Lesson 5: Connecting Representations

How do we decide good value of x for a function graph?
How many ordered pairs do we need to create a good graph of a function?

F.IF.1
A. Understand the concept of a function and use function notation
1. HSF-IF.A.1
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of
the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).

2. HSF-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

B. Interpret functions that arise in applications in terms of the context
5. HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.?

Lesson Description: In this lesson, students learn that a single mathematical relationship can be expressed in four distinct ways, often referred to as the "Rule of Four." Students will practice converting any one representation into the others: Verbal: A written or spoken description of the relationship. Numerical: A table of values or a set of ordered pairs. Algebraic: A symbolic formula or function (e.g., f(x) = 3x - 2).Graphical: A visual representation on the Cartesian plane. Students will be given a starting point (like a graph) and be asked to produce the other three forms (the table, the equation, and the verbal story).
Purpose
The purpose of Section 8.5 is to develop mathematical flexibility. In real-world problem solving, information often arrives in "pieces"—you might have a table of data but need an equation to make a prediction, or you might have a formula but need a graph to present it to a supervisor. By connecting these representations, students stop seeing "tables" and "graphs" as separate topics and start seeing them as different "languages" used to describe the same truth. This synthesis is the hallmark of a successful algebra student.
Depth of Knowledge (DOK) Level
DOK Level 3
Level 3 (Strategic Thinking & Complex Reasoning): This section is a definitive DOK 3. Students must analyze and evaluate which representation is most useful for a specific task. They are required to synthesize information across different formats and explain the connections (e.g., "The 'starting fee' in the verbal description is the y-intercept on the graph and the constant in the equation"). It moves beyond "how" to do the math and into "how" the different parts of math relate to one another.

Function notation is sometimes difficult for student to comprehend. Stress that f(x) does not mean "f times x."

Class and online work

use "you try" pg 186 and practice problems  to assess students' understanding of the lesson concepts