Parallel+and+Perpendicular+Lines

Parallel and Perpendicular Lines



This resource tests student understanding of equations of lines, x-intercepts, y-intercepts, parallel lines, and perpendicular lines. The applet requires users to determine the equation of a line when randomly provided with two pieces of information. For instance, students may be given a line parallel to the line in question, and may be given the x-intercept of the line in question. In this case students would need to have an understanding that the x-intercept comes in the form (x,0), would need to be aware that parallel lines have the same slope, and would need to be able to effective use substitution with the equation of a line (y=mx+b). Other combinations of information would require the students to utilize different concepts, yet the end result of finding the equation of the line remains the same. Students are quizzed on finding the equation of a line when given various pieces of information. Students are finding lines in the form y=mx+b, y=a, and x=b. The applet provides the students with immediate feedback on the accuracy of their equation. When the student is correct the applet with graph the line along with the provided information so that the student receives a visual of how the information provided relates to the equation of the line. Additionally, there is a link at the bottom of the page to a step by step tutorial on how to write the equation of a line under varying circumstances.


 * Grade Level:** 8-11
 * PSSM Content Standard:** Algebra
 * Math Content:** Writing Equations of Lines, Slope, Y-intercepts, Parallel Lines, Perpendicular Lines

Evaluation & Annotations

 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

This resource centers on the ability to write linear equations. Students are given two pieces of information (y-intercept, x-intercept, point, parallel line, or perpendicular line) and are asked to determine the equation for the line. This requires students to understand how to find the equation for a line under varying circumstances and integrates many different concepts into one applet. Students need to utilize their understanding of the slopes of parallel and perpendicular lines, and need to be able to use point slope and/or y-intercept form. While the variance of the questions requires students to adapt their strategies and use different knowledge the applet ultimately requires students to use memorized processes to solve the problems. This demonstrates instrumental understanding. However, there is a focus on relational understanding as well. There is a webpage dedicated to presenting how to determine a linear equation under the varying situations. This adjoining page provides a deeper level of rationale and promotes relational understanding. Furthermore, once the students have answered a problem correctly, the line and other information become visible on the graph. This allows students to understand the connection between the symbolic and the graphical representation.


 * How does learning take place? What are the underlying assumptions (explicit or implicit) about the nature of learning?**

Learning takes place through a written lecture and practice. Students are able to view a detailed written lecture with example problems detailing all of the situations that could arise. Additionally, students are learning through practice with questions being pulled from a large question bank, which eliminates the possibility that students will memorize answers. The applet requires that students truly understand the processes needed to find the equation of a line. The underlying assumption is that students should be able to grasp such a concept without assistance. The written lecture does not have any links for further understanding and does not directly address common misconceptions. Also, the applet itself does not provide hints for struggling students and does not given any constructive feedback when a question is incorrect. This resource is designed for the independent learner.


 * What role does technology play? What advantages or disadvantages does the technology hold for this role? What unique contribution does the technology make in facilitating learning?**

Technology plays a crucial role in the practice aspect of this resource. The dynamic software of the applet provides users with fresh questions that randomize the information presented. Technology provides the students with immediate feedback on the accuracy of their answers, although the feedback is rather shallow. If the user answers correctly the dynamic software of the applet graphs the line and the other information that was given to link the symbolic and graphical representations for users. Without technology it would be impossible for students to receive such a large quantity of diverse questions with immediate feedback and multiple representations.


 * How does it fit within existing school curriculum? (e.g., is it intended to supplement or supplant existing curriculum? Is it intended to enhance the learning of something already central to the curriculum or some new set of understandings or competencies?)**

The topic of writing equations of linear equations is already present in the curriculum. This resource would best be used to supplement existing lessons on writing linear equations. Students would still need direct instruction from a teacher to understand how to write the equation of a line under varying circumstances. The written lecture provided with this resource would best be used as tool to assist student, not to teach them. Students need more detail than the written lecture can provide. This resource could be very beneficial for student practice in a tutorial atmosphere or outside of the classroom. Students would be able to receive immediate feedback while practicing on surplus of diverse questions.


 * How does the technology fit or interact with the social context of learning? (e.g., Are computers used by individuals or groups? Does the technology/activity support collaboration or individual work? What sorts of interaction does the technology facilitate or hinder?)**

This technological tool would best fits with individual practice. The basis of the lesson is not exploration, thus students reflecting and discussing conjectures is not appropriate. Rather this applet would be beneficial for students to review example problems and practice diverse questions. Again, the immediate feedback provides students with an accurate assessment of their progress and understanding. This allows students to practice individually with this tool. Peer collaboration could be integrated into this activity, yet there is the risk that one student would simply answer while the other would stand by and observe. Without prompts built into the resource the teacher may wish to implement reflection questions into the lesson so that students could discuss why certain processes work.


 * How are important differences among learners taken into account?**

Difference is learners are taken into account briefly. When students answer a question correctly the linear equation is graphed and the provided information is also displayed. Yet this graphical representation does not assist students dealing with the symbolic forms of lines. The graph is only displayed at the end, which does not allow students to transition between multiple representations to determine an answer. Also, the applet does not take differences in learning ability into account either. There is no option for hints, difficulty level, or constructive feedback when answers are incorrect. The resource assumes that all learners should be practicing equally without assistance.


 * What do teachers and learners need to know? What demands are placed on teachers and other "users"? What knowledge is needed? What knowledge supports does the innovation provide (e.g., skills in using particular kinds of technology)?**

Entering into the use of this resource, students need to have experience graphing linear equations and solving linear equations. Without a basic understanding of graphing and solving equations students will not be successful with this resource. Also, and need to have a relational understanding of slope, x-intercepts, y-intercepts, parallel lines, and perpendicular lines. This applet should be used as a culminating activity of a unit on linear equations. Finally, students need to have experience writing linear equations before using this applet. The webpage lecture with explanations and examples is not sufficient to educate students about writing the equation of lines. Students need to be able to pose question and see multiple representations. Teachers need to be aware that there are many prerequisites for students and need to be available for support and guidance, depending on the degree of understanding students may possess. To operate the resource there are no technological requirement. In fact, there are no directions, as it is assumed to be self-explanatory.