ALEX Classroom Resource

  

Building Quadratic Functions to Describe Situations (Part 2)

  Classroom Resource Information  

Title:

 Building Quadratic Functions to Describe Situations (Part 2)

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep6-66/building-quadratic-functions-to-describe-situations-part-2/

Content Source:

 PBS
Type: Audio/Video

Overview:

Previously in this video series, students used simple quadratic functions to describe how an object falls over time given the effect of gravity. In this video lesson, they build on that understanding and construct quadratic functions to represent projectile motions. Along the way, they learn about the zeros of a function and the vertex of a graph. They also begin to consider appropriate domains for a function given the situation it represents.

Students use a linear model to describe the height of an object that is launched directly upward at a constant speed. Because of the influence of gravity, however, the object will not continue to travel at a constant rate (eventually it will stop going higher and will start falling), so the model will have to be adjusted (MP4). They notice that this phenomenon can be represented with a quadratic function and that adding a squared term to the linear term seems to “bend” the graph and change its direction.

Content Standard(s):
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: 8
    Accelerated
    30. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and general piecewise functions. [Algebra I with Probability, 28]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a function that models a relationship between two quantities,
    • Produce the graph and table of the function and show the key features (intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. and end behavior) that are appropriate for the function.

    • Given key features from verbal description of a relationship, sketch a graph with the given key features.
    Teacher Vocabulary:
    • Function
    • Intercepts
    • Intervals of Increasing
    • Intervals of decreasing
    • Function is positive
    • Function is negative
    • Relative Maximum
    • Relative Minimum
    • Axis symmetry
    • Origin symmetry
    • End behavior
    Knowledge:
    Students know:
    • Key features of function graphs (i.e., intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. and end behavior).
    • Methods of modeling relationships with a graph or table.
    Skills:
    Students are able to:
    • Accurately graph any relationship.
    • Interpret key features of a graph.
    Understanding:
    Students understand that:
    • The relationship between two variables determines the key features that need to be used when interpreting and producing the graph.
    Diverse Learning Needs:
    Mathematics
    MA2019 (2019)
    Grade: 8
    Accelerated
    33. Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 31]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Engage in the Mathematical Modeling Cycle (Appendix E) to solve contextual problems involving linear, quadratic, exponential, absolute value and linear piecewise function
    Teacher Vocabulary:
    • Mathematical Modeling Cycle
    • Define a problem
    • Make assumptions
    • Define variables
    • Do the math and get solutions
    • Implement and report results
    • Iterate to refine and extend a model
    • Assess a model and solutions
    Knowledge:
    Students know:
    • The Mathematical Modeling Cycle.
    • When to use the Mathematical Modeling Cycle to solve problems.
    Skills:
    Students are able to:
    • Define the problem to be answered.
    • Make assumptions to simplify the problem, identifying the variables in the situation and create an equation.
    • Analyze and perform operations to draw conclusions.
    • Assess the model and solutions in terms of the original context.
    • Refine and extend the model as needed.
    • Report on conclusions and reasonings.
    Understanding:
    Students understand that:
    • Making decisions, evaluating those decisions, and revisiting and revising work is crucial in mathematics and life.
    • Mathematical modeling uses mathematics to answer real-world, complex problems.
    Diverse Learning Needs:
    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.
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Algebra I with Probability
    28. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and linear piecewise functions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given a function that models a relationship between two quantities, produce the graph and table of the function and show the key features (intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. end behavior. and periodicity) that are appropriate for the function.


    Given key features from verbal description of a relationship,
    • Sketch a graph with the given key features.
    • Know periodicity.
    Teacher Vocabulary:
    • Function
    • Periodicity
    • x-intercepts
    • y-intercepts
    • Intervals of Increasing
    • Intervals of decreasing
    • Function is positive
    • Function is negative
    • Relative Maximum
    • Relative Minimum
    • y-axis symmetry
    • Origin symmetry
    • End behavior
    Knowledge:
    Students know:
    • Key features of function graphs (i.e., intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. end behavior. and periodicity).
    • Methods of modeling relationships with a graph or table.
    Skills:
    Students are able to:
    • Accurately graph any relationship.
    • Interpret key features of a graph.
    Understanding:
    Students understand that:
    • The relationship between two variables determines the key features that need to be used when interpreting and producing the graph.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGI.28.1: Define intercepts, intervals, relative maxima, relative minima, symmetry, end behavior, and periodicity.
    ALGI.28.2: For a function that models a relationship between two quantities, find the periodicity.
    ALGI.28.3: For a function that models a relationship between two quantities, find the end behavior.
    ALGI.28.4: For a function that models a relationship between two quantities, find the symmetry.
    ALGI.28.5: For a function that models a relationship between two quantities, find the intervals where the function is increasing, decreasing, positive, or negative.
    ALGI.28.6: For a function that models a relationship between two quantities, find the relative maxima and minima.
    ALGI.28.7: For a function that models a relationship between two quantities, find the x and y intercepts.

    Prior Knowledge Skills:
    • Identify parts of the Cartesian plane.
    • Graph a function given the slope-intercept form of an equation.
    • Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
    • Recall how to plot ordered pairs on a coordinate plane.
    • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.A.AAS.12.28 Given graphs that represent linear functions, identify key features (limit to y intercept, x-intercept, increasing, decreasing) and/or interpret different rates of change (e.g., Which is faster or slower?).
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Algebra I with Probability
    31. Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.

    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Engage in the Mathematical Modeling Cycle (Appendix E) to solve contextual problems involving linear, quadratic, exponential, absolute value and linear piecewise functions.
    Teacher Vocabulary:
    • Mathematical Modeling Cycle
    • Define a problem
    • Make assumptions
    • Define variables
    • Do the math and get solutions
    • Implement and report results
    • Iterate to refine and extend a model
    • Assess a model and solutions
    Knowledge:
    Students know:
    • The Mathematical Modeling Cycle.
    • When to use the Mathematical Modeling Cycle to solve problems.
    Skills:
    Students are able to:
    • Make decisions about problems, evaluate their decisions, and revisit and revise their work.
    • Determine solutions to problems that go beyond procedures or prescribed steps.
    • Make meaning of problems and their solutions.
    Understanding:
    Students understand that:
    • Mathematical modeling uses mathematics to answer real-world, complex problems.
    Diverse Learning Needs:
    Essential Skills:
    Note: One does not need to move through the modeling cycle in the same order, aspects of the cycle may be repeated.
    The Mathematical Modeling Cycle:
    • Define the problem.
    • Make assumptions/Define variables.
    • Do the math/Get solutions.
    • Assess the model and solutions.
    • Iterate to refine and extend model.
    • Implement and report results.


    Prior Knowledge Skills:
      oes not need to move through the modeling cycle in the same order, aspects of the cycle may be repeated. The Mathematical Modeling Cycle:
    • Define the problem.
    • Make assumptions/Define variables.
    • Do the math/Get solutions.
    • Assess the model and solutions.
    • Iterate to refine and extend model.
    • Implement and report results.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.A.AAS.12.31 Choose the graph of the linear function that represents a solution in a real-world scenario. (Ex: Choose the graph that shows a steady increase or decrease rather than a graph with fluctuating data.)
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Mathematical Modeling
    9. Use the Mathematical Modeling Cycle to solve real-world problems involving the design of three-dimensional objects.

    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Can use surface area and volume formulas for three-dimensional figures.
    • Can create a mathematical model and use it to solve a design problem.
    Teacher Vocabulary:
    • Mathematical Modeling Cycle
    • Three Dimensional Object
    Knowledge:
    Students know:
    • the surface area formulas for cylinders, pyramids, cones and spheres.
    Skills:
    Students are able to:
    • Calculate the surface area for cylinders, pyramids, cones and spheres.
    • Calculate volume for cylinders, pyramids, cones and spheres.
    • Use the mathematical modeling cycle
    Understanding:
    Students understand that:
    • Surface area and volume can be used to approximate or solve real-world problems involving three dimensional figures.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    MMOD.9.1: Define three-dimensional, scale factor, and transformations.
    MMOD.9.2: Define the problem to be answered.
    MMOD.9.3: Make assumptions to simplify the situation.
    MMOD.9.4: Identify variables in the situation, and select those that represent essential features in order to formulate a mathematical model.
    MMOD.9.5: Analyze and perform operations to draw conclusions.
    MMOD.9.6: Assess the model and solutions in terms of the original situation.
    MMOD.9.7: Refine and extend the model as needed.
    MMOD.9.8: Report on the conclusions and the reasoning.

    Prior Knowledge Skills:
    • Compare and contrast the random sampling data to the population.
    • Analyze conclusions of the sample to determine its appropriateness for the population.
    • Predict an outcome of the entire population based on random samplings.
    • Justify the mathematical and statistical reasoning.
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Algebra II with Statistics
    22. Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution's feasibility.

    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual situation that may include polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, and piecewise functional relationships in one variable,
    • Model the relationship with equations and solve the problem presented in the contextual situation for the given variable.
    • Interpret solutions in the context of the problem.
    Teacher Vocabulary:
    • Mathematical modeling cycle
    • Feasibility
    Knowledge:
    Students know:
    • When the situation presented in a contextual problem is most accurately modeled by a polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, or general piecewise functional relationship.
    Skills:
    Students are able to:
    • Accurately model contextual situations.
    Understanding:
    Students understand that:
    • There are relationships among features of a contextual problem and a created mathematical model for that problem.
    • Different contexts produce different domains and feasible solutions.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGII.22.1: Define the real-world problem. (i.e., what is the problem asking).
    ALGII.22.2: Make assumptions and define the variables (independent, dependent).
    ALGII.22.3: Assess the model and identify which function will be used (i.e.; polynomial, trigonometric (sine and cosine), logarithmic, radical and general piecewise functions).
    ALGII.22.4: Find the solution.
    ALGII.22.5: Interpret the results.

    Prior Knowledge Skills:
    Note: One does not need to move through the modeling cycle in the same order, aspects of the cycle may be repeated.
    The Mathematical Modeling Cycle:
    • Define the problem.
    • Make assumptions/Define variables.
    • Do the math/Get solutions.
    • Assess the model and solutions.
    • Iterate to refine and extend model.
    • Implement and report results.
    Tags: graph, influence of gravity, linear model, quadratic, vertex, zeros
    License Type: Custom Permission Type
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    For full descriptions of license types and a guide to usage, visit :
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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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