50 Important Math Concepts You Must Know for the ACT

Vocabulary flashcards identifying critical definitions, formulas, and properties for the ACT Mathematics exam as outlined in the lecture notes.

Prime Numbers

Numbers that are only divisible by themselves and one: $1$ is NOT a prime number, and $2$ is the only even prime number.

Even/Odd Rules

The product of two even numbers is always even; the product of two odd numbers is always odd; the sum or difference of two even or two odd numbers is always even.

Mean (Average)

The sum of the values divided by the number of values. The average of two groups of data values is NOT the average of the two averages.

Median

The middle value of a group of values when ordered increasingly. If there is an even number of values, it is the average of the middle two terms.

Mode

The most commonly occurring value in a group; a set of values can have more than one mode.

Remainder

The integer quantity left over when dividing a quantity by a divisor; it is never a decimal and is never bigger than the divisor.

Range

The difference between the highest and lowest values in a data set.

Percent Change

Calculated as $(\text{amount of change}) / (\text{initial value}) \times 100\%$. Note that percent values never add or subtract.

Rational Numbers

Any decimal, repeating or non-repeating, that can be written as the quotient of two integers.

Rational Numbers (Non-repeating)

Any non-repeating decimal that does not have a predictable pattern to it.

Linear Inequalities Rule

When dividing or multiplying by a negative value, you must reverse the sign of the inequality.

Rules of Exponents

Bases must be alike; $x^0 = 1$; for multiplication, add bases $x^a \times x^b = x^{a+b}$; for division, subtract bases $x^a / x^b = x^{a-b}$; for power of exponents, multiply $ (x^a)^b = x^{ab}$.

Roots as Exponents

A square root is the same thing as a $1/2$ power ($25^{1/2} = 5$) and a cube root is the same thing as a $1/3$ power ($125^{1/3} = 5$).

Factoring simple polynomials $x^2 + bx + c$

Identify two factors that must multiply to $c$ and add up to $b$.

Factoring $x^2 - y^2$

The factors of this form are always $(x + y)(x - y)$.

Triangle Side Inequality Rule

The sum of any two sides of a triangle must exceed the measure of the third side.

Isosceles Triangle Properties

Base angles are congruent; sides opposite base angles are congruent; the angle bisector from the vertex angle forms a perpendicular bisector.

Important Pythagorean Triples

Common sets include $3, 4, 5$; $5, 12, 13$; $8, 15, 17$; $7, 24, 25$; and $9, 40, 41$.

Triangle Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Classifying Triangles using Pythagoras

In order from least to greatest $a, b, c$: if $a^2 + b^2 > c^2$, it is acute; if $a^2 + b^2 = c^2$, it is right; if $a^2 + b^2 < c^2$, it is obtuse.

Parallelogram Attributes

A quadrilateral with opposite sides congruent and parallel; adjacent angles add to $180^{\circ}$; opposite angles are congruent; diagonals bisect each other.

Rectangles and Rhombuses

A rectangle has right angles and congruent diagonals; a rhombus has congruent sides and perpendicular diagonals.

Parallel Lines and Transversals

Corresponding angles form an 'F' and are congruent; alternate interior angles form a 'Z' or 'N' and are congruent; same side interior/exterior angles are supplementary.

Circumference and Arc Length

Circumference is $2\pi r$; Arc length formula is $(\frac{n}{360}) \times 2\pi r$, where $n$ is the number of degrees in the arc.

Area of a Circle and Sector

Area is $\pi r^2$; Sector Area formula is $(\frac{n}{360}) \times \pi r^2$, where $n$ is the number of degrees in the central angle.

Inscribed Angles in Circles

Formed by two chords at the edge; equal to half the number of degrees in the intercepted arc. An angle inscribed in a semicircle is always $90^{\circ}$.

Special Area Formulas

Trapezoid: $\frac{1}{2} h(b_1 + b_2)$; Rhombus: $\frac{1}{2} d_1 d_2$ (product of the diagonals).

Surface Area of a Box

Calculation: $2lw + 2lh + 2wh$.

Diagonal of a Box

Represented by the relationship $d^2 = l^2 + w^2 + h^2$.

Polygon Angle Sum

The number of degrees in any $n$-gon is $180(n - 2)$.

Polygon Diagonal Formula

The number of diagonals in any $n$-sided polygon is $\frac{1}{2} n(n - 3)$.

Isosceles Right Triangle Ratios

Base angles are $45^{\circ}$; if given a leg, the hypotenuse is $\text{leg} \sqrt{2}$; if given the hypotenuse, the leg is $\text{hypotenuse} / \sqrt{2}$.

30/60/90 Right Triangle Ratio

The ratio of the sides is $1 : \sqrt{3} : 2$.

Midpoint Formula

Given $(X_1, Y_1)$ and $(X_2, Y_2)$, the midpoint is $(\frac{X_1 + X_2}{2}, \frac{Y_1 + Y_2}{2})$.

Distance Formula

The distance between $(X_1, Y_1)$ and $(X_2, Y_2)$ is $\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$.

Slope Formula and Relationships

Slope is $\frac{Y_2 - Y_1}{X_2 - X_1}$. Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes; reflected lines have negative slopes.

Equation of a Circle

$(x - h)^2 + (y - k)^2 = r^2$ represents a circle of radius $r$ centered at $(h, k)$.

Equation of a Parabola

$y = a(x - h)^2 + k$ with vertex $(h, k)$. If $a > 0$, it turns upward (minimum vertex); if $a < 0$, it turns downward (maximum vertex).

Equation of an Ellipse

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$; major axis is $2a$ and minor axis is $2b$.

Discriminant $D$

Determined by $b^2 - 4ac$: if $D < 0$, roots are imaginary; if $D = 0$, two equal roots; if $D > 0$, roots are real and unequal.

Quadratic Sum and Product of Roots

The sum of the roots is $-b/a$; the product of the roots is $c/a$.

Imaginary Numbers powers of $i$

$\sqrt{-1} = i$; $-1 = i^2$; $-i = i^3$; $1 = i^4$. To simplify $i^n$, reduce $n$ to the largest multiple of $4$ and use the remainder.

Direct and Indirect Proportions

Direct: $y = kx$ or $X_1/Y_1 = X_2/Y_2$. Indirect (Inverse): $y = k/x$ or $X_1 Y_1 = X_2 Y_2$.

Matrix Multiplication Rules

To multiply $r_1 \times c_1$ and $r_2 \times c_2$, $c_1$ must equal $r_2$. The resulting matrix dimension is $r_1 \times c_2$.

Determinant of a $2 \times 2$ Matrix

For matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is $ad - bc$.

Logarithm Rules

$\log(ab) = \log a + \log b$; $\log(a/b) = \log a - \log b$; $\log a^n = n \log a$; $\log_c 1 = 0$.

SOHCAHTOA

Sine = opposite/hypotenuse; Cosine = adjacent/hypotenuse; Tangent = opposite/adjacent.

Reciprocal Trig Relationships

Cosecant = $1/\text{Sine}$; Secant = $1/\text{Cosine}$; Cotangent = $1/\text{Tangent}$.

Pythagorean Trig Relationships

$\sin^2x + \cos^2x = 1$; $\tan^2x + 1 = \sec^2x$; $1 + \cot^2x = \csc^2x$.

ASTC Rule

Quadrants where trig values are positive: I (All functions), II (Sine/Cosecant), III (Tangent/Cotangent), IV (Cosine/Secant).