Breaking Down the Mathematics Section
The TACHS Mathematics section is designed to measure cumulative knowledge from middle school curricula, specifically focusing on the ability to perform operations and solve problems without a calculator. The section is typically divided into two distinct parts: Concepts and Estimation, and Problem Solving. In the first part, students encounter questions that test their grasp of TACHS number properties, such as the distributive property or the order of operations. The second part requires the application of specific math equations for TACHS to derive exact answers for word problems. Scoring is based on the number of correct answers, meaning there is no penalty for guessing, but the pace is rapid—often allowing less than a minute per question. Mastering the formulas beforehand ensures that students do not waste valuable seconds trying to derive a relationship that should be second nature.
The Role of Memorization vs. Application
While memorization is the foundation, the TACHS focuses heavily on application. It is not enough to know that the area of a triangle is 1/2 base times height; a student must recognize when to use this formula when only the hypotenuse and one angle are provided in a multi-step problem. This transition from recall to application is often where high-achieving students separate themselves. The exam utilizes a Standard Score system, where raw points are converted into a scaled score to account for variation in test difficulty. To reach the 90th percentile or higher, a student must demonstrate fluency in applying TACHS algebra review concepts, such as isolating variables in multi-step equations, while maintaining a high level of accuracy in basic arithmetic operations.
Questions on the TACHS rarely ask for a formula directly. Instead, they present a scenario that implies a specific mathematical relationship. For instance, a question might describe a circular garden and ask for the amount of fencing needed, requiring the student to realize that the Circumference formula ($C = 2\pi r$) is the required tool. The exam often includes "distractor" answers—options that result from common formula errors, such as forgetting to divide by two in a triangle area calculation. Understanding the formula sheet for Catholic high school test preparation involves recognizing these cues. Students must be prepared for "Data Interpretation" tasks where formulas must be extracted from tables or graphs, requiring a blend of analytical reading and mathematical computation.
Percent, Decimal, and Fraction Conversions
Fluency in converting between fractions, decimals, and percentages is a cornerstone of the TACHS. Many problems involve finding a percentage of a number or determining a percent increase. The fundamental formula for percentage is $\text{Part} = \text{Percent} \times \text{Whole}$. When dealing with percent change, students must use the Percent Change Formula: $(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100$. Recognizing common equivalents—such as $1/8$ being $12.5%$ or $0.125$—can save significant time. On the exam, you may encounter a question asking for a $15%$ tip on a bill; being able to quickly calculate $10%$ and then add half of that amount is a practical application of these conversion properties that avoids the need for lengthy long-multiplication.
Rules for Operations with Integers
The TACHS frequently tests a student's ability to handle signed numbers and the Order of Operations (PEMDAS). The rules for adding, subtracting, multiplying, and dividing integers are vital. For example, subtracting a negative is equivalent to adding a positive ($a - (-b) = a + b$). The exam often includes complex expressions that require strict adherence to the hierarchy of Parentheses, Exponents, Multiplication/Division (left to right), and Addition/Subtraction (left to right). A common mistake on the TACHS is performing addition before subtraction when subtraction appears further left in the expression. Mastery of these TACHS number properties ensures that students do not lose points on simple computational tasks before they even reach the core problem-solving stage.
Statistical measures are a recurring theme in the arithmetic section. The Arithmetic Mean is calculated using the formula: $\text{Sum of Terms} / \text{Number of Terms} = \text{Mean}$. The TACHS often presents "Average Speed" or "Missing Score" problems, where the mean is provided and the student must solve for a missing value. For example, if a student has scores of 80, 85, and 90, and needs an average of 88, they must set up an algebraic equation to find the fourth score. The Median (the middle value in an ordered list) and the Mode (the most frequent value) are also tested. Students must remember to arrange data in ascending order before identifying the median, a step often overlooked in the rush of the exam environment.
Rate, Time, and Distance Relationships
One of the most common applications of arithmetic on the TACHS is the Distance Formula: $D = R \times T$ (Distance = Rate × Time). These problems may appear straightforward but often include a unit conversion trap—for instance, providing the rate in miles per hour but the time in minutes. Students must be adept at rearranging this formula to solve for any of the three variables ($R = D/T$ or $T = D/R$). Additionally, the TACHS may feature work-rate problems, where the rate of two people working together is the sum of their individual rates. Understanding this additive property of rates is essential for solving complex word problems that involve simultaneous actions or moving objects.
Essential Algebra Equations and Concepts
Solving Linear Equations and Inequalities
Algebraic proficiency is measured by the ability to manipulate symbols to find unknown values. The TACHS focuses on Linear Equations of the form $ax + b = c$. Students must use inverse operations to isolate the variable, maintaining the balance of the equation by performing the same operation on both sides. When dealing with Inequalities, a critical rule to remember is that multiplying or dividing both sides by a negative number reverses the inequality sign (e.g., if $-2x < 10$, then $x > -5$). This specific rule is a frequent target for TACHS question writers. Mastery of these steps is part of a comprehensive TACHS algebra review, allowing students to handle both abstract equations and those derived from word problems.
Working with Exponents and Square Roots
Exponents and roots appear frequently in both the Mathematics and Ability sections. Students must know the Product Rule ($x^a \cdot x^b = x^{a+b}$) and the Power Rule ($(x^a)^b = x^{ab}$). The TACHS typically sticks to perfect squares (1, 4, 9, 16, 25, etc.) and their roots, but students should be comfortable estimating the square root of non-perfect squares to the nearest whole number. For instance, knowing that $\sqrt{50}$ is slightly more than 7 because $7^2 = 49$. These concepts are often integrated into geometry problems, such as finding the side of a square when given the area, where the student must apply the inverse relationship between squaring and taking a square root.
Setting Up and Solving Ratios and Proportions
Proportional reasoning is perhaps the most tested algebraic concept on the TACHS. A Proportion is an equation stating that two ratios are equal ($a/b = c/d$). The primary tool for solving these is Cross-Multiplication, leading to the equation $ad = bc$. This is essential for scale drawing problems, recipe adjustments, and unit conversions. For example, if a map scale is 1 inch to 20 miles, how many inches represent 150 miles? Setting up the proportion $1/20 = x/150$ and solving for $x$ is a standard TACHS task. Students must ensure that the units are consistent across the numerators and denominators to avoid an inverted (and incorrect) result.
Interpreting and Using Variables
The ability to translate a verbal phrase into an algebraic expression is a high-level skill required for the TACHS. Phrases like "three less than twice a number" must be correctly written as $2x - 3$, not $3 - 2x$. This distinction is vital, as the order of operations in subtraction and division is not commutative. The TACHS often uses variables to represent unknown quantities in real-world contexts, such as the cost of $n$ items at $p$ price. Understanding the Distributive Property, $a(b + c) = ab + ac$, is also crucial here, as it allows students to simplify expressions before solving, which reduces the likelihood of computational errors during the later stages of a problem.
Area and Perimeter of Polygons
Geometry on the TACHS focuses on two-dimensional shapes, specifically rectangles, squares, and triangles. The Perimeter is the sum of all sides, while the Area measures the surface within. For a rectangle, $A = l \times w$; for a square, $A = s^2$. A common area of confusion is the triangle, where the formula $A = 1/2 \times b \times h$ must be used. Students often forget the $1/2$ factor. Furthermore, the TACHS may present composite shapes—figures made of two or more simple polygons. In these cases, the student must decompose the shape, calculate the area of each part using TACHS geometry formulas, and then sum them to find the total area.
Circumference and Area of Circles
Circles require the use of the constant Pi (π), which the TACHS usually approximates as $3.14$ or $22/7$. Students must distinguish between the Circumference ($C = 2\pi r$ or $C = \pi d$) and the Area ($A = \pi r^2$). A frequent pitfall is using the diameter in the area formula instead of the radius. If a question provides a diameter of 10, the student must first divide by 2 to find a radius of 5 before squaring it to find the area ($25\pi$). Understanding the relationship between the radius and diameter ($d = 2r$) is fundamental to solving any circle-related problem on the exam. Questions may also ask for the length of an arc or the area of a sector, though these are less common than basic circumference and area.
Volume of Basic 3D Shapes
While the TACHS is primarily 2D-focused, basic 3D geometry is included, particularly the Volume of a Rectangular Prism. The formula $V = l \times w \times h$ is the standard. Students may also see problems involving the volume of a cube ($V = s^3$). It is important to pay attention to units; volume is always expressed in cubic units (e.g., $cm^3$). The exam might ask how many small cubes of a certain size can fit into a larger rectangular box. This requires calculating the volume of both and dividing the larger by the smaller—a multi-step process that tests both formula knowledge and logical reasoning under pressure.
Angles and Properties of Triangles
Understanding the internal properties of shapes is just as important as knowing their dimensions. The most critical rule is the Triangle Sum Theorem, which states that the sum of the interior angles of any triangle is always $180$ degrees. The TACHS also tests the Pythagorean Theorem ($a^2 + b^2 = c^2$) for right triangles. Students should be familiar with "Pythagorean Triples," such as 3-4-5 and 5-12-13, which appear frequently and can be solved without lengthy calculation. Additionally, knowing the properties of isosceles (two equal sides/angles) and equilateral (all sides/angles equal) triangles allows for the quick deduction of missing information in complex geometry problems.
The most challenging aspect of the TACHS is not the math itself, but the translation of words into math. When a problem mentions "the space inside" or "covering a floor," it is a cue for an area formula. If it mentions "the distance around" or "fencing," it signals perimeter or circumference. This stage of the process requires students to filter out irrelevant information (extraneous data) and focus on the goal of the question. Effective test-takers often underline the specific question being asked—such as "What is the radius?" versus "What is the area?"—to ensure they select the correct math equations for TACHS for the final step of their calculation.
Multi-Step Problem-Solving Strategies
Many TACHS questions are designed as multi-step puzzles. For example, a problem might ask for the cost of painting a wall. The student must first calculate the area of the wall ($A = l \times w$), subtract the area of any windows, and then multiply the remaining area by the cost per square foot. This requires a sequential approach: Solve, Substitute, and Simplify. Keeping scrap paper organized is vital here. If a student loses track of a mid-step value, they are likely to choose one of the "trap" answers that corresponds to an incomplete calculation. Developing a habit of labeling each part of the calculation helps maintain clarity during the high-stress environment of the Catholic high school test.
Common Pitfalls in Application
One of the most frequent errors on the TACHS is the "Unit Mismatch." A problem might provide the dimensions of a room in feet but ask for the area in square yards. Since there are 3 feet in a yard, there are 9 square feet in a square yard ($3^2$). Students who simply divide by 3 will arrive at the wrong answer. Another pitfall is the misuse of the Order of Operations in algebraic word problems, particularly when parentheses are implied but not written. Recognizing these common traps and double-checking the units of the final answer against the question's requirements can significantly improve a student's raw score on the Mathematics section.
To prepare effectively, students should construct their own formula sheet for Catholic high school test review. This should not be a passive list but an active study tool. Group formulas by category—Arithmetic, Algebra, Geometry—and include a small diagram for each geometric shape. Beside each formula, write a one-sentence description of when to use it. For example, next to $I = Prt$, write "Use for simple interest problems." Hand-writing these formulas aids in kinesthetic learning and memory retention. As the exam date approaches, students should practice re-creating this sheet from memory on a blank piece of paper, which mimics the "brain dump" technique used by successful test-takers at the start of the exam.
Drill Exercises for Recall Speed
Speed is a major factor in TACHS scoring. Once formulas are memorized, the next step is reducing the time it takes to retrieve them. Flashcards are an excellent tool for this. One side should have the concept (e.g., "Area of a Trapezoid"), and the other side should have the formula ($A = 1/2(b_1 + b_2)h$). Use a timer to see how many formulas can be correctly identified in 60 seconds. This builds the "automaticity" required to handle the more complex problem-solving questions. The goal is to reach a point where the TACHS math formulas are recalled instantly, leaving more cognitive energy available for the logic and reasoning components of the questions.
Using Practice Tests to Gauge Proficiency
Finally, full-length practice tests are essential for understanding how formulas are integrated into the actual exam format. After completing a practice test, students should perform a "Gap Analysis." For every missed question, identify if the error was due to a forgotten formula, a calculation mistake, or a misunderstanding of the question's intent. If a student consistently misses questions involving TACHS geometry formulas, they should dedicate their next study session specifically to that area. This targeted approach ensures that study time is spent on the areas of greatest need, leading to steady improvement in the Standard Score and increasing the likelihood of admission to their first-choice high school.