Classroom Resource

Building Quadratic Functions to Describe Situations (Part 2)

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

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.

Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

MA19.8A.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

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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.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.25

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

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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.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.28

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

UP:MA19.A1.28

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.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.31

Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.

Unpacked Content

UP:MA19.A1.31

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.
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.22

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

UP:MA19.A2.22

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.

Body

  1. 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.
Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.9

Use the Mathematical Modeling Cycle to solve real-world problems involving the design of three-dimensional objects.

Unpacked Content

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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.
Link to Resource
https://aptv.pbslearningmedia.org/resource/im20-math-ep6-66/building-quadratic-…

CR Resource Type

Audio/Video

Resource Provider

PBS
Accessibility
License

License Type

CUSTOM

Custom License Type (URL)

https://creativecommons.org/licenses

Comments or Notes

This resource has student task statements and practice problem worksheets.

Approved Date

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