Systems of Equations An Opening, Work Session, and Closing Using the TI-84 and Navigator

The Opening LearningCheck

Warm-Up Results Are there any misconceptions?

Georgia Performance Standards M8A5. Students will understand systems of linear equations and inequalities and use them to solve problems. –a. Given a problem context, write an appropriate system of linear equations or inequalities. –b. Solve systems of equations graphically and algebraically, using technology as appropriate. –c. Graph the solution set of a system of linear inequalities in two variables. –d. Interpret solutions in problem contexts.

Essential Questions How can I interpret the meaning of a “system of equations” algebraically and geometrically? What does it mean to solve a system of linear equations? How can the solution to a system be interpreted geometrically? Why is graphing a system of inequalities a good way to show the solution set? How can I translate a problem situation into a system of equations or inequalities? What does the solution to a system tell me about the answer to a problem situation?

The Work Session Creating Triangles, Using TI-84 and the Navigator

Activity: “Try”-angles If your calculator number is odd, move to the left side of the room. You will be the RED group. If your calculator number is even, move to the right side of the room. You will be the BLUE group.

“Try”-angles RED Group –Using the red paper, straight edge, and compass in front of you, create right triangles where one of the angles, x is twice as large as the other angle, y. Each member must have unique angle measures for x and y. –Cut out your triangles and place them on the grid paper in the appropriate place. Blue Group –Using the blue paper, straight edge, and compass in front of you, create triangles where two of the angles, x and y, are complementary. Each member must have unique angle measures for x and y. –Cut out your triangles and place them on grid paper in the appropriate place.

“Try”-angles Using your keyboards and complete sentences, make some observations using mathematical lingo about what you see on the grid paper.

Try-angles Using your TI-84, logon to NavNet. Plot your triangle’s “point”. Input an equation that would accurately represent the points your group plotted.

Try-angles Going back to your keyboards, answer the following: –1) What do you notice about the equation, the points, and the initial directions given to your group? –2) Are there any other triangles that could have been “plotted” that are not already on the grid paper? How do you know? –3) Create a triangle that could be placed in the fourth quadrant. What do you notice? –4) Create a triangle that could be plotted in the second quadrant. What do you notice? –5) What is significant about the the red and blue triangles that overlap?

System of Equations Each of the initial instructions for the red and blue group can be translated into equations: –x = 2y (Left or Red Group) –x + y = 90 (Right or Blue Group) You noticed after “plotting” the two points that one of the red triangles and one of the blue triangles overlapped. This overlapping is consequently where the two triangles have degree measures of 30° and 60°, called the solution. Because both equations are true for x = 30° and y = 60° at the same time, they are considered to be a system of equations.

Substitution Method Another method for solving a system of equations is by substituting one variable in one of the equations for the variable expression in the other. –x = 2y –x + y = 90 Because the first equation is solved for x, you can substitute the expression 2y in the second equation for x. (2y) + y = 90 Notice now that the expression on the left can be simplified and y can easily be solved for. 3y = 90 y = 30 Now that y has been determined, to solve for x, substitute the value for y into either equation.

Substitution Method X = 2y X = 2(30) X = 60 X + y = 90 X + 30 = 90 X = 60 Notice that the result for x is the same in both equations.

The Closing LearningCheck

Cool Down Are there any misconceptions?