Algebra

In looking at the PSSM, the key mathematical goals for the middle grades  focus on // understanding patterns, relations, and functions, representing and //// analyzing mathematical situations and structures using algebraic symbols, //// using mathematical models to represent and understand quantitative //  // relationships, and analyzing change in various contexts. // Algebra tends to be one of the most difficult strands to learn according to some students. Providing students with an engaging way to learn these concepts will help them acquire the skills necessary for higher mathematical learning. Here are just a few ways we can utilize interactives to foster their growth:

Algebra Evaluation #1

 **//An interactive lesson in graphing and finding the equation of lines //** || **Algebra: **  · Slope  · Equations of lines  · Coordinate plane || * || //Rationale //: This online interactive allows students to apply the relationship between the plotting points in a coordinate plane and creating a line through those points. Using the (x, y) pairs, students create a table and formulate an equation to determine the slope of the line formed. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Instrumental understanding <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Relational understanding || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="color: #000000; font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';"> : The focus of this applet is on both instrumental and relational understanding. Students practice the procedure of plotting given points and relaying coordinate pairs when given a “planet” or point to put in a table. To get helpful hints, students can click on [|Key Ideas], which reminds them of vocabulary terms and the procedures for plotting on a coordinate plane. Students can also get real time tips in the middle of problem solving if they need help placing points in the correct order. While completing this interactive, students write equations to the lines they have created by connecting their journey through the given planets. Students then can see the relationship between linear lines, positive and negative slope, and how each is reflected in the coordinate plane. ||
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//__<span style="font-family: 'Times New Roman','serif'; font-size: 24pt;">[|Planet Hop] __//**
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Which standard? **
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|understand patterns], relations, and functions;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|represent and analyze] mathematical situations and structures using algebraic symbols;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|use mathematical models] to represent and understand quantitative relationships;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|analyze change] in various contexts. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet focuses on understanding how to plot points in a coordinate plane and use the properties of algebra to devise equations of lines formed by the points. Under the pretense of finding the coordinates of planets, students identify points in the coordinate plane, organize them in a table and use the pattern in the table to write an equation. Students use a model of the solar system within a coordinate plane to discover the relationship between equations and their graphs. After working through several equations, students should be able to see that slope represents the constant rate of change in linear functions.  ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What mathematical content is being learned (or intended to be learned)? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Is the focus on instrumental or relational understanding? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: This technology affords students the opportunity to plot several different functions on the same coordinate plane and compare and contrast the lines each equation creates. Students may refer to the planets they have plotted and the coordinates they represent in the table, all on one screen. Students are also given the virtual path of the line in a smaller screen on the bottom right that helps them determine virtual plots and helps them make conjectures about the slope of the line. When students make incorrect assumptions, they are immediately alerted and given feedback on how to correct their answer.

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">This technology also supports the role of teacher by providing different levels to enhance differentiated instruction. Each level requires a different challenge to the student all surrounding the basic principles of writing equations and finding relationships within the lines those equations create. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">practice <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">direct instruction/explanation <span style="color: #000000; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;"> · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration  || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students practice plotting points and writing them in a table format with x and y respectively. Direct instruction is provided about writing equations, finding slope as well as some basic information about the coordinate plane (see above in key ideas). Students learn throughout the activity by exploring different equations and applying how these equations relate to the slop of the lines. Students begin to realize patterns within the coordinate plane that help them determine positive and negative slope. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What instructional function(s) does the resource serve? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What kinds of representations of the mathematics are used? **

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real-world objects <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet represents mathematics in a variety of ways. Symbolically, the interactive shows numerals that can be manipulated by the user as well as numerals within the coordinate plane. Graphically, math is represented through the use of the coordinate plane. Students are able to visualize the path of the planets they have plotted and the line is created by a rocket ship that travels to and from each planet. Using such real world objects helps students to connect the math they are learning to how it might be used in real-world situations. Within this applet, there are lots of movements, accompanied by sound to reflect the mathematical concepts within creating a line and finding its slope. ||

<span style="color: #800080; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Algebra Evaluation #2

<span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//<span style="font-family: 'Times New Roman','serif'; font-size: 16pt;">An interactive lesson in graphing and finding the intersection of lines //** || **<span style="color: green; font-family: 'Times New Roman','serif';">Algebra: ** <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Solving Systems of Equations || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students utilize the balance feature to simultaneously graph two equations. Students can manipulate the x-values for each equation and determine if there is a point where the two pans balance. They begin to see that if two lines intersect, they will balance at the point of intersection. They also notice that if two lines do not intersect, there is no solution to the set of equations. || <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Relational understanding || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: The focus of this applet is to have students investigate systems of equations and explore how the intersection of two lines helps to solve a system of equations. In order to work the applet, students have to be familiar with graphing simple equations and the properties of equality before beginning the simulation. Rather than teaching students procedures to solve an equation with one variable, or to solve a system of equations, students use the exploration of this applet to discover the balance of the left and right side of an equation, as well as its intersection points. || · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration   || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students begin using the applet in an exploration of simply entering equations and seeing how the two graphs relate to each other. Students then are given specific equations to investigate with specific x-values to change each equation. By guiding students in this way, students are able to discover for themselves how the various x-values change within the line on the graph. By using the slider, students come to understand that the intersection point between the two lines balance the pans thus solving the system of equations. ||
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//<span style="font-family: 'Times New Roman','serif'; font-size: 20pt;">[|Pan Balance] //**
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Which standard? **
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|understand patterns], relations, and functions;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|represent and analyze] mathematical situations and structures using algebraic symbols;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|analyze change] in various contexts. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet focuses on the equivalence concept in algebra and students explore the concept while investigating pairs of lines. The object of the interactive is to have the two pans balance or to find solutions of equations that are equal. Students look for patterns and relationships between two equations to find their “balance”. Students represent and analyze systems of equations and use graphing to evaluate their solutions. Students use a slider to analyze change as the value of x changes in both equations. Students discover that equations will balance when x equals the intersection point. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What mathematical content is being learned (or intended to be learned)? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Is the focus on instrumental or relational understanding? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: Students need an exploration phase to become familiar using this applet before its applications can be utilized. Instead of giving students the procedures for solving equations, this technology affords students the opportunity to discover the rules of solving systems of equations. This technology allows students to represent their knowledge and thinking through finding the balance point between two equations. Graphs are automatically displayed when an equation is placed in the pan, saving students painstaking time of graphing two equations using at least three points on graph paper. Although this activity could be completed individually, it is best served when students are able to communicate their findings and help each other through explorations to determine the balance effect of the interactive. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What instructional function(s) does the resource serve? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What kinds of representations of the mathematics are used? **

<span style="font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real world objects <span style="font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This interactive uses various methods of displaying mathematics. Students type in numerals and algebraic expressions using the keyboard given in the simulation. The equations are automatically graphed in a coordinate plane and students can manipulate the equations within the pan. Students are able to visualize how equations change based on the x-values they assign to each equation and can quickly see the line changes within the graph. Students use a balance pan to compare and contrast equations, a device many of them should be familiar with. The applet is dynamic as it moves and changes lines and graphs, the points on the graph represented in the same colors as the pans, and the slider changes the x-values at the user’s manipulation. ||


 * <span style="color: #800080; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Algebra Annotated Links: **


 * **Screen Shot** ||  **Name and Description**  ||
 * [[image:Lure_of_the_Labyrinth.jpg width="353" height="192" align="center" link="http://labyrinth.thinkport.org/www/index.php"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;"> The [|Lure of the Labyrinth] is a mathematical game that has embedded math into a storyline that has students saving their virtual pet from monsters living within the interactive world. Students need to use algebraic thinking to progress through three levels, each with their own puzzles to work through. The mathematical thinking continues in each of the puzzles. Each puzzle arrives with its own set of variables - and each player gets a different set of numbers to work with. To solve the puzzle successfully, players have to find a strategy that works for them. Teachers can set up students in “teams” to play with each other, or as individuals to unlock each level. Each level comes with standards based lesson plans and a recommended time-line, as well as other resources for teachers found in the “Educators” section. ||
 * [[image:Algebra_vs_Cockroaches.jpg width="367" height="230" align="center" link="http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;"> In [|Algebra vs. Cockroaches], students must find the slope intercept form of an equation before “cockroaches” take over the line. The interactive gives explicit instructions with examples and identifies the mission for finding the slope intercept form of a line. Students are reminded to find the slope of the line by using rise over run and are directed to place this before the x in the equation. Then students identify the y-intercept to complete the slope intercept form. During the interactive, students are timed to build fluency and if they are correct, all the cockroaches are destroyed. If 16 cockroaches appear on a line, students have “lost” to the cockroaches and must complete another slope intercept form of another line. At any time, students may revert back to the instructions to see the examples. ||