# Applied Algebra

**Author(s):**
Marian
Anton
,
Karen
Santoro

**Edition:
**
1

**Copyright:
**
2023

**Pages:
**
286

* Applied Algebra* is designed for a one semester course for students entering college at the developmental level. It enhances students’ algebraic, critical reasoning, and communication skills, and to promote a growth mindset towards learning.

It provides a guided inquiry approach through collaborative exploration, concept formation, and application where students learn algebraic concepts and their meaning in real-world contexts.

The publication focuses on fewer concepts, mainly linear, exponential, and quadratic functions, but treat them conceptually, in more depth, and interconnected for better retention.

The movie Groundhog Day is a good metaphor for our book’s concept. Each day the lead character wakes up on the same day, Groundhog Day, repeating the same routine while struggling with a situation. He starts to predict what is going to happen each day, he learns a little bit more, and gets closer to solving the problem. By the end he understands what he needs to do.

Readers are taken through a similar experience. Each section starts with a launch exploration based on the common threads of the same routine: when given a situation, they identify the variables, make tables, draw graphs, search for patterns, and write equations.

**Introduction****0 Math Foundations in a Nutshell****0.1 Fundamental Arithmetic: The Integers**

Launch Exploration: Grouping and Regrouping

Numeration

Addition of Counting Numbers

Subtraction

Multiplication

Division

Number Zero

Order of Operations

Distributivity of Multiplication over Addition

Wrapping Up

Exercises

**0.2 Fundamental Arithmetic: The Rationals **

Mental Math

Divisibility of Numbers

Launch Exploration: Part of a Whole

Ordinary Fractions

Adding and Subtracting Fractions

Fraction Multiplication

Division by Fractions

Decimal Fractions

Rounding

Percent

Wrapping Up

Exercises

**0.3 Elementary Algebra: Expressions and Equations**

Launch Exploration: Find the ’Heap’

Ratios and Proportions

Powers and Radicals

Intro to Algebraic Expressions

Intro to Algebraic Equations

Wrapping Up

Exercises

**0.4 Elementary Geometry: Shapes and Measures**

Launch Exploration: Perimeter and Area

Primitive Notions

Angles and Circles

Parallel Lines

Triangles

Rectangles and Parallelograms

Perimeter versus Area

Volumes

Wrapping Up

Exercises

**1 Quantitative Analysis, Problem Solving, and Functions ****1.1 Quantitative Analysis: Constants and Variables**

Launch Exploration: Shopping and Fencing

Analyzing a Situation: Quantities and Units

Analyzing a Situation: Constants and Variables

Values of a Variable – Discussion

Wrapping Up

Exercises

**1.2 Algebraic Relations: Translations and Formulas**

Launch Exploration: Building a Patio

Operations and Relations - Key Phrases

Quantitative Relations in Situations

Units Agreement

Wrapping Up

Exercises

**1.3 Problem Solving with Linear and Power Systems**

Launch Exploration: Building a Deck

Solving Linear Systems by Substitution

Linear Systems - More Examples

Power Equation – A Basic Example of a Nonlinear Equation

Mapping Diagrams

Power Systems

Critical Thinking

Wrapping Up

Exercises

**1.4 Introduction to Modeling with Functions**

Launch Exploration: Buying Protein Bars

Function Notation - How to Write About Functions

Real-World Domain and Range

Graphs - How to Visualize Functions

Analyzing a Graph

The Initial Value of a Function

What is a Function?

Function Equations

Wrapping Up

Exercises

**2 Linear Functions ****2.1 Introduction to Linear Relationships**

Launch Exploration: Growing Plants

Linear Function Tables

Another Plant Example

Constant Rate of Change

Linear Function Equations

Positive and Negative Rates of Change

Wrapping Up

Exercises

**2.2 Analyzing Linear Functions**

Launch Exploration: Linear Graphs

Analyzing Linear Graphs

Writing Linear Function Equations from Graphs

Writing Linear Function Equations from Tables

Comparing Slopes

Wrapping Up

Exercises

**2.3 Linear Modeling**

Launch Exploration: Window Washers

Creating and Using Linear Models

More Linear Modeling

Wrapping Up

Exercises

**3 Exponential Functions ****3.1 A New Pattern - Introduction to Exponential Functions**

Launch Exploration: Growing Bacteria

Exponential Function Tables

Exponential Function Equations

Exponential Growth versus Decay

Writing Equations from Two Data Points

Wrapping Up

Exercises

**3.2 Analyzing Exponential Functions**

Launch Exploration: Graphing an Exponential

Domain and Range for Exponentials

Zoom In and Out on Exponential Graphs

Asymptotes and Intercepts

An Exponential Decay Example

Comparing Exponential Graphs

Solving Exponential Equations with Logarithms

Analyzing Exponential Functions in Context

Writing Equations from Exponential Graphs

Wrapping Up

Exercises

**3.3 Problem Solving with Exponentials**

Launch Exploration: A Growing Colony

Relative Rate of Change

Relative Rate of Growth

Relative Rate of Decay

Relative Rates in Equations

Exponential Modeling

Linear versus Exponential

Wrapping Up

Exercises

**4 Quadratic Functions ****4.1 Problem Solving with Quadratic Equations **

Launch Exploration: A Room with a Given Area

The Quadratic Formula

Creating and Using Quadratic Equations

The Number of Real Solutions

Special Cases of Quadratic Problems

Wrapping Up

Exercises

**4.2 Another New Pattern - Intro to Quadratic Functions**

Launch Exploration: A Square Garden

Quadratic Function Equations

Quadratic Function Tables

Quadratic Function Graphs

Why Are Parabolas So Special?

Not All U-Shapes Are Parabolas!

Writing Quadratic Function Equations from Tables

Wrapping Up

Exercises

**4.3 Finding Features of Parabolas Algebraically**

Launch Exploration: Analyzing Parabolas

Symmetry and Vertex

Direction

The Vertex Formula

Intercepts

Finding All the Features of a Parabola

Wrapping Up

Exercises

**4.4 Analyzing Quadratic Functions**

Launch Exploration: A Projectile Motion Graph

Throwing a Ball on the Moon

The Profit of Kandy-n-Kakes

Maximizing Area Inside a Fence

Wrapping Up

Exercises

**Index**

*If you are tired by the usual turgid textbooks on introductory algebra, you have found the antidote - an excellent teaching tool, with successive topics following in seamless flow; this material is visually pleasing, uncluttered and inviting, well grounded in a narrative mode. Anton and Santoro’s accomplishment in offering a rigorous, yet engaging introductory manual, is salutary.***Mircea Pitici, PhD, the editor of "****The Best Writings on Mathematics****" series by Princeton University Press**

* Applied Algebra* is designed for a one semester course for students entering college at the developmental level. It enhances students’ algebraic, critical reasoning, and communication skills, and to promote a growth mindset towards learning.

It provides a guided inquiry approach through collaborative exploration, concept formation, and application where students learn algebraic concepts and their meaning in real-world contexts.

The publication focuses on fewer concepts, mainly linear, exponential, and quadratic functions, but treat them conceptually, in more depth, and interconnected for better retention.

The movie Groundhog Day is a good metaphor for our book’s concept. Each day the lead character wakes up on the same day, Groundhog Day, repeating the same routine while struggling with a situation. He starts to predict what is going to happen each day, he learns a little bit more, and gets closer to solving the problem. By the end he understands what he needs to do.

Readers are taken through a similar experience. Each section starts with a launch exploration based on the common threads of the same routine: when given a situation, they identify the variables, make tables, draw graphs, search for patterns, and write equations.

**Introduction****0 Math Foundations in a Nutshell****0.1 Fundamental Arithmetic: The Integers**

Launch Exploration: Grouping and Regrouping

Numeration

Addition of Counting Numbers

Subtraction

Multiplication

Division

Number Zero

Order of Operations

Distributivity of Multiplication over Addition

Wrapping Up

Exercises

**0.2 Fundamental Arithmetic: The Rationals **

Mental Math

Divisibility of Numbers

Launch Exploration: Part of a Whole

Ordinary Fractions

Adding and Subtracting Fractions

Fraction Multiplication

Division by Fractions

Decimal Fractions

Rounding

Percent

Wrapping Up

Exercises

**0.3 Elementary Algebra: Expressions and Equations**

Launch Exploration: Find the ’Heap’

Ratios and Proportions

Powers and Radicals

Intro to Algebraic Expressions

Intro to Algebraic Equations

Wrapping Up

Exercises

**0.4 Elementary Geometry: Shapes and Measures**

Launch Exploration: Perimeter and Area

Primitive Notions

Angles and Circles

Parallel Lines

Triangles

Rectangles and Parallelograms

Perimeter versus Area

Volumes

Wrapping Up

Exercises

**1 Quantitative Analysis, Problem Solving, and Functions ****1.1 Quantitative Analysis: Constants and Variables**

Launch Exploration: Shopping and Fencing

Analyzing a Situation: Quantities and Units

Analyzing a Situation: Constants and Variables

Values of a Variable – Discussion

Wrapping Up

Exercises

**1.2 Algebraic Relations: Translations and Formulas**

Launch Exploration: Building a Patio

Operations and Relations - Key Phrases

Quantitative Relations in Situations

Units Agreement

Wrapping Up

Exercises

**1.3 Problem Solving with Linear and Power Systems**

Launch Exploration: Building a Deck

Solving Linear Systems by Substitution

Linear Systems - More Examples

Power Equation – A Basic Example of a Nonlinear Equation

Mapping Diagrams

Power Systems

Critical Thinking

Wrapping Up

Exercises

**1.4 Introduction to Modeling with Functions**

Launch Exploration: Buying Protein Bars

Function Notation - How to Write About Functions

Real-World Domain and Range

Graphs - How to Visualize Functions

Analyzing a Graph

The Initial Value of a Function

What is a Function?

Function Equations

Wrapping Up

Exercises

**2 Linear Functions ****2.1 Introduction to Linear Relationships**

Launch Exploration: Growing Plants

Linear Function Tables

Another Plant Example

Constant Rate of Change

Linear Function Equations

Positive and Negative Rates of Change

Wrapping Up

Exercises

**2.2 Analyzing Linear Functions**

Launch Exploration: Linear Graphs

Analyzing Linear Graphs

Writing Linear Function Equations from Graphs

Writing Linear Function Equations from Tables

Comparing Slopes

Wrapping Up

Exercises

**2.3 Linear Modeling**

Launch Exploration: Window Washers

Creating and Using Linear Models

More Linear Modeling

Wrapping Up

Exercises

**3 Exponential Functions ****3.1 A New Pattern - Introduction to Exponential Functions**

Launch Exploration: Growing Bacteria

Exponential Function Tables

Exponential Function Equations

Exponential Growth versus Decay

Writing Equations from Two Data Points

Wrapping Up

Exercises

**3.2 Analyzing Exponential Functions**

Launch Exploration: Graphing an Exponential

Domain and Range for Exponentials

Zoom In and Out on Exponential Graphs

Asymptotes and Intercepts

An Exponential Decay Example

Comparing Exponential Graphs

Solving Exponential Equations with Logarithms

Analyzing Exponential Functions in Context

Writing Equations from Exponential Graphs

Wrapping Up

Exercises

**3.3 Problem Solving with Exponentials**

Launch Exploration: A Growing Colony

Relative Rate of Change

Relative Rate of Growth

Relative Rate of Decay

Relative Rates in Equations

Exponential Modeling

Linear versus Exponential

Wrapping Up

Exercises

**4 Quadratic Functions ****4.1 Problem Solving with Quadratic Equations **

Launch Exploration: A Room with a Given Area

The Quadratic Formula

Creating and Using Quadratic Equations

The Number of Real Solutions

Special Cases of Quadratic Problems

Wrapping Up

Exercises

**4.2 Another New Pattern - Intro to Quadratic Functions**

Launch Exploration: A Square Garden

Quadratic Function Equations

Quadratic Function Tables

Quadratic Function Graphs

Why Are Parabolas So Special?

Not All U-Shapes Are Parabolas!

Writing Quadratic Function Equations from Tables

Wrapping Up

Exercises

**4.3 Finding Features of Parabolas Algebraically**

Launch Exploration: Analyzing Parabolas

Symmetry and Vertex

Direction

The Vertex Formula

Intercepts

Finding All the Features of a Parabola

Wrapping Up

Exercises

**4.4 Analyzing Quadratic Functions**

Launch Exploration: A Projectile Motion Graph

Throwing a Ball on the Moon

The Profit of Kandy-n-Kakes

Maximizing Area Inside a Fence

Wrapping Up

Exercises

**Index**

*If you are tired by the usual turgid textbooks on introductory algebra, you have found the antidote - an excellent teaching tool, with successive topics following in seamless flow; this material is visually pleasing, uncluttered and inviting, well grounded in a narrative mode. Anton and Santoro’s accomplishment in offering a rigorous, yet engaging introductory manual, is salutary.***Mircea Pitici, PhD, the editor of "****The Best Writings on Mathematics****" series by Princeton University Press**