TITLE:  Parabolas and Straight Lines

TASK DEVELOPER: Karen Murray

CONTENT AREA AND GRADE: 12th Grade Algebra II

TARGET TEACHING DATE: February 20,21 2007

SCHOOL: John F. Kennedy High School

STANDARDS:

PATTERNS AND ALGEBRA - GRADE 9-12

STANDARD 4.3 PATTERNS AND ALGEBRA: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Strand B. Functions and Relationships: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.

2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

  • Domain and range
  • Intercepts
  • Intersecting points as solutions of systems of equations

Strand C. Modeling: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Use functions to model real-world phenomena and solve problems that involve varying quantities.

  • Expressions, equations and inequalities
  • Applications in mathematics, biology, and economics (including compound interest)


Strand D. Procedures: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

2. Select and use appropriate methods to solve equations and inequalities.

  • Linear equations - algebraically
  • Quadratic equations - factoring (when the coefficient of X2 is 1) and using the quadratic formula

Back to top

PERFORMANCES:
  1. The students will find the common solutions for a system of equations (a parabola and a straight line).
  2. The students will use algebra to show that a system of equations has no common solution.

Back to top

SETTING:

You are a computer wholesaler. The price that a retailer pays per computer varies based on the number of computers purchased. Using the information that the price of each computer begins at $500 and the price for each computer is reduced by $20 when additional computers are purchased, you need to make a table that shows the pricing structure and revenue. In your table you need to list the number of computers purchased, the price per computer, and the revenue per order. Find the maximum revenue per order. How many computers must be sold per order to attain the maximum revenue?

Back to top

SMARTSKILLS:

Level I: Acquiring Data - Data students will acquire in this standards-based task:

Vocabulary: common solutions, parabola, quadratic equation, linear equation, intersection, complex roots

Images: Graphs of the above functions.

Skills: After reviewing all the examples, point out that a system of equations, point out that a system of equations consisting of a parabola and a line can have zero, one, or two solutions. Draw additional examples to illustrate each situation.

Concepts: In most practical applications, the common solutions are not whole numbers, and the solutions can only be determined algebraically.

Level II: Visualizing Information - Data from Level I that are visualized an information in this standards-based task: Graphs of parabolas and linear equations.

Organizing: The similarities and differences of the graphs and their solutions.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:  Graph and solve algebraically the solutions.

Solving problems: Solve problems finding the common solutions for a parabola and a straight line, or where is no common solution.

Back to top

PREFERENCES:

Student Involvement - The students will complete the task: Some group and some individual

Instruction - Activities will be organized and delivered:

  • As a teacher-facilitated set of hands-on activities
  • In a student booklet during class time

Special Education Accommodations - Students with special needs will require calculators.

Use of Resources - The school will provide classroom time to complete the task.

Use of Resources - The students will provide classroom materials such as pencils, paper and notebooks.

Customer for Student Work - The student will present their work as evidence of task completion to the teacher.

Assessment of Student Work - The student's teacher will be involved in assessing student work generated to complete the task:

Assessment of Student Work - The following forms of assessment will be used to determine progress and results: rubric for student class work and a performance assessment for the test.

Performance assessment - A test which includes 10 questions requiring to give the common solutions, if any, of systems of equations. If there is no common solution, it should be stated.

Reporting Results - The assessment results will be reported as a letter grade for class work and a number grade for the test.

Timeline - The estimated time needed to plan, teach, and score this task is one to three class periods.

Back to top

ACTIVITIES:

Quality-Driven Teaching

Activity 1: Setting High Expectations
Estimated time for this activity: 20 minutes

  • Step 1. Students will construct a table to show the pricing structure and revenue of a computer wholesaler.
  • Step 2. Begin with a price of $500 for one computer, the price for each computer is reduced by $20 when additional computers are purchased. Have groups study the information in the table, then use the table to complete the items.
  • Step 3. Write the function for the price P of x computers in simplified form. Write the function for the revenue R on an order of x computers.
  • Step 4. Answer the following questions: What is the maximum revenue per order? ($3,380) How many computers must be sold per order to attain the maximum revenue? (13)
  • Step 5. Ask groups to share their methods for solving the problem with the class.

Technology for this activity: calculator
Materials for this activity: pencil and paper
Student product or performance for this activity:Completed table and answers
Scoring tool for this activity:Teacher assessment

Activity 2: Activating Prior Knowledge
Estimated time for this activity: 15 minutes

  • Step 1. Warm up activity: Have students evaluate functions for given values. Point out that some of the functions are linear functions and some are quadratic functions.
  • Step 2. Discuss the difference between the graphs of each type of function.
  • Step 3. Review the concepts of slope, y-intercepts, and zeros of a function.

Materials for this activity: Notebooks
Student product or performance for this activity: Students produce their notebooks with sketches of functions.

Activity 3: Acquiring Data
Estimated time for this activity: 20 minutes

Explain to students that a common solution means the y-values of each equation are equal at the points of intersection. Write the equations as equal to one another and solve for the x-values of the points of intersection.

  • Step 1. Review problems on page 130,131 in textbook. Find the common solutions.
  • Step 2. Show the common solutions on a graph on the blackboard.
  • Step 3. Review problems on page 132 in textbook. Show that there are no common solutions.
  • Step 4. Show how no common solution is shown on a graph. Graph on blackboard.

Materials for this activity: chalk and chalkboard, student textbook (2004 AGS Algebra 2)
Student product or performance for this activity: student notebook

Activity 4: Visualizing Information
Estimated time for this activity: 10 minutes

  • Step 1. Ask students to sketch a graph of a parabola and a line that have exactly one common solution.
  • Step 2. Ask a student to volunteer to show their answer on the blackboard.

  • Step 3. Ask students to sketch a graph of two parabolas intersecting.
  • Step 4. Ask a student to volunteer to show their answer on the blackboard.

Activity 5: Applying Knowledge
Estimated time for this activity: 10 minutes

  • Step 1. Ask students to work individually or in pairs to find common solutions. Teacher will provide 5 examples.
  • Step 2. Students will be given 3 problems that have no common solution.
  • Step 3. Teacher will walk around the room and observe class work.

Materials for this activity: student notebook and calculator

Activity 6: Reporting Results
Estimated time for this activity: 15 minutes

  • Step 1. Students are given a worksheet on systems of equations and can work in groups to complete.
  • Step 2. Students may finish worksheet for homework.

Materials for this activity: student worksheet
Student product or performance for this activity: completed worksheet
Scoring tool for this activity: teacher graded

Source: Adapted from: Haenisch, S. (2004) Algebra 2. Circle Pines : AGS.

Back to top

BENCHMARKING:

Student Performance One: Students can create their own sketches of a systems of equations with no solution and with common solutions.

Back to top

SCORING:

New Jersey High School Mathematics Rubric: Brief Constructed Response Items

Mathematics Rubric: Brief Constructed Response
Level 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
Level 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
Level 1

The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
Level 0 The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."
Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
Source: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

Back to top

METACOGNITION:

Cognitive Information: I will collect the following information by asking the questions to students in class and summarizing their verbal responses.

  1. Describe what skills you needed to complete this task.
  2. Explain how you solved the goal, problem, or issue in this task.
  3. Draw a picture that shows how you solved the goal, problem, or issue in this task.
  4. Explain why you completed the task your way.

Attitude Information: I will collect the following information by asking the questions to students in class and summarizing their verbal responses.

  1. Do you feel that you are good in (add content area, e.g., mathematics)?
  2. Did you find this task to be difficult?
  3. Did you see the usefulness of what you were asked to do in real life?

Back to top

RESULTS:

Reflect: I will consider two or more of the following stems to reflect on the results and instructional practices I used and others I might benchmark and apply in the future. Then, I'll write a brief summary about my findings, contributing factors, and implications for improvement.

As I relate my students' results with my lesson activities, I noticed that the students in attendance enjoyed the group activities. According to my selected scoring rubric, I graded each group as level 2. Many of the students required help and were unsure of their answers, but they demonstrated a conceptual understanding of the problem.

Summarize:

The poor attendance in this class had an effect on the test scores. Some of the students tested were absent the previous two days and received very low scores. The class average was only 56 % although some students performed very well. As mentioned earlier, the students enjoyed working in groups.

This task took more time to implement than in the past. Because of the quantity of material that must be taught in this course, it is difficult to spend extra time on a topic that was previously afforded one class period. The results of the test were not higher due to the extra time needed.

Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on:
Title: Word Problems using linear equations
Content Area: Mathematics, Algebra II
Learning Standard(s): 4.3  4.5
Intent: To show alternate methods of solving word problems such as guess and check, substitution, and elimination.

Back to top