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: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
  • 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 $\begin{cases} y &gt; x + 1 \ y &lt; -2x + 5 \end{cases}$
  • 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
  • 10. Congruent triangles.
  • Example: Two triangles are congruent if they have identical side lengths and angle measures (e.g., SSS, SAS, ASA).
  • 11. 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
  • 12. Conditional probability.
  • Example: The probability of drawing a Queen given that the card drawn is a face card.
  • 13. Box plots.
  • Example: Identify the median, quartiles, and range from a given box plot.
  • 14. 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
  • 15. 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.
  • 16. 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 $y = x^2 - 4x + 3$
  • 17. 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: $h = \frac{x<em>1 + x</em>2}{2}$
  • 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: $x<em>1 + x</em>2 = -\frac{b}{a}$
  • Example: For $x^2 - 5x + 6 = 0$, the sum of roots is $-(-5)/1 = 5$
  • 6. Product of a quadratic’s solutions: $x<em>1 x</em>2 = \frac{c}{a}$
  • 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
  • 10. 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
  • 12. Sin and cosine relationship:

$\sin\theta = \cos(90^{\circ} - \theta)<br /> \cos\theta = \sin(90^{\circ} - \theta)$

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

• Unit analysis and area/volume concepts
  • 13. 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$^2$.
  • 14. 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 $2^2 = 4$.
  • 15. 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
  • 16. 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.
  • 17. 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
  • 18. 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: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• 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: $x<em>1 + x</em>2 = -\frac{b}{a}$
  • Product of roots: $x<em>1 x</em>2 = \frac{c}{a}$
  • Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Vertex form: $$ y = a