ALEX Classroom Resource

  

Building Quadratic Functions from Geometric Patterns

  Classroom Resource Information  

Title:

 Building Quadratic Functions from Geometric Patterns

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep3-63/building-quadratic-functions-from-geometric-patterns/

Content Source:

 PBS
Type: Audio/Video

Overview:

Earlier in this video series, students reasoned about visual patterns using different representations and wrote expressions to describe the patterns. In this lesson, they continue to work with patterns but begin to see these relationships as quadratic functions and write equations to define them.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
16. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. [Grade 8, 13, edited for added content]

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [Grade 8, 14, edited for added content]
Note: 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.

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 15]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a functional relationship,
  • Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function,
  • Represent the function with a graph and with functional notation.
  • Evaluate functional notation to produce a range when given a value in the domain,
  • Explain in the original context the meaning of the output when related to the input.
Given a contextual situation that is functional,
  • Model the situation with a graph and construct the graph based on the parameters given in the domain of the context.
  • Distinguish between those that are functions and non-functions.
Teacher Vocabulary:
  • Function
  • Relation
  • Mapping
  • Domain
  • Range
  • Functional notation f(x)
  • Element
  • Input
  • output
  • Quantitative relationship
Knowledge:
Students know:
  • Distinguishing characteristics of functions,
  • Conventions of function notation,
  • Techniques for graphing functions,
  • Techniques for determining the domain of a function from its context.
Skills:
Students are able to:
  • Accurately graph functions when given function notation.
  • Accurately evaluate function equations given values in the domain.
  • Interpret the domain from the context,
  • Produce a graph of a function based on the context given.
Understanding:
Students understand that:
  • Functions are relationships between two variables that have a unique characteristic: that for each input there exists exactly one output.
  • Function notation is useful to see the relationship between two variables when the unique output for each input relation is satisfied.
  • Different contexts produce different domains and graphs.
  • Function notation in itself may produce graph points which should not be in the graph as the domain is limited by the context.
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
    15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

    a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: 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.

    b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given input/output relations between two variables in graphical form, verbal description, set of ordered pairs, or algebraic model, distinguish between those that are functions and non-functions.


    Using functional notation,
    • Evaluate functions for inputs.
    • Interpret statements in terms of context.


    Given a contextual relationship that may be represented as a function,
    • Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function.
    • Relate the domain to its graph and to the quantitative relationship it describes.
    Teacher Vocabulary:
    • Domain
    • Range
    • Function
    • Relation
    • Function notation
    • Set notation
    Knowledge:
    Students know:
    • Distinguishing characteristics of functions.
    • Conventions of function notation.
    • In graphing functions the ordered pairs are (x,f(x)) and the graph is y = f(x).
    Skills:
    Students are able to:
    • Evaluate functions for inputs in their domains.
    • Interpret statements that use function notation in terms of context.
    • Accurately graph functions when given function notation.
    • Accurately determine domain and range values from function notation.
    Understanding:
    Students understand that:
    • A function is a mapping of the domain to the rangeFunction notation is useful in contextual situations to see the relationship between two variables when the unique output for each input relation is satisfied.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGI.15.1: Define domain, range, relation, function, table of values, input, and output.
    ALGI.15.2: Understand the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
    ALGI.15.3: Understand that a function is a rule that assigns to each input exactly one output.
    ALGI.15.4: Identify the equation of a function, given its graph.
    ALGI.15.5: Find the range of a function given its domain.
    ALGI.15.6: Recognize that f(x) and y are the same.
    ALGI.15.7: Recall how to complete input/output tables.
    ALGI.15.8: Recall how to read/interpret information from a table.
    ALGI.15.9: Define function notation.
    ALGI.15.10: Translate a simple word problem into function notation.
    ALGI.15.11: Evaluate function when given x-value.

    Prior Knowledge Skills:
    • Analyze the graph to determine the rate of change.
    • Generate the slope of a line using given ordered pairs.
    • Define linear functions, nonlinear functions, slope, and y-intercept
    • Identify ordered pairs.
    • Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon.
    Alabama Alternate Achievement Standards
    AAS Standard:
    M.A.AAS.12.15 Use the vertical line test to determine if a given relation is a function.
    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 II with Statistics
    13. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given a contextual situation expressing a relationship between quantities with two or more variables, model the relationship with equations and graph the relationship on coordinate axes with labels and scales and use them to make predictions.
    Teacher Vocabulary:
    • Polynomial Functions
    • Exponential Functions
    • Logarithmic Functions
    • Radical Functions
    • Trigonometric Functions
    • Reciprocal Functions
    • Piecewise Functions
    Knowledge:
    Students know:
    • When a particular two variable equation accurately models the situation presented in a contextual problem.
    Skills:
    Students are able to:
    • Write equations in two variables that accurately model contextual situations.
    • Graph equations involving two variables on coordinate axes with appropriate scales and labels, using it to make predictions.
    Understanding:
    Students understand that:
    • There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship which is useful in making predictions.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    ALGII.13.1: Define ordered pair, coordinate plane, polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
    ALGII.13.2: Create equations with two variables (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions).
    ALGII.13.3: Graph equations on coordinate axes with labels and scales (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.).
    ALGII.13.4: Identify an ordered pair and plot it on the coordinate plane.

    Prior Knowledge Skills:
    • Identify X axis.
    • Identify Y axis.
    • Graph points on a coordinate plane.
    • Enter coordinates into a table.
    Mathematics
    MA2019 (2019)
    Grade: 9-12
    Applications of Finite Math
    13. Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.

    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Create fractals using technology or other tools based on a recursive formula.
    • Use population growth models to find population sizes at various times.
    Teacher Vocabulary:
    • Difference equation
    • Recursive process
    • Recursive formula
    • Fractals
    • Population growth models
    • Sequences
    • Series
    Knowledge:
    Students know:
    • How to recognize a pattern.
    Skills:
    Students are able to:
    • Apply recursive formulas in real-world contexts.
    Understanding:
    Students understand that:
    • Models such as population growth should be recognized as recursively developed models.
    • The recursion process can be applied to many situations.
    • A sequence lists the solutions of a set of related problems.
    Diverse Learning Needs:
    Tags: patterns, quadratic functions, relationships, writing equations
    License Type: Public Domain
    For full descriptions of license types and a guide to usage, visit :
    https://creativecommons.org/licenses
    AccessibilityVideo resources: includes closed captioning or subtitles
    Comments

    This resource contains student practice problems and student task statements.

      This resource provided by:  
    Author: Kristy Lacks
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