84 - Brothers or Betties

Lesson Focus / The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
Materials / TI-Navigator, TI-83 Plus/TI-84 Plus, Activity Sheets for students
Grouping / Groups of 3 or 4
Prerequisite Knowledge and Skills / Order of Operations
Linear Functions
Overview of the lesson / The students will write function rules to model the methods of payment used by competing companies, use a system of equations to answer a question, and create a new company to compete with the given companies.
Time / Suggested time for Activity 1: 45 – 55 minutes
Suggested time for Activity 2: 45 – 55 minutes


Activity 1

1. Ask the students toanswer the questions from Activity Sheet 1. For students who are having difficulty starting, you may have them start with question 2 and prompt them with questions such as the following.

  • How would you calculate the cost of having 10 boxes? 20 boxes?
  • What is a constant?
  • What is a variable?
  • What is the independent variable?
  • What is the dependent variable?

Note: the following table or graph could be used to find the solution:

2. Discuss with the students their answers to Activity 1.

  • Are the variables the same for both companies? How did you decide on the independent variable?
  • Describe how you wrote your function rules.
  • What method of solving systems did you use to get your answer? Why did you choose this method?
  • What is your solution? Does it make sense in this problem situation? Why or why not?
  • How can this information help you answer the question even though the solution seems unreasonable?
  • What observations and conclusions can you draw?

Activity 2:

1. Have the students work on the student sheet. If students are having trouble, lead them back to Activity 1, and ask them to follow the models represented there.

2. With the students logged into NavNet, go to the ActivityCenter to the Graph-Equation screen. Before starting the activity, have the students help you set up the appropriate window in order to see the starting points and intersection point. (You may always use the drag (hand) features and the zoom (spyglass) feature in order to see different parts of the functions closer.)

To start the activity:

  • Contribute: Equations
  • Configure:Main Settings:
  • Number of equations per student = 1.
  • Let students view graphs.
  • Send current graph contents as background.
  • Students start with:Empty equations
  • OK
  • Edit the Window Settings (This is where the students should help you based on their calculations and domain and range information).

3. Explain to the students that they should examine their new function compared to the originals before they submit to you. They can do this with the GRPH tab on their calculator controlled by the soft key “GRAPH.” This will allow students to troubleshoot their functions based on the graph before they are asked to submit.

4. Students will submit one group at a time, and present their commercial.Pause the activity after an assigned group has entered their equation. Questions to ask after the commercial presentation:

  • How did you reach the conclusion that your function crosses the intersection of the original functions?
  • How did you choose your y-intercept (starting point)? How did you choose your rate per box?
  • What concepts that we have learned before helped you complete this task?

You can delete the groups function by clicking on their line and pressing delete. This will help keep the graph clear for the next group’s presentation. Resume the activity for the next group to submit and present their commercial.

5. Stop the activity after all the presentations. Have all groups submit their functions at once and vote on who they has created the most reasonable and affordable function. Discuss why they voted the way they did. Be sure to have students consider the number boxes that Conrad might be moving.

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Brothers or BettiesName ______

Conrad is moving across town and needs to determine which moving company to use. He has a choice between Brothers Bulk Movers or Betties Moving Buses. Their representatives give him the following bids:

Brothers will charge $179.50 and $1.27 per box.

Betties charges $175 plus $1.35 per box. Which is the better value when hiring a moving company?

1. Explain why a system of equations will help Conrad decide which company would be best to choose.

2. Define variables to describe Conrad’s situation. Which is the independent variable? The dependent variable?

3. Write a system of two equations using the two variables you defined.

4. Solve the system. Explain how you solved the system

5. Is your solution reasonable? Explain your solution and why it is or is not important to help you answer this question: Which is the better value when hiring a moving company?

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6. Create a function for a competing company. This function must intersect the functions for Brothers and Betties at the intersection point found in Activity 1, #4.


7. Write a verbal description of your company’s pricing.

8. Sketch a graph of the original companies as well as your competing company in the box below. Be sure to label the graph appropriately.

9. Write a commercial for your company using the graphs to support your claim that your company would be a better deal. Be sure to include information about the number of boxes that a customer will be moving.

Your teacher will ask you to submit your function.

TI MathForward™1 © 2008 Texas Instruments Incorporated