Modeling Quadratic Data

PURPOSE: 
The goal of the unit is to have students use technology (Texas Instrument Graphing Calculator -TI-83 and a Calculator-Based Ranger- CBR) to discover connections between a quadratic equation in factored form and its graph. 

OBJECTIVES: 
Students sketch graphs, list x-intercepts, and then convert functions written in the general quadratic form to an equivalent factored form by utilizing graphing technology. As students develop awareness of the patterns in the graph of quadratic functions, the formerly difficult algebraic abstraction of factoring is easier to understand. Students are able to “see” what a factored polynomial looks like and are better able to understand conceptually the mathematics behind the symbol manipulation. The investigation and the data collection experiment in this unit give students the opportunity to model quadratic data and discover real-world meanings for the x-intercepts and the vertex of a parabola. The district curriculum requires students’ understanding of functions.

BRIEF DESCRIPTION: 
The focus of this learning unit is on understanding the importance of factoring. During the first activity, students will be placed in collaborative groups in which they will be asked to evaluate polynomials using both a paper/pencil and a graphing calculator approach. Students will be required to factor a simple polynomial expression. Next, they will enter the general form of the equation into a Texas Instrument TI-83 Graphing Calculator and sketch the resulting graph. After students list the x-intercepts, they will discover connections between the equation in factored form and its graph. Student understanding of the factored form of an equation is extended when students are asked to apply what they have learned to solve a real world problem involving maximum area. The following calculator concepts are taught during the first activity: graphing factored equations, tracing, setting windows, calculating roots, maximum and minimum points (vertex), entering data into lists, and entering equations into lists. 

During the second activity, students use a motion detector (Calculator-Based Ranger called a CBR) to find out how high each can jump. The experiment requires students to jump in front of the motion sensor while it measures the distance to the wall behind the jumper. Although the data collected is linear, a parabola is used to model the time the jumper leaves the ground until the feet return to the ground. Students see the importance of factoring and they have a real appreciation for understanding the real world meaning of x-intercepts.

ACTIVITIES:
(Note: This is a unit plan that may cover several days to several weeks. Not all of the following activities/standards will appear in the video clips used.) 

Procedures:	Curriculum Standards http://www.intime.uni.edu/model/content/cont.html	National Educational Technology Standards (NETS) Performance Indicators http://cnets.iste.org/profiles.htm
Review two forms of the quadratic equation that students have already learned. general form: y=ax2+bx+c vertex form: y=a(x-h)2+k Discuss how changing the constants transforms the graph.	Math 9-12: 2, 9
Introduction to the new form called “factored” form.  Explain worksheet instructions. Encourage students to look for patterns as they are using technology to complete it.	Math 9-12: 2, 9	Grades 9-12: 8, 10
Review group guidelines before allowing students to work collaboratively in groups of four. Remind students that I will be assessing their work as I observe and when I collect the Recorder’s paper from each group.	Math 9-12: 8
Facilitate small group discussion as students complete worksheet.	Math 9-12: 2, 6, 8, 9
Ask students go back to seats. Discuss problems #3 (opening downward), #4 (cubic), #5 & #6 (how to expand), and #9 & #10 (2 ways to find the axis of symmetry)	Math 9-12: 2, 6, 8, 9
Investigation: You have 24 meters of fencing. You want to enclose a rectangular space. What should be the dimensions if you want to have the largest possible area?	Math 9-12: 2, 3, 4, 5, 6, 7, 8, 9, 10
Investigation Questions: Make a table of widths, lengths and areas. Ask students what patterns they notice. Have students complete the table.	Math 9-12: 1, 2, 4, 6, 7, 8, 9
Ask students to make predictions about what the graphs of Length v. Width and Area v. Width look like.	Math 9-12: 7, 8, 9, 10
Review entering data into lists on the graphing calculator. Show students how to enter equations.	Math 9-12: 1, 2, 8	Grades 9-12: 8, 9, 10
Ask students what the solution would be given 40 meters and 100 meters of fencing.	Math 9-12: 2, 3, 4, 5, 6, 7, 8, 9, 10	Grades 9-12: 8, 9, 10
Experiment: How High? Review how the motion sensor (CBR) works. Ask for student demonstration of previous knowledge.	Math 9-12: 2, 4, 6, 8, 9 Science 9-12: B2, B4	Grades 9-12: 8, 9, 10
Give directions to experiment and ask that each student calculate their jump height. Ask for graph predictions.	Math 9-12: 2, 5, 8, 9, 10
Review group guidelines before allowing students to work collaboratively in groups of four. Remind students that I will be assessing their work as I observe and when I collect the Recorder’s paper from each group.	Math 9-12: 8
Students work together to collect and analyze data.	Science 9-12: A1	Grades 9-12: 8, 9, 10
Students are shown two graphs on the overhead and are asked to give real world interpretations of x and y-intercepts and vertex points.	Math 9-12: 2, 5, 6, 7, 8, 9, 10

TOOLS & RESOURCES: 
Books:
Murdock J. Kamischke, E. & Kamischke, E. (2002).Discovering Algebra: An Investigative Approach.Emeryville, CA: Key Curriculum Press.

Murdock J. Kamischke, E. & Kamischke, E. (1998).Advanced Algebra Through Data Exploration. Berkeley, CA: Key Curriculum Press.

Teacher-Created Materials:
From General to Factored Form worksheet
Experiment 1: How High? worksheet
Al’s Jump v. Allison’s Jump overhead

ASSESSMENT:
The students will be assessed on their understanding as I facilitate small group discussion. Comprehension will also be assessed during large group interaction. At the conclusion of the activities, each group will be responsible for turning in one summary report.

CREDITS:
Donna Schmitt, Mathematics Teacher at Dubuque Senior High School, Dubuque, Iowa
dschmitt@dubuque.k12.ia.us

Connie Connolly, Mathematics Teacher at Dubuque Senior High School- Dubuque, Iowa

TIMELINE & COURSE OUTLINE: 
The two activities that were videotaped were part of a unit on quadratic functions. My students are not reenacting this activity. The lessons were taped in one day but would typically require at least three days to complete. 

INSIGHTS:
I have taught the investigation part of the activity the last two years. This was the first time I assigned the “How High” experiment and the first time I allowed all the students to collect data using the CBR in a small group setting 

I have discovered (through this activity) that students enjoy being actively and collaboratively involved in the data collection process. When I videotaped students doing this activity in the past, I noticed that there was a great deal of inquiring about the lesson, taking responsibility for their own learning, understanding the value of working together, and helping one another with the technology. 

I learned from last year’s students that my worksheet “From General to Factored Form” needed improvement. On the original worksheet, students were only given the general form of the equation and then were asked to write the factored form. You will notice that on the revised worksheet, students are sometimes asked to interpret information from a graph or they are asked for the general form given only the x-intercepts. This forces students to make connections. I also changed the worksheet to include the graphs of parabolas that open downward, and I added problems so that my students would make connections between all three forms of the quadratic equation.

TECHNOLOGY RESOURCES:
I chose to use graphing calculators and a motion sensor because these tools are excellent tools to use to investigate concepts and increase mathematical understanding. When used regularly and effectively, these tools enhance learning, especially for the visual learner.

We have a classroom set of twenty TI-83 calculators. The math department has two CBR’s and three CBR’s I borrowed from the science department. 

SCHOOL BACKGROUND INFORMATION:
Population: 58,650
White Collar /Blue Collar Workers
School enrollment: 1516
Number of students with limited English fluency: 8 ESL students (English as a Second Language)
Enrollment breakdown:
1417 White, 53 Black, 21 Asian, 19 Hispanic, 6 American Indian

STUDENT CHARACTERISTICS:
The class featured in the video is my second hour Algebra class composed of thirteen girls and eight boys. There exits no ethnic or linguistic diversity among these students. Their mathematical abilities range from average (73%) to extremely bright (97%) based on their first semester grades. This class represents one of the hardest working grade conscious classes that I have ever taught. Typically these students respond positively with a hand raised high in the air when I ask questions to engage them in learning. All of these relevant features of the class influenced the selection of this unit. I was able to select hands on activities that require them to collaboratively investigate and make sense of a realistic situation because of their willingness to share ideas, learn from one another and talk about math. 

TEACHING STRATEGY:
I chose this particular teaching strategy because it provided my students the opportunity to practice communicating about math, it gave them a chance to verbalize their critical thinking and it helped them increase their understanding of why it may be important to use the factored form to model quadratic data. I think it was an effective way to help students become more responsible for their own learning. 

I feel that I address all three means of classroom management in my teaching. I address Content management most often when I lead large group and small group discussions. Every effort is made daily to connect new lessons to previous knowledge. My choice of small group seating arrangement allows students to easily work together. I have high expectations for my students to be good listeners and contributors during collaborative work time. I address Conduct management when I stop inappropriate behavior, place deadlines on incomplete assignments, and notify parents when assignments are missing. Covenant management can be seen in my room daily. It is a personal goal of mine to provide a positive learning climate that challenges each student to achieve at his or her highest level. I believe it is imperative to develop a trusting relationship with students if I want to help them be successful. Students who feel comfortable and are unafraid to ask me for additional assistance undoubtedly perform better in my classroom. 

Evolution of the Activity: 
The first year that I taught a unit on quadratic models, I focused primarily on teaching students only the “general” form of a parabola. Students were required to do many factoring polynomial problems with very little application. Needless to say, I have changed the way I approach this topic. I first realized that students needed a variety of learning strategies as well as problems with real-world contexts, when I attended my first NCTM conference in 1993. I learned how graphing technology (TI-82) could be used as a tool to help my students deepen their understanding of mathematical concepts. I can still recall attending the conference where the instructor asked us to enter some form of the quadratic equation (y = ax2 + bx + c) into the hand-held technology. I was totally amazed that the calculator produced a picture of this model almost instantaneously. What was even more fascinating was seeing how the graph changed as I changed the parameters a, b and c. I was immediately convinced that graphing technology would better help me engage my students in learning mathematics as well as enrich my unit on quadratics. 

Then, in 1996 I attended a Teachers Teaching with Technology (T3) Conference and become very intrigued by the idea of having students use technology- i.e. Computer Based Laboratories (CBL’s) to facilitate the collection of real time measurement data. When the Computer Based Ranger (CBR) became available at my school, I began looking for opportunities to use it in the classroom. 

Together, the graphing calculator and CBR technologies have allowed me to more effectively teach a unit on quadratic models. I have observed my students using these technologies to connect what they are learning to real-world situations. I have seen students excited about math because the can visually draw conclusions about functions. I have also seen how using technology has motivated students to learn more mathematics.

Technology as Facilitator of Quality Education Model Components Highlighted in This Activity http://www.intime.uni.edu/model/modelimage.html  
(Note:  This is a unit plan that may cover several days to several weeks. Not all of the elements from the Technology as Facilitator of Quality Education Model that are described below will appear in the video clips used.)  

The Principles of Learning that may have been highlighted in the video include Active Involvement, Patterns and Connections, Direct Experience, Compelling Situation, Reflection, Frequent Feedback and Enjoyable Setting. Active Involvement and Patterns and Connections should be observable in the video when the students are asked to gather into small groups to construct their own knowledge. Students are not told how to write the factored form of a quadratic equation. They are required to make Connections and observe Patterns. Numerous times I can be heard asking students, “What patterns do you see? Can you make any connections?” Students are led to discover the Connections between an equation in factored form and its graph. They observe Patterns in the table of values of width, length and area. They make the connection between a graph that opens downward and the value of “a” < 0. They also use critical thinking to analyze and make predictions about the data collected from the CBR.

The Direct Experience and Enjoyable Setting and Compelling Situation components of the Model can be noted during times when students are actively engaged in the learning process and are provided with an opportunity to directly participate. (EX: Enter data into lists / Collect data using the CBR technology). Students were motivated and excited to find out how high they can jump.

Reflection and Frequent Feedback should be easily observed when I ask students to apply an old problem to a new situation. I ask students to tell me what dimensions to use to enclose a rectangular shape given 40 and 100 meters of fencing. This solution requires students to practice what they have previously learned and to draw conclusions based on their ideas.

The Information Processing models most likely observed includes Interpretation and Evaluation. When students are asked to interpret information by drawing conclusions, reflecting and analyzing, they are better able to understand the purpose of modeling data by writing equations in the factored form.

The Critical Thinking and Decision Making components of the Model can be noted by observing student inquiry skills. Students were observers during the data collection phase of the experiment. They were expected to associate an increasing/decreasing table with a projectile motion model. They were asked to predict what the graphs of width vs. length and the width vs. area would look like. And they were asked to apply new learning to a real world problem. Individual Responsibility and Civil Involvement with Others might be observed in the video if the camera crew was able to catch the groups deciding individual roles of reader, recorder, etc.

Content Management should be observed when I lead large group and small group discussions. Students should not be permitted to talk without raising their hand. It should also be observed when students move from whole class instruction into their pre-arranged group seats that the new seating arrangement better allows them to work easily with one another. I’m not sure if Conduct Management(correcting behavior) will appear in the video. I hopeCovenant Management can be observed (students unafraid to ask me for additional assistance). 

Technology: Graphing Calculators are extremely powerful affordable tools that get students excited and interested in learning. They allow students to explore problem solving, pattern recognition and data analysis and they help students understand visually and conceptually the mathematics behind algebra symbol manipulation.

(Learning activity format adapted from National Educational Technology Standards for Students Connecting Curriculum & Technology http://cnets.iste.org/students)