ALEX Classroom Resource

  

Building Quadratic Functions to Describe Situations (Part 1)

  Classroom Resource Information  

Title:

 Building Quadratic Functions to Describe Situations (Part 1)

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep5-65/building-quadratic-functions-to-describe-situations-part-1/

Content Source:

 PBS
Type: Audio/Video

Overview:

This is the first of several video lessons in which students construct quadratic functions to represent various situations. Here they investigate the movement of free-falling objects. Students analyze the vertical distances that falling objects travel over time and see that they can be described by quadratic functions. They use tables, graphs, and equations to represent and make sense of the functions. In subsequent lessons, students build on the functions developed here to represent projectile motions, providing a context to develop an understanding of the zeros, vertex, and domain of quadratic functions.

To express the relationship between distance and time, students need to see regularity in numerical values and express that regularity (MP8).

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher Vocabulary:
  • Linear functions
  • Exponential functions
  • Constant rate of change
  • Constant percent rate of change
  • Intervals
  • Percentage of growth
  • Percentage of decay
  • Slope-intercept form of a line
Knowledge:
Students know:
  • Key components of linear and exponential functions.
  • Properties of operations and equality
Skills:
Students are able to:
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
Understanding:
Students understand that:
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
  • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
27. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). [Algebra I with Probability, 25]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,
  • Create a linear or exponential function that models the situation.
  • Create arithmetic and geometric sequences from the given situation.
  • Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.
Teacher Vocabulary:
  • Arithmetic sequence
  • Geometric sequence
  • Linear function
  • Exponential function
    • Knowledge:
    Students know:
    • That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    • Properties of arithmetic and geometric sequences.
    Skills:
    Students are able to:
    • Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.
    Understanding:
    Students understand that:
    • Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
    • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
    Diverse Learning Needs:
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Algebra I with Probability
    24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

    a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

    b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

    c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a linear or exponential function,
    • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
    • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

    • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
    Teacher Vocabulary:
    • Linear functions
    • Exponential functions
    • Constant rate of change
    • Constant percent rate of change
    • Intervals
    • Percentage of growth
    • Percentage of decay
    Knowledge:
    Students know:
    • Key components of linear and exponential functions.
    • Properties of operations and equality
    Skills:
    Students are able to:
    • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
    • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
    Understanding:
    Students understand that:
    • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
    • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGI.24.1: Define linear function and exponential function.
    ALGI.24.2: Distinguish between graphs of a line and an exponential function.
    ALGI.24.3: Identify the graph of an exponential function.
    ALGI.24.4: Identify the graph of a line.
    ALGI.24.5: Plot points on a coordinate plane from a given table of values. a.
    ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
    ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
    ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers. b.
    ALGI.24.9: Define constant rate of change as slope.
    ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
    ALGI.24.11: Recognize the calculated difference is the constant rate of change.
    ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers. c.
    ALGI.24.13: Define exponential growth and decay.
    ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
    ALGI.24.15: Apply the rules of multiplication and division of integers.

    Prior Knowledge Skills:
    • Recognize ordered pairs.
    • Identify ordered pairs.
    • Recognize linear equations.
    • Recall how to solve problems using the distributive property.
    • Define linear and nonlinear functions, slope, and y-intercept.
    • Analyze the graph to determine the rate of change.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Algebra I with Probability
    25. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,
    • Create a linear or exponential function that models the situation.
    • Create arithmetic and geometric sequences from the given situation.
    • Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.
    Teacher Vocabulary:
    • Arithmetic and geometric sequences
    • Arithmetic sequence
    • Geometric sequence
    • Exponential function
    Knowledge:
    Students know:
    • That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    • Properties of arithmetic and geometric sequences.
    Skills:
    Students are able to:
    • Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.
    Understanding:
    Students understand that:
    • Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
    • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGI.25.1: Define linear function and exponential function.
    ALGI.25.2: Define arithmetic sequence, geometric sequence, and input-output pairs.
    ALGI.25.3: Define sequences and recursively-defined sequences.
    ALGI.25.4: Recognize that sequences are functions whose domain is the set of all positive integers and zero.
    ALGI.25.5: Given a chart, write an equation of a line.
    ALGI.25.6: Given a graph, write an equation of a line.
    ALGI.25.7: Given two ordered pairs, write an equation of a line.

    Prior Knowledge Skills:
    • Given a function, create a rule.
    • Recognize numeric patterns.
    • Recall how to complete input/output tables.
    • Demonstrate how to plot points on a Cartesian plane using ordered pairs.
    • Define function, ordered pairs, input, output.
    • Graph a linear equation given the slope-intercept form of an equation.
    • Graph a function given the slope-intercept form of an equation.
    • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
    • Generate the slope of a line using given ordered pairs.
    • Recall the rules for multiplying integers.
    • Define quotient, divisor, and integer.
    • Solve addition and subtraction of multi-digit whole numbers.
    • Solve addition and subtraction of multi-digit decimal numbers (emphasis on alignment).
    • Recall basic multiplication and division facts.
    • Solve multiplication problems involving multi-digit whole numbers and decimal numbers.
    • Solve division problems involving multi-digit whole numbers and decimal numbers.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
    Tags: analyze distances, domain, equations, graphs, quadratic functions, tables, vertex
    License Type: Custom Permission Type
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    AccessibilityVideo resources: includes closed captioning or subtitles
    Comments

    This resource has student task statements and practice problem worksheets.

      This resource provided by:  
    Author: Kristy Lacks
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