Algebraic Reasoning

Fall Semester

Unit Title

Unit 1: Algebraic Patterns

Unit 2: Analyzing Functions

Unit 3: Solutions of Equations

Unit 4: Algebraic Methods

Time

~ 5 weeks

~ 4 weeks

~ 5 weeks

~ 6 weeks

Understandings

Functions can be identified by analyzing patterns in tables using finite differences or common ratios.

Once a function type is identified, its equation can be generated using its defining characteristics

When modeling real-world situations, restrictions on domain and range must be considered.

Solutions to real-world applications can be represented through tables, equations, and contextual interpretations.

Understanding the relationships among representations strengthens problem-solving and mathematical reasoning.

Families of functions are classified by the distinct relationship between two variables.

Each function family has recognizable graphical characteristics that distinguish it from others.

A function can be represented tabularly, graphically, and symbolically, and each representation communicates the same relationship in a different way.

Domain and range can be represented using inequalities, interval notation, set notation. Each notation communicates the allowable values of a function in a precise mathematical way.

A function assigns exactly one output value to each input value.  Functions can be evaluated by using an input to determine an output — or by using an output to determine the corresponding input.

Solutions to equations are the set of all points that make the equation true.

Data models  help analyze patterns and make informed decisions. Selecting an appropriate model depends on understanding how variables relate and how outputs change over time or input values.

Polynomial functions can often be written as the product of their linear factors.

• The solutions of the linear factors are the x-intercepts of the polynomial.
• A quadratic and its linear factors share the same x-intercepts.
• When real solutions exist, the quadratic’s y-intercept equals the product of the y-intercepts of its linear factors.
   

Factored form reveals critical information about a function’s behavior and intercepts.

Patterns in tables can be used to generate symbolic representations of real-world situations.

Factoring, simplifying, and analyzing polynomial expressions allow us to interpret solutions and model relationships meaningfully.

TEKS

AR.2 A, B, C, D

AR.3 A

AR.7 A

AR.6 A, B, C

AR.3 F

AR.4 A, B, C, D

Process Standards

1 A, B, C, D, E, F, G

Spring Semester

Unit Title

Unit 5: Inverses of Functions

Unit 6: Function Operations

Unit 7: Data Modeling

Unit 8: Matrices

Time

~ 4 weeks

~ 4 weeks

~ 3 weeks

~ 4 weeks

Understandings

An inverse function reverses the roles of the independent and dependent variables.

Inverse functions undo the original relationship by swapping inputs and outputs.

Graphically, the inverse of a function is a reflection of the original graph across the line y=xy = xy=x.

For the inverse to remain a function, the original function may require restricted domains, ensuring that each input still corresponds to exactly one output.

Functions can be combined using addition and multiplication, or separated using subtraction and division, to form new functions.  The resulting function represents a new relationship between variables while maintaining algebraic properties.

In a composition of functions, the output of one function becomes the input of another. This layered relationship creates a new function that represents multiple processes happening in sequence.

Newly formed functions — whether combined or composed — can be represented tabularly, graphically, or symbolically. These relationships are powerful tools for modeling and solving real-world problems that involve multiple dependent processes.

A function serves as a mathematical model for real-world data, helping determine solutions and make predictions.

Patterns in data tables, finite differences, average rates of change, and constant ratios help determine the most appropriate function model.

Rate of change applies the concept of slope in real-world contexts.

For linear models, the solution represents all ordered pairs that make the equation true, and the constant rate of change explains how one variable responds to changes in another.

Matrices provide a structured way to represent and organize data sets, making relationships easier to analyze and manipulate.They serve as a compact and powerful representation of numerical information.

Systems of linear equations can be written and solved using matrices.
This approach offers an organized algebraic strategy, and technology is especially useful when solving systems with three variables.

TEKS

AR.3 B, C

AR.3 D, E

AR.7 A, B, C, D

AR.5 A, B, C, D, E

Process Standards

1 A, B, C, D, E, F, G