ALEX Classroom Resource

  

Algebra I Module 3, Topic D: Using Functions and Graphs to Solve Problems

  Classroom Resource Information  

Title:

 Algebra I Module 3, Topic D: Using Functions and Graphs to Solve Problems

URL:

 https://www.engageny.org/resource/algebra-i-module-3-topic-d-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 3, Topic D, students apply and reinforce the concepts of the module as they examine and compare exponential, piecewise, and step functions in a real-world context (F-IF.C.9). They create equations and functions to model situations (A-CED.A.1, F-BF.A.1, F-LE.A.2), rewrite exponential expressions to reveal and relate elements of an expression to the context of the problem (A-SSE.B.3c, F-LE.B.5), and examine the key features of graphs of functions, relating those features to the context of the problem (F-IF.B.4, F-IF.B.6).

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems.
  • Produce the useful equivalent forms of expressions,
  • Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines and complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines.
  • Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.
Teacher Vocabulary:
  • Quadratic expression
  • Zeros
  • Complete the square
  • Roots
  • Zeros
  • Solutions
  • x-intercepts
  • Maximum value
  • Minimum value
  • Factor
  • Roots
  • Exponents
  • Equivalent form
  • Vertex form of a quadratic expression
Knowledge:
Students know:
  • Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.
Skills:
Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Factor quadratic expressions.
  • Complete the square in quadratic expressions.
  • Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.
Understanding:
Students understand that:
  • Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.6.1: Convert an expression to an alternative format.
ALGI.6.2: Recognize the best format for a specific application.
ALGI.6.3: Match equivalent expressions written in different forms.

a.
ALGI.6.4: Define factor, quadratic expression and zero product property.
ALGI.6.5: Factor a quadratic expression.
ALGI.6.6: Use the zero product property to reveal the zeros in the function.
ALGI.6.7: Solve a one-step equation.
ALGI.6.8: Solve a two-step equation.
ALGI.6.9: Determine the Greatest Common Factor (GCF).

b.
ALGI.6.10: Define maximum and minimum value.
ALGI.6.11: Explain the steps for completing the square.
ALGI.6.12: Given a quadratic expression in which the square has already been completed, determine the maximum or minimum values.

c.
ALGI.6.13: Define roots.
ALGI.6.14: Find the equation using the distributive property.
ALGI.6.15: Locate and identify the roots on a graph using the x-intercepts.
ALGI.6.16: Take given roots and convert into a one-step equation set equal to zero.

Prior Knowledge Skills:
  • Identify how many solutions the linear equation may or may not have.
  • Recall how to solve problems using the distributive property
  • Explain the distributive property.
  • Recall solving one-step equations.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable.
  • Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.
Teacher Vocabulary:
  • Variable
  • Equation
  • Inequality
  • Solution Set
  • Identity
  • No solution for a given domain
  • Approximate solutions
Knowledge:
Students know:
  • When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship.
Skills:
Students are able to:
  • Write equations in one variable that accurately model contextual situations.
Understanding:
Students understand that:
  • Features of a contextual problem can be used to create a mathematical model for that problem.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.11.1: Solve the equation represented by the real-world situation.
ALGI.11.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.11.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.11.4: Define equation, expression, variable, equality and inequality.
ALGI.11.5: Create inequalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.6: Create equalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.7: Solve two-step equations and inequalities.
ALGI.11.8: Solve one-step equations and inequalities using the four basic operations.
ALGI.11.9: Compare and contrast equations and inequalities.
ALGI.11.10: Recognize inequality symbols including greater than, less than, greater than equal to and less than equal to.

Prior Knowledge Skills:
  • Test the found number or number set for accuracy by substitution.
  • Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
  • Define equation, inequality, and variable.
  • Convert mathematical terms to mathematical symbols and numbers.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation.
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?).
Tags: equation, equivalent, expression, function, graph, inequalities, intercepts, quadratic, square, table, variable
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Author: Hannah Bradley
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