Using The Geometer's Sketchpad

To Teach Systems of Linear Equations


Objectives:

1. Students will be able to graph and solve systems of linear equations using GSP.

2. Students will expand their knowledge of intersecting, parallel and coinciding lines and their relation to independent, inconsistent and dependent systems of linear equations.

Materials:

Activity Sheet

Computers equipped with GSP

Discussion of Lesson:

I would use this activity to help students reinforce what they have "learned" about systems of linear equations. Allowing students to use GSP helps them add another tool to their toolbox. GSP also makes it easier for some students to understand and visualize systems of linear equations. Using GSP cuts down on the amount of time it takes students to test conjectures and explore tangents. For example, if the students ask the question, "What if one of the numbers in the equation changes?", they will be able to explore this on GSP much more quickly than if they changed the number, changed the equation to slope-intercept form and graphed it manually.

For this activity students will be arranged in groups. Ideally these would be groups of no more than 3 so that each student would have an equal amount of input and be able to see the screen. You might have to adjust this number according to how many computers are available. However, I would absolutely not put more than 5 students in a group for this activity. So that each student gets an opportunity to operate the computer, students would rotate roles before beginning each activity.

In Activities 1 through 3, students are given an example of an independent, an inconsistent and a dependent system of linear equations. This is done to assure that they explore at least one case of each. In Activity 4 they are asked to create and explore their own system of linear equations (or choose one from their homework in which they had difficulty solving). Once they have solved it using GSP they are asked to modify components of the equations and predict and analyze the results.

Activities:

Activity One

Find the solution of the following system using GSP.

4y=3x-5

x=2y+1

1. Use GSP to graph the above system of linear equations.

2. How many solutions does this system have? Why? What are the solutions (if any)? Is this the same solution you would find if you solved the system "by hand"?

3. What do you notice about the lines?

4. Describe this system. Is it independent, inconsistent or dependent? Why?

Activity Two:

Rotate roles.

Find the solution of the following system using GSP.

4y+8=3x

2y=3/2x+6

1. Use GSP to graph the above system of linear equations.

2. How many solutions does this system have? Why? What are the solutions (if any)? Is this the same solution you would find if you solved the system "by hand"?

3. What do you notice about the lines?

4. Describe this system. Is it independent, inconsistent or dependent? Why?

Activity Three:

Rotate roles.

Find the solution of the following system using GSP.

4/3=1/3y-2/3x

x=2-1/2y

1. Use GSP to graph the above system of linear equations.

2. How many solutions does this system have? Why? What are the solutions (if any)? Is this the same solution you would find if you solved the system "by hand"?

3. What do you notice about the lines?

4. Describe this system. Is it independent, inconsistent or dependent? Why?

Activity Four:

Rotate roles.

1. Investigate your own system of linear equations. You can create a system of your own or choose one from your homework with which you had difficulty solving. Use GSP to graph this system.

2. How many solutions does your system have? Why? What are the solutions (if any)? Is this the same solution you would find if you solved the system "by hand"?

3. What do you notice about the lines?

4. Describe this system. Is it independent inconsistent or dependent? Why?

5. How could you change this system? For example, if it is independent how could you make it inconsistent or dependent.

6. Change the equation and investigate what happens to the solution to the system. Describe the changes you see.

7. Move the line and investigate what happens to the solution to the system. Describe the changes you see.

8. Compare what you see in 6 to what you see in 7.

Assessment:

The main purpose of the group activities is for students to investigate. However, to hold the groups accountable students will receive a group grade based on their write-up of the group activities. Graded on effort, each question will be worth one point each for a total of twenty points.

For individual assessment, students will complete a set of exercises and write in their journals. The exercises are to assess whether or not the students are able to do the computations, a drill and practice kind of thing. The writing is to assess whether or not they understand what is going on behind the computations. The journal graded on completion will be worth five points while the exercises graded on correctness will be worth another five points.

Exercises

Solve each system of linear equations. Describe the system.

1. 2x+3y= -10 2. 5x=10y 3. 3x+4y=1 4. 5x-3=10y

x+2y= -1 3x-6y=7 y-3=x x=2y+3

No doubt in your childhood you heard Aesop's fable of the tortoise and the hare. But did you know that this story was transformed into a math problem in which the warrior Achilles is trying to catch the tortoise? Suppose the tortoise moves at a rate of 100 yards per minute and has a head start of 1000 yards. If Achilles is 10 times faster than the tortoise, when and where will he overtake it? Set up a system of linear equations to find the answer to this question. How fast is Achilles running? Is that very fast? Think of a sport in which Achilles might excel?

Journal

1. Give your own definitions for the following terms: independent system, dependent system, inconsistent system.

2. Give an example (including the equations) of a set of parallel lines, intersecting lines, and coincidental lines. Tell whether each set is an independent, dependent or inconsistent system. The whether each set has no solution, one solution or infinitely many solutions.

3. Discuss the different methods for solving systems of linear equations learned in class. Which method(s) do you like best and why? Did one method seem easier to understand than the others? Explain.



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