer. There are multiple ways to solve a system of equations. Let’s look at some of them using this example:

x + 2y = 8

x + 3y = 9

**Numerical method: **Subtract one equation from the other

Start: x + 3y – (x + 2y) = 9 – (8)

Combine like-terms: y = 1

Substitute: x + 2(1) = 8

Simplify: x + 2 = 8

Solve: x = 6

These lines intersect at (6, 1)

**Algebraic method:** Set equations equal to the same variable and then replace the variable to solve

Start: x + 2y = 8 and x + 3y = 9

Set both equations equal to x: x = 8 – 2y and x = 9 – 3y

Now set the equations equal to each other: 8 – 2y = 9 – 3y

Combine like-terms: -2y + 3y = 9 – 8

Simplify: y = 1

Substitute: x + 3(1) = 9

Solve: x = 6

The intersecting point is (6, 1)

**Using a table:** Create a table for each equation and find the matching point

The matching point for this system is (6, 1).

**Finding the Vertex of a Quadratic Function**

Quadratic equations are written in the format ax² + bx + c = y and form a line that is in the shape of a rounded ‘v’, referred to as a parabola. The vertex is the intersecting point, or where the two lines of the v meet. It may be at the top or bottom depending on which way the parabola opens. To find the vertex (h,k), use the formula: h = (-2b) / a. Then find k using the formula k = ah² + bh + c.

Example:

2x² + 5x + 2 = y

Identify the variables: a = 2, b = 5, c = 2

Find h: h = (-2 x 5) / 2

Solve for h: h = -5

Substitute for k: 2(-5²) + 5(-5) + 2 = k

Simplify: 50 – 25 + 2 = k

Solve: 27 = k

Vertex: (-5, 27)

**Real and Complex Roots**

Real roots occur when a function crosses the *x*-axis, and represent a solution to an equation that is also a real number. It can be positive or negative based on the value of *x *when the axis is crossed. The *y *value is always 0 for a real root. The easiest way to determine the real roots is to graph the function and examine where the line intersects the *x*-axis. There are ways to solve numerically with a graphing calculator, but since one is not provided or allowed for the test, that skill will not be tested.

Complex roots occur when a function with imaginary numbers crosses the *x*-axis, indicated by the symbol *i. *Use the quadratic formula to determine if the function has a complex root, which is proved by a negative square root.

The test may ask how many real roots a function has. To determine this, evaluate the number of times the sign changes.

Example:

f(x) = 4x³ – 2x² – 5x + 14

The sign changes two times: 4x³ – 2x² and -5x + 14

The number of sign changes show there can be at most two real positive roots.

Next find the negative roots by substituting -x for x, or: f(-x) = 4x³ – 2x² – 5x + 14

When the negative sign is distributed, it impacts variables raised to a negative power. The new equation is: -4x³ – 2x² + 5x + 14

The only sign change occurs at -2x² + 5x, so there is one negative real root.

**Connections to Calculus for Middle School Students**

Introducing students to calculus concepts in middle school builds a much stronger foundation for later learning. Introducing functions is one of the best ways to begin this process. By teaching students how to graph a line and linear functions, they prepare for higher-level applications later. This also makes the concept less intimidating as students progress in their knowledge.

## Domain III: Geometry and Measurement

### Overview

**Pythagorean Theorem**

The Pythagorean Theorem is a formula used with right triangles to relate the hypotenuse to the other two sides. The formula is a² + b² = c² where a and b are the legs and c is the hypotenuse.

To solve for c, substitute the numbers into the formula: 3² + 4² = c²

Simplify: 9 + 16 = c²

Add: 25 = c²

Square root: √25 = √c²

Solve: 5 = c

The test generally has more complicated problems, similar to the example shown below:

The flagpole outside of school is 10 feet tall and casts a shadow 8 feet long. The school building casts a shadow 25 feet long. The measurement from the top of the building to the top of the shadow is about 40 feet. To the nearest foot, how tall is the building?

There are two ways to solve this: Pythagorean Theorem or using a ratio.

**Pythagorean Theorem:**

The building’s shadow is 25 feet long and represents one leg. The hypotenuse is 40 feet long.

Substitute the numbers in the formula: 25² + b² = 40²

Simplify: 625 + b² = 1600

Isolate the variable: b² = 975

Square root: √b² = √975

Solve: b = 31.22

Answer = 31 feet

**Set a ratio** **of the flagpole to the building: **10/8 = x/25

Cross-multiply: 10 x 25 = 8x

Simplify: 250 = 8x

Divide: 250/8 = x

Solve: 31.25 = x

Answer: 31 feet

**Properties of Parallel Lines**

Parallel lines are lines that have the same slope and therefore, will never intersect. There are properties associated with parallel lines that occur when the lines are crossed by a transversal.

Congruent angles: Angles that have equal measurement

- Angles 1, 4, 5, and 8 are congruent
- Angles 2, 3, 6, and 7 are congruent

Pairs of angles: Angles that add to 180°.

- Angles 1 and 2, 3 and 4, 5 and 6, 7 and 8
- Angles 1 and 3, 2 and 4, 5 and 7, 6 and 8
- Angles 1 and 7, 2 and 8, 3 and 5, 4 and 6
- Angles 1 and 6, 2 and 5, 3 and 8, 4 and 7

Corresponding angles: Angles that occupy the same place on the parallel lines

- Angles 1 and 5
- Angles 2 and 6
- Angles 3 and 7
- Angles 4 and 8

Alternate angles: Angles opposite each other on the interior or exterior

- Alternate Interior Angles: Angles 3 and 6, Angles 4 and 5
- Alternate Exterior Angles: Angles 1 and 8, Angles 2 and 7

**Properties of Congruent Triangles**

Congruent triangles can be determined by five properties.

- SSS (side, side, side): If all three sides are congruent, then the triangles are congruent.
- SAS (side, angle, side): If two sides and the angle between them are congruent, then the triangles are congruent.
- ASA (angle, side, angle): If two angles and the side between them are congruent, then the triangles are congruent.
- AAS (angle, angle, side): If two angles and the non-corresponding side are congruent, then the triangles are congruent.
- HL (hypotenuse, leg): If the hypotenuse of a right triangle and one leg are congruent, then the triangles are congruent.

**Transformations**

## Domain IV: Probability and Statistics

### Overview

**Range and Interquartile Range**

Data sets can be explained through a variety of measures, most commonly by mean, median, and mode. Another important measure to know is range, or the difference between the largest and the smallest number in the set. This can show if the data is clustered or if there is an outlier. The interquartile range can be found by subtracting the median of the first quartile from the third quartile. Quartiles are identified by the numbers before and after the median of the full set of data. The smaller numbers are Q1, or the first quartile, and the larger numbers are Q3, or the third quartile.

Example: In the data set, find the range and the interquartile range.

Data Set: 43, 52, 24, 35, 83, 57

Arrange in numerical order: 24, 35, 43, 52, 57, 83

Find the range: 83 – 24 = 49

Find the interquartile range:

- Find the median: (43 + 52) / 2 = 95 / 2 = 47.5
- Q1 is 24, 35, 43 and Q3 is 52, 57, 83
- The median of Q1 is 35 and Q3 is 57
- To find the range, subtract: 57 – 35 = 22

On the test, the problems will be more complex but use the same XX. Here is an example of something similar to what can be expected on the test.

Data set: 65, 85, 90, 75, 95, 100, 100, 70, x

Given the data set, identify the value of x. The median of the data is 85 and the interquartile range is 25.

Arrange the data in numerical order: 65, 70, 75, 85, 90, 95, 100, 100, x

The value of x must be less than 85 to satisfy the median requirement.

The Q3 is 97.5 which makes the Q1 72.5, since the interquartile range is 25.

The median of 72.5 comes from the average of 70 and 75 since it is an even set of numbers. Therefore, the value of x must be either 70 or 75.

Consider the answer choices on the exam and choose the correct one. Note, both 70 and 75 would not be choices and only one option should exist.

**Percentiles and Quartiles**

Percentiles allow users to compare numbers. If a score is in the 90th percentile, it is higher than 90% of all scores. If the score is in the 45th percentile, it is higher than 45% of scores and lower than 55%. Quartiles are less specific than percentiles and place scores into groups based on 25, 50, 75 or 100. Scores in the 25th percentile or lower are the first quartile. Scores between the 26th and 50th percentile are the second quartile; 51st to 75th percentile are the third quartile and 76th to 100th percentile are the fourth or top quartile.

In the examples above, the score in the 90th percentile would be in the top quartile or the fourth quartile. The score in the 45th percentile would be in the second quartile.

**Bivariate Data Analysis**

Bivariate data analysis is simply determining the type and strength of the relationship between two variables. The most common graph used for this is a scatter plot as it allows each data set to be placed on a graph to analyze the overall trend. A line is drawn through the points that are close to the center of the data. The closer points are to the line, the stronger the relationship between the two variables. Lines that begin near the origin (0, 0) and trend to the right and upward show a positive correlation. Lines that start high and to the left and move downward to the right show a negative correlation.

## Domain V: Mathematical Processes and Perspectives

### Overview

**Inductive and Deductive Reasoning**

A conjecture is an educated guess based on the presented information. Conjectures are reached in two ways: inductive reasoning or deductive reasoning. Inductive reasoning uses a pattern or trend to make a guess about a future event, but it is not proven true until the event occurs. If a person randomly draws marbles from a bag and the first three are blue, a conjecture may state that the next marble will be blue based on inductive reasoning. Deductive reasoning uses facts to conclude other facts. In the example above, if the person pulling marbles knows there are three blue marbles and seven red marbles, deductive reasoning makes it clear that the next marble pulled will be red.

**Types of Taxes**

Taxes come in a variety of forms: federal income, state income, property, and sales. Federal income tax is levied by the IRS on income earned in the United States by people, businesses, and trusts. The rate paid varies based on the amount of income earned throughout the year. State income tax is levied by the state on all people and businesses that earn money in that state. In Texas, there is no income tax, however, property taxes are much higher than in other states. Property taxes are levied by the state and city governments on owned property. In most states, this includes larger items such as real estate and vehicles. Sales tax is levied on most purchasable items by state and city governments. The rate varies from city to city, but generally runs between 5 and 10%.

## Domain VI: Mathematical Learning, Instruction and Assessment

### Overview

**Manipulatives**

Manipulatives are objects used to represent numbers. They can be small blocks that lock together or small toys used for counting. In grades 4-8, there are common manipulatives used to teach specific concepts.

Algebra Tiles: There are a variety of tiles that represent different variables and powers. This includes large squares, long rods, and small squares.

Fraction Tiles: These are sets that start with one long rod that represents a whole. Additional rods divided into even parts represent fractions and they are the same length as the whole rod. For example, the rod cut into 5 pieces represents ⅕ of the long rod.

Polyhedral dice: These are dice that come in 4, 6, 8, 10, or 20 sided varieties. Students can use these to practice problems with probability.

**Math Instructional Theory**

Instructional theories related to math include behaviorism, cognitivism, and constructivism. Let’s look at each one and its relationship to math instruction.

- Behaviorism: Students learn based on the immediate rewards and consequences related to their actions. In math, students are likely to complete work when they feel successful and less likely when faced with challenges they cannot complete easily or successfully. Giving students problems that progressively increase in difficulty allows for incremental successes that build on prior skills and encourages them to continue working.
- Cognitivism: Students learn by actively participating in the learning process. After receiving input from the teacher, their brains produce a product or new thinking as they assimilate the information. In math, students need opportunities to participate in learning. Teachers should use manipulatives and concrete examples when teaching a new concept. . From there, students can move to more abstract concepts, but must remain engaged in the learning. During a lecture, for example, have students participate in solving problems rather than providing the entire problem and solution on the board.
- Constructivism: Students learn by exploration, such as when the teacher asks a question and allows students to discover the answer for themselves. In math, a teacher might ask students how to measure a car. Rather than presenting students with rulers and measurement steps, students might instead work in groups to discuss ideas about how to measure and what tools to use.

**Formative and Summative Assessments**

Formative assessments are used to inform instruction. They help the teacher plan future lessons and identify what is working and what students understand. A formal assessment could include a weekly quiz or graded assignment that reviews what has been taught. An informal assessment might be giving students an exit ticket or asking them to briefly write an explanation about a concept. The teacher can then individually assess each student.

Summative assessments are formal assessments held at the end of a study unit. They give the teacher an understanding of students’ overall retention level of the covered content.

And that’s some basic info about the test.