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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 1 Cognitive Information Processing Function Families Martha Townsend Georgia Southern University

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Page 1: Technology Based Module CIP Using Mobile Apps

FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 1

Cognitive Information Processing

Function Families

Martha Townsend

Georgia Southern University

Page 2: Technology Based Module CIP Using Mobile Apps

FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 2

Cognitive Information ProcessingFunction Families

Instructional Goals

CIP Goals(Driscoll, 2005,

p. 71-110)

Methods of InstructionTechnology-Based Student Activities

Gain and keep the attention of students.

The students will start each day of class by watching an entertaining SlideRocket Presentation on function families via their iPads or iPhones. This presentation includes a meaningful story with color, audio, and animation.

The teacher will keep students focused on learning using Teacher's Assistant Pro which allows teachers to keep track of student actions, behaviors, and achievements in the classroom. The teacher will view each student’s computer screen on the teacher computer and monitor their activities. The teacher will interact with students through the management software to keep them engaged and on-task.

The students will use applets to explore and experiment with functions. These provide interactive entertainment.

Incorporate relevant prior learning.

The students will complete a warm-up activity each day. The questions on the warm-up will cover the previous lesson on the characteristics of function.

The teacher will analyze student responses and use ClickerSchool Virtual Clicker to communicate feedback to the students.

Highlight and stress important information.

As students watch the presentation they will fill in a note-taking guide with a section labeled “Cues”. In this box they should type in key ideas.

The teacher will use the teacher computer and overhead projector to display key words and important facts.

Present lesson in an organized manner.

The lesson, which is presented in story format, will follow a logical sequence as it introduces new concepts and skills.

Once students have mastered the characteristics of manipulating the example graph, they will recognize the parent graphs and slowly begin to apply those same properties.

Students will take each of the parent graphs and shift them up or down in accordance to their equation. Next students will advance to the more complex concept of reflecting a function over the x-axis. Finally, the students will combine the steps for vertically manipulating a function.

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 3

Instructional Goals (continued)

Show students how to chunk related information.

The graphic organizers and note-taking guides will allow students to group the concepts into categories with manageable steps.

The activities and lesson will teach students inductive reasoning as they construct or evaluate general propositions about functions. These are derived from specific examples.

Provide activities for students which allow them to elaborate on new information.

The students will perform activities that will allow them to connect the graphing of more complex functions to what they already know about the behaviors of basic functions.

The students will perform activities that will allow them to connect the graphing of functions to each other.

The students will look for similarities and differences between the functions, their algebraic equations, and their graphs.

Show students how to use coding when memorizing concepts and facts.

The students will learn what the graph of each function looks like by relating it to an animal that is similar in appearance.

The students will learn the names of the functions by giving the corresponding animals a name that starts with the same letter.

The students will learn how to algebraically manipulate the graph of a function by relating the aspects of their equations to the behaviors of the animal.

Provide students the opportunity for repetition of learning.

The students will receive the information and practice on applying the behaviors of functions several times in various ways.

The students will collaborate with peers and ask questions or help explain the material.

The students will complete homework activities that reinforce the material that was learned that day.

Provide activities for automaticity of fundamental concepts and skills.

The students will continue to review graphing skills and the basic properties of functions through practice worksheets and the test review.

The students will use ClickerSchool Virtual Clicker submit answer to warm-up activities on the basic concepts involved with functions and receive immediate feedback.

The students will access applets on the Internet that will allow them to practice manipulating the graphs of functions in the coordinate plane.

Prior Relevant Learning Focus

Essential Questions Key Definitions

What are the characteristics of a function and how can you implement and apply these characteristics to arrange and illustrate the function in multiple ways?

• Coordinate plane• Relation• Ordered pair• Independent variable• Dependent variable• x-coordinate• y-coordinate

• x|y chart• Abscissa• Ordinate• Input• Output• Table• Function

• Graph• Domain• Range• Vertical

line test

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 4

Learning Focus for Function Family Lesson

Essential QuestionsKey Definitions

(Should be stressed and highlighted)

What are the properties of the parent graphs of linear, quadratic, cubic, absolute value, and square root functions and how can you implement and apply these properties to identify, compare, and illustrate the functions?

• Maximum• Minimum• Absolute Value• Parent Graph• Zero• Linear

• Quadratic• Parabola• Radical• Infinity• Cubic

How can you recognize if the parent graphs and equations of linear, quadratic, cubic, absolute value, and square root functions have been vertically affected?How can you distinguish among, implement, and demonstrate vertical transformations of linear, quadratic, cubic, absolute value, and square root functions?

• Coefficient• Reflection• Solution• Increasing• Shrink• Decreasing

• Average rate of change

• Slope• End behavior• Stretch• Shift

Prior Relevant Learning Objectives

Students should

have already:

At the beginning of this lessonstudents should be able to:

Comprehended and interpreted the characteristics of relations and functions.

• Develop and illustrate a relation in multiple ways without the use of Quick Graph but by using tables, x|y charts, listing of ordered pairs, or graphs in the coordinate plane.

• Identify and arrange the domain and range of a relation by looking at a table, x/y chart, a listing of ordered pairs, or graph.

• Deduce and classify a relation as being a function by looking at a table, x/y chart, or a listing of ordered pairs.

• Perform and apply the vertical line test to determine if a graph is a function.

• Calculate to develop x|y charts of functions from their equations and illustrate these functions in the coordinate plane.

• Demonstrate graphs of functions using Quick Graph.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 5

Learning ObjectivesRelated to Cognitive Information Process

Students completingthis lesson will:

Students that successfully completethis lesson will be able to:

Understand, recognize, evaluate, and analyze the properties of the parent graph equations of linear, quadratic, cubic, absolute value, and square root functions.

• Identify, define, describe, and label the parent graph equations and organize, classify, and summarize this information.

• Determine if an equation is one of the parent graph equations.

• Differentiate between and compare the equations of parent graph functions.

• Develop and produce the parent graph equations from memory.

• Recall and recognize the characteristic of the equations of parent graphs.

Understand, recognize, evaluate, and analyze the characteristics of the parent graphs of linear, quadratic, cubic, absolute value, and square root functions.

• Identify, define, describe, and label the graphs of functions that are linear, quadratic, cubic, absolute value or square root and organize, classify, and summarize this information.

• Determine if a graph is one of the parent graphs.• Differentiate between and compare the parent graphs.• Develop and produce illustrations of the parent graphs

from memory.• Recall and recognize the characteristics of parent graphs.

Synthesize, interpret, evaluate, and apply the vertical transformations of linear, quadratic, cubic, absolute value, and square root function equations.

• Identify, define, describe, and label the vertical shift, vertical stretch, vertical shrink, and vertical reflection of linear, quadratic, cubic, absolute value, and square root function equations and organize, classify, and summarize this information.

• Determine if an equation is a parent graph equation that has been vertically transformed.

• Recall and recognize the vertical transformations that can be applied to parent graph equations.

• Differentiate between and compare the different vertical transformations of parent graph equations.

• Compare and contrast parent graph equations to equations that have been vertically transformed.

• Develop and produce equations of linear, quadratic, cubic, absolute value, and square root functions by combining, implementing, and executing the rules of transforming graphs.

• Explain the different transformations that can be applied to a parent graph equation.

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 6

Learning objectives (continued)

Synthesize, interpret, evaluate, and apply the vertical transformations of the graphs of linear, quadratic, cubic, absolute value, and square root functions.

• Identify, define, describe, and label the vertical shift, vertical stretch, vertical shrink, and vertical reflection of linear, quadratic, cubic, absolute value, and square root graphs and organize, classify, and summarize this information.

• Determine if a graph is simply a parent graph that has been vertically transformed.

• Recall and recognize the vertical transformations that can be applied to parent graphs.

• Differentiate between and compare different graphs of functions that have been vertical transformed.

• Compare and contrast parent graphs to graphs that have been vertically transformed.

• Develop and produce illustrations of the graphs of linear, quadratic, cubic, absolute value, and square root functions by combining, implementing, and executing the rules of transforming graphs.

• Explain the different transformations that can be applied to a parent graph equation.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 7

Standards(Common Core Georgia Performance Standards)

(CCGPS)

Students will analyze functions using different representations.

https://www.georgiastandards.org/Common-Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_Accel-GeometryB-AdvAlgebra_Standards.pdf

MCC9‐12.F.IF.7Characteristics of functions

Students will graph functions expressed symbolically and show key features of the graph. They will do this by hand in simple cases and using technology for more complicated cases.

MCC9‐12.F.IF.7a MCC9‐12.F.IF.7b MCC9‐12.F.IF.7c

Students will graph linear and quadratic functions.

Students will graph square root and absolute value functions.

Students will graph cubic functions.

Lesson Management

Day Lesson Activities Homework

1-3

• Warm-up with immediate feedback from ClickerSchool Virtual Clicker

• Transition of Buzz Presentation• Graphic organizer• Note-taking guide• Practice worksheet• Applet activity• Post on discussion board

• Review lesson presentation• Review notes• Homework practice sheet

4

• Review activity with immediate feedback from ClickerSchool Virtual Clicker

• Applet activity for review• Post on discussion board• Extra activity if time

• Review all presentations• Review all notes• Review all graphic organizers

5• Final assessment with immediate

feedback from ClickerSchool Virtual Clicker

Audience

The audience for this lesson is an Accelerated CCGPS Analytic Geometry B / Advanced Algebra Mathematics class. There are 24 students, 16 female and 8 males. This class is taught on a block schedule and meets 1.5 hours every day. The class has been together since the beginning of the second semester of the school year (approximately 14 weeks), and they are comfortable working collaboratively. They are already familiar with Google Groups, Google Drive, and Quick Graph. They have a Google account. They have already set up a folder that is shared with the teacher which is used for submitting their completed assignments.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 8

Learning Environment

Student Activities UsingTechnology-Based Instruction

(Samples of some of the activities are attached.)

• Warm-ups• Note-taking guides• Graphic organizers• Student practice activities• Additional-time activity• Manipulative activity• Online discussions• Multiple-choice assessment

Function Families Lesson Setting Offers:

(Samples of some of the presentation slides are attached.)

• Student access to individual computer• Common starting point for all students• Incorporation of prior learning• Technology-based instruction• All material accessible through Google Drive• All material turned in electronically• Audio and video for delivery of content• Differentiated sources (if students do not an iPad or an

iPhone for their own personal use, the school will issue them an iPad for classroom use.)o ClickerSchool Virtual Clicker activitieso Teacher's Assistant Proo Google Groupso Quick Grapho Google Driveo Internet researcho Applets

• Repetition of learning through a variety of practice• The use of imagery to retain new information• Communication with peers• Additional activity for those that finish early• Appropriate alignment of learning activities with objectives

and assessment

Assessments

Students will be assessed on their attention to the lesson, their ability to complete their note-taking guides, charts, and graphic organizers, practice worksheets, and their collaboration with their peers and teacher (see rubric). They will receive separate grades on their warm-ups and homework assignments based on correctness. Students will take a multiple choice assessment and submit their answers using Google Forms and ClickerSchool Virtual Clicker. These final assessments cover all objectives on the lesson.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 9

Justification for a Technology-Based ModuleStudents will have a complete online experience

Technology Class Set-up and Activities

Google Drive:Shared Documents

• The teacher will share a Google document with the class that will have the instructions and requirements for completing the lesson. It will contain all web-links used in the lesson.

Google Drive: Shared Folders

• The teacher will share a lesson folder with the class. It will include the following documents: objectives, rubrics, charts, graphic organizers, practice worksheets, and note-taking guides. These will be read-only documents, but students will make copies of them so that they can edit their own version.

• The students should already have a “turn-in folder” set up that is shared with the teacher. To turn in an assignment, the student will add the completed material to the turn-in folder and it will be available to the teacher.

Google Groups

• Students will respond to others ideas.• Students will provide new resources and ideas related to functions.• Students will elaborate on the material.• Students will offer help to other students.• Students will post responses to teacher questions.

Google Drive:Google Forms

• Students will submit answers to multiple choice assessments

SlideRocket Presentation Software

• Students will observe a lesson presentation

Internet • Students will perform research required for an “additional time” activityOn-line Mathematical Applets

• Students will explore characteristics of functions.o Multi-Function Data Flyer:

http://www.shodor.org/interactivate/activities/o Slider Graph:o http://math.hws.edu/javamath/basic_applets/SliderGraph.html

ClickerSchool Virtual Clicker

• Teachers and students will use this interactive response system app using their iPads or iPhones to submit responses to warm-ups and assessments.

Teacher's Assistant Pro

• The teacher will keep track of student actions, behaviors, and achievements in the classroom using this iPhone or iPad app.

Quick Graph: Your Scientific Graphing Calculator

• The student will use this app to graph functions. These results will then be emailed to the teacher.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 10

Assessment of Lesson RubricJustification of CIP

CIPActivity

Limited1

Developing2

Proficient3

Exemplary4

CLASSROOM ATTENTION

Focuses attention on all aspects of the lesson.

Focusing attention allows for the processing of important information while ignoring impertinent information.

Cannot focus attention

♦ Does not evaluate material when promptedby teacher♦ Cannot maintain a focus of attention despite cognitive efforts which may cause some level of anxiety

Requires redirection

♦ Evaluates material only when promptedby teacher♦ Requires considerable degree of cognitive effort to focus attention

Requires minimal redirection

♦ Recognizes teachercues toevaluate material♦ Requires little cognitive demand for successful completion

Maintains strong focus and attention on all components of the lesson

♦ Systematicallyevaluates materialwithout needing the cues from the teacher♦ Demands self-monitoring or self-regulation of own level of attention

PRACTICE WORKSHEETS

Makes connections among mathematical ideas and concepts and procedures.

When students can make connections among multiple representations they can develop meaning and solve problems.

♦ Has no connection of the concepts or meaning to the facts, rules, formulas, or definitions being learned♦ Partially identifiessimilarities anddifferences between concepts but only by saying they are opposites♦ Cannot solve problems using procedures

♦ Has no connection of the concepts or meaning to the procedure being used ♦ Identifies somesimilarities anddifferences betweenconcepts♦ Struggles with general procedures and solving problems

♦ Has some ability to make connections between concepts or meaning to the procedure being used♦ Identifiesspecific similarities anddifferences betweenconceptsand makesgeneralizations♦ Follows general procedures but requires a lot of thought

♦ Experimentswith new information and proceduresand uses them indifferent ways♦ Identifiesspecific similarities anddifferences betweenconcepts and can makegeneralizationsandidentify behaviors♦ Follows general procedures and can do so with ease

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 11

Functions Families Lesson Rubric (continued)

PRACTICE WORKSHEETS

Synthesizes or brings together information.

When students can retrieve learned concepts, rules, formulas, processes, or definitions it helps them to gain meaning and understanding of new information for final interpretation.

Cannot synthesize information

♦ Misunderstands the material

Synthesizes a majority of the information incorrectly

♦ Focuses on generating correct responses rather than establishing mathematical understanding

Synthesizes the information somewhat precisely

♦ Understands the material, though there may be a few minor errors

Provides a highly imaginative and individualized synthesis of the information

♦ Understands all concepts in the material

GRAPHIC ORGANIZERSANDCHARTS

Organizes and charts so that data and ideas are correctly categorized and represented

When students represent information in multiple ways it is easier for them to encode and store information.

Cannot organize information

♦ Createsgraphic organizers and charts that are not complete and do not correctly represent the material♦ Displays no sense of organization and categorization

Requires assistance to select an appropriate organizational strategy

♦ Requires assistance to create graphic organizers and charts thataccurately or clearly represent the material♦ Attempts to show organization and categorization but struggles

Completes organizers with minor problems with organization

♦ Creates organizers and charts thataccurately representthe concepts andunderstanding of the topic♦ Displays a relatively good sense of organization and categorization

Independently selects an appropriate organizational strategy

♦ Creates organizers and charts thataccurately depictthe concepts andunderstanding of the topic but can also do it indifferent ways♦ Creates highly organized and effective organizers and charts

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 12

Function Families Lesson Rubric (continued)

GRAPHIC ORGANIZERSANDCHARTS

Effectively interprets

When students analyze the usefulness of information and reflect to develop personal meaning and knowledge.

♦ Grossly misinterprets the information♦ Makes little or no attempt to determine what information is necessary to complete given task

♦ Makes some significant errors in interpreting the information♦ Cannot determine effectively what information is needed to complete a given task

♦ Understands and gives interpretation of most of the information♦ Makes some errors in determining what information is needed for a given task

♦ Interprets all of the information in accurate and highly intuitive ways♦ Accurately determines what information is reliable and relevant to complete a given task

NOTE-TAKING GUIDES: SUMMARY

Summarizes information

When students process new information it is easier for concepts and ideas to be stored into their memory.

♦ Retells findings♦ Does not put the material in own words♦ Does not have key ideas nor accurate phrases♦ Does not setup the material with original examples

♦ Summarizesinformation♦ Puts some of the material in own words♦ Has few key ideas and accurate phrases♦ Sets up material but without examples

♦ CreatesRelationships;drawsconclusions♦ Rephrases most all of the material♦ Has most key ideas and accurate phrases♦ Sets up material and has some original examples

♦ Applieslearning tonew, different situations♦ Condenses and rephrases all of the material♦ Has all key ideas and accurate phrases♦ Sets up material with original examples

NOTE-TAKING GUIDES: CUES

Identifies and labels key ideas, asks questions that connect these ideas.

When students focus on key ideas they determine the importance of new information.

♦ Does not accurately seek out needed information♦ Fails to use the most critical questioning techniques connecting information

♦ Makes little or to assess beneficial information♦ Fails to ask significant questions for connecting information

♦ Accurately assesses the lesson to identify beneficial information♦ Uses appropriate questioning techniques to connect information

♦ Insightfully determines beneficial information and effectively seeks it out♦ Effectively uses a variety of questioning techniques to connect information resources

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 13

(continued)

Function Families Lesson Rubric (continued)

NOTE-TAKING GUIDES: NOTES

Labels and categorizes notes

When students adapt and reorganize information it is easier for them to process new facts.

Sorts data

♦ Lists information inaccurately, includes unimportant information, and leaves out important details

Orders and ranks data

♦ Processes the information incompletely or includes many errors

Classifies information

♦ Processes the information completely with few errors

Classifiesinformation andmakes comparisons

♦ Accurately and insightfully processes information

POSTS

Communicates

When students collaborate with other students and teachers, it creates situations where they can repeat, elaborate, explain, and self-question.

Ignores any form of communication from teacher and other students

♦ Does not offer any feedback♦ Does not acknowledge ideas and comments♦ Explainsinformation related to the concepts but only with assistance

Responds briefly to teacher and other students comments and ideas

♦ Pays little attention to the details of others ideas and comments♦ Offers very little feedback with no discussion♦ Explainstheinformation from the concepts but it may notbe clear or relatedto the procedures

Explains and gives reasons for agreement and/ordisagreementof comments and ideas of teacher and other students

♦ Critiques or supports ideas and comments♦ Asks simple questions♦ Explains theinformation related to the conceptsin a clear waythat iseasily understood

Actively seeks out ideas and comments of teacher and other students

♦ Asks questions to clarify ideas and comments and suggests possibilities♦ Suggests support of ideas and comments ♦ Explainsthe informationrelated to the conceptsand drawsconclusions from itwhich areunderstandable toanother

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 15

Justification of Cognitive Information Processing(Processing, storage, and retrieval of knowledge in the mind)

(Driscoll, 2005, p. 71-110)

CIPProcess

Technology-BasedTeaching Strategies

Student Strategies

Selective attention

Actively focusing on certain important stimuli while excluding others

♦ Signaling students: The teacher sends messages through ClickerSchool Virtual Clicker classroom management software alerting students that are off-task.♦ Moving around: The teacher periodically walks around the computer lab.♦ Utilizing a variety of materials: The lesson requires students to access several different types of technology:

• Google Groups• Quick Graph• Computers• Social networking• Google Drive• Internet research• Applets

♦ Capturing student interest: The presentation contains animation and audio which is captivating to the students. The animals that represent the functions are meaningful, colorful, and attractive.♦ Highlighting stimulus features: The note-taking guide has students define and highlight the key words and definitions.♦ Giving material meaning: The lesson presentation makes the material meaningful to the students by relating the functions to animals.♦ Keeping the classroom in order: The lesson mandates that students work independently on the computer. The collaboration with peers is totally online via Google Groups. This setting helps to keep the classroom free of noise and distractions.♦ Encouraging the reduction of student impulsiveness: Teachers reassure students that they have the time necessary to complete the lesson because it is broken down into five days.♦ Assigning learning objectives: The lesson provides teachers with a new set of learning objectives that they assign to the material and then share with students.

♦ Adhering to cues from the teacher which alerts attention to lesson activities♦ Recognizing the features of the presentation that signal important information♦ Recognizing the complexity of new information and devoting a greater amount of attention to learning it

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 16

Justification of Cognitive Information Processing (continued)

Automaticity

Devoting little attention to a task because the basic skills that are involved in completing it are habitual

♦ Questioning students about material: The teacher asks students individual questions on the basics of functions through ClickerSchool Virtual Clicker and Google Groups.♦ Providing students with practice of basic skills: Teachers assign warm-up practice exercises on the basic concepts related to the lesson.♦ Encouraging students to practice extensively: Teachers encourage students to complete worksheets and provide homework that continues this practice.

♦ Overlearning the material by constantly studying♦ Continuing to practice even though the problems are correct♦ Reworking problems presented in the lesson♦ Completing homework practice assignments

Pattern recognition and perception

Recognizing familiar patterns as examples of concepts and principles already stored in memory

♦ Providing opportunities for linking information: The lesson requires students to compare and match behaviors and characteristics of the new functions to the corresponding behaviors and characteristics of the simple functions that were previously stored in memory.♦ Requiring that students generate unique examples and non-examples: This lesson supports feature analysis meaning that it shows an example of a function then requires the students to generate a counter-example. Students use their note-taking guides to develop original and alternate examples. They use a graphic organizer to create and fill-in their own non-examples.This activity helps students understand what is meaningful and critical when analyzing a function.♦ Offering extensive practice: The lesson offers an extensive amount of practice on the graphing of functions using a variety of techniques. This ensures that students overcome any past experiences or prior learning that might interfere with their ability to process this new knowledge.♦ Overcoming predetermined perceptions: Teachers monitor students’ achievements and give immediate feedback ClickerSchool Virtual Clicker so that they do not judge students based on preconceived notions.

♦ Focusing on the material in the review warm-ups and linking it to information about the new functions♦ Paying close attention to the examples in the presentation in an effort to create original counter-examples♦ Taking advantage of the extensive practice so that any past learning experiences will not interfere with linking previously learned material to new information

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 17

Justification of Cognitive Information Processing (continued)

♦ Offering opportunities for guided discovery: The lesson uses charts and graphic organizers to help students identify the relevant and irrelevant characteristics of parent functions and their related vertical transformations.♦ Presenting situations to analyze conditional situations: The lesson presents students with related sets of concepts and procedures dealing with functions and their transformations. It then provides them with worksheets where they analyze graphs to determine if certain transformations have been applied to them (if/then).

Chunking

1. Increasing working memory capacity through creating larger bits or taking small bits of information and chunking them together

♦ Breaking down the material: The lesson is broken down into understandable and manageable parts per day to counteract the limitations of short-term memory. Students learn the characteristics of functions bit by bit over the course of five days.♦ Organizing learning tasks: This lesson organizes the learning tasks according to complexity. First of all students recognize important information and respond to this information by filling in a note-taking guide. Next they follow procedures and apply definitions as they complete their graphic organizers and charts. Finally they form concepts and apply rules as they solve the problems on their practice worksheets.♦ Promoting inductive reasoning: The lesson provides students with examples of parent graphs and the charts have students derive and discuss their general characteristics (inductive reasoning).

♦ Following the guidelines of the note-taking guide and filling it in appropriately♦ Taking the complex information from the lesson presentation and breaking it down into manageable parts♦ Completing the tasks of the lessons in the appropriate order: note-taking, charts and organizers, then practice worksheets

(continued)

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Justification of Cognitive Information Processing (continued)

Rehearsal

Repeating information in an effort to maintain it so it can be applied

♦ Providing students with charts and graphic organizers: This lesson has a variety of charts and graphic organizers that categorizes the same information about functions in several different ways.♦ Providing opportunities for student practice: The teacher arranges for a variety of practice opportunities. The goal is to help the learner generalize the concepts, principles, and skills related to function families so that they can be applied to different problem situations.♦ Prompting discussions: The lesson provides for discussion posts where students have the opportunity to explain or to help clarify the problems that others are having. This interaction creates situations for students to repeat learned information which will helps them to maintain it.♦ Asking students redundant questions: Because of the use of Teacher's Assistant Proclassroom management software, teachers have the domain to periodically ask students questions throughout the course of the lesson. This allows students to reiterate the material and express their knowledge.

♦ Filling in several charts and graphic organizers to continually work with the information♦ Completing a variety of practice problems after each lesson presentation♦ Completing homework practice assignments♦ Completing a review assignment♦ Helping others by asking and answering questions♦ Answering teacher- asked questions about lesson content

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 19

Encoding

Relating incoming information to concepts and ideas already in memory so that the material is more memorable

♦ Organizing instruction: The teacher provides students with graphic organizers that help them group related information into categories and lists. Using these aids, students name and define the function, reference it to a category, and define its specific attributes.♦ Mediating: The lesson uses the story about Buzz and the way he behaves to teach vertical transformations of the graphs of functions and how they relate to specific equations. This way of presenting the material makes the steps and procedures involved in transforming graphs more meaningful and easier to remember.♦ Incorporating mnemonics: The lesson relates the names of the functions to the names of the animals representing the functions by starting the names with the same letter.♦ Incorporating imagery: The lesson provides students with the means to remember what the graph of each function looks like by connecting it to an animal that is similar in appearance. Students also use imagery tricks to help remember how each animal behaves under certain circumstances and relates this back to its corresponding function.♦ Providing mental scaffolding: The lesson uses charts and graphic organizers to provide students the opportunity to link new material about function families with relevant prior knowledge about basic functions♦ Providing a frame of reference: The lesson acknowledges parent graphs as being a frame of reference. Characteristics and behaviors of these graphs are learned first. Then students apply complex techniques, including procedures and visualizations, to these graphs to create the more complex structures.

♦ Using graphic organizers to help work practice problems♦ Encoding information in more than one way by using mnemonic devices, acronyms, and imagery♦ Elaborating on new information by providing unique examples of each type of graph on note-taking guides♦ Self-questioning♦ Making information meaningful by relating it to personal life (This activity is prompted by discussions on the discussion board.)♦ Using note-taking guides to take notes using unique words and phrasing♦ Using note-taking guides to summarize information♦ Imposing subjective organization♦ Posting inferential questions on discussion board♦ Including prompts in notes while filling in the note-taking guide

Justification of Cognitive Information Processing (continued)

(continued)

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Justification of Cognitive Information Processing (continued)

Retrieval

Bringing back previously learned material

♦ Offering opportunities of recall: This lesson makes recall easier by offering student a multitude of practice exercises that present questions in a variety of ways.

• Cued recall: The students complete practice exercises with the aid of graphic organizers which provide cues that help students remember the desired information.

♦ Offering opportunities for recognition:• Yes/no type of recognition: On sections that

require students to interpret the vertical reflection of a graph, students must determine if graph is in the normal position or flipped over the x-axis.

• Forced choice recognition: On warm-ups, reviews, and the assessments, this lesson requires that students only recognize which answer is correct by presenting them with multiple-choice questions. They must only make a decision or judgment.

♦ Offering opportunities for encoding specificity: The lesson is entirely technology-based with many multiple choice practice exercises. All assessments are multiple choice and technology-based; therefore, cues that are used by students to facilitate encoding will also serve as the best retrieval cues at test time.♦ Offering opportunities to counteract interference: The lesson provides a lot of practice on previously learned material through the warm-up activities; therefore, retroactive interference (interference from newly learned material) is not of concern to the student.♦ Offering opportunities to counteract proactive interference: The lesson has students signal key ideas and definitions which counteract proactive interference (interference from previously learned material); therefore, it is not of concern to the student.

♦ Accessing the lesson at home for further study♦ Reviewing the presentation as needed to ensure that the important concepts will be remembered♦ Practicing all material using the graphic organizer for guidance♦ Using the note-taking guide to study and complete practice exercises♦ Being a state-dependent learners by not taking a drug that might cause drowsiness during learning

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 21

Justification of Alignment

CIPLearning Objective

Instructional Strategy Types of Assessment

Retrieve

Students will be able to:• recall• duplicate• memorize• recognize• relate• reproduce• matches• identifies• selects• indicate• choose

The Web-based instruction presentation will:• Explain how learners will be expected to

recall the information• Draw attention to key features of the

material• Provide a meaningful context for

effective encoding of information• Provide memorization tricks• Provide cues for effective recall and

generalization of information

Formative AssessmentsQuestions are convergent

• Multiple-choice warm-ups• Homework assignments• Practice worksheets• Teacher questioning

Summative AssessmentsExam items of the form: reproduce, identify, or selectItems are convergent with limited answers.

• Multiple-choice objective testUnderstand

Students will be able to:• classify• summarize• compare• explain• restate• paraphrase• define• label• list• name• order• describe• discuss• generalizes• assemble• organize• collaborate• communicate• individualize

Google Groups will:• Provide an environment for question-

and-answer sessions• Provide an environment where students

can review material with others• Provide an environment where students

can teach others

The Web-based instruction presentation will:• Present a variety of observable, concrete

examples related directly to the information

• Encourage learners to recall previously learned information

• Give examples that illustrate concepts or rules

• Relate the information to preexisting knowledge

• Present cues to the organization of concepts

• Break down the process of performing or applying rules into steps

• Present all terms clearly using the fewest number of words to convey the meaning

Formative AssessmentsQuestions are convergent

• Class discussions• Graphic organizers• Charts• Note-taking guides• Teacher questioning• Posts

Summative AssessmentItems are convergentExam items of the form:, label, list, describe, define, explain, or summarize

• Multiple-choice objective test

(continued)

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Justification of Alignment (continued)

• Define categories• Group related terms or units• Present the material into manageable

parts• Present information so that it can be

made into chunks• Present a variety examples• Present rule applications and highlight

the distinctive features of examples, definitions, and procedures.

• Encourage the development of graphic organizers and charts

• Encourage the writing of well-developed notes

• Provide activities for summary and closure

Apply

Students will be able to:• execute• implement• engage• execute• practice• demonstrate• employ• solve• use• chart• compute• discover• establish• inform• participate• predict• provide• relate• transfer• utilize

Lesson format will:• Schedule occasions for practice and

spaced review• Encourage the recall of relevant rules

and concepts• Provide practice in multiple contexts• Provide guided practice with immediate

feedback ClickerSchool Virtual Clicker• Present a variety of practice that allow

the students to practice applying the rules or identifying/describing concepts

• Encourage guidance throughout early stages of practice using graphic organizers

• Provide learners with opportunities to engage with concepts and rules using applets

• Encourage the use of mental practice• Arrange repeated practice• Require students to demonstrate

problem-solving• Require students to demonstrate

application of rules, methods, or procedures

Formative AssessmentsQuestions may be convergent or divergent.

• Warm-ups• Problem worksheets• Homework sheets

Summative AssessmentsQuestions are convergentExam items of the form: apply, use, solve, demonstrate, employ

• Multiple-choice objective test

(continued)

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Justification of Alignment (continued)

Analyze

Students will be able to:• differentiate• determine• consider• investigate• analyze• calculate• compare• contrast• differentiate• distinguish• examine• experiment• question• categorize

Google Groups will:• Provide an environment for students to

respond to others ideas• Provide an environment where students

can elaborate on material• Provide an environment where students

can analyze on material

Lesson format will:• Activities that require the analysis of

important information• Activities that determine how

procedures involved in the material work together

• Activities that require analyzing a final outcome to see how it was produced

• Activities for students to analyze content using computer applets

Formative AssessmentsQuestions may be convergent or divergent.

• Posts• Practice Worksheets• Note-taking guide

Summative AssessmentQuestions are convergentExam items of the form: analyze, compare, distinguish, examine, test

• Multiple-choice objective test

Evaluate

Students will be able to:

• assess• rate• criticize• rank• decide• assess• calculate• decide• argue• defend• conclude• justify

Google Groups will:• Provide an environment for students to

self-question• Provide an environment where students

critique the comments of others• Provide an environment where students

discuss the appropriateness of procedures

Lesson format will:• Provide activities for evaluating material

to determine if it fulfills given purpose• Provide activities to demonstrate

processes for evaluating material• Provide activities to rate and rank

material

Formative AssessmentsQuestions may be convergent or divergent

• Posts• Problem worksheets• Charts

Summative AssessmentsQuestions are convergentExam items of the form: evaluate, argue, assess, defend, judge, predict, rate, support

• Multiple-choice objective test

Create

Students will be able to:• develop

illustrate• produce• construct• discover• establish

Google Groups will:• Provide an environment where students

can provide new resources and ideas• Provide information feedback as to the

creativity or originality of comments

Lesson format will:• Provide activities for generating and

constructing new outcomes based on rules and procedures

Formative AssessmentsSmall group discussions Questions may be convergent or divergent

• Problem worksheets• Posts• Note-taking guide

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 24

(continued)

Justification of Alignment (continued)

• formulate• generate• initiate• originate• perform• give

examples• sketch• show• express

• Encourage learners to identify and select their own examples and non-examples of concepts and rule applications if possible.

Summative AssessmentsQuestions may be convergent or divergentExam items of the form: develop, plan, prepare, propose, construct, design, formulate, create, assemble

• Multiple-choice objective test

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Figure A

Sample Slide from SlideRocket Presentation

MEET BUZZ

Buzz is a freshman in High School and lives with his parents. The street that he lives on is the y-axis.

f(x) = Buzz

Buzz is at his parents' home when his stinger is on the origin. We call this the “origin point”.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 26

Figure B

Sample Slide from SlideRocket Presentation: Vertical Shift

BUZZ VISITING FRIENDSVertical Shift

Buzz likes to visit his friends on the street: (y-axis).

Buzz visits Bill when he moves his origin point 5 blocks up.

f(x) = Buzz + 5

Buzz visits Nancy when he moves his origin point 7 blocks down (-7).

f(x) = Buzz – 7

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Figure C

Sample Slide from SlideRocket Presentation: Vertical Reflection

SLEEPING BUZZ Vertical Reflection

When Buzz sleeps, he sleeps upside (vertically flips over his origin point). When we refer to Buzz in this position, we write f(x) = -Buzz.

The negative sign in front of his name indicates that he is in the sleeping position. If there is not a sign at all or a positive sign is in front this means thathe is in the awake position.

This is Buzz asleep at his parent's house.

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Figure D

Sample Slide from SlideRocket Presentation: Vertical Shrink

YOUNG BUZZVertical Shrink

When Buzz was young he was short and fat. When we refer to Buzz at this age, we put a number between 0 and 1 in front of Buzz. The smaller the number the fatter Buzz gets.

f(x) = (¼)Buzz.

This is a young Buzz at his parents’ house.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 29

Figure E

Sample Slide from SlideRocket Presentation: Vertical Stretch and Shift

YOUNG BUZZVertical Stretch and Shift

When Buzz gets old he will be tall and thin. When we refer to Buzz at this age, we put a number greater than 1 in front of Buzz. The larger the number the taller and thinner Buzz gets.

f(x) = (5)Buzz + 3.

This is an older Buzz at a friend’s house that lives 3 blocks up.

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 30

Figure F

Sample Slide from SlideRocket Presentation Introducing “Linus”

Introduction of Function Animals

Animal Name: “Linus” Function Name: Linear FunctionGraph Name: LineMath Symbol: xAnimal Symbol: see below

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Figure G

Sample Slide from SlideRocket Presentation Introducing “Abby”

Introduction of Function Animals

Animal Name: “Abby” Function Name: Absolute Value FunctionGraph Name: Absolute ValueMath Symbol: |x|Animal Symbol: see below

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 32

Figure H

Sample Slide from SlideRocket Presentation Introducing “Curt”

Introduction of Function Animals

Animal Name: “Curt” Function Name: CubicGraph Name: CubicMath Symbol: x3

Animal Symbol: see below

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 33

Figure I

Sample Slide from SlideRocket Presentation Introducing “Squirm”

Introduction of Function Animals

Animal Name: “Squirm” Function Name: Square Root FunctionGraph Name: RadicalMath Symbol: √ xAnimal Symbol: see below

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 34

Figure J

Sample Slide from SlideRocket Presentation Introducing “Rachel”

Introduction of Function Animals

Animal Name: “Rachel” Function Name: Rational FunctionGraph Name: Rational

Math Symbol:1x

Animal Symbol: see below

2

4

-3

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 35

Figure K

Sample Slide from SlideRocket Presentation Introducing “Quinton”

Introduction of Function Animals

Animal Name: “Quinton” Function Name: Quadratic FunctionGraph Name: ParabolaMath Symbol: x2

Animal Symbol: (see below)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 36

Figure L

Screen Shot of Applets for Manipulation of Functions

Multi-Function Data

Slide Graph

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 37

Figure M

Sample Warm-up Quiz on the Basic Behaviors of Functions

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 38

Figure N

Note-taking Guide

Name: Class: Date: Lesson Topic:

Learning Objectives:

CUES

(After class)

• Key ideas

• Questions that connect ideas

• Diagrams and charts

• Prompts for studying

PRESENTATION NOTES

(During class)

• Short and concise phrases

• Symbols and abbreviations

• Labels and categories

• Lists and tables

• Spaced out facts

SUMMARY(After Class)

• In own words

• Original examples

• Most important key ideas

• Accurate phrases for quick reference

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 39

Figure O

Sample Chart Fill in the organizer according to the information you viewed during

the presentation.

Animal Name

AnimalSymbol

Math Symbol

Function Name

Name of Graph

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 40

Figure P

Sample Chart

According to the condition of Buzz below, fill in the cells using the directions at the top of each column. Follow the example done for you.

Description of How Buzz

changed

Mathematical Operation

That Create This Change

Mathematical Name of the

Change That Took Place

Asleep and youngerUpside

down and fatter

Negative number

between 0 and 1

multiplied by Buzz

Vertical reflection and

vertical shrink

Awake at his parents

Visiting a friend and awake

Younger at his parents house and

asleep

Older at his friend’s house and awake

Younger at his friend’s house

Older and awake at his parents’ house

Asleep at his friends house and younger

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 41

Definition Facts/Characteristics

Examples

Non-ExamplesDefinition Facts/Characteristics

Examples

Non-ExamplesDefinition Facts/Characteristics

Examples

Non-ExamplesFigure Q

Sample Graphic Organizer

Linear Function

QuadraticFunction

Cubic Function

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 42

Figure R

Sample Practice Worksheet on Writing EquationsLook at each graph below and fill in each cell according to the heading at the top of each column. An example is given.

Animal NameAwake

or asleep

Age of Animal

MathSymbol

PossibleEquation

Quadratic + 6 X2 f(x) =

6x2

Figure S

Sample Practice Worksheet on Writing Equations

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 43

Look at each graph below and fill in each cell according to the heading at the top of each column. An example is given.

Function

Notation=

AwakeOr

Asleep

(+ or -)

Possible

Age

Function

Symbol

+-

or“nothi

ng”

Vertical

Position

f(x)

= + 5 x2 - 6

Figure T

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 44

Look at each graph below and fill in each cell according to the heading at the top of each column. An example is given.

Awake + Asleep -

Age of Animal

MathSymbol

Up +Down –

# of units

PossibleEquation

- 7 √x - 5 f(x) = -7√x - 5

Sample Practice Worksheet on Writing Equations

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 45

Figure U

Sample Practice Worksheet Drawing Graphs

Place a box around each of the 4 parts of the equations below.Name the animal in the blank provided. Fill in the chart according to the directions at the top of each column.

Box 1 Box 2 Box 3 Box 4

Sketch ofPossibleGraph

(in 1,2,3,4 order)

Draw Graph

Awake or Asleep

Draw AnimalAt Age

DrawParentGraph

How manyUnits Up or

Down?

3rd 2nd 1st 4th

(example)

animal: Quinton .

Down 7

y = - (1/4) x3 + 2

animal_________

y = -5 x + 2

animal_________

y = (1/3) √x - 5

animal_________

Figure V

Sample Review Activity on Shifting Parent Graphs

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 46

GraphAwake or

asleepPossible Age of

AnimalShape ofthe graph

Shift up or down

1. A. + B. -

2. A. 1/2

B. 2 C. 1

3. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

4. A. + 3 B. - 2 C. no shift

5. A. + B. -

6. A. 1/2

B. 2 C. 1

7. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

8. A. + 3 B. - 2 C. no shift

9. A. + B. -

10. A. 1/2

B. 2 C. 1

11. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

12. A. + 3 B. - 2 C. no shift

13. A. + B. -

14. A. 1/2

B. 2 C. 1

15. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

16. A. + 3 B. - 2 C. no shift

17. A. + B. -

18. A. 1/2

B. 2 C. 1

19. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

20. A. + 3 B. - 2 C. no shift

21. A. + B. -

22. A. 1/2

B. 2 C. 1

23. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

24. A. + 3 B. - 2 C. no shift

25. A. + B. -

26. A. 1/2

B. 2 C. 1

27. A. x D. x3

B. |x| E. √x C. x2 F. 1/x

28. A. + 3 B. - 2 C. no shift

Figure W

Activity on Linear Regression Which Serves as an Additional Activity

“Extra Time” Activity

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 47

Basics of Linear RegressionPage 1

If you complete all components of the lesson on Function Families, use the extra

time to complete this activity.

1. Before you perform any Internet research, estimate the ages of each of the famous people above. Fill in the appropriate column in the chart (x-value).

2. Using the Internet, research each of the famous people and record their ages in the appropriate column in the chart (y-value).

How well were you able to estimate the ages of these famous people?

3. To help answer this question, first of all make a scatterplot using the estimate of their age as the x-value and the actual age as y-value. Use the graph paper attached. Be sure to give it a title, use appropriate increments on the axes, and label each axis.

4. If you estimated the age correctly, what is true about this point?

5. Sketch a graph of the line of estimates that were 100% accurate.

6. What is the equation of this line?

7. What does it mean if a point on your scatterplot is above this line?

(continued)

Figure W (continued)

Activity on Linear Regression Which Serves as an Additional Activity

“Extra Time” ActivityBasics of Linear Regression

Page 2

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 48

8. In general, did you overestimate or underestimate the ages of your famous people?

9. How can you tell this from your scatter plot?

10. Plot the points on Quick Graph.

11. Have Quick Graph calculate linear regression and draw this line. What is the equation of this line?

12. Sketch this line on your graph paper.

13. Write a paragraph that discusses the slopes of your lines and the characteristics of your data. What conclusions can you draw?

(continued)

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 49

Figure W (continued)

Activity on Linear Regression Which Serves as an Additional Activity

“Extra Time” ActivityBasics of Linear Regression

Page 3

Famous PersonEstimated Age

x-valueActual Age

y-value

Bill Clinton

Tom Cruise

Hillary Duff

Miley Cirus

Carrie Underwood

Abraham Lincoln (age at death)

Babe Ruth (age at death)

“The Rock”

George Washington

Tim Allen

Oprah Winfrey

Cleopatra (age at death)

Barney

Einstein (age at death)

Figure X

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FUNCTION FAMILIES: COGNITIVE INFORMATION PROCESSING 50

Sample Assessment Questions

(continued)

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Figure X (continued)

Sample Assessment Questions

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References

Driscoll, M.P. (2005). Psychology of learning for instruction (3rd ed.). Boston: Allyn and Bacon