All of SAT Math Explained in 26 Minutes

Summary: All of SAT Math Explained in 26 Minutes

The presenter organizes SAT Math into five levels, moving from foundational skills to uncommon, challenging problems. The central advice is to understand how mathematical ideas connect, use Desmos strategically, memorize a limited set of key formulas, and practice difficult questions—especially in areas where you have made mistakes before.

Level 1: Foundations

• Basic algebra: Be comfortable isolating variables, combining fractions, and following the order of operations.
• Desmos: The presenter recommends using Desmos to solve many single-variable equations, including ones involving fractions, square roots, and absolute values. On a graph, solutions correspond to the relevant intersections.
• Linear functions: A function takes an input and produces an output. Understand how an equation, a table of values, and a graph describe the same function.
• Slope and intercept: Slope measures the change in (y) for a change in (x). To find a line’s equation from two points:
  1. Calculate the slope from the points’ changes in (y) and (x).
  2. Substitute the slope and one point into (y = mx + b).
  3. Solve for the (y)-intercept (b).

Level 2: Core SAT Math

• Variables, constants, and coefficients: Variables can change; constants stay fixed; a coefficient is a number multiplying a variable.
• Systems of equations: Think of each equation as a constraint. A solution satisfies both constraints and is the point where their graphs intersect. Parallel distinct lines have no solution; identical lines have infinitely many.
• Parallel and perpendicular lines: Parallel lines have equal slopes. Perpendicular lines have negative-reciprocal slopes.
• Systems of inequalities: These can be approached similarly, including graphically with Desmos.
• Translating words into math: Recognize phrases such as “is” (equals) and “of” (multiplication), and convert statements into equations.
• Exponent rules: Know the standard exponent rules. For example, multiplying powers with the same base means adding their exponents.
• Geometry basics:
  • Angles forming a right angle sum to (90^\circ); a straight angle sums to (180^\circ); a full circle is (360^\circ).
  • Vertical angles are equal, as are corresponding angles formed by parallel lines.
  • A triangle’s angles sum to (180^\circ); a quadrilateral’s sum to (360^\circ). A polygon with (n) sides has an interior-angle sum of (180(n - 2)^\circ).
  • Use the Pythagorean theorem for missing sides in right triangles.
  • Know sine, cosine, and tangent.
  • Know that density is mass divided by volume, and distinguish isosceles and equilateral triangles.
  • Understand perimeter, area, and volume. The presenter says area and volume formulas are on the SAT formula sheet, so focus on applying them.
• Statistics: The mean is the sum divided by the number of values; the median is the middle value; the range is maximum minus minimum. Outliers can pull the mean away from the median.

Level 3: Intermediate Topics

• Integers and transformations: Know that integers are whole numbers, including zero and negative whole numbers. Understand horizontal and vertical shifts of functions; some transformation questions can be checked with Desmos.
• Percentages: Convert percentages to multipliers:
  • (20\%) of (x) is (0.2x).
  • Increasing (x) by (20\%) gives (1.2x).
  • Decreasing (x) by (20\%) gives (0.8x).
  • Be careful to distinguish an amount that is 150% of a value from one that is 150% more than that value.
• Circles:
  • Know the equation ((x - h)^2 + (y - k)^2 = r^2). It follows from applying the Pythagorean theorem to the horizontal and vertical distances from the center.
  • A tangent to a circle is perpendicular to the radius at the point of tangency.
  • Find arc length and sector area by taking the corresponding fraction of the circle’s circumference or area.
  • An inscribed angle measures half the central angle subtending the same arc.
  • Remember (180^\circ = \pi) radians and convert using the appropriate proportion.
• Triangle relationships:
  • Congruent triangles are identical in size and shape; similar triangles have equal angles and proportional side lengths.
  • Similarity can be established using angle-angle, side-side-side, or side-angle-side—not side-side-angle alone.
• Conditional probability: A probability “given” a condition is calculated using only the subset that meets that condition.
• Data displays and sampling: Interpret box plots, including their minimum, quartiles, median, and maximum. Read scatter plots and lines of best fit, and use random, representative samples to avoid bias.
• Quadratics: Be able to factor quadratic expressions and use Desmos to find roots and the vertex.

Level 4: Advanced and Applied Topics

• Quadratic forms and properties:
  • In standard form (ax^2 + bx + c), the sign of (a) determines whether the parabola opens up or down, and (c) is the (y)-intercept.
  • In vertex form (a(x - h)^2 + k), the vertex is ((h, k)).
  • The vertex’s (x)-coordinate is (-b/(2a)).
  • The vertex lies midway between the two roots. The sum and product of roots can also be found from the coefficients, which can provide shortcuts.
  • For a quadratic intersecting a line, the presenter recommends Desmos when appropriate.
• Polynomial functions: Understand how solutions relate to the graph: at roots with odd multiplicity, the graph crosses the (x)-axis; at roots with even multiplicity, it touches the axis and turns back.
• Exponential functions: Build an exponential model from repeated multiplication. For example, money growing by (4\%) annually can be represented as (1000(1.04)^t). Understand how changing or scaling the exponent affects the model.
• Right-triangle trigonometry: The sine of an angle equals the cosine of its complement, and vice versa.
• Scaling and units: Area scales with the square of a length multiplier; volume scales with its cube. For example, doubling a square’s side length multiplies its area by four, while doubling a cube’s side length multiplies its volume by eight. Apply squared conversion factors when converting area units.
• Surface area: Be able to construct an equation for unusual shapes, such as combined boxes, rather than relying only on memorized formulas.
• Statistics: Read frequency tables. In a symmetric data set, the mean and median are equal. Standard deviation measures how spread out the data is.
• Margin of error: A sample estimate and its margin of error describe a plausible interval, not every value that is mathematically possible. Larger samples generally reduce the margin of error.

Level 5: Hardest and Less Common Problems

• Altitude to the hypotenuse: In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles. All three triangles are similar, so their angles and side proportions can be used to solve for missing lengths.
• Quadratics with integer conditions: When a quadratic factors using unknown integer constants, set up relationships between the factors and the coefficients, then use the integer constraints to determine which answer must be an integer.
• Pyramid surface area: Distinguish the pyramid’s vertical height from the slant height of its triangular faces. Use a right triangle relating the vertical height, the distance across the base, and the slant height.
• Hand-solving for top scores: When complicated constants prevent effective use of Desmos, be ready to solve by hand. The presenter recommends knowing the quadratic formula, the discriminant (b^2 - 4ac), and how to complete the square for circle problems.
• Additional useful facts: The presenter also highlights standard-form line slopes, arc length as radius times central angle in radians, common right-triangle side patterns such as (3!-!4!-!5) and (5!-!12!-!13), and the triangle inequality (the sum of any two sides must exceed the third).
• Practice strategy: Work on the hardest problems in each category, with particular attention to mean questions involving integers, exponential functions with complicated fractional exponents, and quadratic questions involving combinations of coefficients. Review past mistakes and ask how a problem could be made harder.

Speakers and Sources Featured

• Main speaker: The presenter from LearnSATMath, who teaches the SAT Math topics and refers to his own Desmos video and tutoring.
• Acy: A sponsor featured in a demonstration of using its tutoring tool to explain a cylinder problem and generate practice questions.
• Desmos: Referenced as a graphing tool for solving equations, systems, and quadratic problems.
• The SAT/College Board: Referenced as the source of the tested content and example question styles; no separate College Board speaker appears in the subtitles.