Rather than teaching students procedures to solve an equation with one variable, or to solve a system of equations, students may begin with an exploration of Balance Pans to discover the balance of the left and right side of an equation.
Project the Pop Balloon Overhead for the students.
They can use Balance Pans - Expressions Tool to help them solve the problems on the overhead. (At the beginning of the lesson, you may wish to demonstrate for students how equations must be entered into the pans. The required syntax can be confusing, especially for students who are already wrestling with algebraic concepts. For your own benefit, you should attempt to use the Pan Balance on your own before trying it in the classroom, to ensure that you have enough familiarity with it to answer student questions.)
Maria tossed a small, full water balloon in the air. This situation can be represented by the equation y = -x² + 6x. (x represents time in seconds, y represents the distance from the ground in feet.)
Her brother tossed a dart at the water ballon. This situation can be represented by the equation y = 2x. Assuming Maria and her brother both started at the same exact time, from the exact same place, when would the dart reach the water balloon? [4 seconds] What is the distance from the ground when they meet? [8 feet]
When students put these expressions into the applet and move the slider, they should note two points of intersection, namely (0,0) and (4,8). (0,0) represents the problem at the very beginning, before any time has elapse. (4,8) represents the solution to the above problem.
Show students how to use the balance pan; they can place expressions in each side of the balance pan to solve two equations simultaneously. For example, place -(x^2) + 6*x. Note: It is important to show students that the tool requires the * for multiplication, and exponents, such as the power of 2, can be shown by ^2. Next, place 2*x in the right pan. Since both expressions equal y, by the Transitive Property of Equality, they should be equal to each other. Slowly move the slider to find out when the pans are balanced. The value for x may also be typed in the blue. Students should notice that when x = 4, the pans are balanced. They should also notice that the value of the expression, 8, is displayed above the pans, indicating the height.
Show students that each equation can also be graphed to find the intersection after the initial intersection at (0,0). Click and drag the mouse while on the coordinate plane. Point out to students how the contents of the red pan are graphed in red on the (x,y) coordiate plane (y = -x² + 6x) and the contents of the blue pan are graphed in blue (y = 2x). Have students determine where these two graphs intersect. Students should notice that the point of intersection is (4,8), which is the same solution as previously discovered. The slider and balance pans move to help find the point of balance, which is also the intersection.
