ALEX Classroom Resource

  

Productivity and Graphing Linear and Quadratic Functions

  Classroom Resource Information  

Title:

 Productivity and Graphing Linear and Quadratic Functions

URL:

 https://www.econedlink.org/resources/productivity-and-graphing-linear-and-quadratic-functions/

Content Source:

 EconEdLink
Type: Lesson/Unit Plan

Overview:

In this lesson, students interpret key features of graphs for both linear and quadratic functions in the context of total and marginal production. The lesson begins with a short video about a young entrepreneur who designed his own line of bowties. Students then predict the relationship between the number of workers and the production of bowties. Students test their predictions by participating in a production activity making paper bowties. Next, they sketch graphs of their total and marginal product and describe the key features of their graphs. The lesson closes with students graphing two datasets and deciding which dataset most realistically describes the relationship between the number of workers and production.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
23. Construct a function to model the linear relationship between two variables.

a. Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship from two points in a table or graph. [Grade 8, 16]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Create the graphical representation of linear function, given a linear equation in slope-intercept form, using the initial value and rate of change.
  • Give meaning rates of change and the initial values of linear functions in different contexts.
Teacher Vocabulary:
  • Function
  • Linear
  • Non-linear
  • Slope
  • y-intercept
Knowledge:
Students know:
  • That the rate of change of a function is the ratio of change in the output to the change in the input.
  • how to find the rate of change/slope as well as the initial value/y-intercept.
Skills:
Students are able to:
  • Construct the graph of a linear function.
  • Identify the slope and y-intercept of functions in different contexts.
Understanding:
Students understand that:
  • Terms such as slope and y-intercept describe a graphical.
  • Representation of a linear function and correlate their meaning to the rate of change and initial value, where the input is 0.
  • Using the units from a context appropriately is needed to make their description of rate of change and initial value accurate.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
16. Construct a function to model a linear relationship between two variables.

a. Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Create the graphical representation of linear function, given a linear equation in slope-intercept form, using the initial value and rate of change.
  • give meaning rates of change and the initial values of linear functions in different contexts.
Teacher Vocabulary:
  • Function
  • Linear
  • Non-linear
  • Slope
  • y-intercept
Knowledge:
Students know:
  • that the rate of change of a function is the ratio of change in the output to the change in the input.
  • how to find the rate of change/slope as well as the initial value/y-intercept.
Skills:
Students are able to:
  • construct the graph of a linear function.
  • Identify the slope and y-intercept of functions in different contexts.
Understanding:
Students understand that:
  • terms such as slope and y-intercept describe a graphical representation of a linear function and correlate their meaning to the rate of change and initial value, where the input is 0.
  • Using the units from a context appropriately is needed to make their description of rate of change and initial value accurate.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.16.1: Define function, rate of change, and initial value.
M.8.16.2: Recall how to complete an input/output function table.
M.8.16.3: Recall how to find the rate of change (slope) in a linear equation.
M.8.16.4: Recall how to name points from a graph (ordered pairs).
M.8.16.5: Analyze real-world situations to identify the rate of change and initial value from a table, graph, or description.

Prior Knowledge Skills:
  • Solve an equation by substituting a value to find an output.
  • Find the coordinates of an ordered pair.
  • Recognize how the steepness of a graphed line changes vertically and horizontally.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
18. Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a system of a linear equation and a quadratic equation,
  • Solve the system algebraically by substitution.
  • Graph the linear equation and the quadratic equation on the same Cartesian plane, and identify the intersection point(s).
  • Make sense of the existence of 0, 1, or 2 solutions to the system by explaining the relationship of the solutions to the graph.
  • Verify that the proposed solutions satisfy both equations.
Teacher Vocabulary:
  • Solving systems of equations
  • System of equations
  • Substitution method
  • Elimination method
  • Cartesian plane
Knowledge:
Students know:
  • Appropriate use of properties of equality.
  • Techniques to solve quadratic equations.
  • The conditions under which a linear equation and a quadratic equation have 0, 1, or 2 solutions.
  • Techniques for producing and interpreting graphs of linear and quadratic equations.
Skills:
Students are able to:
  • Accurately use properties of equality to solve a system of a linear and a quadratic equation.
  • Graph linear and quadratic equations precisely and interpret the results.
Understanding:
Students understand that:
  • Solutions of a system of equations is the set of all ordered pairs that make both equations true simultaneously.
  • A system consisting of a linear equation and a quadratic equation will have 0,1, or 2 solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.18.1: Use the substitution method to replace a variable in the quadratic equation.
ALGI.18.2: Solve for the variables in a system of equations. (Algebraically).
ALGI.18.3: Graph a quadratic equation.
ALGI.18.4: Graph a linear equation.
ALGI.18.5: Identify the point(s) of intersection when given graphs.
ALGI.18.6: Use digital tools to defend solutions to authentic problems.
ALGI.18.7: Use digital tools to formulate solutions to authentic problems (Ex: electronic graphing tools, probes, spreadsheets).

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, and output.
  • Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
  • 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.
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
  • Analyze the graph to determine the rate of change.
  • Generate the slope of a line using given ordered pairs.
  • Show how to plot points on a Cartesian plane.
  • Define ordered pairs.
  • Show how to graph on Cartesian plane.
  • Substitute for the variable to find the value of a given expression.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Define ordered pairs.
  • Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
  • Discover that the opposite of the opposite of a number is the number itself.
  • Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
  • Identify the parts of a table of equivalent ratios (input, output, etc.).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.18 Interpret the meaning of a point on the graph of a line. (Ex.: On a graph of football ticket purchases, trace the graph to a point and tell the number of tickets purchased and the total cost.)
Tags: financial literacy, function, graphing, linear, quadratic
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Author: Hannah Bradley
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