SAT Math: Lessons & Preparation Route
Short lessons, worked examples, and practice that connects.
Linear equations
An equation is a balance. Applying the same reversible operation to both sides keeps its solutions unchanged.
Lines & slope
Slope measures change in y per unit change in x. In y = mx + b, m is that rate and b is the value when x = 0.
Systems of equations
A solution of a system satisfies both equations simultaneously. For two lines it is their intersection, if one exists.
Inequalities & constraints
An inequality describes a set of values rather than a single value. A context may restrict that set to nonnegative integers.
Exponents & expressions
Exponent rules describe repeated multiplication. For a nonzero base, products add exponents and quotients subtract them.
Quadratic structure
A quadratic can be written in standard, factored, or vertex form. Each form makes a different feature easy to read.
Exponential models
An exponential model changes by a constant factor over equal intervals. In A · rᵗ, A is the initial value and r is the factor per interval.
Ratios & units
A ratio compares quantities. Proportions preserve that comparison as the scale changes. Units help you choose the direction of each conversion.
Percent change
A percent is a fraction out of 100. Percent change compares the difference with the original amount.
Data & statistics
The mean balances all observations; the median locates the middle after sorting. An outlier can affect these summaries differently.
Probability & samples
Probability compares favorable outcomes with the relevant sample space when outcomes are equally likely. A condition changes that sample space.
Geometry essentials
Geometric formulas relate lengths, areas, and volumes. Similarity scales lengths by k, areas by k², and volumes by k³.
Parameters & feasible regions
A parameter describes a whole collection of equations. The goal is to decide which coefficient or constant creates the requested behavior. Expanding first reveals whether the variable survives, cancels to an identity, or cancels to a contradiction. A special value may change the number of solutions, so dividing by an expression containing the parameter can lose exactly the case the question is testing.
Restrictions before solutions
A simplified formula can hide an excluded input. For a quotient, division by zero is forbidden before any cancellation. For a principal square root, the radicand and the root must be nonnegative. These are conditions on the original expression, not optional checks added after algebra.
Transforming a function
In g(x)=Af(x−h)+k, the inner expression chooses an input for f. Multiplication by A and addition of k then change the output. Label these two stages instead of treating every visible minus sign as a movement in the same direction. A table supplies old outputs, so only transformations that use listed inputs can be evaluated from it.
Where nonlinear models meet
A nonlinear system still requires simultaneous agreement. When both equations express y, equate their right sides to find candidate inputs. Then recover the outputs and apply domain or sign conditions. A graph can suggest intersection locations, but algebra gives exact values and distinguishes one coordinate from the other.
Polynomial & absolute-value reasoning
Polynomial form can expose a coefficient, zero or remainder without a full solution. The remainder when P(x) is divided by x−r is P(r). In a product of three monic linear factors, the x² coefficient is the sum of the three written constants, because each x² term uses one constant and two x terms.
Data pairs & model errors
A data point records what happened. A fitted model estimates an output for a supplied input. These values need not match. Use units and name the explanatory and response variables before interpreting a slope or an error. An intercept describes the predicted response at zero, which may lie outside the observed range.
Samples, uncertainty & claims
A sample estimate, a causal comparison and a guarantee about an individual are different claims. Ask who could be selected, how treatment was assigned, and what variation remains. A random sample supports inference to its sampling population under appropriate response and measurement conditions. It does not promise that one realized sample exactly reproduces that population.
Comparing distributions
The mean uses every observation, the median uses ordered position, and spread measures variation. A frequency table weights a value by how often it occurs. A boxplot marks a five-number summary; tied observations and quartile conventions mean one should not automatically assign exactly one quarter of a small data set to every drawn segment.
Two-way tables & conditions
A two-way table separates observations by two categorical variables. Each cell belongs to one row and one column. A conditional probability restricts the denominator before counting favorable outcomes. A joint probability instead divides a cell by the grand total. These are different requests even when they use the same numerator.
From areas to volumes
Lengths use one power of a length unit, areas use two and volumes use three. An enlargement scales each independent length dimension. Formula selection comes before substitution: a circumference is not a base area, and a visible diameter is not the radius required by many formulas.
Angles, similarity & cosine
A diagram alone does not establish parallel lines, a right angle or similarity. Once those conditions are given, the corresponding theorems connect algebra to geometry. Corresponding angles at parallel lines are equal; a straight-line pair is supplementary. Choosing the wrong relation can create a neatly solved but irrelevant equation.
Circle equations & sectors
The equation (x−h)²+(y−k)²=r² describes points at distance r from the center (h,k). The shifts determine location; the right side determines squared radius. A positive radius is obtained by taking the nonnegative square root. Do not treat a coordinate of the center as a distance from that center.
Foundation pilot, not complete SAT coverage.