I.  INTRODUCTION

            The purpose of this project is to help students in traditionally-rigorous intermediate algebra courses appreciate the power, beauty and utility of their own knowledge of quadratic equations.

Parabolas exist everywhere. For ages, people longed to define these shapes more precisely than words could describe. Math is the language that made this possible. Precisely defined by quadratic equations, parabolas found in nature could now be recreated in the built world—in architecture, in bridges, in sculpture. Consequently, this exciting discovery has been passed down through the ages as part of the standard algebra curriculum.

For today’s students, however, quadratic equations are but a sterile artifact of that once-exciting quest for a language that could describe natural form and prescribe its construction. For them, quadratic equations are just another thing they have to learn in algebra class. The connection between quadratic equations and the real world has been lost and, for many students, so has any intrinsic motivation to learn to build and calculate them.

A constructivist approach to learning holds that we care about learning those things that hold meaning for us, and nothing is truly meaningful until we have experienced and discovered it for ourselves (Karagiorgi and Symeou, 2005; March 2003). Discovery and insight involve making connections with what we already know (Piaget 1969), and is enriched through interaction with others.  According to David Jonassen (2003), a constructivist approach involves designing educational experiences that support meaningful learning by clearly defining the problem and activity space, by providing opportunities to construct knowledge, compare divergent perspectives, and use real-world problems or tools, and by encouraging collaboration, conversation, group consensus building, complex intellectual engagement, mindful thinking, and reflection.  This teaching guide specifically targets each of these components.

            In the lessons that follow, students engage in semi-structured experiences designed to support accurate and meaningful knowledge construction.  Students are asked to locate photographs of parabolas in art and architecture, and then are encouraged to work collaboratively to develop and confirm accurate equations describing the forms using the language and structure of mathematics.  Problem space is defined by the instructor/facilitator by guiding student energies toward location of parabolas (as opposed to other conic forms). Students spend several days engaged in experimenting with, creating, refining, and confirming accurate equations through group collaboration, conversation, consensus building, peer assistance, and peer review.  Throughout this collaborative process, students consider divergent opinions and perspectives, ultimately settling on the solution that they believe best describes the initial selection of architectural art.  Creating and verifying original equations certainly qualifies as an intellectually-challenging task in line with the constructivist philosophy of complex intellectual engagement. At all stages of the project, students use real-world tools of their own choosing—CAD software, drawings and sketches of graphs, and 2D or 3D graphing calculators---to validate the equations they have created.  At the conclusion of the project, students and instructors individually and collectively engage in mindful reflection about the value of what they have experienced and learned.

            In Constructivism: Theory, Perspectives, and Practice, Catherine Twomey Fosnot (2005) reflects upon lessons learned from attempts at constructivist reforms in mathematics education.  Fosnot describes several examples of mathematics teachers who use a technique of asking neutral questions (“Can you tell me why that is?”) to allow students to enter an unsteady state of disequilibrium as they attempt to solve real-world, “messy” mathematical problems.  Fosnot coins the term mathematizing to describe learners’ processes for creatively finding solutions in real-world contexts.  She reminds us that professional mathematics is never a process of transmitting facts and knowledge from teacher to student—the essence of the work of mathematicians is to define a problem, notice patterns, develop hypotheses, and experiment with equations (the language of mathematics) to accurately describe a real-world phenomenon or suggest a mathematical theoretical solution. 

Pilot results

            The lessons described in this Guide were originally developed for intermediate-algebra high school students and then refined and piloted with intermediate algebra students at a community college.  By allowing our students several days to “mathematize,” we were able to capture the struggle, frustration, and heady success of creating original mathematic work. They wanted to experiment; they were so engaged and excited that they lingered after class; one student was inspired to learn AutoCad just for fun.

            Following the Piagetian model of intellectual development, the following lesson plans are divided into two types: high school and college. Although the spirit of discovery and inquiry learning is still present in both sets of lesson plans, college age students are asked to engage in different and slightly more sophisticated type of reflection activities than high school age students. In addition, the lesson plans for the high school unit are divided into three lesson plans rather than the four lesson plans for the college age students. This difference reflects more the practical differences between high schools and colleges (high schools frequently ban the use of cell phones during school hours).

II.  Sample Constructivist Lesson Plan: High School

The Parabola Project:  

Connecting Art, Architecture and Mathematics

LESSON 1:      Recognizing Parabolas in Art and Architecture

Duration:     50 minutes

Intended Audience

Audience:                     High School Algebra Faculty

Mathematics Level:       Intermediate Algebra   (or accelerated Elementary Algebra)

Misconception Addressed

Parabolas are confined to mathematics textbooks and have no real-life applications.

Learning Objectives

·        Students will visually recognize parabolic forms in photographs of art and architectural landmarks.

·        Students will practice and refine internet-based research methodologies.

·        Students will conceptually connect quadratic equations with applications of architecture.

·        Students will develop an increased appreciation of the utility of algebra in the world around them.

Materials

• Teacher computer and projector

• projector

• internet access

• student computers

Assumptions/Prerequisite Skills

• Students have completed introductory linear graphing projects.
• Students are able to recognize the Standard Form of a quadratic equation: y = ax² + bx + c 
• Students know that a quadratic equation produces the graph of a parabola.
• Students are members of small groups for ongoing group activities.

Activity #1: What do you know about  parabolas?   (10 mins)                              

Part A (Establishing the Misconception)

As class begins, students are seated in their established small groups (three to four members). The instructor begins the class by asking:  “What do you know about parabolas?”   The instructor should write down all germane responses on the overhead projector or the whiteboard.

Many students will give mathematical answers such as “They are quadratic equations” or “They have an x-squared.” The instructor should note all these responses on the overhead without comment.  Other students may provide real-world examples “They are the MacDonald Arches” or “They are used in bridges.” Again, note all responses on the overhead.

In practice, students are very enthusiastic about showing off their knowledge of parabolas, however, many of their responses reveal a fragmented and isolated set of observations. In educational parlance, many of these responses reveal incomplete or poorly formed schema; "holes" in an overall understanding of how the different concepts relate to one another. If possible, the instructor should save responding to these misconceptions in the hope that the students come to correct their deficiences by the constructivist activities of this project.

Activity #2 Examples of Parabolas in Architecture – Student Suggestions                                                                                                                    (10 minutes)

After student discussion of the responses in Activity 1, the teacher should guide the discussion by asking students for examples of parabolas appearing in architecture.

Students will typically answer “McDonalds”, “Golden Gate Bridge”, “St. Louis Arch” or “Roman Aqueduct”.  As students call out suggestions, the teacher uses the teacher’s computer and projector to do quick internet searches for the suggestions made by students, and points out the parabolic forms in the suggestions made by students.

The Roman Aqueduct is a popular response.

                        

Activity #3:

Examples of Parabolas in Architecture: Teacher Guided                                                     (10 minutes)

_______________________________________________ 

After student discussion of the responses in Activity 2, the teacher should extend the lesson by showing students other examples of parabolas appearing in architecture.

Useful examples from modern architecture appear on Santiago Calatrava’s website.  Calatrava is a modern-day Spanish architect whose designs show beautiful integration of geometric form , sculpture, and architecture:

Calatrava museum in Milwaukee (exterior view)

 Calatrava museum in Milwaukee (exterior view)

Calatrava Website  http://www.calatrava.com/

Other show and tell examples that have been used successfully with students are the Duomo in Florence and the Roman Coliseum in Rome.

                

Activity #4: Research and Identifying Parabolas (20 mins

_______________________________________________________________________

After identifying parabolas and other geometric forms in multiple examples of architecture, students are given the mission to search the internet for interesting examples of parabolic form.  If students have classroom access to iPads, computers,  and or printers, tell them to do the following:

1)       Find a good example of parabolas used in art or architecture.

2)       Print the example.  Try to make the printout of the picture rather large – almost   taking up a full sheet of paper if possible.

3)       Find two or three examples. Instruct the students that examples with parabolas that are not “slanted” work best.

4)       Have graph paper available for students to trace the parabola onto a graph.

5)       Whatever is not accomplished in class can be assigned for homework that evening.

The Parabola Project:  

Connecting Art, Architecture and Mathematics

LESSON 3: 

Confirming Invented Equations Through the Use of Graphing Software 

Duration: 50 minutes                                              

Intended Audience/Grade Level

Audience:                     High School Algebra Teachers

Mathematics Level:       Intermediate Algebra (or accelerated Elementary Algebra I) 

Misconception Addressed by Proposed Activities

Parabolas are abstract mathematical constructs with no real-life applications. 

Learning Objectives 

• Students will learn how to use graphing software (apps or programs) to confirm computational solutions for invented quadratic equations. 
•  Students will visually recognize parabolic forms in photographs of art and architectural landmarks.
• Students will practice and refine internet-based research methodologies.
• Students will conceptually connect quadratic equations with applications of architecture. 
• Students will develop an increased appreciation of the utility of algebra in the world around them 
• Students will identify components and characteristics of a quadratic equation, including coefficients (a,b,c), dependent variable, independent variable, quadratic terms, linear terms, constant terms, standard form, x-intercepts, y-intercepts, vertex, and axis of symmetry. 
• Students will identify three data points and create a 3x3 system of equations. 
• Students will solve a 3x3 system of equations without use of a graphing calculator. 
• Students will increase fluency in the language of mathematics by writing quadratic forms describing real-world architecturally-based parabolic forms. 
• Students will develop a sense of empowerment and self-efficacy in their ability to creatively construct mathematical representations. 
• Students will develop stronger analytic skills, including identification of underlying constructs central to the problem, sourcing information necessary for procedural accuracy, and learning through inquiry.

Materials

• computer and projector

• printer

• internet access

• graph paper

Assumptions/Prerequisite Skills

• Students should have a working knowledge of solving quadratic equations using graphing software (calculators, apps, online programs)

• Students should have a working knowledge of how to solve a 3x3 system of equations

• Students should have a working knowledge of how to graph a parabola either by recognizing characteristics of a standard quadratic equation or by using a graphing calculator

• Students are members of small groups for ongoing group activities

Activity #1: Reflection and Check In                                  (10 minutes)

To start the class, students should immediately go to their small groups. Teacher should have a question on the overhead projector: "What questions and/or insights did you have from last night's assignment?" The instructor should solicit and write down on the overhead the major issues, insights, and solutions arising from the students' attempts at completing the assignment. At this point, do not offer any solutions of your own. Students will then have an opportunity to compare work and check each others answer questions and offer insights to other members of their group. Teacher should circulate around the classroom, offer suggestions, guide any students or groups that getting too far off topic or are having difficulty.

Activity #2: Confirming Self-Generated Equations Using Online Calculator                                                                             (30 minutes)

Students will then log on to computers or iPads and use the program called QUICK GRAPH or some other online calculator (DESMOS or FOOSPLOT) capable of graphing two or three dimensional equations. Students should continue working in their small groups. Specifically, students will enter their respective equations into the graphing function of the calculator and confirm if their self-generated equation matches the equations generated by the graphing program.  

If students have made errors in computation, they will discover that the shape of the graph and the orientation (upward, downward) may be very different that the photographs of the architectural designs.  If so, they should first go back and carefully heck their computations to identify errors in solving the 3x3 system.  At this point, encourage students to work together to identify the errors in their own and each others' computations.

  Graphing Calculator Example

   

Parabola in Architecture Example

Students can compare whether their self-generated quadratic equation matches the computer-generated parabola. At this point, the teacher can guide individual group discussions by suggesting the following questions: 

•  How does your parabola differ from the computer generated parabola?

•  If you were to place your parabola on an x-y axis, is there a better place to place the vertex? Why?

•  If you were to place your parabola on an x-y axis, where would you place the x-intercepts? Is there a better place to place them? Why?

• Where would you place the y intercept?

• Why does your parabola open upwards while the parabola in the image opens downwards?

Activity #3: Wrap Up and Final Project Questions            (10 minutes)

In an overall class discussion, the instructor should ask the question: "What did we learn today?" Write down germane responses on the overhead projector. Teacher should also remind students that Group Portfolios are due next class with the following requirements.

1. The photographs of architectural arches
2. Graph paper sketch showing the transfer of the parabola onto the graph paper, with three (x,y) coordinates correctly labeled and identified.
3. Pencil-and-paper computations showing (a,b, c) solutions for 3x3 system.
4. Print of the graph from online graphing calculator.  Printout should show the entry for the equation and should be detailed enough so that the three chosen coordinates are confirmed.

III.    Sample Constructivist Lesson Plan: College

The Parabola Project:  

Connecting Art, Architecture and Mathematics

LESSON 1: 

Recognizing Parabolic Forms in Art and Architecture

 ________________________________________________

Duration: Approximately 40 minutes

Intended Audience/Grade Level

Audience: Community College Developmental Mathematics Faculty.

Mathematics Level: Intermediate Algebra

Parabolas are confined to mathematics textbooks and have no real-life applications.

Misconception Addressed by Proposed Activities

Students will visually recognize parabolic forms in photographs of art and architectural landmarks.

Learning Objectives

• Students will practice and refine internet-based research methodologies.
• Students will conceptually connect quadratic equations with applications of architecture.
• Students will develop an increased appreciation of the utility of algebra in the world around them.

Materials

• Classroom “teacher” computer and projector
• Student Response System capable of open-ended responses (Poll Everywhere or CPS Clickers)

• Students have completed introductory linear graphing projects.

• Students are able to recognize that an equation in the form y = ax2 + bx + c will produce the graph of a parabola.
• Students are members of small groups for ongoing group activities.

________________________________________________________________________________

Activity #1 What do you know about parabolas?

Part A (Establishing the Misconception):

As students arrive in the classroom, they are greeted with a Poll Everywhere question. The teacher asks the students to text message their responses to the question “What do you know about parabolas?”

Student responses will appear in real time on the response screen. As students submit their responses via text, the teacher acknowledges each response as it appears on the response screen. In the spirit of constructivism, the teacher encourages discussion but does not provide solutions or suggestions for “correct” responses.

Some of the students may provide real-world examples (path of a rocket, bridges, waveforms). Other students will focus on problem sets in textbooks.

In practice, students are very enthusiastic about the poll everywhere text messaging response system. The messaging system guarantees anonymity for students, and students seem to feel more willing to contribute responses and participate in virtual discussion.

_____________________________________________________________________

Activity #2 Examples of Parabolas in Architecture – Student Suggestions

After student discussion of the responses in Activity 1, the teacher should guide the discussion by asking students for examples of parabolas appearing in architecture. Students will typically answer “McDonalds”, “Golden Gate Bridge”, “St. Louis Arch” or “Roman Aqueduct”. As students call out suggestions, the teacher uses the classroom “teacher computer” and projector to do quick internet searches for the suggestions made by students, and points out the parabolic forms in the suggestions made by students.

_______________________________________________________________________________________________

Activity #3 Examples of Parabolas in Architecture – Teacher Guided Discussion

After student discussion of the responses in Activity 2, the teacher should extend the by showing students other examples of parabolas appearing in architecture. Useful examples from modern architecture appear on Santiago Calatrava’s website. Calatrava is a modernday

Spanish architect whose designs show beautiful integration of geometric form , sculpture, andarchitecture: http://www.calatrava.com/

Other “show and tell” examples that have been used successfully with students are St. Peter’s Basilica (Vatican City), the Roman Coliseum, and the Duomo (cathedral) in Florence, Italy.

_______________________________________________________________________________

Activity #4 Practice and Extend – Student Homework

After identifying parabolas and other geometric forms in multiple examples of architecture, students are given the mission to search the internet for interesting examples of parabolic form.

The homework assignment:

1. Find a good example of parabolas used in art or architecture.
2. Print the example. Try to make the printout of the picture rather large – almost taking up a full sheet of paper if possible.
3. Bring a couple of copies of the printout to class.
4. Also bring graph paper for tomorrow’s assignment.

    _________________________________________________________________
   
   

   ____________________________________________________
   

   LESSON 2: 

   Creating Sketches of Graphs and Developing Equations for Parabolas in Art and Architecture 

   
   

   Duration: Two one-hour class periods (spanning two days)                                               

   Intended Audience/Grade Level

   Audience: Community College Developmental Mathematics Faculty.

   Mathematics Level:  Intermediate Algebra  

   Misconception Addressed by Proposed Activities

   Parabolas are confined to mathematics textbooks and have no real-life applications.

   
   

   Learning Objectives 

   • Students will visually recognize parabolic forms in photographs of art and architectural landmarks.   

   • Students will practice and refine internet-based research methodologies. Students will conceptually connect quadratic equations with applications of architecture.
   • Students will develop an increased appreciation of the utility of algebra in the world around them.
   • Students will develop an increased appreciation of the utility of algebra the world around them. 
   • Students will identify components and characteristics of a quadratic equation, including coefficients (a,b,c), dependent variable, independent variable, quadratic terms, linear terms, constant terms, standard form, x-intercepts, y-intercepts, vertex, and axis of symmetry. 
   • Students will identify three data points and create a 3x3 system of equations. 
   • Students will solve a 3x3 system of equations without use of a matrix calculator.
   • Students will develop an increased appreciation of the utility of algebra in the world around them. 
   • Students will increase fluency in the language of mathematics by writing quadratic forms describing real-world architecturally-based parabolic forms. 
   • Students will develop a sense of empowerment and self-efficacy in their ability to creatively construct mathematical representations. 
   • Students will develop stronger analytic skills, including identification of underlying constructs central to the problem, sourcing information necessary for procedural accuracy, and learning through inquiry.

   Materials

   
   

   • Classroom “teacher” computer and projector

   
   

   • Student Response System capable of open-ended responses (Poll Everywhere or CPS Clickers)

   
   

   • Graph paper

   
   

   Assumptions/Prerequisite Skills

   
   

   • Students have completed introductory linear graphing projects.

   
   

   • Students are able to recognize that an equation in the form y = ax² + bx + c  will produce the graph of a    parabola.

   
   

   • Students know how to solve a 3x3 system.

   
   

   • Students are members of small groups for ongoing group activities.

   
   

   ____________________________________________________________________

   
   

   Activity #1:
   
   
   Where is Your Parabola?
   
   

   
   

   
   Part A (Review of first lesson and extension to computational mathematics):

   
   

   As students arrive in the classroom, they are greeted with a Poll Everywhere question.  The teacher asks the students to text message their responses to the question “Where is your parabola?”   Student responses will appear in real time on the response screen.

   
   

   

   
   

   
   

    As students submit their responses via text, the teacher acknowledges each response as it appears on the response screen.  The teacher should comment on responses and show excitement and interest in the various locations discovered by the students.

    Students are asked to sit with their groups (3-4 students per group).  They are asked to show each other the photographs that they brought to class, and discuss locations and the appearance or patterns that they can identify.

   Activity #2:

   Sketching the Graphs of Parabolas in Architecture                                                 (20 minutes)

   After student discussion of the responses in Activity 1, the teacher should guide the discussion toward the mathematical computational exercises.

   
   

   First, students are asked to trace the parabola on graph paper.  Students may darken the outline of the parabola on their internet picture by using a black marker and/or hold the graph paper up to the window so that the shape of the parabola can be more easily traced on the graph paper.

   
   

   The teacher should give the following set of instructions:

   
   

   1. Start by sketching the x-axis and the y-axis on a sheet of graph paper.
   
   2. Lay your graph paper over the picture of your building, and trace the shape of the parabola onto your graph paper.
   3. Eventually , you will need to identify three points, It will be easier to do this if you try the following:
    
   

   a.)     Make sure your parabola intersects the y-axis in one point.

   b.)     Make sure your parabola intersects the x-axis in two points.

   c.)     Try to orient your parabola so that it intersects the “cross-hairs” of the graph paper, so that you are dealing with whole numbers and not fractions when you select points.  (All students hate generating equations from fractional coordinates, so there is no difficulty in understanding this instruction.)

   
   

   After the parabolas have been transferred to graph paper, students are asked to identify three points on the graph.  Students in groups collaborate and communicate to confirm each other’s accurate identification of points.

   When students have successfully identified the points, they are asked to “send” the coordinates to the text messaging system so that the teacher is aware of the progress of each group.

   

   
   

   Activity #3:

   
   

   Creating the Equations of Parabolas in Architecture (significant time segments of two class periods)

   After groups have successfully identified and verified the coordinates for the 3-4 examples per group, the most difficult part of the activity begins.  Each group must use the (x,y) coordinates of the three points to set up a 3x3 system and solve for the coefficients of a, b, and c.

   For example, using the students’ coordinates of (-13,0), (0.9), and (14,0) projected on the text messaging screen above, the group would set up and solve the following 3x3 system:

   General form:  y = ax² + bx + c

   0 = 169a -13b + c

   9 = 0a + 0b + c

   0= 196a + 14b + c

   Solutions for a, b, and c are substituted back into the general form – and voila! The students have generated the equations for the parabolas shown in their photographs.

   In reality, this part of the exercise can be quite frustrating for students.  Allow plenty of time for groups to solve their 3-4 problems, typically two class periods.

   I constantly reminded students that in real-life, mathematicians do not do 25 math problems a day.  Instead, real mathematicians take 25 days to do one problem!

   After two days of group work, and moments of extreme struggle and frustration, students were able to arrive at solutions and generate their parabolic equations.  It is amazing to see the frustration evolve into excitement and heady feelings of success!

   __________________________________________________________________ 

   LESSON 3: 

   Confirming Invented Equations Through the Use of Graphing Software 

   _____________________________________________________________________________

   Duration: 50 minutes                                              

   Intended Audience/Grade Level

   ______________________________________________________________________________

   Audience: Community College Developmental Mathematics Faculty.
   Mathematics Level:  Intermediate Algebra  
   

   Misconception Addressed by Proposed Activitie

   • Parabolas are confined to mathematics textbooks and have no real-life applications.

   
   

   Learning Objectives 

   •  Students will develop proficiency in using graphing software to confirm computational solutions for invented  quadratic equations. 
   • Students will visually recognize parabolic forms in photographs of art and architectural landmark
   • Students will practice and refine internet-based research methodologies.
   • Students will conceptually connect quadratic equations with applications of architecture. 
   •  Students will develop an increased appreciation of the utility of algebra in the world around them. 
   • Students will identify components and characteristics of a quadratic equation, including coefficients (a,b,c), dependent variable, independent variable, quadratic terms, linear terms, constant terms, standard form, x-intercepts, y-intercepts, vertex, and axis of symmetry. 
   • Students will identify three data points and create a 3x3 system of equations. 
   • Students will solve a 3x3 system of equations without use of a matrix calculator. 
   • Students will develop an increased appreciation of the utility of algebra in the world around them.
   • Students will increase fluency in the language of mathematics by writing quadratic forms describing real-world architecturally-based parabolic forms. 
   • Students will develop a sense of empowerment and self-efficacy in their ability to creatively construct mathematical representations. 
   • Students will develop stronger analytic skills, including identification of underlying constructs central to the problem, sourcing information necessary for procedural accuracy, and learning through inquiry.

   ______________________________________________________

   Materials

   
   

   • Classroom “teacher” computer and projector

   
   

   • Student Response System capable of open-ended responses (Poll Everywhere or CPS Clickers)

   
   

   • Access to 2D and 3D graphing software (www.runiter.com)

   
   ___________________________________________________________________

   Assumptions/Prerequisite Skills

   • Students have completed introductory linear graphing projects using the runiter graphing software.

   
   

   • Students have solved their 3x3 systems, identifying tentative solutions for coefficients a, b, and c in the general form of the equation of a parabola.

   
   

   • Students are members of small groups for ongoing group activities.

   
   ___________________________________________________________________

   Activity #1:

   What is Your Current Project Status?

   Part A (Review of first three days and extension to graphing software)

   
   

   As students arrive in the classroom, they are greeted with a Poll Everywhere question.  The teacher asks the students to text message their responses to the question “What is your current project status?”   Student responses will appear in real time on the response screen.

   
   

   

   
   

   
   

   As students submit their responses via text, the teacher acknowledges each response as it appears on the response screen.  The teacher should answer questions as they appear.  By this stage of the project (day 4), students will feel very comfortable asking for help by posting questions on the screen.

   Within groups, students should work together to finalize the equations for each member (parabola) in the group.  The teacher should circulate and assist where needed.

   ________________________________________________________________________________________ 

   Activity #2:

   Graphing the Invented Equations Using Runiter Software

   As groups collaborate to finalize and confirm member’s equations, they are ready to move to the next phase of the project.

   
   

   Using the classroom projected computer (loaded with Runiter software) or their own laptops, the next task is to enter their equations in the graphing software and try to confirm the accuracy or identify errors.

   If students have made errors in computation, they will discover that the shape of the graph and the orientation (upward, downward) may be very different that the photographs of the architectural designs.  If this is so, they go back to the drawing board and check their computations to identify errors in solving the 3x3 system.  Groups work together to identify the errors, and seem determined to solve the equations themselves without input from the teacher!

   
   

   Here is an example of a parabola that “matches” the original architectural form of the Gateway Arch in St. Louis, Missouri.  Note that difficulty level of the fractional coefficients far exceeds the difficulty level of similar “textbook” problems!

   
   

   

   
   

    

   
   

   The excitement in the class runs quite high as students try out their equations on the projected teacher computer.  All eyes are on each attempt, and students trying out their equations get cheers or moans from their classmates as the graphs appear on the projected screen show “success” or “need more work”.

   
   

   Throughout this process, the instructor may guide discussion toward identifying what’s wrong (fractions are entered without parens in the software, a-coefficient should always be negative for parabolas opening downward, etc.)

   _____________________________________________________________________

   Activity #3:

   Homework:  Assemble Your Final Portfolio 

   
   

   As students finalize their projects by completing the graphing calculator activity, the are asked to assemble a portfolio packet that will be graded.

   
   

   The group portfolios should include:

   
   

   1.)     The photographs of architectural arches (one for each student in the group).

   2.)     Graph paper sketch showing the transfer of the parabola onto the graph paper, with three (x,y) coordinates correctly labeled and identified.

   3.)     Pencil-and-paper computations showing (a,b, c) solutions for 3x3 system.

   4.)     Print of 2D (or 3D) graph from http://www.runiter.com/ software.  Printout should show the entry for the equation and should be detailed enough so that the coordinates of the points (in 2, above) are confirmed.

   
   

   
   

   _____________________________________________________________________

   
   

   LESSON 4:  Conclusions and Reflections 

   Duration: 20 minutes                                              

   Intended Audience/Grade Level

   Audience: Community College Developmental Mathematics Faculty.

   Mathematics Level:  Intermediate Algebra  

   Misconception Addressed by Proposed Activities

   Parabolas are confined to mathematics textbooks and have no real-life applications.

   
   

   Learning Objectives 

   
   

   • Students will develop a sense of empowerment and self-efficacy in their ability to creatively construct mathematical representations.

   Materials

   
   

   • Classroom “teacher” computer and projector

   
   

   • Student Response System capable of open-ended responses (Poll Everywhere or CPS Clickers)

   
   

   Assumptions/Prerequisite Skills

   
   

   • Students have completed the parabola project and have assembled portfolios.

   
   

   __________________________________

   Activity #1:

   Reflections

   
   

   As students arrive in the classroom, they are greeted with a final Poll Everywhere question.  The teacher asks the students to text message their responses to the “Reflections?” screen.

   
   

   In order to elicit mindful reflection, students are asked to think about and respond to the following five

    questions:

   
   

   1.)      What did you learn?

   2.)     What surprised you?

   3.)     What bothered you?

   4.)     Should I (the teacher) do this again with other classes?

   5.)     Any other thoughts?

   
   

   Student responses to all five questions were text-messaged to the reflections screen.  The teacher reflected on the student responses, and much enthusiastic discussion occurred.  In the first class where I piloted these lessons, students asked me for MORE projects, and many said that they had never had so much fun in a math class –EVER.

   
   

   I will let the student responses speak for themselves:

   
   

   
   

   

   
   

   
   

   
   

   

   
   

   
   

   

   
   

   
   

Assumptions/Prerequisite Skills

Recognizing Parabolic Forms in Art and Architecture

http://www.calatrava.com/

        

Other “show and tell” examples that have been used successfully with students are St. Peter’s Basilica (Vatican City)

Activity #4  Research and Identification of Parabolas         (20 minutes)

So is the Roman Coloseum.

 

Activity #3: Parabolas in Art and Architecture - Teacher Guided

                                                                                                        (10 minutes)

In practice, students are very enthusiastic about showing off their knowledge of parabolas, however many of their responses reveal a fragmented and isolated set of observations.  In educational parlance,  many of the responses reveal incomplete or poorly formed schema; “holes” in an overall understanding of how the different concepts relate to one another. If possible, the instructor should save responding to these misconceptions in the hope that the students come to correct their deficiencies by constructivist methods.

Activity #2:

Parabolas in Art and Architecture - Student Examples     (10 minutes) 

Recognizing Parabolic Forms in Art and Architecture 

Duration: 50 minutes                                              

Intended Audience/Grade Level

Audience:                     High School Algebra Teachers

Mathematics Level:       Intermediate Algebra (or accelerated Elementary Algebra I) 

Misconception Addressed by Proposed Activities

Parabolas are abstract mathematical constructs with no real-life applications. 

Learning Objectives 

• Students will visually recognize parabolic forms in photographs of art and architectural landmarks.
• Students will practice and refine internet-based research methodologies.
• Students will conceptually connect quadratic equations with applications of architecture. 
• Students will develop an increased appreciation of the utility of algebra in the world around them 

Materials

• computer and projector

• printer

• internet access

• graph paper

Assumptions/Prerequisite Skills

• Students should have a working knowledge of solving quadratic equations using graphing software (calculators, apps, online programs)

• Students should have a working knowledge of how to solve a 3x3 system of equations

• Students should have a working knowledge of how to graph a parabola either by recognizing characteristics of a standard quadratic equation or by using a graphing calculator

• Students are members of small groups for ongoing group activities

Activity #1: Reflection and Check In                                  (10 minutes)

Duration: 50 minutes                                              

Intended Audience/Grade Level

Audience:                     High School Algebra Teachers

Mathematics Level:       Intermediate Algebra (or accelerated Elementary Algebra I) 

Misconception Addressed by Proposed Activities

Parabolas are abstract mathematical constructs with no real-life applications. 

Learning Objectives 

• Students will learn how to use graphing software (apps or programs) to confirm computational solutions for invented quadratic equations. 
•  Students will visually recognize parabolic forms in photographs of art and architectural landmarks.
• Students will practice and refine internet-based research methodologies.
• Students will conceptually connect quadratic equations with applications of architecture. 

.

Materials

• computer and projector

• printer

• internet access

• graph paper

Assumptions/Prerequisite Skills

• Students should have a working knowledge of solving quadratic equations using graphing software (calculators, apps, online programs)

• Students should have a working knowledge of how to solve a 3x3 system of equations

• Students should have a working knowledge of how to graph a parabola either by recognizing characteristics of a standard quadratic equation or by using a graphing calculator

• Students are members of small groups for ongoing group activities

Activity #1: "What do you know about parbolas?"            (10 minutes)

Duration: 50 minutes                                              

Intended Audience/Grade Level

Audience:                     High School Algebra Teachers

Mathematics Level:       Intermediate Algebra (or accelerated Elementary Algebra I) 

Misconception Addressed by Proposed Activities

Parabolas are abstract mathematical constructs with no real-life applications. 

Learning Objectives 

·        Students will visually recognize parabolic forms in photographs of art and architectural landmarks.

·        Students will practice and refine internet-based research methodologies.

·        Students will conceptually connect quadratic equations with applications of architecture.

·        Students will develop an increased appreciation of the utility of algebra in the world around them.

• computer and projector

• printer

• internet access

• graph paper

Assumptions/Prerequisite Skills

• Students should have a working knowledge of solving quadratic equations using graphing software (calculators, apps, online programs)

• Students should have a working knowledge of how to solve a 3x3 system of equations

• Students should have a working knowledge of how to graph a parabola either by recognizing characteristics of a standard quadratic equation or by using a graphing calculator

• Students are members of small groups for ongoing group activities

Materials

Activity #1: What do you know about parabolas?               (10 minutes)

Part A (Establishing the Misconception):

The Parabola Project:  

Connecting Art, Architecture and Mathematics

Materials

Learning Objectives 

Misconception Addressed by Proposed Activities

 

Duration: 50 minutes                                               

Intended Audience/Grade Level

Assumptions/Prerequisite Skills

Activity #1 What do you know about parabolas?               (10 minutes)