PSAT 10 Math — Concepts, Skills, and Problem-Solving Methods

Algebraic Expression

A mathematical phrase like $3x+2$ that can be evaluated but is not true or false until values are assigned.

Equation

A mathematical statement that sets two expressions equal, such as $3x+2=11$(it has solutions that make the statement true).

Inequality

A comparison of two expressions using symbols like $<$ or $>$, showing a set of possible solutions.

Like Terms

Terms that have the same variable part (same variables raised to the same powers), such as $7x$ and $-2x$.

Distributive Property

A law used to multiply a single term by terms within parentheses: $a(b+c)=ab+ac$.

Linear Equation

An equation in which the variable is to the first power and there are no products of variables, such as $2x-7=9$.

Inverse Operations

Operations that undo each other, such as addition and subtraction or multiplication and division.

Solving Inequalities

Finding values of the variable that satisfy the inequality, where multiplying or dividing by a negative reverses the inequality sign.

Vertex

The highest or lowest point of a parabola defined by a quadratic function, given in the form $y=a(x-h)^2+k$.

Rational Expression

An expression that is a fraction with polynomials in the numerator and denominator, e.g., $\frac{x+1}{x-2}$.

Domain

The set of allowable inputs for a function.

Range

The set of possible outputs from a function.

Slope

The measure of how much $y$ changes with a unit increase in $x$, typically represented as $m$ in equations.

Systems of Equations

A set of equations with the same variables. Solutions are the ordered pairs that satisfy all equations.

Point-slope Form

An equation format $y-y_1=m(x-x_1)$ representing a line through point $\left(x_1,y_1\right)$ with slope $m$.

Slope-intercept Form

The equation of a line written as $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

Standard Form of a Line

An equation of a line expressed as $Ax + By = C$, where A, B, and C are integers.

Factoring

The process of breaking down a polynomial into simpler multiply-able terms.

Quadratic Function

A function that can be expressed in the form $y = ax^2 + bx + c$, which graphs as a parabola.

Greatest Common Factor (GCF)

The largest factor that divides all terms in an expression uniformly.

Zero-product Property

If $ab=0$, then either $a=0$ or $b=0$; used to solve polynomial equations.

Exponential Growth

A situation where a quantity increases by a constant factor over equal intervals of time.

Weighted Average

An average that takes into account the different weights of factors or data points being averaged.

Mean

The average of a set of numbers, calculated by dividing the sum by the count of values.

Median

The middle value of a dataset when arranged in numerical order; if even, the average of the two middle values.

Outlier

A value that is significantly higher or lower than most of the other values in a dataset.

Pythagorean Theorem

A theorem stating that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: $a^2 + b^2 = c^2$.

Linear Relationship

A relationship between two variables that can be graphically represented as a straight line.

Condition Probability

The likelihood of an event occurring given that another event has already occurred.

Scatterplot

A graphical representation of two quantitative variables where each point represents an observation.

Distance Formula

A formula used to determine the distance between two points in the coordinate plane: $d=\text{sqrt}{(x_2-x_1)^2 +(y_2-y_1)^2}$.

Midpoint Formula

A formula to determine the midpoint between two points: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Percentage Increase

The amount of increase expressed as a percentage of the original amount, calculated as $\frac{new - original}{original} \times 100$.

Percentage Decrease

The amount of decrease expressed as a percentage of the original amount, calculated as $\frac{original - new}{original} \times 100$.

Probability of an Event A

The likelihood of event A occurring, calculated as $P(A)=\frac{number~of~favorable~outcomes}{number~of~total~outcomes}$.

Common Mistake: Distributing Incorrectly

Failing to distribute a term correctly across all terms in parentheses, especially with negative signs.

Common Mistake: Mixing Units

Combining different units (e.g., hours with minutes) without proper conversion, leading to incorrect conclusions.

Common Mistake: Using Incorrect Slope Formula

Computing slope with mismatched subtraction order, which can lead to the wrong sign for the slope.

Common Mistake: Confusing Domain and Range

Misidentifying allowable inputs (domain) versus possible outputs (range) for functions.

Common Mistake: Ignoring Context Constraints

Failing to apply real-world limitations (like time cannot be negative) to answers derived mathematically.

Common Mistake: Not Checking Restrictions

Overlooking domain restrictions in rational expressions, resulting in invalid answers.

Common Mistake: Rearranging Equations Incorrectly

Making sign errors or not applying inverse operations correctly when isolating variables.

Common Mistake: Incorrect Interpretation of Results

Failing to connect a computed result back to the context of a problem, leading to irrelevant conclusions.

Common Mistake: Misreading Vertex Form

Interpreting vertex as $(-h,k)$ instead of $(h,k)$ in the vertex form of a quadratic.