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Design Project: Quadratic Modeling with the TI-Nspire

## Project Topic

In this course students will be working with parabolas. Students will not only be solving quadratic equations but investigating the nature of quadratics. This course will include lessons on completing the square, the quadratic formula, vertex form, standard form, comparing and contrasting the two forms, nature of the roots.

## Intent of Project

The purpose of this project is introduce students to quadratic equations and their graphs: parabolas. Students will first work with quadratic equations and solve them using a variety of methods. Once students are comfortable solving quadratic equations, they will begin to analyze their graphs and find relations between graphs and their corresponding quadratic equations. Representing these parabolas in two forms students will begin to appreciate the appropriate context for each form. Learning objectives are listed below according to the five major categories of human performance (Gagne, 2005, p.48).

**At the conclusion of this course students should be able to:**

- solve quadratic equation using the quadratic formula. (intellectual)
- solve quadratic equations by completing the square. (intellectual)
- find maximum and minimum values of a quadratic equation. (intellectual)
- come up with a method to aid in the memorization of quadratic equation. (cognitive)
- determine the cues to decide which form to use. (cognitive)
- use prior knowledge of absolute value equations and translations to aid in standard form of parabolas. (cognitive)
- memorize the quadratic formula. (verbal)
- state the value two forms of quadratic equations. (verbal)
- express the value of quadratic equations in real world context. (attitude)
- understand the use of parabolas in the real world.
- graph parabolas in the Cartesian plane. (motor)
- use the TI-Nspire to graph parabolas. (motor)

## Needs Assessment

### Instructional Problem

The students need to meet curriculum objectives provided by New York State namely: A2.A.24 Know and apply the technique of completing the square, A2.A.25 Solve quadratic equations, using the quadratic formula, and A2.A.46 Perform transformations with functions and relations: *f*(*x* + *a*), *f*(*x*) + *a*, *f(-*x*), -*f(*x*), *af*(*x*). Also, students also need to be familiar with various forms of technology as they prepare for the constantly changing job market that awaits them at the completion of their education.

### What is to be Learned

Students will learn a variety of skills based on the New York State Standards. These skills will be categorized into two main skills: solving quadratic equations and graphing parabolas. When these skills are mastered students can begin to appreciate parabolas in real-world applications. Students will also learn how to use technology to help them understand the nature of parabolas.

### About the Learners and Instructional Context

This course is designed for students taking Algebra II and Trigonometry, usually individuals of Junior class standing in high school. Students in this class have already successfully passed Algebra I and Geometry in ninth and tenth grades respectively and have taken the corresponding Regents examinations for the courses. Students in my current course excel at basic algebra and can easily isolate variables in equations. They have difficulty in understanding the reasoning and relationships in the mathematics they are completing, however. For this reason, students will be working in this course to develop a deeper understanding of the relationships between solving equations algebraically and graphically. My students are from varying economic backgrounds. Due to the economic climate of the area, technology excites most of the students since they cannot experience recreational technology that others can afford. This being said, various forms of instructional technology can be powerful intrinsic motivators for the students who otherwise would not have such experiences.

Students will be learning in cooperative learning communities of the instructor’s choosing throughout the course. These learning communities will be heterogeneous in nature and will require the students to work with students of various academic levels. In addition to varying academic skill, students will also have various experiences with computers. As the TI-Nspire is document-based and has commands similar to that of Microsoft Word, students who are familiar with Word will have an advantage over others. Students will not only be responsible for their own learning but assisting others in their groups as well.

### Instructional Problems and Solution

The students need to meet curriculum objectives provided by New York State in regards to solving quadratic equations. In addition to solving quadratic equations, student must know how to graph a parabola and how to translate the parent function of parabolas. With the students coming from various backgrounds students will be grouped together so that they can learn from each other. This ensures that there is at least one “expert” in each group. This expert may be the academic and technology expert or there may be two different “experts” in each group. Through collaboration student will be able to use technology to graph and understand parabolas.

### Goals

- Have students work in groups
- Have each group member understand how to solve quadratic equations
- Provided general instruction on quadratics
- Provide instruction on using the TI-Nspire
- Have groups work independently on equations of varying difficulty
- Have students relate absolute value equations with parabolas
- Have each students convert parabolas form vertex form to standard form
- Have each students convert parabolas form standard form to vertex form
- Have each student understand the importance of parabolas in real world applications

## Performance Objectives

At the conclusion of this mini-course, students should be able to do the following:

- [Situation] Given a quadratic equation in standard form, [LCV] demonstrate [object] how to find the zeros [action] by using the quadratic formula [tools and constraints] without any errors on 4 out of 5 problems with the use of the TI-Nspire.
- [Situation] Given a quadratic equation in standard form, [LCV] demonstrate [object] how to find the zeros [action] by using the method of completing the square [tools and constraints] without any errors on 4 out of 5 problems with the use of the TI-Nspire.
- [Situation] Given a quadratic equation in standard form, [LCV] demonstrate [object] how to transform the equation to vertex form [action] by using the method of completing the square [tools and constraints] without any errors on 4 out of 5 problems with the use of the TI-Nspire.
- [Situation] Given a quadratic equation in vertex form, [LCV] demonstrate [object] how to transform the equation to standard form [action] by using FOIL and basic algebraic rules [tools and constraints] without any errors on 4 out of 5 problems with the use of the TI-Nspire.
- [Situation] Given a quadratic equation in standard form, vertex form, or a graphic, [LCV] identify [object] the roots of the equation [action] by looking at a graph [tools and constraints] without any errors on 4 out of 5 problems with the use of the graphic capabilities of the TI-Nspire.
- [Situation] Given a situation with a real world context, [LCV] choose [object] to reflect on the situation [action] using quadratic equations [tools and constraints] with various forms of technology.
- [Situation] Given a quadratic equation in any form [LCV] execute [object] a proper parabolic function on a Cartesian graph [action] by graphing the equation [tools and constraints] both with and without the aid of the TI-Nspire.

## Task Analysis

**Objective:** Given a quadratic equation in standard form, demonstrate how to find the zeros by using the method of completing the square without any errors on 4 out of 5 problems with the use of the TI-Nspire. (Learning Hierarchy for Objective 1)

*Enabling Prerequisites:*

- Simplify radicals
- Factor trinomials of degree two in one variable
- Identify perfect square trinomials.
- Solve linear equations in one variable.

*Supportive Prerequisites*

- Understand that every number has two square roots.
- Understand the difference between constants and variables

**Objective:** Given a quadratic equation in standard form, demonstrate how to find the zeros by using the quadratic formula without any errors on 4 out of 5 problems with the use of the TI-Nspire. (Learning Hierarchy for Objective 2)

*Enabling Prerequisites:*

- Simplify radicals
- Substitute values into an equation
- Identify perfect square trinomials.
- Simplify fractions with an expression in the numerator

*Supportive Prerequisites*

- Understand that every number has two square roots.

**Objective:**Given a quadratic equation in standard form, demonstrate how to transform the equation to vertex form by using the method of completing the square without any errors on 4 out of 5 problems with the use of the TI-Nspire. (Learning Hierarchy for Objective 3)

*Enabling Prerequisites:*

- Understand the process of completing the square.
- Know and remember the vertex form of a parabola.
- Determine if a quadratic equation is in vertex form.

*Supportive Prerequisites*

- Understand how to graph a absolute value equation.
- Find similarities between standard form or an absolute value equations and vertex form of quadratic equations.

**Objective:**Given a quadratic equation in vertex form, demonstrate how to transform the equation to standard form by using FOIL and basic algebraic rules without any errors on 4 out of 5 problems with the use of the TI-Nspire. (Learning Hierarchy for Objective 4)

*Enabling Prerequisites:*

- FOIL binomials.
- combine like terms.
- Determine if a quadratic equation is in standard form.

*Supportive Prerequisites*

- Understand you can not "distribute" the exponent but multiply the expression by itself.
- Find similarities between slope-intercept form of a line and standard form of quadratic equations.

**Objective:**Given a quadratic equation in standard form, vertex form, or a graphic, identify the roots of the equation by looking at a graph without any errors on 4 out of 5 problems with the use of the graphic capabilities of the TI-Nspire. (Learning Hierarchy for Objective 5)

*Enabling Prerequisites:*

- Graph functions in the TI-Nspire
- Navigate through the TI_Nspire's menus
- Find the zeros of an equation given a picture of its graph.

*Supportive Prerequisites*

- Create documents on the TI-Nspire
- Create graphs of parabolas
- Learn shortcuts on the TI-Nspire

**Objective:**Given a situation with a real world context, choose to reflect on the situation using quadratic equations with various forms of technology. (Learning Hierarchy for Objective 6)
*Enabling Prerequisites:*

- Utilize the TI-Nspire to create quadratic regressions.
- Navigate through the TI-Nspire's menus
- Understand how maximums, minimums, and zeros can aid in modeling with quadratics.

*Supportive Prerequisites*

- Understand the use of the gravity constant in parabolas
- Appreciate the use of mathematics in an architectural and motion context.
- Visualize parabolas in a real world context.

**Objective:**Given a quadratic equation in any form execute a proper parabolic function on a Cartesian graph by graphing the equation both with and without the aid of the TI-Nspire. (Learning Hierarchy for Objective 7)

*Enabling Prerequisites:*

- Create a table of values to plot points on a parabola.
- Translate graphs both vertically and horizontally.
- Graph parent functions.
- Utilize trace tools in the TI-Nspire.

*Supportive Prerequisites*

- Understanding what influences whether a graph has a maximum or minimum.
- Check so see if algebraic formulas for zeros and axis of symmetries are refuted or confirmed by graph.

## Instructional Curriculum Map

The following link will show you a Link Between the Target Objectives

The following link will show you a ICM for Creating a Deeper Understanding of Quadratics with the TI-Nspire

## Resources

New York State Department of Education. (2005). Mathematics Common Core Curriculum: Algebra 2 and Trigonometry. Retrieved from http://www.p12.nysed.gov/ciai/mst/math/standards/a2trig.html

Gagne, R. M., Wagner, W. W., Golas, K. C., & Keller, J. M., (2005). Principles of Instruction Design. Belmont, California: Wadsworth, Cengage Learning.