ALEX Classroom Resource

Algebra II Module 3, Topic D: Using Logarithms in Modeling Situtation

Classroom Resource Information

Title:

Algebra II Module 3, Topic D: Using Logarithms in Modeling Situtation

URL:

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

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

Module 3, Topic D opens with a hands-on simulation and modeling activity in which students gather data and apply the analysis of Lesson 22 in Topic C to model it with an exponential function (A-CED.2, F-LE.5). Students use logarithms to solve exponential equations analytically and express the solution as a logarithm (F-LE.4). Students study the relationship between exponential growth and decay and geometric series (F-IF.3), and students must use properties of exponents to interpret expressions for exponential functions (F-IF.8b). Armed with a more thorough understanding of exponential functions and equations, students revisit the topic of Newton’s Law of Cooling that was introduced in Algebra I (F-BF.1b).

Content Standard(s):
Mathematics
MA2015 (2016)
Algebra II
35 ) Find inverse functions. [F-BF4]

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. [F-BF4a]

Example: f(x) =2x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

Mathematics
MA2015 (2016)
Algebra II
20 ) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1]

Mathematics
MA2015 (2016)
Algebra II
27 ) Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11]

Mathematics
MA2015 (2016)
Algebra II
31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

Mathematics
MA2015 (2016)
Algebra II
36 ) For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [F-LE4]

Mathematics
MA2019 (2019)
Algebra I with Probability
16. Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).
Unpacked Content
Evidence Of Student Attainment:
Students:
• Can identify similarities and differences between relations and functions represented by equations, graphs or tables.
• Determine if a relation is a function and explain that a function is a special kind of relation.
Teacher Vocabulary:
• Function
• Relation
Knowledge:
Students know:
• How to represent relations and functions by equations, graphs or tables and can compare and contrast the different representations.
• A function is a special kind of relation.
Skills:
Students are able to:
• Compare and contrast relations and functions given different representations.
• Identify which relations are functions and which are not.
Understanding:
Students understand that:
• All functions are relations, but that some relations are not functions.
• Equations, graphs, and tables are useful representations for comparing and contrasting relations and functions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.16.1: Define functions, relations (ordered pairs), input, output.
ALGI.16.2: Recall how to complete input/output tables.
ALGI.16.3: Recall how to read/interpret information from a table.
ALGI.16.4: Identify algebraic expressions.
ALGI.16.5 Recall how to name points from a graph (ordered pairs).
ALGI.16.6: Recall how to name points on a Cartesian plane using ordered pairs.

Prior Knowledge Skills:
• Recall how to read a graph or table.
• Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane.
• 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.12.15 Use the vertical line test to determine if a given relation is a function.

Mathematics
MA2019 (2019)
Algebra I with Probability
17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function Input Output
G Amount of studying: s Grade in course: G(s)
S Grade in course: g Amount of screen time: S(g)
T Amount of screen time: t Number of follers: T(t)
Unpacked Content
Evidence Of Student Attainment:
Students:
Given different types of standard functions
• Use arithmetic operations to combine functions in context.
• Use function composition to combine functions in context.
• Write, evaluate, and interpret combined functions in context.
Teacher Vocabulary:
• Function composition
Knowledge:
Students know:
• Techniques to combine functions using arithmetic operations.
• Techniques for combining functions using function composition.
Skills:
Students are able to:
• Accurately develop a model that shows the functional relationship between two quantities.
• Accurately create a new function through arithmetic operations of other functions.
• Present an argument to show how the function models the relationship between the quantities.
Understanding:
Students understand that:
• Arithmetic combinations of functions may be used to improve the fit of a model.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Prior Knowledge Skills:
• Explain the distributive property.
• Give examples of the properties of operations including distributive.
• Combine like terms of a given expression.
• Recognize the correct order to solve expressions with more than one operation.
• Calculate a numerical expression (Ex. V=(4x4x4).
• Choose the correct value to replace each variable in the algebraic expression (Substitution).
• Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
Mathematics
MA2019 (2019)
Algebra I with Probability
21. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given relations or functions represented in various ways(algebraically, graphically, numerically in tables, or by verbal descriptions),
• Use key features to compare the functions.
• Explain and justify the similarities and differences of the functions.
Teacher Vocabulary:
• Linear function
• Exponential function
• Absolute value function
• Linear Piecewise
Knowledge:
Students know:
• Techniques to find key features of functions when presented in different ways.
• Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).
Skills:
Students are able to:
• Accurately determine which key features are most appropriate for comparing functions.
• Manipulate functions algebraically to reveal key functions.
• Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.
Understanding:
Students understand that:
• Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
• Different representations of functions may aid in comparing key features of the functions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.21.1: Define function, function notation, (linear, polynomial, rational, absolute value, exponential, piecewise, and logarithmic) functions, and transitive property.
ALGI.21.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGI.21.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator.
ALGI.21.4: Solve equations for y.
ALGI.21.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations.

Prior Knowledge Skills:
• Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
• Graph a function given the slope-intercept form of an equation.
• Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.
• Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
• 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.12.21 Given a function table, identify the missing number.

Tags: analysis, equivalent, exponential equation, exponential function, Geometric series, inverse functions, logarithms, modeling, Newtons Law of Cooling