Free TSIA2 Math Exam Flashcards

A unit rate expresses a quantity per one unit of another measure, such as miles per hour or price per ounce. To scale to a new input, multiply the rate by that input instead of re-deriving the relationship each time.

A proportion is an equation stating two ratios are equal, a/b = c/d. Cross-multiplying preserves equivalence, so ad = bc lets you isolate an unknown when three values are known.

Percent change equals (new - original) / original times 100. The denominator is the original baseline, not the new value, which is the most common error in increase and decrease problems.

A discount is the dollar reduction; the sale price is the original price minus that reduction. For 25% off $64, the discount is $16 and the sale price is $48; do not multiply the original price by 25.

Simple interest is I = P * r * t, with rate as a decimal and time in years. It grows linearly, so doubling time doubles the interest only when principal and rate stay fixed; it differs from compound interest, which grows on the accumulated balance.

A part-to-part ratio compares two groups (3 boys to 2 girls); a part-to-whole ratio compares one group to the total (3 boys to 5 students). Switching between them requires the total, which is the sum of the parts.

A proportional estimate scales a known ratio to a new total when a full count is impractical. The error comes from assuming the sample ratio matches the population, so it works best with a representative sample.

a(b + c) = ab + ac spreads multiplication across a sum or difference inside parentheses. A classic sign error is 5(2x - 3) = 10x - 3; the correct result is 10x - 15, because the -3 must also be multiplied by 5.

Like terms have identical variable factors, including exponents, so only their coefficients add. 3x and 3x^2 are not like terms and cannot combine into a single term, no matter how tempting it looks.

x^2 + bx + c factors as (x + p)(x + q) where p * q = c and p + q = b. Signs matter: x^2 - 5x + 6 = (x - 2)(x - 3), not (x + 2)(x + 3), because -2 and -3 multiply to 6 and add to -5.

a^2 - b^2 = (a - b)(a + b) factors in one step with no middle term. Recognizing the pattern, such as x^2 - 49 = (x - 7)(x + 7), saves time compared to expanding and refactoring.

A radical is simplified when the radicand has no perfect-square factor. Extract squares by writing sqrt(50) = sqrt(25 * 2) = 5*sqrt(2); leaving sqrt(50) is not wrong but is not the simplified form TSIA2 expects.

A rational expression is a fraction of polynomials. It is undefined where the denominator is zero, not where the numerator is zero, and you may cancel a common factor only when it is shared by both numerator and denominator and is nonzero at the value of interest.

A linear equation in one variable has no variable raised to a power other than 1. Isolate the variable by applying inverse operations to both sides; each operation must act on both sides to preserve equality.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. So 3 - 2x > 7 becomes -2x > 4, and dividing by -2 gives x < -2, not x > -2.

Elimination solves a system by adding the equations, or scaled versions, so one variable cancels. Multiply one equation so a coefficient pair becomes opposites, then add to eliminate that variable in one step.

A linear system has one solution if the lines intersect, zero if they are parallel, and infinitely many if they are the same line. For a single equation, a false statement like 2 = 5 means no solution, while an identity like x = x means infinitely many.

If (x - a)(x - b) = 0, then x = a or x = b, because a product is zero only when a factor is zero. Set each factor equal to zero separately; this is why factoring solves quadratics that factor cleanly.

x = (-b +/- sqrt(b^2 - 4ac)) / (2a) solves any ax^2 + bx + c = 0. The discriminant b^2 - 4ac gives two real solutions when positive, one when zero, and none when negative.

If a^x = a^y, then x = y, provided the base a is positive and not 1. When bases differ but share a power, rewrite them: 4^x = 2^6 becomes 2^(2x) = 2^6, so 2x = 6 and x = 3.

|x| = c has two solutions x = c and x = -c when c > 0, one solution when c = 0, and no solution when c < 0. Split the equation into two cases before solving each branch.

A literal equation has multiple variables; solving for one means isolating it in terms of the others using the same inverse operations as one-variable equations. The result is an expression, not a number.

Slope is (change in y) / (change in x) = (y2 - y1) / (x2 - x1). A positive slope rises left to right, a negative slope falls, vertical lines have undefined slope (zero denominator), and horizontal lines have slope zero.

y = mx + b gives slope m and y-intercept b directly. Parallel lines share m; perpendicular lines have slopes that multiply to -1, so a line perpendicular to y = 2x + 1 has slope -1/2.

y - y1 = m(x - x1) builds a line from one point and the slope, avoiding the need to compute a second point first. It is the fastest form when a table or graph gives a point and a rate.

The x-intercept is where y = 0 and the y-intercept is where x = 0; set the other variable to zero to find each. Do not confuse them with the vertex of a parabola, which is a maximum or minimum point.

Perimeter is 2L + 2W, a length around the figure; area is L * W, a squared quantity. Units reveal the difference: perimeter is linear (cm) and area is square (cm^2).

Triangle area is (1/2) * base * height, and the height must be perpendicular to the chosen base, not a side length. Choosing a different base changes the matching height but never the area.

Circumference is 2 * pi * r, a linear measure; area is pi * r^2, a squared measure. Mixing them up usually comes from forgetting which formula has r versus r^2.

Trapezoid area is (1/2) * (b1 + b2) * h, where the two bases are the parallel sides and the height is the perpendicular distance between them. It equals the average of the bases times the height, not the average of all four sides.

For similar figures with linear scale factor k, side lengths and perimeter scale by k, but area scales by k^2. Doubling all side lengths quadruples the area, a frequent TSIA2 trap.

Prism volume is base area times height, and cylinder volume is pi * r^2 * h, which is the same formula because the base is a circle. The height must be perpendicular to the base, not a slant length.

Surface area is the sum of every face area, including bases; for a cylinder it is 2 * pi * r^2 + 2 * pi * r * h. Confusing surface area with volume shows up as wrong units: square versus cubic.

Translations, reflections, and rotations preserve side lengths and angles, so the image is congruent to the preimage. Dilations change size and are not rigid, though they do preserve angle measures.

For a right triangle, a^2 + b^2 = c^2 where c is the hypotenuse. The theorem holds only for right triangles; using it on an arbitrary triangle gives a false value.

SOH-CAH-TOA: sin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent. The hypotenuse is fixed, while the labels opposite and adjacent depend on the chosen reference angle.

An angle of elevation is measured upward from horizontal; an angle of depression is measured downward from horizontal. Both use the same right triangle trig ratios; only the direction of measurement differs.

Mean is the sum divided by the count; median is the middle value when data are ordered. The mean shifts toward outliers, while the median resists them, so a skewed distribution is better summarized by the median.

Range is maximum minus minimum; IQR is Q3 minus Q1. IQR ignores the extremes, so it is more resistant to outliers than range.

MAD is the average of the absolute deviations from the mean, and a larger MAD means more spread. Unlike variance, MAD uses absolute values, so it stays in the original units of the data.

Adding an extreme value changes the mean, range, and MAD, but often leaves the median unchanged. Knowing which statistics resist outliers is a frequent TSIA2 trap.

Categorical data place items into groups (eye color, class period); quantitative data are numerical measurements (height, score). Counts of categories are quantitative, but the underlying labels are categorical.

A box plot shows the five-number summary (min, Q1, median, Q3, max) with the box spanning Q1 to Q3. The whiskers extend to the extremes, so a long whisker indicates skew in that direction.

A scatterplot shows a possible relationship between two quantitative variables, described as positive, negative, or no association. Clustering is not causation; it only signals a trend worth modeling.

The probability of an event equals favorable outcomes divided by total equally likely outcomes. The denominator must be the full sample space, not just the favorable outcomes.

The complement of an event has probability 1 - P(event), and together they cover the whole sample space. Use the complement when at least one is hard to count but its complement, none, is easy.

For independent events, multiply individual probabilities; for dependent events, multiply by the conditional probability after the first occurs. Drawing without replacement makes events dependent, while drawing with replacement keeps them independent.

TSIA2 Math starts with the 20-question College Readiness Classification (CRC) test; students below the benchmark take a 48-question Diagnostic Test. The Diagnostic reports a level per content category rather than a single pass or fail.

A Mathematics CRC score of 950-990 meets Texas college readiness in math. Below 950, readiness can still be met with a Diagnostic level of 6. The 950 is a scaled score, not a percentage of correct answers.

TSIA2 Math is untimed, so pacing should favor accuracy over speed. Handheld calculators are not allowed; some questions provide an on-screen calculator when the item requires one.