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Vector-Valued Functions

Maps scalar inputs to vectors to define curves in two and three-dimensional space. Calculates derivatives and integrals to analyze velocity, acceleration, arc length, and curvature.

Math Numbers & Counting Counting Objects Number Names Comparing Numbers Number Operations Counting Sequence Numbers 0-10 Place Value Understanding and System Add and Subtract Within 20 Addition and Subtraction Concepts Addition and Subtraction Equations Addition and Subtraction Problems Foundations for Multiplication Multiplication and Division Properties Multiply and Divide Within 100 Multiplication and Division Problems Factors and Multiples Properties of Operations Patterns and Relationships Generate and Analyze Patterns Multi-Digit Arithmetic Place Value Operations Multi-Digit and Decimal Operations Numerical Expressions Four Operations and Patterns Four Operations Problem Solving Measurement & Data Measurable Attributes Measuring Lengths Measure and Estimate Lengths in Standard Units Relate Addition and Subtraction to Length Classifying and Counting Objects Time and Money Area Concepts and Measurement Perimeter and Area Measures Angle Concepts and Measurement Time, Volume, and Mass Measurement Unit Conversions Geometry Identifying Shapes Shapes and Attributes Shape Attributes and Reasoning Shapes and Composition Classifying 2D Figures Geometric Figures and Relationships Lines, Angles, and Shapes Angle, Area, and Volume Geometric Measurement Volume of 3D Shapes Coordinate Plane Applications Transformations in the Plane Congruence and Similarity Understand congruence in terms of rigid motions Similarity and Transformations Prove Theorems Involving Similarity Pythagorean Theorem Trigonometry for General Triangles Make Geometric Constructions Prove Geometric Theorems Theorems About Circles Find Arc Lengths And Areas of Sectors of Circles Volume Formulas Relationships Between Two-Dimensional and Three-Dimensional Objects Prove Simple Geometric Theorems Algebraically Translate Between Geometric Description and Equation for Conic Section Apply Geometric Concepts in Modeling Situations Fractions & Decimals Fractions as Numbers Building Fractions Fraction Equivalence and Ordering Adding and Subtracting Fractions Multiplying and Dividing Fractions Dividing Fractions Decimal Fractions Multi-Digit Computation and Factors Add, Subtract, Multiply, and Divide Rational Numbers Rational Number System Irrational Numbers and Approximations Ratios Ratios and Proportions Proportional Relationships Unit Rate Algebra Algebraic Expressions Generate Equivalent Expressions Quantitative Relationships Proportional Relationships and Linear Equations Equations and Inequalities Equation Solving and Reasoning Linear Equations and Systems Graph Equations and Inequalities Systems of Equations Real-World Algebraic Problems Quantitative Reasoning with Units Expression Structure Equivalent Expression Forms Radicals and Integer Exponents Rational Exponents Rational and Irrational Numbers Polynomial Operations Polynomial Identities Polynomial Zeros and Factors Rational Expressions Complex Number Operations Complex Numbers in Polynomials Complex Numbers on Plane Statistics & Probability Represent and Interpret Data Data Distributions Statistical Variability Probability Models Compound Event Probabilities Statistical Sampling Interpret Categorical and Quantitative Data Bivariate Data Patterns Interpret Linear Models Comparing Two Populations Random Processes in Statistics Independence and Conditional Probability Expected Values Probability-Based Decision Making Statistical Inference and Conclusions Functions Function Concepts and Notation Define and Compare Functions Interpret Functions in Context Analyze Function Representations Model Relationships with Functions Identify Linear vs Exponential Growth Distinguish Between Function Types Compare Growth Rates Interpret Function Expressions Build Functions from Relationships Construct and Model Functions Transform and Combine Functions Model Comparison and Selection Solve Exponential Equations Trigonometry Trigonometric Ratios Involving Right Triangles Trigonometric Functions and Unit Circle Model with Trigonometric Functions Trigonometric Identities Vectors & Matrices Introduction to Vectors and Matrices Vector Quantities Vector Operations Matrix Operations Calculus Limits and Continuity Derivative Concepts and Notation Derivative Rules and Techniques Applications of Derivatives Optimization Problems Related Rates Curve Sketching and Analysis Integration Concepts and Notation Antiderivatives and Indefinite Integrals Definite Integrals and Area Fundamental Theorem of Calculus Integration Techniques Applications of Integration Differential Equations Sequences and Series Parametric and Polar Functions Vector-Valued Functions

Sequences

Curvilinear Blueprint Slides

Curvilinear Coordinates and Vector Analysis

This mathematical physics sequence explores the coordinate systems necessary for solving problems involving complex shapes, moving beyond Cartesian coordinates to General Curvilinear systems. Students derive scale factors, volume elements, and differential operators, culminating in solving Laplace's equation and understanding metric tensors.

Flight Path Slides

Vector Velocity

An advanced exploration of vector-valued functions and their applications in modeling 2D motion and force, preparing students for multivariable calculus.

Coordinate Chronicles Worksheet

Vector Voyages

A comprehensive unit on parametric equations and their applications in modeling motion. Students move from the basics of parametric curves to advanced calculus concepts like derivatives, concavity, vectors, and arc length.

Tracking the Path Slides

Simulating Motion with Parametric Equations

This sequence introduces students to parametric equations through the lens of particle motion and physics simulations. Students progress from basic plotting and parameter elimination to advanced calculus applications involving derivatives, vectors, and arc length.

Mapping Motion Slides

Kinematic Analysis with Vector-Valued Functions

A comprehensive advanced calculus unit exploring the use of vector-valued functions to model and analyze motion in 2D and 3D space. Students will master differentiation, integration, and arc length calculations within a kinematic context, culminating in complex projectile modeling.

Path Pilot Slides

Applied Modeling and Path Optimization

A project-based calculus sequence for 12th grade students focusing on the engineering applications of vector-valued functions, including path optimization, differentiability, and arc length.