Completing+the+Square

Completing the Square



This resource takes students through the process of completing the square. Problems are presented with varying difficulty from when a=1 to when a>1. Written directions are presented and students are required to fill in the missing numbers. First students are required to “move the constant term to the right”. Immediate feedback is provided on the accuracy of the step. If the user inputs the correct answer the next step is presented. If the answer is not correct there is specific feedback to the error and students are given another attempt to correct the mistake. As the students progress they will perform the following steps with directions from the applet: “add the magic number to both sides”, “factor the left hand side and simplify the right hand side”, “take the root of both sides”, and “finally solve for x”. On all of the steps there is immediate feedback which is specific to the step. When finding the magic number the applet will inform students if they forgot to square the number or to divide by two. The feedback is very detailed. For students to better understand the process there is a link after the applet to another page, which describes the rationale for each step.


 * Grade Level:** 9, 11
 * PSSM Content Standard:** Algebra
 * Math Content:** Quadratics

Evaluation & Annotations
This resource focuses on solving quadratic equation by the process of completing the square. The applet takes the students through the steps needed to complete the square and provides the name of the steps, while requiring students to fill in the missing information in the problem. This aspect of the applet encourages an instrumental understanding. Students are memorizing the steps and the necessary procedures without any deep understanding of the reasoning behind them. However, below the applet there is a link to another page, which explains the process of completing the square in depth. Several times I have visited the page it has been down so its accessibility is in question. Still, when up and running, the page provides a relational understanding of the process to the students. Students are then able to understand where the "magic number" is derived from and why each step is essential in solving the equation. In summary, the applet itself is centered around instrumental understanding, yet the resource with additional pages enables students to build a relational understanding.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

Through this applet learning takes place through a webpage lecture on the rationale behind completing the square and practice with the process itself. The underlying assumptions are twofold. First and foremost, it is assumed that students' needs to understand why each step is essential in completing the square, as provided by the webpage. Second, it is assumed that practice makes perfect. The applet provides an enormous bank of practice questions, each with hints and immediate feedback on the accuracy of student answers. This applet was created based on the assumption that students are capable of learning the rationale and process of completing the square on their own with a brief lecture and immediate feedback and guidance from the applet.
 * How does learning take place? What are the underlying assumptions (explicit or implicit) about the nature of learning?**


 * 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 an essential role with this resource. The webpage which provides a relational understanding of completing the square is not interactive and could have been presented without the use of technology. However, the applet itself uses the power to technology to provide extensive practice opportunities to students. The applet draws questions from a large test bank, which halts students from practicing the same problems and memorizing answers. Next, the resource guides students through the necessary steps by the names of the steps, while requiring students to fill in the blanks. The most useful feature about the technology is its ability to provide immediate and question specific information. If the students are stuck, they are able to request detailed, question specific hints. Once the students complete a step and hit enter the applet either progresses them to the next step if they were correct, or provides them with specific information on how their answer was incorrect. This level of feedback is astounding and is allows students to receive more assistance than a teacher could provide in a room of students.


 * 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?)**

This resource could be utilized in a number of ways. For independent learners it could supplant the existing curriculum. Students with a sound foundation of mathematics may be able to understand the webpage explanation of completing the square and use the feedback from the applet to fine tune their ability to complete the process. This would allow the teacher to provide assistance to those struggling, and would enable students to practice grasping concepts on their own. Conversely, this applet could be used with students who are more dependent and who do not have a sound mathematical base. In this case the applet itself could be used after a lesson and guided practice on completing the square. The applet would be a stepping stone for students, as they completed problems from the applet for homework with the opportunity for assistance. In both cases the applet would be covering material that is already a crucial aspect of the curriculum, yet would provide a more supported way to do so.


 * 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 technology would best be used in an individual setting. The webpage with a detailed description of completing the square enables students to look back on a lesson for guidance. The applet itself provides hints and detailed immediate feedback. Students are able to receive information on why their answer was incorrect and use the information to individually correct their error. This resource provides all of the necessary tools for an individual to practice and review completing the square independently. It may be a good idea for a teacher to conduct a lesson prior to the practice of the concept, yet once the students have a base knowledge of completing the square there is no need for group work.


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

Difference in learners are definitely taken into account with this resource. The implementation of the hint feature allows students with difficulty of the process the opportunity to receive a push in the right direction, while users more confident with their ability to complete the square can proceed without the feature. The webpage with notes on completing the square also offers those needing assistance a detailed explanation of the process and several sample problems. While the resource address potential differences in learning ability, it does not address visual learners' needs. The applet providing a graphical representation of the zeros at the end of the problem would be very helpful for students to see and would further ingrain in their minds the purpose of completing the square.


 * 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)?**

This resource requires that students are capable of manipulating equations, working with exponents, and taking square roots. Without these basic skills students will become bogged down in the smaller steps of the process and will be unable to focus on the progression of completing the square as a whole. Having a basic understanding of completing the square prior to using the resource is not a requirement, yet it would be highly beneficial for students. Not all students are going to be able to understand the webpage explanation without the resource of an instructor. There are no specific technological requirements of the applet. There are directions provided for students, yet the process of using the applet is self explanatory. Teachers need to be aware that there is not the connection made between the process and the graph of the quadratic function. Therefore, teachers need to stress the connection to students in other aspects of the lesson.