"We asked students to invent procedures, such as the one of the inequality, by looking at a variety of examples, conjecturing and justifying them by using symbolic manipulations when available. Furthermore, we tried to pose problems that cannot be answered directly by reading the screen, but rather that require interpretations of the visible outcomes. The role of the technology we used was to furnish examples and representations to analyze and from which to reason."
Solving Systems of Linear Equations Graphically
Lesson Objective:
Students will be able to interpret and solve systems of linear equations by identifying the intersect visually on a graph.
Launch:
Begin class with the questions:
- Has anyone watched the summer olympics?
- Does anyone like watching the sprinters?
- Has anyone ever been in a track and field?
- What would happen if we raced an olympian and a high schooler?
- What kind of suggestions do you have to make the race more fair if they must start at the same time?
Lay out the parameters:
- Olympian, Usain Bolt, beat Olympian record for the 100m dash at an avg. speed : 10.44 m/s
- Curtis Godin holds the Orange County record for the 100m dash at an avg. speed: 9.49m/s
- Start by giving Godin a 10 meter head start
- Draw a picture, so students know how to
Initial question:
- Who will win the 100 meter dash? (with a 10 meter head start)
- Will Bolt ever pass Godin? If so, at what time will Bolt and Godin be right next to each other? If not, is there ever a point when they will be next to each other?
Note:
If students are comfortable with coming up with equations given a set of information, give them time to translate this problem into mathematics independently.
If students struggle with creating equations, walk through the process as a class. This may be a good time to introduce kinematic equations. However, try not to spend too much time with them trying to figure out the equation. The graph and systems of equations portion is the emphasis for this lesson.
Explore:
Instructions for students:
- Have students use Illuminations' interactive called, "Pan Balance--Expressions" Click Here for Pan Balance Applet
- Have students input their two equations: one for Usain Bolt, the other for Curtis Godin.
- Have students try to see where the two sprinters will meet
- Manipulate the head start values, and even the speeds to see how this changes the point where they are at the same distance away from the starting line
- Encourage students to ask "What if.." questions during the explore
- What does the x value/slider in the applet represent in our scenario?
- What do the numbers above the red/blue box represent?
- What does it mean when the balance is down? up?
- What happens on the graph as the balance changes position?
- What does the change in the balance's position represent in our Bolt/Godin race?
- What is happening to the graph at the distance where they meet?What happens if we change Godin's head start to a different number? Who wins this time?
- What if Godin gets no head start? Will there be a point where they meet? What does the graph look like?
- What if Bolt gets a head start? Will there be a point where they meet? What does the graph look like?
- Who will win the 100 meter dash? (with a 10 meter head start)
- How do you know?
- Did you look at the numbers? Scale? Graph?
- Will Bolt ever pass Godin?
- Does this make sense logically (Remember Bolt is an olympian athlete, and Godin a high school athlete)
- Does this make sense mathematically?
- What proof can you show me mathematically?
- If so, at about what time will Bolt and Godin be right next to each other?
- How did you determine this? Why did you choose that method to determine it?
- Did you look at the numbers? Scale? Graph?
- What is happening to the graph at the point?
- What happens if we change Godin's head start to a different number? Who wins this time?
Make the connection between mathematics and the sprinter scenario concrete for students:
Give definitions of:
- Systems of Linear Equations: 2 or more linear equations
- Solution(s) to a System of Linear Equations: a point which satisfies 2 or more linear equations
- Intersections: a point on a graph where two functions "cross"--more precisely, where the x and y values are the same on both of the graphs
Relate back to the sprinter scenario, and identify the different pairs of linear equations, the solutions, and the intersections.
Make it explicit that one way to solve a system of linear equations is graph, and find the point of intersection, much like was done in the activity.
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