ACT Math: Concepts, Skills, and Problem-Solving Methods

Number Sense, Expressions, and First-Step Algebra

ACT Math rewards two things at once: accurate computation and good decision-making about what to compute. That starts with number sense—understanding how numbers behave—because many questions are designed so that a small simplification or a smart rewrite saves time and prevents mistakes.

The real number system and place value

Real numbers include all the numbers you can place on a number line: integers, fractions, terminating and repeating decimals, and irrational numbers like square roots of non-perfect squares.

Why it matters: When you simplify expressions, compare quantities, or solve equations, you rely on properties that are true for real numbers (like distributing multiplication over addition). Also, ACT questions often hide easy comparisons inside fractions, negatives, or radicals.

A few practical reminders:

• Integers are whole numbers and their negatives.
• Rational numbers can be written as a fraction \frac{a}{b} where a and b are integers and b \ne 0.
• Irrational numbers cannot be written as a fraction of integers (for example \sqrt{2}).

Common “number line” traps:

• Negative numbers: -5 is less than -2.
• Absolute value: absolute value is distance from zero, so it is never negative.

Definition:
|x| = \text{the distance of } x \text{ from } 0

So | -7 | = 7 and | 7 | = 7.

Fractions, decimals, percent, ratios, and proportions

These are multiple “languages” for the same idea: a part compared to a whole.

Percent means “per 100.” Converting between forms is often the fastest path.

• Percent to decimal: divide by 100.
• Decimal to percent: multiply by 100.

For example, 35\% = 0.35 = \frac{35}{100} = \frac{7}{20}.

Ratio compares quantities, often written a:b or \frac{a}{b}. A proportion is an equation stating two ratios are equal.

Example (proportion reasoning): If \frac{x}{12} = \frac{5}{8}, you can solve by cross-multiplying:

8x = 12 \cdot 5
8x = 60
x = \frac{60}{8} = 7.5

Why it matters: Many word problems—scale drawings, similar figures, mixture problems, and unit rates—are proportion problems in disguise.

A key caution: Cross-multiplying is valid only when you have an equation of two fractions, like \frac{a}{b} = \frac{c}{d}, with denominators not equal to zero.

Order of operations and simplifying expressions

When you simplify, you are rewriting an expression in an equivalent form—same value for all allowed inputs—usually to make it easier to evaluate or solve.

Order of operations (often remembered as PEMDAS) means:
1) Parentheses (grouping)
2) Exponents
3) Multiplication and division left-to-right
4) Addition and subtraction left-to-right

Example: Evaluate 6 - 2(3^2).

• Exponent first: 3^2 = 9
• Multiply: 2 \cdot 9 = 18
• Subtract: 6 - 18 = -12

Distributive property is a workhorse:

a(b + c) = ab + ac

This is especially important for removing parentheses and combining like terms.

Example: Simplify 3(2x - 5) + 4x.

• Distribute: 3(2x - 5) = 6x - 15
• Combine like terms: 6x - 15 + 4x = 10x - 15

Common mistake: combining unlike terms. You can combine 6x and 4x because both are “x-terms,” but you cannot combine 10x and -15 into a single term.

Exponents, radicals, and scientific notation

Exponents describe repeated multiplication. The ACT commonly tests rules that let you rewrite expressions efficiently.

Key exponent rules (for nonzero bases):

a^m \cdot a^n = a^{m+n}
\frac{a^m}{a^n} = a^{m-n}
(a^m)^n = a^{mn}
(ab)^n = a^n b^n

Negative exponents mean “reciprocal”:

a^{-n} = \frac{1}{a^n}

Radicals are roots. The most common radical is the square root. A critical simplification skill is pulling out perfect-square factors:

\sqrt{ab} = \sqrt{a}\sqrt{b}

Example: Simplify \sqrt{72}.

• Factor out a perfect square: 72 = 36 \cdot 2
• Then \sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}

Common mistake: assuming \sqrt{a+b} = \sqrt{a} + \sqrt{b}. This is not generally true.

Scientific notation writes large/small numbers as:

a \times 10^n

where 1 \le a < 10 and n is an integer.

Example: 0.00056 = 5.6 \times 10^{-4} because you move the decimal 4 places to the right to make 5.6.

Solving linear equations and inequalities

Solving an equation means finding the value(s) of the variable that make the statement true.

For a linear equation, your goal is to isolate the variable using inverse operations.

Example: Solve 5x - 7 = 18.

• Add 7 to both sides: 5x = 25
• Divide by 5: x = 5

Inequalities are similar, with one extra rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example: Solve -3x \le 12.

• Divide by -3 and flip: x \ge -4

Why it matters: ACT word problems often translate into inequalities (at least, at most, no more than, minimum, maximum).

Translating word problems into algebra

Many ACT questions test whether you can turn English into math.

Useful translation anchors:

• “sum” means add
• “difference” means subtract (watch order)
• “product” means multiply
• “quotient” means divide
• “is” means equals

Example: “Five less than twice a number x is 17.”

• Twice a number: 2x
• Five less than that: 2x - 5
• Is 17: 2x - 5 = 17
• Solve: 2x = 22, so x = 11

Common mistake: reversing “less than.” “Five less than twice x” is 2x - 5, not 5 - 2x.

Exam Focus

• Typical question patterns:
  • Simplify an expression by distributing and combining like terms.
  • Solve a linear equation/inequality, sometimes with fractions.
  • Convert between fraction/decimal/percent in a real-world context (discounts, tax, interest).
• Common mistakes:
  • Forgetting to flip the inequality when dividing by a negative.
  • Combining unlike terms or mishandling negative signs when distributing.
  • Using incorrect radical “rules,” especially \sqrt{a+b}.

Functions, Linear Models, and Systems

A function is a rule that takes an input and produces exactly one output. ACT Math uses functions to connect algebra to graphs and to model relationships between quantities.

Function notation, domain, and interpretation

Function notation like f(x) means “the output of function f when the input is x.”

Why it matters: ACT questions often hide simple substitution inside function notation.

Example: If f(x) = 2x^2 - 3x, find f(4).

• Substitute 4 for x:

f(4) = 2(4^2) - 3(4)
f(4) = 2(16) - 12
f(4) = 32 - 12 = 20

Domain is the set of allowed inputs. On the ACT, domain restrictions usually come from:

• Denominators cannot be zero.
• Even roots require the radicand to be nonnegative (in real-number contexts).

Example: Domain of g(x) = \frac{1}{x-5} excludes x = 5.

Linear relationships: slope and intercept

A linear function has a constant rate of change. Its graph is a straight line.

The common form is:

y = mx + b

• m is the slope (change in y per 1 unit change in x)
• b is the y-intercept (the value of y when x = 0)

Slope between two points (x1, y1) and (x2, y2) :

m = \frac{y2 - y1}{x2 - x1}

Why it matters: slope connects algebra, geometry, and data. It is also the backbone of many “rate” word problems.

Example: Find the line through (2, 3) and (6, 11).

• Compute slope:

m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2

• Use point-slope form:

y - 3 = 2(x - 2)

• Simplify:

y - 3 = 2x - 4
y = 2x - 1

Common mistake: flipping x and y differences, or forgetting parentheses around subtraction (especially with negatives).

Parallel and perpendicular lines

These relationships show up when you’re given one line and asked to build another.

• Parallel lines have the same slope.
• Perpendicular lines (non-vertical) have slopes that are negative reciprocals:

If m1 m2 = -1, the lines are perpendicular.

Example: A line perpendicular to y = \frac{1}{3}x + 2 has slope -3.

Caution: Vertical lines have undefined slope; horizontal lines have slope 0. A vertical line is perpendicular to a horizontal line.

Systems of linear equations

A system is two (or more) equations that must be true at the same time. Graphically, you are looking for intersection point(s).

Common solving methods:

• Substitution: solve one equation for one variable, substitute into the other.
• Elimination: add/subtract multiples of equations to eliminate a variable.

Example (elimination):

2x + y = 11
3x - y = 4

Add equations to eliminate y:

5x = 15
x = 3

Substitute into 2x + y = 11:

2(3) + y = 11
y = 5

So the solution is (3, 5).

Why it matters: Systems model situations with two constraints (tickets sold, mixture problems, comparing two plans).

Function transformations (graph shifts and stretches)

You don’t always need to draw a full graph. Often, you recognize how a known graph changes.

Starting with y = f(x):

• y = f(x) + k shifts up by k.
• y = f(x - h) shifts right by h.
• y = af(x) vertically stretches by factor |a| (and reflects over the x-axis if a < 0).

Common mistake: mixing up f(x-h) and f(x)+h. The shift inside the function goes the “opposite” direction you expect: x - h moves right.

Exam Focus

• Typical question patterns:
  • Evaluate or interpret f(x) at given inputs; identify domain restrictions.
  • Find slope from two points or from a graph; write a line equation.
  • Solve a system (often two linear equations) using elimination/substitution.
• Common mistakes:
  • Confusing b (y-intercept) with x-intercept.
  • Sign errors in slope and point-slope form, especially with negative coordinates.
  • Mishandling shifts: f(x-h) vs f(x)+h.

Polynomials, Factoring, and Quadratic Equations

Polynomials are expressions built from variables with nonnegative integer exponents (like x^2 or x^3). Quadratics are especially important on the ACT because they connect algebraic solving, graphs, and geometry.

Polynomial structure and operations

A polynomial looks like a sum of terms such as a_n x^n. The highest exponent is the degree.

Operations you should be comfortable with:

• Adding/subtracting polynomials: combine like terms.
• Multiplying polynomials: distribute every term.

Example: Multiply (x + 3)(x - 5).

(x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15

Why it matters: Factoring and expansion show up in solving equations, simplifying rational expressions, and modeling area.

Factoring as “reverse distributing”

Factoring rewrites an expression as a product. This is powerful because equations become easier to solve when written as products.

Core factoring patterns:

1) Greatest common factor (GCF):

ax + ay = a(x + y)

2) Difference of squares:

a^2 - b^2 = (a - b)(a + b)

3) Perfect square trinomial:

a^2 + 2ab + b^2 = (a + b)^2

and

a^2 - 2ab + b^2 = (a - b)^2

Example: Factor x^2 - 9.

x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)

Common mistake: trying to factor a sum of squares a^2 + b^2 over the real numbers. It does not factor into real linear factors.

Solving quadratic equations

A quadratic equation can often be written in standard form:

ax^2 + bx + c = 0

There are three main ACT-friendly methods.

1) Factoring and the zero-product property

If you can factor into two binomials:

(px + q)(rx + s) = 0

then you use the zero-product property: if a product is zero, at least one factor is zero.

Example: Solve x^2 - 5x + 6 = 0.

• Factor:

x^2 - 5x + 6 = (x - 2)(x - 3)

• Set factors to zero:

x - 2 = 0 \Rightarrow x = 2
x - 3 = 0 \Rightarrow x = 3

2) Completing the square

Completing the square rewrites a quadratic into vertex form. It’s especially useful when factoring is hard.

Example: Solve x^2 + 6x + 5 = 0 by completing the square.

• Move constant:

x^2 + 6x = -5

• Add the square of half of 6 (which is 3):

x^2 + 6x + 9 = -5 + 9

• Factor left side:

(x + 3)^2 = 4

• Take square roots:

x + 3 = 2
x + 3 = -2

So x = -1 or x = -5.

Common mistake: forgetting the \pm when taking square roots.

3) Quadratic formula

When a quadratic is in ax^2 + bx + c = 0, the solutions are:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the radical is the discriminant:

D = b^2 - 4ac

• If D > 0: two real solutions
• If D = 0: one real solution (a repeated root)
• If D < 0: no real solutions (complex solutions, usually not emphasized in basic ACT contexts)

Why it matters: The quadratic formula is a reliable fallback when factoring is not obvious.

Quadratic graphs and key features

The graph of y = ax^2 + bx + c is a parabola.

• If a > 0 it opens upward; if a < 0 it opens downward.
• The vertex is the turning point.

A very useful form is vertex form:

y = a(x - h)^2 + k

where (h, k) is the vertex.

Connections you should notice:

• Solutions to ax^2 + bx + c = 0 are x-intercepts of the graph.
• Factoring gives intercepts quickly.

Example: For y = (x - 2)^2 - 9, the vertex is (2, -9) and it opens upward.

Rational expressions (algebraic fractions)

A rational expression is a fraction with polynomials.

Example:

\frac{x^2 - 9}{x - 3}

You simplify by factoring and canceling common factors—but only factors, not terms.

\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} = x + 3

Important: the original expression is undefined at x = 3, even though the simplified form looks fine there. On the ACT, this shows up as “holes” in graphs or excluded values in domains.

Common mistake: canceling across addition, like trying to cancel the x in \frac{x+2}{x+5}. You can only cancel a common factor, not a piece of a sum.

Exam Focus

• Typical question patterns:
  • Factor a quadratic or special product; solve by setting factors equal to zero.
  • Use the quadratic formula or discriminant to determine number of real solutions.
  • Simplify rational expressions and identify excluded values.
• Common mistakes:
  • Dropping the \pm when solving equations like (x-h)^2 = k.
  • Canceling terms instead of factors in rational expressions.
  • Sign errors in the quadratic formula, especially -b and 2a.

Geometry Fundamentals: Angles, Triangles, and Polygons

Geometry on the ACT is less about formal proofs and more about fluent use of relationships: how angles add up, how triangles behave, and how measurements scale.

Angles and lines

Key angle facts that appear constantly:

• A straight line is 180^\circ.
• A full circle is 360^\circ.
• Vertical angles are equal.
• Adjacent angles that form a straight line are supplementary (sum to 180^\circ).
• When parallel lines are cut by a transversal, corresponding angles are equal and interior angles on the same side are supplementary.

Why it matters: Many problems give a diagram with minimal labels. Recognizing these relationships lets you set up equations quickly.

Example: If two supplementary angles are (3x + 10)^\circ and (5x - 2)^\circ , then:

(3x + 10) + (5x - 2) = 180
8x + 8 = 180
8x = 172
x = 21.5

Triangles: angle sum, special triangles, and the Pythagorean theorem

For any triangle, the interior angles sum to 180^\circ.

A right triangle has one 90^\circ angle. In right triangles, the side opposite the right angle is the hypotenuse.

The Pythagorean theorem relates the side lengths of a right triangle:

a^2 + b^2 = c^2

where c is the hypotenuse.

Example: A right triangle has legs 9 and 12. Find the hypotenuse.

9^2 + 12^2 = c^2
81 + 144 = c^2
225 = c^2
c = 15

Common mistake: using the theorem on a non-right triangle, or mixing up which side is c.

Special right triangles

These come up so often that memorizing them saves time.

1) 45^\circ - 45^\circ - 90^\circ triangle:

• legs equal
• hypotenuse is \text{leg} \cdot \sqrt{2}

2) 30^\circ - 60^\circ - 90^\circ triangle:

• short leg (opposite 30^\circ) is smallest
• long leg is \text{short} \cdot \sqrt{3}
• hypotenuse is 2 \cdot \text{short}

Example: In a 30^\circ - 60^\circ - 90^\circ triangle with short leg 7, hypotenuse is 14 and long leg is 7\sqrt{3}.

Similarity, scale factors, and proportional reasoning in geometry

Two figures are similar if they have the same shape (equal angles) and proportional side lengths.

Why it matters: Similarity turns geometry into algebra via proportions. It also explains how area and volume scale when lengths scale.

If similar figures have a linear scale factor k:

• Perimeters scale by k.
• Areas scale by k^2.
• Volumes (for similar solids) scale by k^3.

Example: If a triangle is enlarged by factor 3, its area becomes 3^2 = 9 times as large.

Common mistake: scaling area by k instead of k^2.

Polygons: interior angles and regular polygons

A polygon is a closed figure with straight sides. A regular polygon has all sides and angles equal.

Sum of interior angles of an n-gon:

(n - 2) \cdot 180^\circ

For a regular n-gon, each interior angle is:

\frac{(n - 2) \cdot 180^\circ}{n}

Example: Each interior angle of a regular hexagon ( n = 6 ):

\frac{(6 - 2) \cdot 180}{6} = \frac{720}{6} = 120^\circ

Exam Focus

• Typical question patterns:
  • Use supplementary/vertical/corresponding angles to find unknown measures.
  • Apply triangle angle sum, Pythagorean theorem, and special right triangles.
  • Use similarity to find missing lengths or compare areas.
• Common mistakes:
  • Forgetting that triangle angles sum to 180^\circ (or mixing with straight-line 180^\circ).
  • Mislabeling the hypotenuse in Pythagorean problems.
  • Scaling area or volume incorrectly when given a scale factor.

Coordinate Geometry: Graphing, Distance, Midpoint, and Circles

Coordinate geometry blends algebra with geometry by placing figures on the coordinate plane. ACT questions here often feel “formula-heavy,” but the real skill is recognizing which formula applies and setting it up cleanly.

Distance and midpoint

The distance formula comes from the Pythagorean theorem. For points (x1, y1) and (x2, y2):

d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Example: Distance between (1, 2) and (5, 5):

d = \sqrt{(5-1)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5

The midpoint formula gives the point halfway between two points:

M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)

Example: Midpoint of (1, 2) and (5, 5):

M = \left(\frac{1+5}{2}, \frac{2+5}{2}\right) = (3, 3.5)

Common mistake: forgetting to divide both coordinates by 2.

Lines: slope-intercept and point-slope revisited

In coordinate geometry, lines show up in several equivalent forms.

Notation reference (same line, different forms):

Example: Convert 2x + 3y = 12 to slope-intercept.

3y = -2x + 12
y = -\frac{2}{3}x + 4

Circles on the coordinate plane

A circle is defined as all points a fixed distance (radius) from a center.

Standard equation of a circle with center (h, k) and radius r:

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

Why it matters: ACT problems often give you the center and radius directly, or they give you an equation and ask you to identify the center/radius.

Example: For (x - 2)^2 + (y + 1)^2 = 25:

• Center is (2, -1)
• Radius is 5

Common mistake: forgetting that r^2 is on the right side. If the equation ends with = 25, then r = 5, not 25.

Parabolas as quadratic graphs

Even when conic sections are not labeled explicitly, the ACT expects you to understand parabolas from quadratics.

If you have:

y = ax^2 + bx + c

you can often find intercepts by solving ax^2 + bx + c = 0, and you can find the vertex by completing the square or using:

x = -\frac{b}{2a}

Then substitute back to get the corresponding y.

Example: For y = x^2 - 4x + 1, the vertex x-coordinate is:

x = -\frac{-4}{2 \cdot 1} = 2

Then:

y = 2^2 - 4(2) + 1 = 4 - 8 + 1 = -3

Vertex is (2, -3).

Exam Focus

• Typical question patterns:
  • Compute distance, midpoint, or slope between two points.
  • Identify circle center/radius from an equation, or write the equation from center/radius.
  • Use a quadratic graph’s intercepts/vertex to answer a question without graphing fully.
• Common mistakes:
  • Dropping parentheses in x2 - x1 or y2 - y1 leading to sign errors.
  • Confusing center coordinates in circles with signs in (x-h)^2 and (y-k)^2.
  • Using x = -\frac{b}{2a} but forgetting to compute the matching y-value.

Measurement: Perimeter, Area, Surface Area, Volume, and Units

Measurement questions test your ability to connect formulas to geometry and to keep units consistent. Many ACT errors come from unit confusion rather than “hard math.”

Perimeter and area of common 2D shapes

Perimeter measures boundary length; area measures surface coverage.

Rectangle:

• Perimeter: P = 2l + 2w
• Area: A = lw

Triangle:

• Area: A = \frac{1}{2}bh

where b is a base and h is the perpendicular height to that base.

Parallelogram:

• Area: A = bh

Trapezoid:

• Area: A = \frac{1}{2}(b1 + b2)h

Why it matters: The ACT often gives a diagram where the “height” is not a side length. You must use the perpendicular distance.

Example: A triangle has base 10 and height 7. Area is:

A = \frac{1}{2} \cdot 10 \cdot 7 = 35

Common mistake: using a slanted side as the height.

Circles: circumference, area, arcs, and sectors

Circle formulas:

• Circumference:

C = 2\pi r

• Area:

A = \pi r^2

Arc length and sector area are “fraction of the circle” problems. If the central angle is \theta in degrees:

\text{Arc length} = \frac{\theta}{360} \cdot 2\pi r

\text{Sector area} = \frac{\theta}{360} \cdot \pi r^2

Example: A circle has radius 6 and a sector angle 120^\circ. Sector area:

\text{Sector area} = \frac{120}{360} \cdot \pi \cdot 6^2 = \frac{1}{3} \cdot 36\pi = 12\pi

Common mistake: using 180 instead of 360 (mixing with semicircle thinking) or forgetting to square r in area.

Surface area and volume of common solids

3D problems require you to distinguish:

• Surface area: total area of all outer faces.
• Volume: space inside.

Rectangular prism:

V = lwh

Cylinder:

V = \pi r^2 h

Why it matters: Many real-world contexts (tanks, cans, boxes) are cylinders or prisms.

Example: Cylinder with r = 3 and h = 10:

V = \pi \cdot 3^2 \cdot 10 = 90\pi

Common mistake: using 2\pi r (circumference) instead of \pi r^2 (base area) in cylinder volume.

Unit conversions and dimensional reasoning

Unit problems are about multiplying by “forms of 1.” For example, to convert inches to feet, you use:

\frac{1\ \text{ft}}{12\ \text{in}}

because it equals 1 but changes units.

Example: Convert 30 inches to feet.

30\ \text{in} \cdot \frac{1\ \text{ft}}{12\ \text{in}} = \frac{30}{12}\ \text{ft} = 2.5\ \text{ft}

A strong habit: write units at each step. If units cancel correctly, your setup is likely correct.

Exam Focus

• Typical question patterns:
  • Apply area formulas in composite figures (add/subtract simpler shapes).
  • Use circle arc/sector relationships with degree measures.
  • Compute volume from a real situation and convert units.
• Common mistakes:
  • Using a non-perpendicular “height” in triangle/parallelogram areas.
  • Confusing circumference with area, or forgetting r^2.
  • Unit mismatches (e.g., mixing inches and feet without converting).

Trigonometry: Right-Triangle Ratios and Applications

ACT trigonometry is primarily about right triangles and the basic trig functions. The deeper idea is that trig ratios connect angles to side ratios, letting you solve for distances that are hard to measure directly.

Sine, cosine, and tangent in right triangles

In a right triangle relative to an acute angle \theta:

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Why it matters: Many applications (height of a tree, angle of elevation, ramp incline) reduce to these ratios.

Example: In a right triangle, relative to angle \theta, opposite side is 9 and adjacent side is 12. Then:

\tan(\theta) = \frac{9}{12} = \frac{3}{4}

Common mistake: picking the wrong “opposite” side. Opposite/adjacent depend on which angle \theta you are referencing.

Using inverse trig to find an angle

If you know a trig ratio value and want the angle, you use inverse trig (often via calculator on the ACT).

Example: If \tan(\theta) = 0.75, then:

\theta = \tan^{-1}(0.75)

On a calculator, this gives an angle in degrees (if your calculator is in degree mode). A frequent ACT pitfall is being in radian mode when the test expects degrees.

Special-angle trig values via special triangles

You can quickly find common trig values using special right triangles.

From a 45^\circ - 45^\circ - 90^\circ triangle with legs 1 and hypotenuse \sqrt{2}:

\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

\cos(45^\circ) = \frac{\sqrt{2}}{2}

\tan(45^\circ) = 1

From a 30^\circ - 60^\circ - 90^\circ triangle with short leg 1, long leg \sqrt{3}, hypotenuse 2:

\sin(30^\circ) = \frac{1}{2}

\cos(30^\circ) = \frac{\sqrt{3}}{2}

\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

And:

\sin(60^\circ) = \frac{\sqrt{3}}{2}

\cos(60^\circ) = \frac{1}{2}

\tan(60^\circ) = \sqrt{3}

Why it matters: These values appear directly and also indirectly in geometry problems involving equilateral triangles cut in half.

Angle of elevation and depression

These are real-world framing terms:

• Angle of elevation: you look upward from horizontal.
• Angle of depression: you look downward from horizontal.

In both cases, you typically form a right triangle where the angle is at the observer and the vertical/horizontal legs represent height and distance.

Example: From a point on the ground, the angle of elevation to the top of a building is 35^\circ. The horizontal distance to the building is 50 ft. Approximate height h:

\tan(35^\circ) = \frac{h}{50}

h = 50\tan(35^\circ)

You would evaluate \tan(35^\circ) on a calculator.

Common mistake: using sine/cosine when tangent is the ratio you actually have (opposite and adjacent).

Exam Focus

• Typical question patterns:
  • Compute \sin, \cos, \tan from side lengths in a right triangle.
  • Use special-angle values 30^\circ, 45^\circ, 60^\circ.
  • Model a height/distance situation using angle of elevation with tangent.
• Common mistakes:
  • Choosing opposite/adjacent incorrectly for a given angle.
  • Calculator in radian mode when answers are in degrees.
  • Forgetting to rationalize or simplify trig expressions when needed (like \frac{1}{\sqrt{3}}).

Statistics, Data Interpretation, and Probability

ACT Math statistics and probability are about interpreting information, not just calculating. A common theme is that you must read graphs/tables carefully and decide what computation (if any) is required.

Center and spread: mean, median, mode, range

Given a data set:

• Mean is the average:

\text{mean} = \frac{\text{sum of values}}{\text{number of values}}

• Median is the middle value after sorting (or the average of the two middle values if there are an even number).
• Mode is the most frequent value.
• Range is max minus min.

Why it matters: The ACT tests whether you choose the correct measure for the situation. Median is resistant to outliers; mean is pulled by extreme values.

Example: Data: 2, 2, 3, 4, 14.

• Mean:

\frac{2+2+3+4+14}{5} = \frac{25}{5} = 5

• Median is 3.

Notice how the outlier 14 pulls the mean upward.

Common mistake: finding the median without sorting first.

Reading graphs: scatterplots, trends, and slope as rate

Scatterplots show paired data (x, y). You may be asked about:

• Direction (positive/negative association)
• Strength (points close to a line vs widely scattered)
• Interpreting slope in context

Why it matters: Many questions are “interpretation” questions—no heavy algebra required, but careful reading is.

Example idea: If a trend line has slope 2 and x is hours, y is dollars earned, then slope means 2 dollars per hour.

Common mistake: mixing up which variable is on which axis. Always read axis labels.

Basic probability

Probability is the fraction of outcomes that satisfy an event.

P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total equally likely outcomes}}

Probabilities are between 0 and 1.

Complement rule:

P(A^c) = 1 - P(A)

Example: If the probability of rain is 0.3, then probability of no rain is 0.7.

Compound probability: “and” vs “or”

For independent events (like flipping a fair coin twice), the probability of both happening is the product:

P(A \text{ and } B) = P(A)P(B)

Example: Probability of getting heads twice:

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

For “or” situations, you must avoid double-counting overlap. A general formula is:

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Why it matters: ACT questions often test whether you understand overlap.

Common mistake: adding probabilities for “and” situations instead of multiplying (when appropriate).

Counting: permutations and combinations

Counting questions ask “how many ways?” The two big ideas:

• Permutation: order matters.
• Combination: order does not matter.

Permutations of n distinct items:

n!

Permutations of choosing r items from n (order matters):

P(n, r) = \frac{n!}{(n-r)!}

Combinations of choosing r items from n (order does not matter):

C(n, r) = \frac{n!}{r!(n-r)!}

Example: How many ways to choose 3 students from 10 (order does not matter)?

C(10, 3) = \frac{10!}{3!7!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120

A useful reasoning shortcut: if you catch yourself listing arrangements and you don’t care about order, you probably need combinations.

Exam Focus

• Typical question patterns:
  • Compute mean/median/range from a table or list; interpret an outlier’s effect.
  • Read slopes and trends from graphs; identify which statement matches a scatterplot.
  • Probability of combined events; count outcomes with combinations/permutations.
• Common mistakes:
  • Using mean when the problem context suggests median (or vice versa), especially with outliers.
  • Treating “and” like “or” in probability, or forgetting to subtract overlap.
  • Using permutations when combinations are needed (or forgetting order matters).