TEAS Math Comprehensive Study Guide

Core Principles of TEAS Math Preparation

• Conceptual Priority Structure:     * Highest Priority Topics: foundational, easiest to master, and virtually guaranteed to appear on every test (e.g., percentages to decimals, mean/median/range, simple area, order of operations, estimation, basic unit conversions).     * High Priority Next Steps: moderately more challenging topics that build on foundations (e.g., dependent/independent variables, ratios/proportions, writing basic algebra equations, right triangle geometry).     * Challenge Topics: advanced problems involving multi-step algebra, complex graph reading, or advanced logic (e.g., rate problems, system-style logic puzzles, inequalities with sign flipping). These are primarily for students seeking maximum scores; passing scores can often be achieved by mastering the first two categories.
• Test Mechanics:     * Calculator Limitations: The TEAS on-screen calculator handles basic operations (addition, subtraction, multiplication, division, square roots) but lacks parentheses and exponent buttons. This necessitates a manual understanding of the Order of Operations (PEMDAS).     * Unscored Questions: Every TEAS exam includes four experimental math questions that do not affect the final score. These are often high-difficulty "challenge" topics used for data collection.     * Rounding and Precision: The test frequently requires rounding to specific place values (e.g., hundredths vs. hundreds). Precise identification of place value is critical.

Proportional Reasoning and Percentages

• Converting Percentages to Decimals:     * Definition: Percent means "out of 100."     * Method 1 (Division): Divide the percentage by 100 (e.g., $15.2\% = \frac{15.2}{100}$).     * Method 2 (Shortcut): Move the decimal point two spots to the left (e.g., $15.2\% \rightarrow 0.152$).
• Percent Change:     * Formula: $\text{Percent Change} = \frac{\text{Amount of Change}}{\text{Old Value}} \times 100\%$     * Critical Error Prevention: Always divide by the old (original) value, never the new value.     * Example: A stock rising from $54 to $60 has an amount of change of $6. The calculation is $\frac{6}{54} \approx 0.111 \times 100\% = 11.1\%$.
• Calculating Original Price from Discount:     * Method: Set up a proportion where the current price corresponds to the remaining percentage (e.g., a $25\%$ discount means $75\%$ of the original remains).     * Equation: $\text{Original Price (X)} - (0.25 \times X) = \text{Discounted Price}$ or $\frac{\text{Current Price}}{\text{Remaining Decimal}} = \text{Original Price}$.     * Example: If a dress is $600 after a $25\%$ discount, $X = \frac{600}{0.75} = \$800$.

Statistics and Data Interpretation

• Descriptive Statistics for Data Sets:     * Prerequisite Step: Always rewrite data sets in order from lowest to highest before calculating median or range.     * Mean: The average. Sum all values and divide by the count ($n$).     * Median: The middle number in an ordered set. If the set has an even number of values, the median is the average of the two middle numbers.     * Range: The difference between the highest and lowest values ($\text{Highest} - \text{Lowest} = \text{Range}$).
• Visual Data Analysis:     * Pie Charts (Circle Graphs): Represent parts of a whole ($100\%$). When evaluating statements (e.g., "University treats more than all others combined"), check if the slice exceeds $50\%$.     * Bar Graphs: Used for comparing discrete categories or votes. When identifying a graph from a table without Y-axis numbers, look for the relative proportions between bars (e.g., if Category A is double Category B, the bar must be twice as high).     * Line Graphs: Show trends over time or how one variable responds as another changes.     * Scatter Plots: Show relationships between two variables to identify correlations.
• Correlation Types:     * Positive (Direct) Correlation: Both measurements move in the same direction (e.g., time studied and test scores).     * Negative (Inverse) Correlation: Measurements move in opposite directions (e.g., driving speed and travel time; as speed increases, time decreases).

Algebra and Geometry

• Order of Operations (PEMDAS):     * P: Parentheses (perform internal operations first, following PEMDAS inside parentheses).     * E: Exponents (e.g., $(-2)^2 = -2 \times -2 = 4$).     * M/D: Multiplication and Division (perform left to right).     * A/S: Addition and Subtraction (perform left to right).
• Geometry Formulas:     * Area of a Rectangle: $\text{Area} = \text{Length} \times \text{Width}$. Results are always in square units ($ft^2$, $m^2$).     * Perimeter: The distance around the boundary ($2 \times (L + W)$).     * Area of any Triangle: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$.     * Pythagorean Theorem (Right Triangles): $A^2 + B^2 = C^2$, where $C$ is the hypotenuse (diagonal side). To find $C$, sum the squares of the legs and take the square root.     * Volume (Rectangular Prism): $\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$.
• Variables:     * Independent Variable: The variable the researcher controls or changes (the "cause").     * Dependent Variable: The variable that changes in response (the "effect"). It "depends" on the independent variable.
• Inequalities:     * Solve similarly to equations, but with one critical rule: If you multiply or divide by a negative number, you must flip the inequality sign (e.g., $-10x < -30 \rightarrow x > 3$).

Measurement and Estimation

• Standard Unit Sizes (Heuristic):     * Centimeter ($cm$): Width of a pinky fingertip.     * Inch ($in$): Diameter of a quarter.     * Foot ($ft$): Length of a sheet of paper.     * Meter ($m$): Length of a guitar.     * Micrometer ($\mu m$): Size of a bacterial cell.
• Unit Conversions:     * Metric Volume: $1,000\,mL = 1\,L$. To convert $mL$ to $L$, divide by $1,000$ or move the decimal 3 spots left.     * Dimensional Analysis: Multiply by a conversion fraction where the unit to be cancelled is on the opposite side (e.g., to convert miles to km: $\text{miles} \times \frac{1.6\,km}{1\,mile}$).
• Estimation Strategy:     * When asked to estimate, round all figures to the specified place value (e.g., nearest ten) before performing the calculation to avoid compounding smaller errors.

Probability and Logic

• Basic Probability:     * Formula: $P = \frac{\text{Identified Desired Outcomes}}{\text{Total Possible Outcomes}}$     * Example (Not Orange): In a bag with 8 red, 7 green, 6 orange, and 9 blue candies, the probability of "not orange" is $\frac{8+7+9}{8+7+6+9} = \frac{24}{30}$. Simplify to $\frac{4}{5}$.
• Logic Problems (Mixed Ages/Pricing):     * Strategy 1 (Algebra): Define variables ($T$ for Tommy, $M$ for Mark) and set up sequential equations.     * Strategy 2 (Variable Substitution): Plug in a random easy number (e.g., 5 or 10) for one person's age and work through the logic to see how other values relate.

Questions & Discussion

• Participant Observation: The speaker (Tyler) emphasizes that the math section is often frightening for students, but manageable with a step-by-step approach from easy to hard.
• Common Trap (Calculator Usage): Tyler warns: "If you do not have it written out beforehand, don't touch that damn calculator." He advises writing every plus and minus sign on paper to prevent data entry errors, especially in population change problems (addition/subtraction chains).
• Common Trap (Right Triangle Geometry): Students often forget that $A^2 + B^2$ equals the hypotenuse squared. They frequently select the squared value as the answer instead of taking the final square root.
• Common Trap (Reading the Question): In multi-part chemistry/lab problems (e.g., Beaker A vs. Beaker B), students often solve for variable $X$ (Beaker A) and select it as the answer, even if the question specifically asks for Beaker B ($3.5X$).