Pre-Algebra Skills

Number Sense and Operations

• Whole Numbers:

  • Place value (ones, tens, hundreds, etc.)

  • Addition and subtraction (single-digit, double-digit, multi-digit)

  • Multiplication (single-digit by single-digit, single-digit by multi-digit, multi-digit by multi-digit)

  • Division (short division, long division)

• Decimals:

  • Place value (tenths, hundredths, thousandths, etc.)

  • Addition and subtraction of decimals

  • Multiplication and division of decimals

  • Rounding decimals

• Fractions:

  • Understanding fractions (numerator, denominator)

  • Equivalent fractions

  • Simplifying fractions

  • Adding and subtracting fractions with like and unlike denominators

  • Multiplying and dividing fractions

  • Converting between mixed numbers and improper fractions

• Integers:

  • Understanding positive and negative numbers

  • Comparing and ordering integers

  • Adding and subtracting integers (number line model, chip model)

  • Multiplying and dividing integers

  • Absolute value

• Order of Operations:

  • Understanding PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

  • Applying PEMDAS to solve problems

Algebra Basics

• Variables and Expressions:

  • Understanding variables as unknown quantities

  • Writing algebraic expressions (e.g., 2x + 5)

  • Evaluating algebraic expressions (substituting values for variables)

  • Combining like terms

  • Using the distributive property

• Equations and Inequalities:

  • Solving one-step equations (addition, subtraction, multiplication, division)

  • Solving two-step equations

  • Writing and solving equations from word problems

  • Understanding inequalities (greater than, less than, greater than or equal to, less than or equal to)

  • Solving one-step and two-step inequalities

  • Graphing inequalities on a number line

Ratios, Proportions, and Percents

• Ratios and Proportions:

  • Understanding ratios (part-to-part, part-to-whole)

  • Writing ratios in different forms (fraction, colon, word form)

  • Finding equivalent ratios

  • Solving proportions using cross-multiplication

• Percents:

  • Understanding the concept of percent (per hundred)

  • Converting between fractions, decimals, and percents

  • Finding a percent of a number

  • Finding the percent increase or decrease

  • Solving percent word problems (e.g., discount, tax, tip)

Geometry Basics

• Basic Shapes:

  • Identifying and classifying shapes (e.g., triangles, quadrilaterals, circles)

  • Understanding properties of shapes (e.g., sides, angles)

  • Perimeter and area of rectangles, squares, triangles, parallelograms, and circles

• Pythagorean Theorem:

  • Understanding the Pythagorean Theorem (a² + b² = c²)

  • Using the Pythagorean Theorem to find missing side lengths in right triangles

Data Analysis and Probability

• Data Analysis:

  • Collecting and organizing data (tables, charts, graphs)

  • Finding the mean, median, mode, and range of a data set

  • Interpreting data from graphs (bar graphs, line graphs, pie charts)

• Probability:

  • Understanding probability as the likelihood of an event

  • Finding the probability of simple events

  • Experimental probability vs. theoretical probability
    
    Number Systems and Operations

• Number Systems:

  • Natural numbers, whole numbers, integers

  • Rational numbers (fractions, decimals, percents)

  • Real numbers (including irrational numbers)

• Operations:

  • Addition, subtraction, multiplication, and division of whole numbers, integers, and rational numbers

  • Order of operations

  • Properties of operations (commutative, associative, distributive)

• Exponents and Roots:

  • Exponents and powers

  • Square roots and cube roots

  • Scientific notation

Algebra

• Variables and Expressions:

  • Algebraic expressions and their evaluation

  • Combining like terms

  • Distributive property

• Equations and Inequalities:

  • One-step and two-step equations

  • Multi-step equations

  • Inequalities (solving and graphing)

• Functions and Graphs:

  • Functions and their representations (tables, graphs, equations)

  • Linear equations and their graphs

  • Slope-intercept form

  • Systems of linear equations

Geometry

• Basic Geometric Concepts:

  • Points, lines, line segments, rays, angles

  • Polygons (triangles, quadrilaterals, etc.)

  • Circles

• Measurement:

  • Perimeter and area of plane figures

  • Volume of solids (prisms, cylinders, pyramids, cones)

  • Pythagorean Theorem

• Transformations:

  • Translations, rotations, reflections

  • Dilations

Data Analysis and Probability

• Data Analysis:

  • Collecting, organizing, and displaying data (tables, charts, graphs)

  • Measures of central tendency (mean, median, mode)

  • Measures of dispersion (range, quartiles, interquartile range)

• Probability:

  • Basic probability concepts

  • Experimental and theoretical probability

  • Independent and dependent events

Additional Topics (Optional)

• Ratios and Proportions:

  • Ratios, rates, and unit rates

  • Solving proportions

• Percents:

  • Percent of a number

  • Percent increase and decrease

  • Simple interest

1. Foundational Concepts

• Real Number System:

  • Properties of real numbers (commutative, associative, distributive)

  • Order of operations (PEMDAS/GEMDAS)

  • Absolute value and its applications

  • Square roots and radicals

• Algebraic Expressions:

  • Variables and constants

  • Writing and evaluating expressions

  • Combining like terms

  • Distributive property

2. Linear Equations and Inequalities

• Solving Equations:

  • One-step and two-step equations

  • Multi-step equations

  • Equations with variables on both sides

  • Literal equations (solving for a specific variable)

• Solving Inequalities:

  • One-step and two-step inequalities

  • Multi-step inequalities

  • Compound inequalities (AND and OR)

  • Graphing inequalities on a number line

3. Functions and Their Graphs

• Introduction to Functions:

  • Definition of a function

  • Domain and range

  • Function notation

  • Evaluating functions

• Linear Functions:

  • Slope-intercept form (y = mx + b)

  • Standard form (Ax + By = C)

  • Point-slope form

  • Graphing linear equations

  • Writing linear equations from given information (slope and y-intercept, two points, etc.)

  • Parallel and perpendicular lines

  • Linear inequalities (graphing solutions)

• Non-Linear Functions:

  • Absolute value functions

  • Piecewise functions

  • Step functions

4. Systems of Equations and Inequalities

• Solving Systems of Linear Equations:

  • Graphing method

  • Substitution method

  • Elimination method

• Systems of Linear Inequalities:

  • Graphing systems of inequalities

  • Identifying solutions to systems of inequalities

5. Polynomials and Factoring

• Polynomials:

  • Adding, subtracting, multiplying polynomials

  • Degree and leading coefficient

• Factoring Polynomials:

  • Greatest common factor (GCF)

  • Difference of squares

  • Trinomial factoring

  • Factoring by grouping

6. Quadratic Functions and Equations

• Quadratic Functions:

  • Graphing quadratic functions

  • Vertex form, standard form, and factored form

  • Axis of symmetry and vertex

  • Domain and range

• Solving Quadratic Equations:

  • Factoring method

  • Square root method

  • Quadratic formula

  • Completing the square

7. Exponential Functions and Data Analysis

• Exponential Functions:

  • Exponential growth and decay

  • Graphing exponential functions

  • Solving exponential equations (basic)

• Data Analysis:

  • Scatter plots

  • Line of best fit

  • Correlation and causation

  • Interpreting data from graphs and tables

Additional Topics (Depending on Curriculum):

• Radical Expressions and Equations

• Rational Expressions and Equations

• Sequences and Series

• Matrices and Determinants

Fundamental Concepts

• Points, Lines, and Planes:

  • Definitions and properties of points, lines, and planes

  • Intersection of lines and planes

  • Collinear and coplanar points

• Angles:

  • Types of angles (acute, obtuse, right, straight)

  • Angle measurement and classification

  • Angle pairs (complementary, supplementary, vertical, adjacent)

  • Angle postulates and theorems (Angle Addition Postulate, Linear Pair Theorem, etc.)

Geometric Proofs and Reasoning

• Inductive and Deductive Reasoning:

  • Distinguishing between inductive and deductive reasoning

  • Using inductive reasoning to make conjectures

  • Using deductive reasoning to prove statements

• Two-Column Proofs:

  • Writing formal geometric proofs using a two-column format

  • Using postulates, theorems, and definitions as reasons

• Paragraph Proofs:

  • Writing informal proofs in paragraph form

  • Explaining logical reasoning in a clear and concise manner

Transformations

• Rigid Transformations:

  • Translations (slides)

  • Reflections (flips)

  • Rotations (turns)

  • Properties preserved under rigid transformations (distance, angle measure, congruence)

• Dilations:

  • Scale factor and center of dilation

  • Properties preserved under dilations (angle measure, similarity)

• Symmetry:

  • Line symmetry

  • Rotational symmetry

Congruence and Similarity

• Congruent Figures:

  • Definition of congruent figures

  • Congruence postulates and theorems (SSS, SAS, ASA, AAS)

  • Proving triangles congruent using proofs

• Similar Figures:

  • Definition of similar figures

  • Similarity postulates and theorems (AA, SAS, SSS)

  • Finding missing side lengths and angle measures in similar figures

Right Triangles and Trigonometry

• Pythagorean Theorem:

  • Using the Pythagorean Theorem to find missing side lengths in right triangles

  • Converse of the Pythagorean Theorem

• Special Right Triangles:

  • 45-45-90 triangles

  • 30-60-90 triangles

• Trigonometric Ratios:

  • Sine, cosine, and tangent ratios

  • Using trigonometric ratios to find missing side lengths and angle measures in right triangles

• Applications of Trigonometry:

  • Solving real-world problems involving right triangles (e.g., height of a tree, distance across a lake)

Polygons and Quadrilaterals

• Polygons:

  • Definition of polygons (convex, concave, regular, irregular)

  • Angle sum theorem for polygons

  • Interior and exterior angles of polygons

• Quadrilaterals:

  • Properties of parallelograms, rectangles, squares, rhombuses, trapezoids, and kites

  • Proving quadrilaterals are special types using properties

Circles

• Parts of a Circle:

  • Center, radius, diameter, chord, secant, tangent

• Angles and Arcs:

  • Central angles, inscribed angles, intercepted arcs

  • Arc length and sector area

• Circles and Lines:

  • Tangent lines and their properties

  • Secant-tangent and secant-secant theorems

Coordinate Geometry

• Distance and Midpoint Formulas:

  • Finding the distance between two points

  • Finding the midpoint of a line segment

• Equations of Lines:

  • Slope-intercept form

  • Point-slope form

  • Standard form

• Graphs of Linear Equations:

  • Graphing lines using slope-intercept form and point-slope form

• Parallel and Perpendicular Lines:

  • Slopes of parallel and perpendicular lines

  • Writing equations of parallel and perpendicular lines

Three-Dimensional Geometry

• Solid Figures:

  • Prisms, pyramids, cylinders, cones, spheres

  • Surface area and volume formulas for solid figures

• Cross Sections:

  • Identifying cross sections of three-dimensional figures

Applications and Problem Solving

• Real-World Applications:

  • Using geometry to solve real-world problems (e.g., architecture, engineering, design)

• Proofs and Constructions:

  • Writing formal and informal proofs

  • Using geometric constructions (e.g., bisecting an angle, constructing a perpendicular bisector)

• Problem-Solving Strategies:

  • Using problem-solving strategies (e.g., draw a diagram, work backwards, guess and check)

Unit 1: Functions

• Functions and Their Graphs:

  • Domain and range

  • Function notation

  • Vertical line test

  • Graphing functions

  • Transformations of functions (translations, reflections, stretches, shrinks)

• Linear Functions:

  • Slope-intercept form

  • Point-slope form

  • Standard form

  • Parallel and perpendicular lines

  • Linear inequalities

• Systems of Linear Equations and Inequalities:

  • Solving systems by graphing

  • Solving systems by substitution

  • Solving systems by elimination  

  • Systems of linear inequalities

Unit 2: Quadratic Functions

• Quadratic Functions and Their Graphs:

  • Parabolas

  • Vertex form

  • Standard form

  • Axis of symmetry

  • Vertex

  • Maximum and minimum values

• Solving Quadratic Equations:

  • Factoring

  • Square root method

  • Completing the square

  • Quadratic formula

  • Discriminant

• Complex Numbers:

  • Imaginary unit (i)

  • Operations with complex numbers

• Quadratic Inequalities:

  • Solving quadratic inequalities

  • Graphing quadratic inequalities

Unit 3: Polynomial Functions

• Polynomial Functions and Their Graphs:

  • Degree of a polynomial

  • Leading coefficient

  • End behavior

  • Zeros of a polynomial

  • Multiplicity of zeros

  • Turning points

• Polynomial Operations:

  • Adding, subtracting, multiplying, and dividing polynomials

  • Synthetic division

  • Remainder Theorem

  • Factor Theorem

• Factoring Polynomials:

  • Factoring by grouping

  • Factoring special products (difference of squares, sum/difference of cubes)

• Solving Polynomial Equations:

  • Using the Rational Root Theorem

  • Using the Fundamental Theorem of Algebra

Unit 4: Rational Functions

• Rational Functions and Their Graphs:

  • Domain and range

  • Vertical asymptotes

  • Horizontal asymptotes

  • Slant asymptotes

  • Holes in the graph

• Operations with Rational Expressions:

  • Simplifying rational expressions

  • Multiplying and dividing rational expressions

  • Adding and subtracting rational expressions  

• Solving Rational Equations and Inequalities:

  • Finding the LCD

  • Clearing fractions

  • Solving rational equations

  • Solving rational inequalities

   

Unit 5: Exponential and Logarithmic Functions

• Exponential Functions:

  • Exponential growth and decay

  • Graphing exponential functions

• Logarithmic Functions:

  • Definition of a logarithm

  • Properties of logarithms

  • Solving logarithmic equations

  • Graphing logarithmic functions

• Exponential and Logarithmic Equations and Inequalities:

  • Solving exponential equations

  • Solving logarithmic equations

  • Solving exponential and logarithmic inequalities

Unit 6: Conic Sections

• Circles:

  • Equation of a circle

  • Graphing circles

• Parabolas:

  • Standard form and vertex form

  • Graphing parabolas

• Ellipses:

  • Standard form

  • Graphing ellipses

• Hyperbolas:

  • Standard form

  • Graphing hyperbolas

Unit 7: Sequences and Series

• Sequences and Series:

  • Arithmetic sequences and series

  • Geometric sequences and series

  • Infinite geometric series

  • Sigma notation

• Binomial Theorem:

  • Binomial expansion

  • Pascal's Triangle

Unit 8: Data Analysis and Probability

• Data Analysis:

  • Measures of central tendency and dispersion

  • Box plots

  • Histograms

  • Scatter plots

  • Correlation and regression

• Probability:

  • Basic probability

  • Conditional probability

  • Independent and dependent events

  • Binomial probability

Unit 9: Trigonometry

• Trigonometric Functions:

  • Right triangle trigonometry

  • Unit circle

  • Trigonometric identities

  • Graphing trigonometric functions

• Trigonometric Equations and Identities:

  • Solving trigonometric equations

  • Verifying trigonometric identities