SCHOOLinSITES
Algebra II—Gifted Diane Griffis---PCHS
|What is a Function? |
|Assessment Description/Performance Task |
|The unit should be designed around the performance tasks, which include Exploring Functions, Fences and Functions, From Wonderland to Function land, Walking, Falling, and Making Money, Sequences and Functions, Southern |
|Yard and Garden, Painted Cubes, and Logo Symmetry. |
|Suggested use for the culminating task, Ye Old Village Shoppes, is to assign it at the beginning of the unit for students to complete outside of class (individually or in partners) as you progress through the unit. If it |
|is due the day before the unit test, it serves as a medium for reviewing for the test via class presentations and/or discussion. |
|What is a function? |
|Characteristics of Functions |Logical Equivalence |Average rate of change |Sequences |Transformations |
|Functions |Conditional statements | |Whole number domains |Vertical and horizontal shifts |
|Parent graphs of 6 basic functions |Negation | |Closed and Recursive definitions of |Stretches and shrinks |
|Discrete vs. Continuous functions |Converse | |functions |Reflections |
|Domain and Range |Inverse | |Restricted domains |Even and odd functions |
| |Contrapositive | | | |
|STANDARDS. |MA1G2. Students will understand and use |MA1A1. Students will explore and interpret|MA1A1. Students will explore and |MM1A1. Students will explore and interpret the |
|MA1A1. Students will explore and |the language of mathematical argument and |the characteristics of functions, using |interpret the characteristics of |characteristics of functions, using graphs, |
|interpret the characteristics of |justification. |graphs, tables, and simple algebraic |functions, using graphs, tables, and |tables, and simple algebraic techniques. |
|functions, using graphs, tables, and |a. Use conjecture, inductive reasoning, |techniques. |simple algebraic techniques. |Graph transformations of basic functions |
|simple algebraic techniques. |deductive reasoning, counterexamples, and |g. Explore rates of change, comparing |b. Represent functions using function |including vertical shifts, stretches, and |
|Represent functions using function |indirect proof as appropriate. |constant rates of change (i.e. slope) |notation. |shrinks, as well as reflections across the x- and|
|notation. |b. Understand and use the relationships |versus variable rates of change. Compare |e. Relate to a given context the |y-axes. |
|Graph the basic functions [pic] where n = |among a statement and its converse, |rates of change of linear, quadratic, |characteristics of a function, and use |Investigate and explain the characteristics of a |
|1 to 3, [pic], [pic], and [pic]. |inverse, and contrapositive. |square root, and other function families. |graphs and tables to investigate its |function: domain, range, zeros, intercepts, |
|Investigate and explain the | | |behavior. |intervals of increase and decrease, maximum and |
|characteristics of a function: domain, | | |f. Recognize sequences as functions with |minimum values, and end behavior. |
|range, zeros, intercepts, intervals of | | |domains that are whole numbers. |Determine graphically and algebraically whether a|
|increase and decrease, maximum and minimum| | | |function has symmetry and whether it is even, |
|values, and end behavior. | | | |odd, or neither. |
|e. Relate to a given context the | | | | |
|characteristics of a function, and use | | | | |
|graphs and tables to investigate its | | | | |
|behavior. | | | | |
|Unit Vocabulary |inverse |
|average rate of change |logical equivalence |
|conditional statement |odd function |
|converse |negation |
|contrapositive |range |
|domain |recursive |
|end behavior |shrink |
|even function |stretch |
|function notation |symmetry |
|horizontal shift |transformation |
|intervals of increase |vertical shift |
|intervals of decrease |zeros |
|Unit Outline Title: Characteristics of Functions |
|Name of Lesson: Characteristics of Functions (#1) |
|Suggested Time: 270 minutes |
|Standards: |
|MA1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. |
|Represent functions using function notation. |
|Graph the basic functions [pic] where n = 1 to 3, [pic], [pic], and [pic]. |
|d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. |
|e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. |
|Essential Question(s): |
|Unit: What is a function? |
|Lesson: |
|What are the characteristics of a function? |
|What are the symbols we use to represent a function and its characteristics? |
|Assessment Description/Performance Task: |
|Constructed response Informal assessment |
|Performance task Selected response |
| |
|Brief Description: |
|Ongoing informal/formal assessments created by teacher such as exit cards, quizzes, and warm-ups. |
|Use of performance tasks below provided at Math 1 Framework. |
|Instructional Methods |
|Design your lesson to include a combination of performance tasks and direct instruction that address all relevant standards. The following suggestions and information about each task can assist you in designing|
|your lesson. |
| |
|Exploring Functions with Fiona Learning Task: This unit works well as an exploratory task to introduce the unit. After parts 1 and 2, summarize by asking “What does h(3) = 39 mean?” and “What other |
|representations mean the same thing as h(3) = 39?” After parts 3 and 4, highlight discrete vs. continuous, College Board would say S(2.4) in this context would be impossible, and discuss extrapolation (where it |
|is appropriate and where it is not appropriate). Follow the lesson with instruction/summary over definitions of function, dependent variable and independent variable, and domain and range. |
|Standard(s): MA1A1a |
|Textbook: Mathematics 1 Text – section 1.7 |
|Alternate Resources: Activity Generator CD/Classzone – TI-Navigator: Using Function Notation |
|Introduce the parent graphs of the basic functions (MA1A1b). Plot points to allow students to discover the shapes of these graphs. Develop patterns for these parent graphs. Identify critical points on graphs |
|(vertices, asymptotes, turning points, etc.). Discuss intervals of increase and decrease, maximums and minimums, zeros, and end behavior of graphs. |
|Suggested domain and range worksheet is included (MA1A1d). See section 1.2 in the Mathematics 1 text. |
|Fences and Functions Do not give formulas for area and perimeter. Encourage students to make models of gardens using graph paper or algebra tiles. Emphasize difference between discreet and continuous |
|situations. Emphasize limiting cases. This task will need to be split between classwork and homework. |
|Standard(s): MA1A1a, d, e |
|Textbook: Mathematics 1 Text – sections 1.2, 1.3, 1.4, 2.10 |
|Alternate Resources: |
|Differentiation: |
|Both differentiation and extensions are provided within the tasks. |
|For this Lesson: |
|You will need graph paper and algebra tiles (for modeling possible garden configurations). |
|Vocabulary: |
|notation |
|domain and range function |
|intervals of increase and decrease |
|maximum and minimum |
|end behavior |
|x-intercepts |
|zeros |
|limiting case |
|Unit Outline Title: Characteristics of Functions |
|Name of Lesson: Logical Equivalence (#2) |
|Suggested Time: 180 minutes |
|Standards: |
|MA1G2. Students will understand and use the language of mathematical argument and justification. |
|a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate. |
|b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive. |
|Essential Question(s): |
|Unit: What is a function? |
|Lesson: |
|If I know what I like, do I like what I know? |
|What are conditional statements and related conditionals? |
|Which related conditionals are logically equivalent? |
|Assessment Description/Performance Task: |
|Constructed response Informal assessment |
|Performance task Selected response |
| |
|Brief Description: |
|Ongoing informal/formal assessments created by teacher such as exit cards, quizzes, and warm-ups. |
|Use of performance tasks below provided at Math 1 Framework. |
|Instructional Methods |
|Design your lesson to include a combination of performance tasks and direct instruction that address all relevant standards. The following suggestions and information about each task can assist you in designing|
|your lesson. |
| |
|You may want to begin the lesson with a video clip of the scene from Alice in Wonderland addressed in this task. |
|Introduce the concepts of mathematical statements and truth-values of statements. This may be done using direct instruction or the following task or a combination. The task could be done as exploratory or as |
|teacher-guided instruction. |
|From Wonderland to Functionland Learning Task: This task thoroughly introduces the language of mathematical argument and justification and defines/explains all relevant vocabulary for the standard including |
|related conditionals. The important connection between conditional statements and function notation begins with part 9. |
|Standard(s): MA1G2a, b |
|Textbook: Mathematics 1 Text – sections 4.2 through 4.6 |
|Alternate Resources: @ HomeTutor CD Lesson: Geometry 2.1, 2.2, 2.3, 2.6, 2.7; Activity Generator: Use Inductive Reasoning, Analyze Conditional Statements, Logic Puzzles; Geometry Game: Completing Proofs |
|Differentiation: |
|Both differentiation and extensions are provided within the tasks. |
|For this Lesson: |
|You will need graph paper. |
|Optional: copy of Alice in Wonderland for video clip |
|Vocabulary: |
|biconditional |
|compound statement |
|converse/ inverse/ contrapositive |
|hypothesis / conclusion |
|logical equivalence |
|negation |
|statement / conditional statement |
|truth value |
|Unit Outline Title: Characteristics of Functions |
|Name of Lesson: Average Rates of Change (#3) |
|Suggested Time: 90 minutes |
|Standards: |
|MA1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. |
|g. Explore rates of change, comparing constant rates of change (i.e. slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. |
|Essential Question(s): |
|Unit: What is a function? |
|Lesson: |
|Does a non-linear function have slope? |
|Assessment Description/Performance Task: |
|Constructed response Informal assessment |
|Performance task Selected response |
| |
|Brief Description: |
|Ongoing informal/formal assessments created by teacher such as exit cards, quizzes, and warm-ups. |
|Use of performance tasks below provided at Math 1 Framework. |
|Instructional Methods |
|Design your lesson to include a combination of performance tasks and direct instruction that address all relevant standards. The following suggestions and information about each task can assist you in designing|
|your lesson. |
| |
|Walking, Falling, and Making Money Learning Task: This task introduces the concept of average rate of change as a tool for describing and understanding characteristics of functions. It begins with an |
|exploration of constant speed versus variable speed. It also allows for student role-play. It finishes by using rates in other contexts. |
|Standard(s): MA1A1g |
|Textbook: Mathematics 1 Text – sections 1.5 and 3.12 |
|Alternate Resources: |
|Possible exit card question: “What does the football announcer mean when he says the losing team won the second half? Use ‘average rate of change’ in your answer.” |
|Differentiation: |
|Both differentiation and extensions are provided within the tasks. |
|For this Lesson: |
|You will need graph paper. |
|Optional: measuring tape and masking tape for role-play |
|Vocabulary: |
|average rate of change |
|Unit Outline Title: Characteristics of Functions |
|Name of Lesson: Sequences as Functions (#4) |
|Suggested Time: 270 minutes |
|Standards: |
|MA1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. |
|b. Graph the basic functions [pic] where n = 1 to 3, [pic], [pic], and [pic]. |
|e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. |
|f. Recognize sequences as functions with domains that are whole numbers. |
|Essential Question(s): |
|Unit: What is a function? |
|Lesson: |
|What kinds of functions use whole numbers? |
|Assessment Description/Performance Task: |
|Constructed response Informal assessment |
|Performance task Selected response |
| |
|Brief Description: |
|Ongoing informal/formal assessments created by teacher such as exit cards, quizzes, and warm-ups. |
|Use of performance tasks below provided at Math 1 Framework. |
|Instructional Methods |
|For this lesson, you will need to be selective as there is not enough time to complete all of the tasks. Keep in mind that all standards need to be addressed! Design your lesson to include a combination of |
|performance tasks and direct instruction that address all relevant standards. The following suggestions and information about each task can assist you in designing your lesson. |
| |
|Sequences as Functions Learning Task: Previously, students have seen functions as modeling problem situations. In this task, students will study sequences, which is a more general representation of functions. |
|Parts of this task are better used as a guide for teacher-driven instruction rather than an exploratory lesson. Discuss the domain, emphasizing the use of whole numbers as the domain. |
|Standard(s): MA1A1f |
|Textbook: Mathematics 1 text – section 3.13 |
|Alternate Resources: |
|Southern Yard and Garden Learning Task: In this task, students will investigate functions whose formulas involve the algebraic expressions [pic] and [pic]. Continue discussion of Unit 1 vocabulary (domain, |
|range, continuous, etc.) as you complete this task. Transformations will be discussed more thoroughly in the next lesson. |
|Standard(s): MA1A1b, f |
|Textbook: |
|Alternate Resources: |
|Painted Cubes Learning Task: This task provides a hands-on opportunity for students to develop patterns (i.e. sequences). It also introduces and explores[pic]. |
|Standard(s): MA1A1b, e (MA1A1c is forshadowed) |
|Textbook: Mathematics 1 Text – sections 1.7, 2.10, 3.1. Also Investigating Math Activity 1.7 and 2.10 |
|Alternate Resources: @ HomeTutor Lesson: Alg 1 10.2 Activity Generator: |
|Differentiation: |
|Both differentiation and extensions are provided within the tasks. |
|For this Lesson: |
|May want to use Cuisenaire rods or other types of tiles. |
|Vocabulary: |
|domain |
|range |
|recursive |
|Unit Outline Title: Characteristics of Functions |
|Name of Lesson: Transformations & Symmetry (#5) |
|Suggested Time: 360 minutes |
|Standards: |
|MA1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. |
|Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. |
|Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. |
|Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. |
|Essential Question(s): |
|Unit: What do solutions of equations represent? |
|Lesson: |
|What does the equation tell me about the characteristics of the graph? |
|Assessment Description/Performance Task: |
|Constructed response Informal assessment |
|Performance task Selected response |
| |
|Brief Description: |
|Ongoing informal/formal assessments created by teacher such as exit cards, quizzes, and warm-ups. |
|Use of performance tasks below provided at Math 1 Framework. |
|Instructional Methods |
|Design your lesson to include a combination of performance tasks and direct instruction that addresses all relevant standards. The following suggestions and information about each task can assist you in |
|designing your lesson. |
| |
|Provide direct instruction for various transformations, including reflections and provide practice. (MA1A1c) |
|An exit card is included to use as a formative assessment. |
|Logo Symmetry Learning Task: This task investigates many required standards and provides a thorough exploration of symmetry, transformation, and domain restriction by analyzing many familiar logos. Before |
|starting part 2, students may require brief instruction on the result of restricting the variables of a function. You may want to thoroughly preview this task before beginning. Some parts can be used as |
|discovery-based, some parts can be used as teacher guided instruction, some parts can be used as independent work for classwork or homework, and some parts might be used for extension. |
|Standard(s): MA1A1c, d, h |
|Textbook: Mathematics 1 test – section 1.7; Investigating Math Activity 1.7, 1.9, 2.10, 2.11, 3.1, 3.3, 3.6 |
|Alternate Resources: @HomeTutor: Algebra 1 (4.7, 10.1, 10.2, 11.1); Activity Generator: Exploring Graphs of Absolute Value Functions, Graphing a Quadratic Function, Investigating Graphs of Quadratic Functions, |
|Exploring Pendulum Motion, Graphing y=a/(x-h)+k |
|Suggested use for the culminating task: Assign Ye Olde Shoppe at the beginning of the unit for students to complete outside of class (individually or in partners) as you progress through the unit. If it is due |
|the day before the unit test, it serves as a medium for reviewing for the test via class presentations and/or discussion. |
|Differentiation: |
|Differentiation is provided within each task. |
|Extensions are included within tasks. |
|For this Lesson: |
|Avoid using linear equations as example for transformations unless using point-slope form of the equation. |
|You will use graphing calculators for many of the tasks. You may also find the transformation application on the TI-84 helpful. |
|Vocabulary: |range |
|function |transformation |
|quadratic function |vertical shift |
|even function |horizontal shift |
|odd function |shrink |
|symmetry |stretch |
|domain | |
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