Algebra and Geometry TEKS Review & Essential Formulas

Assessment & Student Details

• Student Name: Campbell Moore
• Estimated Test Date: Sep. 1

Review Algebra TEKS Standards

• Standard A.5A: Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are on both sides of the equation.
• Standard A.3A: Determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including:
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Standard Form: $ax + by = c$
  • Slope-Intercept Form: $y = mx + b$
• Standard A.3B: Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in the context of mathematical and real-world problems.
• Standard A.12E: Solve mathematical and scientific formulas, as well as other literal equations, for a specified variable.

Geometry TEKS Standards

• Standard G.2B: Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.
• Standard G.2A: Determine the coordinates of a point that is a given fraction of the distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint.
• Standard G.4C: Verify that a conjecture is false using a counterexample.
• Standard G.4A: Distinguish between undefined terms, definitions, postulates, conjectures, and theorems.
• Standard G.4B: Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse.
• Standard G.4D: Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

Essential Formulas and Logic Rules

Coordinate Geometry Formulas

• Distance Formula:$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

  • Calculates the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a two-dimensional Cartesian coordinate system.

• Midpoint Formula:$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

  • Determines the midpoint coordinates $M$ equidistant from two endpoints $(x_1, y_1)$ and $(x_2, y_2)$.

• Slope Formula:$m = \frac{y_2 - y_1}{x_2 - x_1}$

  • Measures the slope or rate of change $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$.

Logic and Conditional Statements

• Conditional Statement:

  • Verbal Structure: "If $p$, then $q$"
  • Symbolic Notation: $p \rightarrow q$

• Converse:

  • Verbal Structure: "If $q$, then $p$"
  • Symbolic Notation: $q \rightarrow p$

• Inverse:

  • Verbal Structure: "If not $p$, then not $q$"
  • Symbolic Notation: $\sim p \rightarrow \sim q$

• Contrapositive:

  • Verbal Structure: "If not $q$, then not $p$"
  • Symbolic Notation: $\sim q \rightarrow \sim p$

• Biconditional Connection:

  • A biconditional statement represents the logical connection between a true conditional statement ($p \rightarrow q$) and its true converse ($q \rightarrow p$).