3.2 Applications of Systems of Linear Equations

Lesson Plan 3.2: Applications of Systems of Linear Equations

Objectives:

• Students should be able to organize information in word problems involving two or more variables, and use this information to construct a system of linear equations.

Suggested Lecture Breaks:

• MWF: You have two days. Try to get through problem 5 on the first day.
• MW/TR: You have one day.

Introduction:

In this section, most students should recognize this as an extension of the word problems in one variable. However, many students skate by on their memory of how to solve word problems in one variable, and when they get to word problems in more than one variable they look at it and say "I don't know what to do." The literature indicates a student's proclivity to put minimal effort into the critical thinking skills necessary to solve problems due to either a lack of desire to do so or a low level of self-competency, either of which may be evidenced by the aforementioned behavior. This lesson plan is meant to help guide students in organizing their approach to solving word problems in two variables. To this end, the lesson plan is designed to give students a structure for how to approach each problem using a modified version of the steps to solve word problems from the textbook.

Additionally, students often forget to declare their variables and indicate the units in their setup of the problem and sometimes in the final answer, so this is also emphasized in the lesson plan.

Suggested Lecture Notes:

• Emphasize that these word problems are just an extension of the word problems done in Section 2.3
• (Note: the most essential part of this task is the use of diagrams and tables to organize information as well as the final presentation of the answer to a word problem) List out the following modification of the "guidelines for word problems" steps from the textbook

1. Read the problem carefully
   1. Identify information given with appropriate units
   2. Identify information that needs to be found with appropriate units
2. Assign variables to represent unknown values
   1. Draw diagrams and/or tables to organize your information
3. Write a system of equations using both variables
4. Solve the system
5. State the answer in a complete sentence in terms of how the problem was stated
6. Check answer

• Emphasize declaring variables
• Work through two different examples using this method
  • Suggestion: Solve a mixing problem as the first problem. For the second problem, you might do something like problem 6 of the reading guide.
  • For the first problem, maybe work through that with some student interaction; the second problem should involve heavy class-teacher dialogue
  • Heavily emphasize the importance of understanding and writing the correct units

Comments on the handout: (50-55 minutes)

• Check to make sure they are declaring variables
• Question 3: This one is tricky - remind students that when we define variables, we need units (in this case dollars). This hints at the fact that we need to be using the dollar value of nickels and quarters.
• Question 4: This problem looks long, but each piece is straightforward and students seem to prefer it over others.
• Question 8: This requires a system of three linear equations, which the students might not realize. Encourage them to define all the variables they see necessary.
• Question 11: This requires some pondering - what does it mean for a canoe to be going upstream vs. downstream in the context? Well, downstream implies the current is helping, so we add the current rate. Otherwise, we subtract the current for going upstream.