Linear+Inequalities

Linear Inequalities



This applet enables users to graph systems of linear inequalities. Students are able to graph up to four linear inequalities in standards form by inputting the coefficients, the constant value, and the inequality sign. Once the linear inequalities are entered users are able to view the equation when y is solved for and when x is solved for. This enables students to determine if they correctly manipulated the equation to standard form. The resource graphs the linear inequalities and shades with multiple colors to demonstrate the solution points. Students are also able to enter coordinate pairs and the applet will graph the points. This enables users to visually test is a point is a solution to the system of linear inequalities.


 * Grade Level:** 9-11
 * PSSM Content Standard:** Algebra
 * Math Content:** Linear Inequalities, Systems of Linear Equations, Systems of Linear Inequalities

Evaluation & Annotations

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

This resource is designated to educate students about systems of linear inequalities. Students are able to enter linear inequalities and equations in standard form by entering a, b, c, and the inequality or equal sign. Students then are able to view the subsequent graphical representations, which deepens their understanding through multiple representations. The applet also allows students to plot points to visually see if the point is a solution to the system of linear inequalities or equations. One of the perks of this tool is that students can easily change the inequity sign of a linear inequality, update the graph, and view the variation in the inequality and the system. This could be used as an exploration tool for students beginning or progressing through a lesson on linear inequalities. Another aspect of mathematics that is being reviewed through this activity is manipulating equations. Students are able to input an equation or inequality in standard form and view the equation or inequality in slope intercept form to verify that they solve for it correctly. With the wide array of possibilities for this applet, relational understanding is definitely furthered. Students are able to place an equation or inequality in standard form and check the accuracy of their manipulation, view the graphical representations of equations and inequalities, and check solution points. Tinkering with the inequality signs and analyzing the results also creates a relational understanding. The adjoining web pages on graphing linear inequalities supports student understanding on a symbolic level. The one drawback of this resource is that it does not prompt students to check solution points by substitution, does not encourage student to attempt to manipulate equations, and does not create prompts for students to analyze the relevance of different inequality signs. This is a powerful tool, which can create a relational understanding, yet teacher input and development of supplemental resources is essential.


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

Through this resource learning could take place through exploration. Students can be prompted to test different linear inequalities and determine patterns between the inequality signs. Additionally, students can be encouraged to substitute solution points into systems of linear inequalities and make connections between the points that yield true statements and their placement on the graph. The underlying assumption of this resource is that students will develop a deeper understanding of linear inequalities through multiple representations and through exploring how different aspects of linear inequalities appear on a graphical representation. This tool can also be utilized for practice. Students can use the features of the tool to verify their manipulation and graphing of equations and inequalities. Students can also use the tool to check their ability to graph lines, plot points, and find the solution to systems of equations. However, this is all contingent upon teacher direction.


 * 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 integral role in the use of this applet. Dynamic software allows linear inequalities and equations to be graphed and viewed in standard form and slope intercept form. The dynamic software additionally allows for points to be plotted and the window of the graph to be easily adjusted. Without such technology, students would be unable to verify if their manipulation of linear inequalities and equations was accurate, and they would not be able to verify their graphs and solution points. The exploration opportunity of this resource would not be possible without dynamic software. Students would not be able to analyze how different inequality signs affect solutions and how solution and non solution points appear on the graphical representation without such technology.


 * 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 would best be used as a supplement to existing curriculum. It could be integrated into lessons with a few different topics: systems of liner inequalities, systems of equations or linear equations. If used with systems of inequalities the applet could be presented to students in order for them to explore the connection between solution points symbolically and graphically. Students could be prompted to make the connection that a solution point which yields a correct statement through substitution will also appear in the shaded area of the graph. The connection between the various inequality signs could also be explored. Student could be prompted to graph various different equation and be asked how including the "equal to" with an inequality affect the graph or how the "greater than" and "less than" inequalities differ on the graph. If using this with systems of equation or simply graphing linear equations, students would be able to check their answers. This would provide a stand-alone instructional resource that students could use to verify their work for accuracy outside of the classroom.


 * 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?)**

If this applet is used for the discovery of systems of linear inequalities, collaborative work would be beneficial. This would encourage students to reflect and discuss the connections between the symbolic and graphical representations deeply. If this applet is used for practice, as a means to verify answers, students could easily work independently. In this case, students would need to have to be provided with sufficient notes because the applet does not explain the process of manipulating equations or graphing linear equations and inequalities; the answers simply appear.


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

This applet focuses on the connection between the graphical and symbolic representations of systems of linear inequalities and equations. The dynamic software provides students with detailed graphs that can easily be manipulated by altering the a, b, and c values, as well as the inequality signs. While the applet meets the needs of visual learners, there is little detail with the symbolic nature of systems. The applet only displays the equation without explaining the manipulation and how the slope and y-intercept are displayed. There is a general lack written explanation as well. This tool requires that students complete the symbolic aspect of the problems individually, which requires teacher supervision. This resource also does not provide different avenues for learners with varying abilities. Given the nature of the applet and its fill in the blank option for equation, there is no way it could. Instead, responsibility of ensuring each learner's needs are met falls on the teacher.


 * 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)?**

Upon entering into this applet, assuming the aim is systems, students need to be able to plot points, and manipulate equations. The points are automatically plotted when entered and the equations are automatically manipulated from standard to slope-intercept form without explanation. Students also need an understanding of graphing linear equations. This is not touched upon by the applet or the resource's web pages. Students will be fine without an understanding of graphing linear inequalities and solving systems, as the applet has the potential to foster discovery on these topics. Teachers need to be aware that this is not a stand-alone resource. It is very powerful in its ability to allow students to build a relational understanding through discovery, yet it needs to be supplemented with prompts and exercises. There are discussions provided on the website, however teacher's creating their own lesson specific to their students' needs would be preferred. For students to discover the relationship between a point that creates a true statement when substituted and its placement on the graph of a linear equation, there needs to be very carefully constructed questions and prompts. The same is true for students analyzing inequality signs. One other aspect of the applet with which teachers need to be cognizant is the lack of direction on the applets features. Altering the window, plotting points, altering the thickness of lines, and changing between connected lines and scatter plots are aspects of the applet that the teacher will need to explain to students prior to the implementation of the resource.