Illinois Licensure Testing System: Mathematics Test Framework Study Guide

Mathematical Practices, Technology, and Disciplinary Literacy

Mathematical Practices

This topic covers the foundational skills and strategies used in mathematical problem solving and reasoning. It emphasizes the importance of applying various approaches, generalizing processes, and communicating mathematical ideas effectively.

• Problem-Solving Strategies: Includes drawing diagrams, working backward, identifying mathematical structures, and recognizing patterns.

• Generalization: Extending problem-solving processes to new situations.

• Model Selection: Choosing appropriate mathematical models to analyze and understand situations.

• Justification: Explaining the application of concepts, procedures, and theorems.

• Conjectures and Reasoning: Developing and evaluating conjectures, critiquing reasoning.

• Communication: Using precise vocabulary, visual representations, and symbols.

• Connections: Linking mathematical domains and connecting mathematics to other disciplines (e.g., rate of change in physics).

Technology in Mathematics

Understanding and integrating technology is essential for modern mathematics instruction and assessment.

• Capabilities of Technology: Recognizing the benefits of tools such as spreadsheets, graphing calculators, and manipulatives.

• Technology Selection: Choosing appropriate technology for analyzing concepts and solving problems.

• Integration: Incorporating technology into instruction and assessment.

• Algorithm Development: Using technology to develop and apply algorithms.

• Differentiation: Individualizing instruction and assessment with technology.

Disciplinary Literacy in Mathematics

Promoting reading development and comprehension in the mathematics classroom is vital for student success.

• Reading Process: Understanding how meaning is constructed from text and prior knowledge.

• Integration of Language Arts: Providing opportunities for reading, writing, and discussion to enhance understanding.

• Vocabulary Development: Strategies for building mathematics vocabulary, including context clues and reference materials.

• Comprehension Strategies: Teaching and modeling strategies for effective reading and critical analysis of texts.

• Monitoring Progress: Using observations and assessments to differentiate instruction based on reading needs.

Number Systems and Operations

Real Number System and Operations

This topic explores the structure and properties of the real number system, including arithmetic operations and problem-solving techniques.

• Structure and Properties: Understanding real numbers, including integers, fractions, decimals, percents, powers, and rational exponents.

• Prime and Composite Numbers: Solving problems involving least common multiples and greatest common factors.

• Modular Arithmetic: Applying modular arithmetic in problem-solving.

• Ratios and Proportional Reasoning: Using ratios to represent and solve problems.

• Dimensional Analysis: Modeling and solving problems with appropriate units.

Complex Numbers and Linear Algebra

Complex numbers and basic linear algebra concepts are introduced, including arithmetic operations and applications.

• Complex Numbers: Performing arithmetic operations and representing complex numbers in rectangular and polar forms.

• Quadratic Equations: Applying the fundamental theorem of algebra to solve equations with complex solutions.

• Vectors and Matrices: Using vector and matrix arithmetic to model and solve problems, such as systems of linear equations.

Functions and Algebra

Algebraic Techniques

Algebraic techniques are essential for translating, manipulating, and solving mathematical expressions and equations.

• Translation: Converting word situations to algebraic sentences.

• Patterns and Sequences: Identifying and analyzing patterns, sequences, and series.

• Properties of Real Numbers: Applying properties to simplify expressions and solve equations.

Linear Relations and Functions

Linear functions and relations are fundamental in algebra, with applications in graphing and modeling real-world situations.

• Function Notation: Using notation to analyze domain and range.

• Representation: Expressing linear relations in symbolic, numeric, graphic, and verbal forms.

• Slopes and Intercepts: Constructing and interpreting graphs of linear equations and inequalities.

• Systems of Equations: Solving systems graphically and algebraically.

Quadratic Functions and Conic Sections

Quadratic functions and conic sections are explored through their properties, graphs, and applications.

• Representation: Expressing quadratic functions in various forms.

• Properties: Analyzing symmetry, roots, intercepts, and vertices.

• Solving Equations: Solving quadratic equations and systems.

• Conic Sections: Applying properties of hyperbolas, parabolas, circles, and ellipses.

Polynomial, Absolute Value, Radical, and Rational Functions and Inequalities

This topic covers the representation, analysis, and solution of various types of functions and inequalities.

• Representation: Expressing functions in symbolic, numeric, graphic, and verbal forms.

• Properties: Analyzing symmetry, roots, intercepts, critical points, asymptotes, and vertices.

• Solving Equations: Solving equations and inequalities graphically, algebraically, and numerically.

• Factoring and Partial Fractions: Applying algebraic properties to solve problems.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are important for modeling growth, decay, and other phenomena.

• Relationship: Understanding the connection between exponential and logarithmic functions.

• Representation: Expressing functions in various forms.

• Properties: Analyzing symmetry, roots, intercepts, critical points, and asymptotes.

• Applications: Modeling compound interest, exponential growth and decay.

• Solving Equations: Solving equations and inequalities graphically and algebraically.

Differential and Integral Calculus

Differential Calculus

Differential calculus focuses on the concept of limits, derivatives, and their applications in analyzing functions.

• Limits: Analyzing properties of functions and series.

• Derivatives: Finding and interpreting derivatives.

• Applications: Modeling rates of change, optimization, and related rates.

Integral Calculus

Integral calculus deals with the concept of integrals and their applications in finding areas, volumes, and modeling motion.

• Integrals: Calculating integrals of functions.

• Fundamental Theorem of Calculus: Applying the theorem to solve problems.

• Applications: Finding areas, volumes, and describing motion.

Measurement and Geometry

Principles and Applications of Measurement

Measurement principles are applied to solve problems involving various systems and units.

• Conversion: Converting measurements within English and metric systems.

• Problem Solving: Solving problems involving time, temperature, angles, weight, and mass.

• Nonlinear Scales: Using scales such as Richter, decibel, and pH.

• Formulas: Applying formulas for length, area, and volume.

• Indirect Measurement: Using trigonometric ratios and proportional reasoning.

Euclidean Geometry

Euclidean geometry covers points, lines, angles, planes, and their applications in proofs and constructions.

• Definitions and Theorems: Applying axioms and principles to develop proofs.

• Constructions: Justifying geometric constructions.

• Properties: Analyzing circles and other figures.

• Similarity and Congruence: Comparing measures of figures.

• Representations: Translating between two- and three-dimensional figures.

Coordinate and Transformational Geometry

Coordinate and transformational geometry use algebraic methods to analyze geometric figures and transformations.

• Coordinate Methods: Using distance formula and slope to prove theorems.

• Parallel and Perpendicular Lines: Finding equations and solving problems.

• Transformations: Analyzing reflections, rotations, translations, and dilations.

• Representations: Using coordinate systems for geometric analysis.

Statistics, Probability, and Discrete Mathematics

Collecting, Organizing, and Representing Data

Data collection and representation are key for statistical analysis and interpretation.

• Sampling Techniques: Collecting representative data and avoiding bias.

• Measures of Central Tendency: Calculating mean, median, and mode.

• Measures of Variation: Calculating range and standard deviation.

• Graphical Representations: Using bar graphs, scatterplots, box plots, stem-and-leaf diagrams, and histograms.

• Normal Distributions: Estimating population percentages.

• Data Analysis: Interpreting two-way tables and scatterplots, identifying lines of best fit.

• Evaluating Claims: Assessing inferences based on sample data.

Probability and Discrete Mathematics

Probability and discrete mathematics involve set theory, logic, counting strategies, and probability techniques.

• Set Theory and Logic: Understanding unions, intersections, and logical statements.

• Counting Strategies: Using combinations and permutations.

• Probability Techniques: Applying addition and multiplication rules, determining sample spaces.

• Expected Value: Analyzing situations for fairness and probability.

• Graphic Representations: Using Venn diagrams and trees.

• Simulations: Modeling theoretical and experimental probabilities.

Example: Solving a Linear Equation

To solve the equation $2x + 3 = 7$:

• Subtract 3 from both sides: $2x = 4$

• Divide both sides by 2: $x = 2$

Example: Calculating the Mean

Given data: 4, 6, 8

• Mean: $\frac{4 + 6 + 8}{3} = 6$

Example: Slope of a Line

Given points (1,2) and (3,6):

• Slope: $m = \frac{6 - 2}{3 - 1} = 2$

Example: Quadratic Formula

To solve $ax^2 + bx + c = 0$:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Probability of a Single Event

Probability of rolling a 3 on a fair six-sided die:

• $P(3) = \frac{1}{6}$

Example: Area of a Circle

Given radius $r$:

• Area: $A = \pi r^2$

Example: Exponential Growth

Population grows according to $P(t) = P_0 e^{rt}$

• $P_0$ = initial population, $r$ = growth rate, $t$ = time

Example: Standard Deviation

For data $x_1, x_2, ..., x_n$ with mean $\mu$:

• $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}$

Example: Permutations

Number of ways to arrange $n$ objects:

• $n!$

Example: Combination Formula

Number of ways to choose $r$ objects from $n$:

• $C(n, r) = \frac{n!}{r!(n-r)!}$

Example: Distance Formula

Distance between $(x_1, y_1)$ and $(x_2, y_2)$:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example: Matrix Solution to Linear System

Solving $Ax = b$ where $A$ is a matrix:

• If $A$ is invertible, $x = A^{-1}b$

Example: Fundamental Theorem of Algebra

Every non-zero, single-variable polynomial of degree $n$ has exactly $n$ roots (including complex roots).

• For $x^2 + 1 = 0$, roots are $x = i$ and $x = -i$

Example: Trigonometric Identity

$\sin^2 x + \cos^2 x = 1$

• This identity holds for all real $x$.

Example: Indirect Measurement Using Trigonometry

To find the height $h$ of a tree using angle $\theta$ and distance $d$:

• $h = d \tan \theta$

Example: Box Plot Representation

A box plot visually represents the median, quartiles, and extremes of a data set.

• Median: Middle value

• Quartiles: Divide data into four equal parts

Example: Venn Diagram for Sets

Visual representation of unions and intersections of sets.

• Union: $A \cup B$

• Intersection: $A \cap B$

Example: Compound Interest Formula

$A = P(1 + \frac{r}{n})^{nt}$

• $A$ = final amount, $P$ = principal, $r$ = annual rate, $n$ = number of times compounded per year, $t$ = years

Example: Slope Criteria for Parallel and Perpendicular Lines

• Parallel lines: Same slope

• Perpendicular lines: Slopes are negative reciprocals

Example: Transformations in the Coordinate Plane

• Reflection: Over $y$-axis, $f(x) \rightarrow f(-x)$

• Translation: $f(x) \rightarrow f(x + a)$

• Rotation: About origin by angle $\theta$

Example: Normal Distribution

For mean $\mu$ and standard deviation $\sigma$, approximately 68% of data falls within $\mu \pm \sigma$.

• Standard normal curve is bell-shaped.

Example: Solving a System of Linear Equations

Given $x + y = 5$ and $x - y = 1$:

• Add equations: $2x = 6 \Rightarrow x = 3$

• Substitute: $3 + y = 5 \Rightarrow y = 2$

Example: Factoring a Polynomial

Factor $x^2 - 5x + 6$:

• $(x - 2)(x - 3)$

Example: Rational Expression Simplification

Simplify $\frac{x^2 - 4}{x - 2}$:

• Factor numerator: $\frac{(x - 2)(x + 2)}{x - 2} = x + 2$ (for $x \neq 2$)

Example: Absolute Value Equation

Solve $|x - 3| = 5$:

• $x - 3 = 5 \Rightarrow x = 8$

• $x - 3 = -5 \Rightarrow x = -2$

Example: Radical Equation

Solve $\sqrt{x} = 4$:

• $x = 16$

Example: Logarithmic Equation

Solve $\log_{10} x = 2$:

• $x = 100$

Example: Sequence Formula

Arithmetic sequence: $a_n = a_1 + (n - 1)d$

• $a_1$ = first term, $d$ = common difference

Example: Binomial Theorem

$(a + b)^n = \sum_{k=0}^n C(n, k) a^{n-k} b^k$

• $C(n, k)$ = combination formula

Example: Conic Section Equation

Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Center $(h, k)$, radius $r$

Example: Matrix Multiplication

For matrices $A$ and $B$:

• Product $AB$ is defined if columns of $A$ match rows of $B$

Example: Partial Fraction Decomposition

Decompose $\frac{1}{x(x+1)}$:

• $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$

Example: Compound Probability

Probability of two independent events $A$ and $B$:

• $P(A \cap B) = P(A) \times P(B)$

Example: Expected Value

For outcomes $x_i$ with probabilities $p_i$:

• $E = \sum x_i p_i$

Example: Solving an Inequality

Solve $2x - 5 < 7$:

• $2x < 12 \Rightarrow x < 6$

Example: Absolute Value Inequality

Solve $|x - 4| \leq 3$:

• $-3 \leq x - 4 \leq 3 \Rightarrow 1 \leq x \leq 7$

Example: Roots and Radicals

Square root of 25:

• $\sqrt{25} = 5$

Example: Inverse Function

If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$

• Inverse undoes the original function.

Example: Series Summation

Sum of first $n$ natural numbers:

• $S = \frac{n(n+1)}{2}$

Example: Table: Comparison of Function Types

Function Type	General Form	Key Properties
Linear	$y = mx + b$	Slope, y-intercept
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry
Exponential	$y = a b^x$	Growth/decay, asymptote
Logarithmic	$y = \log_b x$	Inverse of exponential
Rational	$y = \frac{p(x)}{q(x)}$	Asymptotes, domain restrictions
Absolute Value	$y = |x|$	Vertex at origin, symmetry
Radical	$y = \sqrt{x}$	Domain $x \geq 0$

Additional info: This study guide summarizes the objectives and content areas from the Illinois Licensure Testing System Mathematics Test Framework, covering all major topics relevant to Beginning & Intermediate Algebra and related mathematical domains.