The PRAXIS II Middle School: Mathematics Exam (5169) is designed for individuals, who would like teach mathematics at the middle school level. You will be given two hours to complete this 55 multiple-choice and numeric entry questions exam. There will be 34 questions regarding arithmetic and algebra with the 21 other questions being about geometry and data.

Discrete Mathematics, Statistical Concepts, Probability and Data

In this section of the exam you will be required to solve problems by: finding and interpreting common measures of dispersion, standard deviation, outliers, range and spread of data; finding and interpreting common measures of central tendency such as mean, median and mode. You will also be asked to solve probability problems by using counting techniques, by using geometric probability, and solve probability problems that involve finite sample spaces. Your ability to analyze data in charts, line bars, circles, histograms, tables, graphs, scatterplots, stemplots, and draw conclusions from that data.

Functions and their Graphs

This section of the exam will assess your knowledge of algebraic expressions, equations, domain values in ranges of a function, usage of a mathematical model for two different situations, and notations and functions. You will also be given a set of conditions and required to identify if a graph in the plane is the graph of a function. Your ability to utilize a graph and to choose an equation to represent the graph will be assessed. Questions regarding sketching a graph representing exponential or quadratic function, absolute value, step, or linear function will also be assessed.

Measurement and Geometry

You will be required to solve problems in metric and English systems in this section of the exam. Your ability to solve problems regarding the properties of polygons in the plane, points, lines, translations, rotations, reflections, angles, sectors, arcs, circles, squares, rectangles, trapezoid’s, parallelograms, triangles, perpendicular lines, and two and three dimensional objects will be assessed in this portion of the exam. Your ability to compute the parameter and area of circles, triangles, quadrilaterals and regions within these figures and; compute the surface area and volume of spheres, solids, cylinders, cones, prisms and the combination of these figures will be assessed. Questions regarding the application of the Pythagorean theorem to solve problems that include isosceles and equilateral triangles will be included in this section of the exam.

Basic Algebra and Mathematics

In this section of the exam you will be required to solve and graph linear equations and inequalities with one or two variables, solve and graph nonlinear algebraic equations, solve equations and inequalities involving absolute value, solve quadratic equation problems, and determine line equations. Problems that require solving algebraic expression and problems that include formulas, equations, adding, subtracting, multiplying, and dividing polynomials, and algebraic fractions will be included in this section of the exam. Your knowledge of standard algebraic operations that involve negative exponents, fractional exponents, radicals, exponents, and complex numbers will be included in this section of the exam. Addition, subtraction, multiplication, and division of rational numbers, identification of basic mathematical operations, identification of the multiplicative and additive in versus of a number, and the order and finite set of real numbers will be included in this section of the exam. Your knowledge of classification of real numbers, complex, rational, or irrational, decimals, powers, roots, and exponents will be assessed in this section of the exam. Number system operation, counting numbers, factors, divisibility, multiples, ratio, proportion, and percentages will also be included in this section of the exam.

Problem Solving Exercises

Questions in the section of the exam will assess your understanding of the learning of mathematical content and learning to think in a mathematical manner. The following topics will be covered in a section of the exam: mathematical problem-solving, mathematical reasoning and proof, mathematical connections, mathematical representation, and the usage of technology. Mathematical problem solving questions will assess your ability to solve mathematical problems, and apply mathematics to solve problems in other contexts. In the mathematical reasoning and proof section you will be asked to utilize serious reasoning and proof methods, create and investigate mathematical conjectures, and create and evaluate mathematical proofs and arguments. Under the mathematical connections section you will be asked to connect mathematical ideas, apply mathematics in the context of other areas, and show an understanding of the interrelationship between mathematical ideas. Under the mathematical representation category you will be asked to apply and translate mathematical representations to solve problems. Your ability to use mathematical representations for modeling and interpretation of social, physical, and mathematical phenomena will be assessed. Your ability to create and utilize representations for the organization, recording, and communication of mathematical ideas will also be included in this section of the exam. In the use of technology section your ability to use technology to solve problems will be assessed.

**PRAXIS II Middle School: Mathematics Practice Questions**

**1. Mathematics is the formal science of:**

A. structure

B. order

C. relationships

D. All of the above

**2. Which of the following is not a common mathematics term?**

A. Integers

B. Theory

C. Place value

D. Percentage

**3. Basic mathematics concepts include:**

A. repeating patterns

B. growth patterns

C. equivalence and balance

D. All of the above

**4. Which of the following is not a type of statistics?**

A. Descriptive

B. Exploratory

C. Analytical

D. Confirmatory

**5. Which of the following is not a general rule of the probability theory?**

A. All probabilities must be a negative number.

B. All probabilities must be a nonnegative number.

C. All possible outcomes equal one.

D. The possibility of non-overlapping outcomes is the total of the individual outcomes.

**Answer Key For Middle School Mathematics**

**1. Answer: D**

The American Heritage College Dictionary defines mathematics as “the study of the measurement, properties, and relationships of quantities using numbers and symbols.” It is a formal science of structure, order, and relationship and is considered the basic language and foundation of all the other sciences. It is especially critical in the development of technology. It evolved from counting, measuring, and describing shapes. Some areas of mathematics and their definitions include:

ARITHMETIC: a system to count numbers using addition, subtraction, multiplication, and division

ALGEBRA: an abstract form of arithmetic using symbols to represent numbers

GEOMETRY: the relationship of points, lines, angles, surfaces, and solids

PROBABILITY: the chance random events will occur

STATISTICS: the collection, organization, and analysis of data

TRIGONOMETRY: the relationship of the sides and angles of triangles, as well as their functions and applications

CALCULUS: the limits, differentiation, and integration of the functions of variables.

**2. Answer: B**

Number concepts are the building blocks of all mathematical calculations and representations. Students must understand what a number means, in what ways it can and cannot be used, and its relationship to other numbers. They need to be able to depict numbers concretely, pictorially, and symbolically. Students need to understand the basic definitions of number concepts in order to use numbers properly in whatever math discipline in which they are working.

These definitions of some common math terms are from The American Heritage College Dictionary:

INTEGERS are the positive and negative whole numbers plus zero.

NATURAL NUMBERS or COUNTING NUMBERS are the positive integers.

FRACTIONS are the result of dividing one quantity by another quantity.

A PRIME NUMBER is only divisible by one and itself.

A PERCENTAGE is a fraction or ratio expressed as part of one hundred.

A RATIO is the relation between two quantities expressed as the result (quotient) of one divided by the other.

PLACE VALUE is the position of a figure in a numeral or series.

**3. Answer: D**

There are basic concepts students need to understand in order to begin to think algebraically, so they can use what they see to make generalizations about unknowns. Patterns and functions represent change and relationships. Repeating patterns show the same unit over and over again. In growth patterns, each unit is dependent upon the one before it, as well as its position in the pattern. The function is the relationship between values; i.e., the second depends on the first. Using concrete examples helps students visualize what the function is describing. As students begin to understand functional relationships, symbols can be used as an abstract stand-in for the relationships.

Equivalence and balance are critical concepts in understanding algebraic equations. It is important teachers explain the equal sign represents some type of relationship between the numbers and symbols on each side of the sign, and if a calculation is performed on one side, the same calculation must be performed on the other side. Each side is equal and they must balance.

**4. Answer: C**

Statistics is the collection, organization, and interpretation of data. The data can be facts or isolated bits of information, but it all relates in one way or another to a specific topic. This precise, analytical system is used to identify, study, and solve various problems in many industries. Statistics can help people interpret events and make decisions in uncertain and difficult situations. Healthcare professionals, financial analysts, scientists, engineers, and insurance actuaries all use statistics to infer relationships, measure interactions, and predict outcomes among variables.

Descriptive statistics is the foundation for the entire system. It is used to define and explain the basic components in a study. Exploratory statistics tries to figure out what the collected data are saying. This method involves averages and percentages, which are usually displayed on a graph or in a table of some sort. Since by definition it relies on information from previous experiments, this data is sometimes called secondary research. Confirmatory statistics is the method that applies general ideas and concepts to an issue or a problem in an effort to answer specific questions.

**5. Answer: A**

The probability theory is the study and analysis of random events and whether those events can predict the behavior of a defined system. A probability is the numerical measure of the likelihood the event will happen. It is a number from zero to one. Zero means it will definitely not happen, one means it definitely will happen, and point five (0.5) means it is a draw, i.e., just as likely to happen as not happen (in other words, it has a fifty/fifty chance of happening).

Probability is the possibility of an event happening or something being true. It is used to explain events that do not happen with any certainty. Probabilities must meet these general rules:

All probabilities must be a nonnegative number.

The collection of all possible outcomes is equal to one.

If there are two possible outcomes that cannot happen at the same time (non-overlapping), the possibility either will happen is the total of the individual outcomes.