SAT Math Level Notes - Vocabulary Flashcards (Video)

Introduction

The SAT Math content is organized into 5 levels based on score, with the idea that math builds on itself. You should master everything in a level before moving to the next.
Score ranges reflect a mapping to actual SAT scores (estimated): about 20 points lower than Bluebook 5-6 and 40 points lower than Bluebook 1-4. Example: a 650 on Bluebook 1

\approx

610 on the actual SAT.

Every formula you see should be memorized. Formulas not explicitly stated are on the provided formula sheet (included at the end).
Terminology: “algebraically” means solving on paper; “graphically” means solving with DESMOS.
DESMOS is used for solving multi-step problems involving graphs/tables; Level references to when DESMOS is specifically recommended are included in the bullets below.

Level 1 (200–450)

Algebra- Key fundamentals to be comfortable with:
- Order of operations (PEMDAS)
- Example: Solve 3 + 2 \times (5 - 1) \rightarrow 3 + 2 \times 4 = 3 + 8 = 11
- Fractions
- Example: Add \frac{1}{2} + \frac{1}{3} \rightarrow \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
- Factoring
- Example: Factor x^2 - 4x + 3 \rightarrow (x-1)(x-3)
DESMOS- A video covers all needed DESMOS basics; the most important skill emphasized is solving single-variable equations.
- Example: Use DESMOS to solve 2x + 7 = 15
- Reference to video timestamp: 0:54–1:53 for solving single-variable equations.
Linear Functions
- (a) Functions map inputs to outputs.
- Example: For f(x) = 2x + 1, an input of 3 maps to an output of f(3) = 7
- (b) Understand the relationship between a table, a graph, and an equation of a linear function.
- Example: Given points (1,3) and (2,5) in a table, find the linear equation.
- (c) Equation of a line in slope-intercept form: y = mx + b
- Example: Identify slope (m=3) and y-intercept (b=-2) in y = 3x - 2
- (d) Find the slope and the y-intercept from 2 points: m = \frac{y2 - y1}{x2 - x1}
- Example: Find the slope of a line passing through (1,5) and (3,11).
- (e) Find the x-intercept of a line algebraically and graphically.
- Example: Find the x-intercept of y = 2x - 4 (set y=0, so 0 = 2x - 4 \rightarrow x=2).

Level 2 (450–550)

Core concepts
- 1. Variables vs. constants (what they are).
- Example: In y = 3x + 5, x and y are variables, while 3 and 5 are constants.
- 2. Coefficients (what they are).
- Example: In 3x^2 + 2x - 1, 3 and 2 are the coefficients of x^2 and x, respectively.
- 3. Systems of equations (use DESMOS).
- (a) What does it mean if a system has a solution?
  - Example: The solution to \begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases} is their intersection point.
- (b) What about no solutions?
  - Example: Parallel lines like y=2x+3 and y=2x-1 have no solutions.
- (c) What about infinite solutions?
  - Example: Two identical equations like y=2x+3 and 2y=4x+6 have infinite solutions.
- 4. Parallel & perpendicular lines
- (a) If two lines have no solution, they are parallel.
- (b) Parallel lines have the same slope.
  - Example: y = 3x + 2 and y = 3x - 5
- (c) Perpendicular lines have slopes that are negative reciprocals.
  - Example: y = 2x + 1 and y = -\frac{1}{2}x + 3
- 5. Systems of inequalities (use DESMOS).
- Example: Graph the solution for \begin{cases} y > x + 1 \ y < -2x + 5 \end{cases}
- 6. Number of solutions of a polynomial function based on its graph.
- Example: A parabola that intersects the x-axis twice has two real solutions.
- 7. Translating English to math
- (a) “of” means multiplication.
- (b) “is” means equals.
- Example: "5 less than twice a number is 10" translates to 2x - 5 = 10
- 8. Exponent rules
- (a) Exponent product rule: x^a \times x^b = x^{a+b}
  - Example: x^3 \times x^2 = x^{3+2} = x^5
- (b) Exponent quotient rule: \frac{x^a}{x^b} = x^{a-b}
  - Example: \frac{x^7}{x^3} = x^{7-3} = x^4
- (c) Exponent power rule: (x^a)^b = x^{ab}
  - Example: (x^3)^2 = x^{3 \times 2} = x^6
- (d) Fractional exponents: x^{1/a} = \sqrt[a]{x}
  - Example: x^{1/2} = \sqrt{x} (square root)
- (e) Negative exponents: x^{-a} = \frac{1}{x^a}
  - Example: x^{-2} = \frac{1}{x^2}
- (f) Zero power: x^0 = 1
  - Example: 5^0 = 1

Level 3 (550–650)

Integers: what they are.
- Example: \text{…}, -3, -2, -1, 0, 1, 2, 3, \text{…} are integers.
Function translations
- (a) Horizontal shift: f(x) \to f(x - h) moves the graph h units to the right.
- Example: f(x) = x^2 \to f(x-2) = (x-2)^2 shifts the graph of x^2 2 units to the right.
- (b) Vertical shift: f(x) \to f(x) + k moves the graph up by k units.
- Example: f(x) = x^2 \to f(x) + 3 = x^2 + 3 shifts the graph of x^2 3 units up.
Percentage language
- 3. Percentage of vs. percentage increase vs. percentage decrease
- Example: 30% of x (\rightarrow ) 0.3x
- Increasing x by 30% (\rightarrow ) 1.3x
- Decreasing x by 30% (\rightarrow ) 0.7x
General circle equation
- 4. General equation of a circle: (x - h)^2 + (y - k)^2 = r^2
- Example: The circle (x-1)^2 + (y+2)^2 = 9 has center (1, -2) and radius r=3
- Interpretation: Pythagorean theorem applied to a circle with center at (h, k) and radius r.
Tangent line to a circle
- 5. Find the slope of a line tangent to a circle.
- Example: Find the slope of the line tangent to x^2+y^2=25 at point (3,4).
Arc length
- 6. Arc length: let a be arc length, C be circumference, and \theta be the central angle in degrees.
- Formula: a = \frac{\theta}{360} \times C
- Example: Find the arc length for a 60^{\circ} central angle in a circle with radius 3. (Here, C = 2\pi(3) = 6\pi).
- Note: more intuitive with an example.
Sector area
- 7. Sector Area: same idea as arc length but for area; use circle’s area A.
- Formula: A_{\text{sector}} = \frac{\theta}{360} \times A = \frac{\theta}{360} \times (\text{Area circle} = \pi r^2)
- Example: Find the area of a sector with a 60^{\circ} central angle in a circle with radius 3.
Inscribed angle theorem
- 8. Inscribed angle theorem (consider the case where the central angle is 180^{\circ}).
- Example: If an inscribed angle intercepts an arc of 80^{\circ}, the measure of the inscribed angle is 40^{\circ}.
Degree–radian conversion
- 9. Converting degrees

\leftrightarrow

radians:
- d = r \times \frac{180}{\pi}
- r = d \times \frac{\pi}{180}
- Example: Convert 90^{\circ} to radians (\pi/2) and \pi/2 radians to degrees (90^{\circ}).

Congruent and Similar triangles
- 1. Congruent triangles.
- Example: Two triangles are congruent if they have identical side lengths and angle measures (e.g., SSS, SAS, ASA).
- 1. Similar triangles
- (a) Angles are the same.
- (b) Sides are proportional.
- (c) Therefore, trig ratios (sin, cos, tan) are the same.
- (d) Similarity is proven by AA, SSS, SAS, but not SSA.
- Example: A triangle with angles 30^{\circ}, 60^{\circ}, 90^{\circ} and sides (3,4,5) is similar to a triangle with the same angles and sides (6,8,10).
Probability and data visualization
- 1. Conditional probability.
- Example: The probability of drawing a Queen given that the card drawn is a face card.
- 1. Box plots.
- Example: Identify the median, quartiles, and range from a given box plot.
- 1. Scatterplots
- (a) How to interpret a scatterplot.
- (b) How to find the line of best fit.
- Example: Determine if a scatterplot shows a positive, negative, or no correlation between two variables.
Statistics concepts
- 1. Sample vs. population (what makes a sample representative?).
- Example: A survey of 100 students (sample) from an entire school (population) must be random to be representative.
- 1. Quadratics with DESMOS
- (a) Find solutions of a quadratic.
- (b) Find the vertex of a quadratic.
- Example: Use DESMOS to find the solutions and vertex of y = x^2 - 4x + 3
- 1. Factoring a quadratic and finding solutions algebraically.
- Example: Factor x^2 - 5x + 6 = 0 to (x-2)(x-3) = 0, so x=2 or x=3.

Level 4 (650–730)

Quadratics in standard and vertex form
- 1. Standard form of a quadratic: ax^2 + bx + c = 0
- Example: 2x^2 + 5x - 3 = 0
- (a) A determines upward/downward shape.
- (b) c is the y-intercept.
- 2. Vertex form of a quadratic: y = a(x - h)^2 + k
- Example: y = 2(x-1)^2 + 3 has a vertex at (1,3).
- (a) a is the same as in standard form.
- (b) The vertex is (h, k) .
Vertex information
- 3. The vertex given the x-intercepts: h = \frac{x1 + x2}{2}
- Example: If x-intercepts are 1 and 5, then h = \frac{1+5}{2} = 3
- 4. The vertex given a and b: h = -\frac{b}{2a}
- Example: For y = x^2 - 4x + 3, h = -\frac{-4}{2(1)} = 2
Sum and product of roots
- 5. Sum of a quadratic’s solutions: x1 + x2 = -\frac{b}{a}
- Example: For x^2 - 5x + 6 = 0, the sum of roots is -(-5)/1 = 5
- 6. Product of a quadratic’s solutions: x1 x2 = \frac{c}{a}
- Example: For x^2 - 5x + 6 = 0, the product of roots is 6/1 = 6
Intersections and graphs
- 7. Find where a quadratic intersects a horizontal line (use DESMOS).
- Example: Find the intersection points of y = x^2 - 4 and y = 0
- 8. Identifying the graph of a polynomial function.
- Example: Match an equation like y = x^3 - x to its corresponding graph based on roots and end behavior.
Exponential functions
- 9. Exponential functions
- (a) Construct an exponential function given an example.
  - Example: A population starts at 100 and grows by 4% per year: P(t) = 100(1.04)^t
- (b) Find the y-intercept.
  - Example: In P(t) = 100(1.04)^t, the y-intercept is 100 (when t=0).
- (c) How shifting the exponent changes interpretation? Example: 100(1.04)^{x-1}
  - Example: In 100(1.04)^{x/12}, the growth is broken down per month.
- (d) How scaling the exponent changes interpretation? Example: 100(1.04)^{2x}
  - Example: In 100(1.04)^{2x}, the growth can be interpreted as (1.04^2) = 1.0816 per x unit, or 8.16\% growth.
Radicals and fractions
- 1. Algebra problems with radicals and fractions (as in examples).
- Example: Solve the equation \sqrt{x+2} = 3 (square both sides to get x+2=9 \rightarrow x=7).
Trigonometric relationships
- 1. Sin and cosine relationship:

\sin\theta = \cos(90^{\circ} - \theta)
\cos\theta = \sin(90^{\circ} - \theta)

- Example: If \sin(30^{\circ}) = 0.5, then \cos(60^{\circ}) = 0.5.

Unit analysis and area/volume concepts
- 1. Square units: If unit u1 converts to u2 by a scale factor, then
- Example: 1 ft = 12 in \Rightarrow 1 ft^2 = 12^2 in^2 = 144 in^2.
- 1. Scaling up units (e.g., how doubling side lengths affects area or volume).
- Example: Doubling the side length of a square multiplies its area by a factor of 2^2 = 4.
- 1. Surface area (construct them; formulas are not memorized here).
- Example: Calculate the surface area of a cube with side length 2 units.
Measures of central tendency and variability
- 1. Mean = median in a symmetric data set.
- Example: For the data set \text{1, 2, 3, 4, 5}, the mean and median are both 3.
- 1. Comparing standard deviations (without computing them).
- Example: A data set \text{5, 6, 7} has a smaller standard deviation than \text{1, 6, 11} because its values are more tightly clustered around the mean.
Margin of error
- 1. Margin of error
- (a) Definition: Margin of error is the expected—but not guaranteed—deviation of the population mean from the sample mean. This distinction is subtle.
- (b) Margin of error is reduced by increasing sample size.
- Example: A poll states a candidate has 52% support with a margin of error of \pm 3\% mean the true support is likely between 49% and 55%.

Level 5 (730–800)

Similar triangles and pyramids
- 1. Similar triangles created from a triangle’s altitude.
- Example: An altitude drawn to the hypotenuse of a right triangle creates three similar right triangles.
- 2. Pyramids
- (a) Surface area of a square right pyramid.
  - Example: Calculate the surface area of a square right pyramid with base side length 4 and slant height 5.
- (b) The height of a pyramid is not equal to the slant height.
  - Example: In a square pyramid, the height is perpendicular to the base, while the slant height is the height of a triangular face.
Integer factors and proofs
- 3. Integer factors (examples of factoring/prime decomposition).
- Example: Find all integer factors of 60 (including its prime factorization: 2^2 \times 3 \times 5).
Quadratics, discriminant, and completing the square
- 4. Constant-proofing
- (a) Quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  - Example: Solve x^2 + 2x - 3 = 0 using the quadratic formula.
- (b) Discriminant of a quadratic: d = b^2 - 4ac
  - Example: For x^2 + x + 1 = 0, the discriminant is 1^2 - 4(1)(1) = -3, indicating no real solutions.
- (c) Completing the square (for circle problems):
  - Start: y = x^2 + bx + c
  - Complete square: y = (x + \frac{b}{2})^2 + c - (\frac{b}{2})^2
  - Example: Convert x^2 + 6x + 5 to vertex form by completing the square: (x+3)^2 - 4.
Miscellaneous useful relationships
- 5. Nice to know
- (a) Slope of a line from standard form Ax + By = C \Rightarrow m = -\frac{A}{B}
  - Example: For 2x + 3y = 6, the slope is m = -2/3
- (b) Arc length for radians: a = r\theta where \theta is in radians.
  - Example: If a circle has radius 2 and a central angle of \frac{\pi}{3} radians, the arc length is 2 \times \frac{\pi}{3}.
- (c) 3-4-5 and 5-12-13 triangles (classic Pythagorean triples).
  - Example: A right triangle with legs 6 and 8 will have a hypotenuse of 10 (a 3-4-5 multiple).
- (d) Triangle inequality theorem.
  - Example: Sides of lengths 3, 4, 10 cannot form a triangle because 3+4 \ngtr 10 (the sum of any two sides must be greater than the third).
Mean problems and integers
- 6. Mean problems with the term “integer” in them.
- Example: The average of 3 integers is 10. If two of the integers are 8 and 12, what is the third integer? ( (8+12+x)/3 = 10 \rightarrow 20+x=30 \rightarrow x=10).
Exponential functions with fractional exponents
- 7. Exponential functions with complicated fractional exponents.
- Example: Evaluate 8^{2/3} (which equals (^3\sqrt{8})^2 = 2^2 = 4).
Quadratics with parameter relationships
- 8. Quadratics problems solving for some combination of a, b, and c.
- Example: If the quadratic x^2 + bx + c = 0 has solutions 2 and 3, find the values of b and c.
Strategies for harder problems
- 9. Review your weakest problems and create variations of them. Consider how the College Board could make this harder.

Bluebook Formula Sheet

A note: Some formulas were excluded from the earlier levels because you can access them during the test on the formula sheet.
The content above references these formulas; consult the official Bluebook Formula Sheet for the complete, allowed list during the SAT.

Connections and practical takeaways

The material emphasizes building from foundational algebra to more advanced topics: solving equations, understanding functions and their representations (table, graph, equation), and mastering key geometric relationships (circles, triangles, trigonometry).
DESMOS is a practical tool for visualizing and solving problems involving graphs, systems of equations, and inequalities.
Memorization of formulas is stressed, but understanding the derivations and how changes to a function (shifts, stretches, compressions) affect outputs is equally important.
For data and statistics, be comfortable with mean/median concepts, variability notes (standard deviations, margin of error), and interpreting data visualizations (box plots, scatterplots).
The progression includes both pure algebra (polynomials, quadratics, exponents) and geometric/trigonometric contexts (circles, sector/arc length, congruent/similar triangles).

Notation quick reference (LaTeX)

Slope form: y = mx + b
Slope between two points: m = \frac{y2 - y1}{x2 - x1}
Circle: (x - h)^2 + (y - k)^2 = r^2
Arc length: a = \frac{\theta}{360} \times C, \quad C = 2 \pi r
Sector area: A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2
Degrees to radians: d = r \times \frac{180}{\pi}, \quad r = \frac{d \pi}{180}
Quadratic relationships:
- Sum of roots: x1 + x2 = -\frac{b}{a}
- Product of roots: x1 x2 = \frac{c}{a}
- Quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
- Vertex form: $$ y = a