ALEX Classroom Resources

ALEX Classroom Resources  
   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 17 :
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)
Subject: Mathematics (9 - 12)
Title: Algebra II Module 3, Topic A: Real Numbers
URL: https://www.engageny.org/resource/algebra-ii-module-3-topic-a-overview
Description:

In Module 3, Topic A, students both review what they already know about exponential expressions and functions with integer exponents and extend the meaning of an exponential expression to allow for first rational and then irrational exponents (N-RN.1). Students practice applying properties of exponents to expressions with non-integer exponents (N-RN.2). Students use the average rate of change and repeated reasoning to discover Euler’s number, e (F-IF.6). 



   View Standards     Standard(s): [MA2015] AL2 (9-12) 20 :
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]

[MA2015] AL2 (9-12) 36 :
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]

[MA2019] AL1-19 (9-12) 17 :
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)
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra II Module 3, Topic B: Logarithms
URL: https://www.engageny.org/resource/algebra-ii-module-3-topic-b-overview
Description:

At the beginning of Module 3, Topic B, students apply the properties of exponents to solve exponential equations numerically (F-BF.1a) as a way to motivate the need for logarithms, which are first introduced by the more intuitive name “WhatPower." In the intermediate lessons, students discover the logarithmic properties by creating and examining logarithmic tables and answering sets of directed questions. In the final lessons in Topic B, they solve logarithmic equations by applying the inverse relationship between exponents and logarithms (F-LE.4).



   View Standards     Standard(s): [MA2015] AL2 (9-12) 35 :
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.

[MA2015] AL2 (9-12) 29 :
29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

[MA2015] AL2 (9-12) 36 :
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]

[MA2019] AL1-19 (9-12) 17 :
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)
[MA2019] AL1-19 (9-12) 23 :
23. Identify the effect on the graph of replacing f(x) by f(x)+k,k·f(x), f(k·x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.
[MA2019] AL1-19 (9-12) 30 :
30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra II Module 3, Topic C: Exponential and Logarithmic Functions and Their Graphs
URL: https://www.engageny.org/resource/algebra-ii-module-3-topic-c-overview
Description:

In Module 3, Topic C, students graph logarithmic functions, identifying key features (F-IF.4, F-IF.7) and discover how the logarithmic properties are evidenced in the graphs of corresponding logarithmic functions. The inverse relationship between an exponential function and its corresponding logarithmic function is made explicit (F-BF.3).  In the final lesson in Topic C, students synthesize what they know about linear, quadratic, sinusoidal, and exponential functions to determine which function is most appropriate to use to model a variety of real-world scenarios (F-BF.1a).



   View Standards     Standard(s): [MA2015] AL2 (9-12) 35 :
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.

[MA2015] AL2 (9-12) 20 :
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]

[MA2015] AL2 (9-12) 27 :
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]

[MA2015] AL2 (9-12) 31 :
31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

[MA2015] AL2 (9-12) 36 :
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]

[MA2019] AL1-19 (9-12) 16 :
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).
[MA2019] AL1-19 (9-12) 17 :
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)
[MA2019] AL1-19 (9-12) 21 :
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.
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
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
Description:

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).



   View Standards     Standard(s): [MA2015] AL2 (9-12) 14 :
14 ) Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* [A-SSE4]

Example: Calculate mortgage payments.

[MA2015] AL2 (9-12) 31 :
31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

[MA2019] AL1-19 (9-12) 16 :
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).
[MA2019] AL1-19 (9-12) 17 :
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)
[MA2019] AL1-19 (9-12) 21 :
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.
[MA2019] AL1-19 (9-12) 30 :
30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra II Module 3, Topic E: Geometric Series and Finance
URL: https://www.engageny.org/resource/algebra-ii-module-3-topic-e-overview
Description:

Module 3, Topic E is a culminating series of lessons driven by MP.4, Modeling with Mathematics. Students apply what they have learned about mathematical models and exponential growth to financial literacy while developing and practicing the formula for the sum of a finite geometric series (A-SSE.4). Throughout this set of lessons, students study the mathematics behind car loans, credit card payments, savings plans, and mortgages, developing the needed formulas from summing a finite geometric series in each case. Key features of tables and graphs are used to answer questions about finances (F-IF.7e).



   View Standards     Standard(s): [MA2015] AL1 (9-12) 11 :
11 ) (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. [A-APR7]

[MA2015] ALT (9-12) 30 :
30 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b]

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c]

c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e]

[MA2015] ALT (9-12) 33 :
33 ) Write a function that describes a relationship between two quantities.* [F-BF1]

a. Combine standard function types using arithmetic operations. [F-BF1b]

Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

[MA2015] PRE (9-12) 19 :
19 ) (+) Compose functions. [F-BF1c]

Example: If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

[MA2015] PRE (9-12) 18 :
18 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b]

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c]

c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [F-IF7d]

d. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e]

[MA2019] AL1-19 (9-12) 16 :
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).
[MA2019] AL1-19 (9-12) 17 :
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)
[MA2019] AL1-19 (9-12) 21 :
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.
[MA2019] AL1-19 (9-12) 30 :
30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Precalculus and Advanced Topics Module 3, Topic B: Rational Functions and Composition of Functions
URL: https://www.engageny.org/resource/precalculus-and-advanced-topics-module-3-topic-b-overview
Description:

In Module 3, Topic B, students explore the composition of functions in-depth and notice that a composition of a polynomial function with the function f(x) = 1/x gives functions that can be written as ratios of polynomial functions. A study of rational expressions shows that these expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression. Students apply these operations to simplify rational expressions and go on to graphing rational functions, identifying zeros and asymptotes, and analyzing end behavior.

Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to Algebra I, Algebra II, and Algebra II with Trigonometry Alabama Courses of Study.



ALEX Classroom Resources: 6

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