Reading Graphs

Abstract

This lesson is designed to introduce students to graphing functions and to reading simple functions from graphs. Many of the examples are motivated by a situation described by the graph.

These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.

Objectives

Upon completion of this lesson, students will:

  • have practiced plotting functions on the Cartesian coordinate plane
  • have seen several categories of functions, including lines and parabolas
  • be able to read a graph, answering questions about the situation described by the graph

Standards Addressed

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Functions

  • Building Functions

    • Build a function that models a relationship between two quantities
    • Build new functions from existing functions

  • Interpreting Functions

    • Understand the concept of a function and use function notation
    • Interpret functions that arise in applications in terms of the context
    • Analyze functions using different representations

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems
    • Interpret expressions for functions in terms of the situation they model

  • Trigonometric Functions

    • Model periodic phenomena with trigonometric functions

Grades 6-8

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

Algebra I

  • Algebra

    • Competency Goal 4: The learner will use relations and functions to solve problems.

Technical Mathematics II

  • Data Analysis and Probability

    • Competency Goal 2: The learner will use relations and functions to solve problems.

Advanced Functions and Modeling

  • Data Analysis and Probability

    • Competency Goal 1: The learner will analyze data and apply probability concepts to solve problems.

3rd Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of numeric patterns, symbols as representations of unknown quantity, and situations showing increase over time.

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of organizing, interpreting, analyzing and making predictions about data, the benefits of multiple representations of a data set, and the basic concepts of probability.

6th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of shape, location, and movement within a coordinate system; similarity, complementary, and supplementary angles; and the relationship between line and rotational symmetry.

5th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of the use of patterns, relations, functions, models, structures, and algebraic symbols to represent quantitative relationships and will analyze change in various contexts.

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of investigation design, the effect of data-collection methods on a data set, the interpretation and application of the measures of central tendency, and the application of b
    • The student will demonstrate through the mathematical processes an understanding of investigation design, the effect of data-collection methods on a data set, the interpretation and application of the measures of central tendency, and the application of basic concepts of probability.

7th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of proportional relationships.

8th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation
    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.
    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
    • Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.
    • Standard EA-6: The student will demonstrate through the mathematical processes an understanding of quadratic relationships and functions.

Intermediate Algebra

  • Algebra

    • The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • The student will demonstrate through the mathematical processes an understanding of quadratic equations and the complex number system.
    • The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

5th Grade

  • Probability and Statistics

    • 5.17c The student will create a problem statement involving probability and based on information from a given problem situation. Students will not be required to solve the created problem statement.

Secondary

  • Algebra II

    • AII.08 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.
    • AII.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.
    • AII.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.
    • AII.19 The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions.
    • AII.8
    • AII.10
    • AII.15
    • AII.19

Textbooks Aligned

7th

  • Module 1 - Search and Rescue

    • Section 4: Function Models
      • Reason for Alignment: The Reading Graphs lesson focuses on interpreting functions from graphs. It also deals with building graphs from situations and vice versa. This entire lesson would take some time, but parts of it are usable with the entire class and would strongly support the textbook work on functions and graphs.

8th

  • Module 6 - Visualizing Change

    • Section 5: Exploring Quadratic Functions
      • Reason for Alignment: This lesson is at an introductory level for students to graph functions and to read simple functions from graphs. This lesson would be good for a back-up or reinforcement to the textbook. Studetns should find the discussions interesting and thorough.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer and fractional arithmetic
    • plot points on the Cartesian coordinate system
    • read the coordinates of a point from a graph
  • Algebraic: Students must be able to:
    • work with very simple algebraic expressions
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

concave up

A curve is "concave up" when it is a concave shape, meaning curved like the inside of a bowl, with the two ends of the curve pointing up

constant functions

Functions that stay the same no matter what the variable does are called constant functions

constants

In math, things that do not change are called constants. The things that do change are called variables.

coordinate plane

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

coordinates

A unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the x-axis while the second number identifies the position on the y-axis

function

A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

origin

In the Cartesian coordinate plane, the origin is the point at which the horizontal and vertical axes intersect, at zero (0,0)

velocity

The rate of change of position over time is velocity, calculated by dividing distance by time

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Can anyone give me an example of a function? Can anyone give me an example of an everyday situation that a function can be applied to?

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn more about functions.
    • We are going to use the computers to learn more about functions, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    • Lead a discussion on gathering information from graphs.
    • Lead a discussion on making new graphs from old ones: graphs involving distance, velocity, and acceleration.

  4. Guided Practice

    • Have the students try to build graphs from several situations using graph paper. Teams work best for the story-telling activities.

  5. Independent Practice

    • Have the students try to build formulas from several situations, and then graph them using Graph Sketcher. Teams work best for the story-telling activities.

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways.

  • Omit the discussion on distance, velocity and acceleration.

Suggested Follow-Up

After these discussions and activities, students will have more experience with functions and relationship between the English description, graphical and algebraic representations. The next lesson, Impossible Graphs, shows the students that not all graphs make sense in certain situations.