Solving Systems of Equations
Activity 1: Rocket Activity
CCGPS:
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Standards for Mathematical Practice:
4. Model with mathematics
5. Use appropriate tools strategically
Rationale:
This standard is being introduced with a contextualized problem. I chose this problem because the graph is not difficult to interpret. The graph is what the student may visualize after reading the problem since x is the horizontal distance from the starting point and y is the vertical distance from the starting point. Both the quadratic equation and the linear equation show paths (the path of the rocket and the path of the laser light). This activity introduces students to solving by graph because most students will intuitively hypothesize that the intersection is the solution.
Materials:
- Computers
- Geometer’s Sketch Pad program
- GSP Activity
- Worksheets
Previous knowledge:
- Solving quadratic equations
- Finding the slope of a line given two points
Pre-Activity:
This activity is meant for students to work alone, but questioning peers could be allowed.
No introduction is necessary, but there are suggested post-activity discussion questions.
Post-Activity Discussion Questions:
- On question #2, how many students changed their original answer from #1?
- What was your original answer?
- Why did you change it?
- What did you change it to?
- What is special about the spot when the parachute is activated?
- Answer: It’s the intersection of the two equations.
- Are there any other intersections on the graph? What does this intersection represent?
- In Part 2, how did you determine the coordinates of your point P?
- How close was your estimate of the slope to your answer in #6?
Assessment:
The worksheets should be collected as the assessment for the activity.
Activity 2
CCGPS:
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Standards for Mathematical Practice:
5. Use appropriate tools strategically
Rationale:
This activity is a good follow-up to Activity 1 because it reenforces the concept of intersections as solutions to a system of equations. Students focus on how to solve a system of equations graphically. Also, this activity goes deeper into systems of equations by showing students all the possible end results of solving the system (one solution, two solutions, or no solutions).
Materials:
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Standards for Mathematical Practice:
5. Use appropriate tools strategically
Rationale:
This activity is a good follow-up to Activity 1 because it reenforces the concept of intersections as solutions to a system of equations. Students focus on how to solve a system of equations graphically. Also, this activity goes deeper into systems of equations by showing students all the possible end results of solving the system (one solution, two solutions, or no solutions).
Materials:
- Computers
- Geometer’s Sketch Pad program
- GSP Activity
- Worksheets
- LearnZillion video
Previous Knowledge:
Pre-Activity:
The night before the activity, students should be instructed to watch the following LearnZillion video and write down any questions they had:
- Solving quadratic equations
- Solving linear equations
Pre-Activity:
The night before the activity, students should be instructed to watch the following LearnZillion video and write down any questions they had:
Before starting the activity, the teacher should ask if anyone had questions from the video. The teacher should address these questions, and then implement the activity.
Post-Activity Discussion:
Post-Activity Discussion:
- Ask three students to sketch their graphs for Part A on the board and to explain why they chose that particular linear equation or how their graph has one, two, or no solutions.
- Did anyone have to change their answer for #3 after doing Part B?
- Why did you have to change it? Why did you think there was only one solution?
Assessment:
Take up the worksheets and check their graphs.
Activity 3: Graphing Calculator and Pencil-and-Paper Techniques
CCGPS:
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Standards for Mathematical Practice:
5. Use appropriate tools strategically.
6. Attend to precision.
Rationale:
For the wrap up of this standard, students must recall the methods used to solve systems of linear equations and use similar methods to solve a system of equations with a quadratic and a linear equation. I saved this activity for last because at this point they should have a solid visual understanding of solving the systems, which will help with solving algebraically.
Materials:
MCC9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Standards for Mathematical Practice:
5. Use appropriate tools strategically.
6. Attend to precision.
Rationale:
For the wrap up of this standard, students must recall the methods used to solve systems of linear equations and use similar methods to solve a system of equations with a quadratic and a linear equation. I saved this activity for last because at this point they should have a solid visual understanding of solving the systems, which will help with solving algebraically.
Materials:
- Graphing calculators
- Worksheets
- Computer access (if the assessment is done in class)
Previous Knowledge:
Pre-Activity:
Review the different methods used for solving systems of linear equations. Discuss with the students if any or all of these methods could still be used to solve a system of equations with a quadratic and a linear function.
Post-Activity:
Discuss the estimating process with the students.
Assessment:
Students are instructed to go to the link (a Khan Academy activity) at the "Assessment" button below and correctly answer 5 of the questions. Once they have done this, they must take a screenshot showing that they correctly answered the questions and send the screenshot to the teacher.
- Solving systems of linear equations
- Solving quadratic equations
Pre-Activity:
Review the different methods used for solving systems of linear equations. Discuss with the students if any or all of these methods could still be used to solve a system of equations with a quadratic and a linear function.
Post-Activity:
Discuss the estimating process with the students.
- Did anyone have an estimate that was off by a lot?
- Why do you think that happened?
- What methods did you use to solve the system on paper?
- Did any of the systems have no solutions?
- How do you know that?
- Did anyone guess wrong for the number of solutions?
- Why do you think your guess was wrong?
Assessment:
Students are instructed to go to the link (a Khan Academy activity) at the "Assessment" button below and correctly answer 5 of the questions. Once they have done this, they must take a screenshot showing that they correctly answered the questions and send the screenshot to the teacher.