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Learning Activities (4) Building blocks of a lesson plan that include before, during, and after strategies to actively engage students in learning a concept or skill. Classroom Resources (8)


ALEX Learning Activities  
   View Standards     Standard(s): [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)
Title: Radical Graphs Matter Sequel
Description:

Students will graph functions on a piece of large graph paper as a group. The teacher will display the finished product in the classroom or hallway.

This activity results from the ALEX Resource Development Summit.




   View Standards     Standard(s): [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)
Title: Radical Graphs Matter Prequel
Description:

Students can graph the parent functions. They will use paper and pencil to graph the basic parent functions and list the key features of each function.  

This activity results from the ALEX Resource Development Summit.




   View Standards     Standard(s): [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)
Title: Radical Graphs Matter
Description:

Students will graph radical functions in the coordinate plane and identify key features. Students will be sure to label the axes and scale of their graphs. Students will identify the domain and the range of each function from the graph of the function.  

This activity results from the ALEX Resource Development Summit.




   View Standards     Standard(s): [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)
Title: The Keys to Trig Graphs
Description:

The trigonometric graphing applet will allow students to investigate the changes in the period and amplitude on the graph of a trigonometric function.




ALEX Learning Activities: 4

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ALEX Classroom Resources  
   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) 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.



   View Standards     Standard(s): [MA2015] AL1 (9-12) 16 :
16 ) Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1]

[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 I Module 3, Topic C: Transformations of Functions
URL: https://www.engageny.org/resource/algebra-i-module-3-topic-c-overview
Description:

In Module 3, Topic C, students extend their understanding of piecewise functions and their graphs including the absolute value and step functions. They learn a graphical approach to circumventing complex algebraic solutions to equations in one variable, seeing them as f(x) = g(x) and recognizing that the intersection of the graphs of f(x) and g(x) are solutions to the original equation (A-REI.D.11). Students use the absolute value function and other piecewise functions to investigate transformations of functions and draw formal conclusions about the effects of a transformation on the function’s graph (F-IF.C.7, F-BF.B.3).



   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 12 :
12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.
[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) 29 :
29. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Limit to linear, quadratic, exponential, and absolute value 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)
Title: Algebra I Module 4, Topic C: Function Transformations and Modeling
URL: https://www.engageny.org/resource/algebra-i-module-4-topic-c-overview
Description:

In Module 4, Topic C, students explore the families of functions that are related to the parent functions, specifically for quadratic (f(x) = x2), square root (f(x) = the square root of x), and cube root (f(x) = cube root of x), to perform horizontal and vertical translations as well as shrinking and stretching (F-IF.C.7b, F-BF.B.3). They recognize the application of transformations in vertex form for a quadratic function and use it to expand their ability to efficiently sketch graphs of square and cube root functions. Students compare quadratic, square root, or cube root functions in context and represent each in different ways (verbally with a description, as a table of values, algebraically, or graphically). In the final two lessons, students examine real-world problems of quadratic relationships presented as a data set, a graph, a written relationship, or an equation. They choose the most useful form for writing the function and apply the techniques learned throughout the module to analyze and solve a given problem (A-CED.A.2), including calculating and interpreting the rate of change for the function over an interval (F-IF.B.6).



   View Standards     Standard(s): [MA2015] AL2 (9-12) 13 :
13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

[MA2015] AL2 (9-12) 16 :
16 ) Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). [A-APR2]

[MA2015] AL2 (9-12) 17 :
17 ) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. [A-APR3]

[MA2015] AL2 (9-12) 19 :
19 ) Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or for the more complicated examples, a computer algebra system. [A-APR6]

[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 1, Topic B: Factoring--Its Use and Its Obstacles
URL: https://www.engageny.org/resource/algebra-ii-module-1-topic-b-overview
Description:

Module 1, Topic B focuses on factoring polynomials and the advantages of factored form of a polynomial to both solve equations and sketch graphs of polynomial functions. Students solve problems involving real-world situations and develop fluency with creating equations and functions given a verbal description, visual representation, or graph. This topic concludes with a discussion of polynomial division with remainder, further strengthening the connection between the remainder, the factors and zeros of a polynomial equation, and graphs of polynomial functions.  



   View Standards     Standard(s): [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)
Title: Algebra II Module 2, Topic A: The Story of Trigonometry and Its Context
URL: https://www.engageny.org/resource/algebra-ii-module-2-topic-a-overview
Description:

In Module 2, Topic A, students develop an understanding of the six basic trigonometric functions as functions of the amount of rotation of a point on the unit circle (F-TF.A.1), and then translate that understanding to the trigonometric functions as functions on the real number line (F-TF.A.2). Students study graphs of the functions to discover properties of the trigonometric functions (F-IF.C.7e).



   View Standards     Standard(s): [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)
Title: Algebra II Module 2, Topic B: Understanding Trigonometric Functions and Putting Them to Use
URL: https://www.engageny.org/resource/algebra-ii-module-2-topic-b-overview
Description:

Students will use trigonometric functions to model periodic phenomena (F-TF.B.5) by fitting sinusoidal functions to data (S-ID.B.6a). Students use the properties of the graphs of sinusoidal functions to help fit functions to data to solve problems in the context of the data (F-IF.C.7e).  We end the module with the study of trigonometric identities and how to prove them (F-TF.C.8). 



ALEX Classroom Resources: 8

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