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Light It Up

  • Lesson
9-12
2
Algebra
Joanne Nelson
Location: unknown

In this cooperative learning activity, students are presented with a real-world problem: Given a mirror and laser pointer, determine the position where one should stand so that a reflected light image will hit a designated target.

This investigation allows students to develop several rational functions that models three specific forms of a rational function. Students explore the relationship between the graph, the equation, and problem context.

Preparation

Look through the Light It Up Activity Sheet.

pdficon Light It Up Activity Sheet 

Do the activity sheet in advance to familiarize yourself with the questions and possible answers before you use it with your students. (Answers to some of the questions on the activity sheet are provided at the very end.)

Set-Up

The class should be divided into teams of three to four students. Each student should have a graphing calculator and a copy of the Light It Up Activity Sheet. Each team will need a mirror, a laser pointer (or flashlight), a block (or a thick book), tape, and two tape measures.

This lesson is designed to guide the students to discover a connection between the problem situation, its graphical representation, and the algebraic representation of rational functions. In particular, students should be able to connect the graph of a rational function to a rational function in standard form. The standard form of a rational function is:

y = \frac{a}{{x - b}} + c.

In this lesson, the role of the teacher is to assess student understanding and listen, moving from team to team. Encourage students to work cooperatively. Refrain from answering individual student questions, especially those that can be easily answered by the team.

Main Lesson

Distribute the Light It Up Activity Sheet to all students. Note that the activity sheet consists of five parts; although the five parts are related and build on one another, you may wish to distribute them one at a time, conducting a brief discussion between parts.

A description of the problem situation appears on the first page of the activity sheet. Read the problem out loud, and then ask a student to describe the problem in her own words. Be sure that all students understand the problem that they will be solving before moving on.

In their groups, students will work on each section of the activity sheet. As mentioned above, the activity sheet consists of five parts, as follows:

Climbing the Wall: Students investigate the inverse relationship between time and speed on a climbing wall, and they represent a problem situation with a basic rational function.

Trip to the Fair: With another rational function, students describe the relationship between the number of people who go on the trip and the cost per person.

A Closer Look at the Trip to the Fair: An extension of the last situation, students consider what happens when a fee is added to the cost per student. The result is a vertical shift in the graph.

An Even Closer Look at the Trip to the Fair: When several non-paying students are added, the rational function changes again, this time with a horizontal shift.

The Light It Up Game: Students return to the problem discussed on the opening sheet. With a laboratory experiment, students search for a rational function to describe the situation.

Cooling Down: Students consider a different situation that can also be represented with a rational function. However, they are not given as much guidance in analyzing this situation.

Allow students to work collectively on the activity sheet, pausing them at certain points to discuss what they have discovered. These intermittent discussions are a good opportunity to assess student understanding and to make sure that all students are participating within their groups.

When all students have completed Question 3, have a whole class discussion about asymptotes. Talk about the definition of an asymptote and its affect on the domain and range of a function. Describe other functions, such as exponential functions and trigonometric functions, which contain asymptotes. Then allow students to return to the activity sheet.

After completion of Question 7, students should be able to write a rational function in standard form. They should also be able to state the connection between a rational function (written in standard form) and the equations of the vertical and horizontal asymptotes. After Question 8, they should be able to determine the value of a. As you circulate around the classroom, randomly ask different teams to explain the relationship between the graph and the equation. You should ask the following questions:

  • Explain how you and your group arrived at your response to questions 1(b), 2(b), 4(c), and 6(c).
  • What information do the vertical and horizontal asymptotes reveal about the equation?
  • If a is negative, what affect does it have on the graph?
  • How can you determine the value of a?

Continue to question each group until you feel that they are making connections between the graphs and the equations. Visit each group at least once.

When all teams have completed Questions 1 through 8, conduct a whole-class discussion that covers the following two questions, as well as any other questions that students may have:

  • From the graph, how can you determine the values of a, b, and c in the equation?
  • From the problem context, how can you determine the values of a, b, and c in the equation?

After completion of Question 8, students will investigate the Light It Up game. This is the main activity of the lesson. The previous questions and discussion were meant to prepare students to recognize rational functions. As necessary, help students set up the Light It Up experiment described in Question 9. You may wish to demonstrate how this investigation works by showing one example at the front of the room before students begin their own experiments.

The benefit of the hands-on experiment is that each student group will gather data by holding the laser pointer at a different height. This will result in different rational functions. On the other hand, you may wish to have all students explore the situation using the same height and therefore gathering data about the same situation. This will allow for a whole-class discussion to occur in which students have a common set of data.

Regardless of how students investigate the problem from Question 9, allow them to complete the activity and the remainder of the activity sheet.


Selected Solutions to the Activity Sheet 

1b. y = \frac{{100}}{x}.

2b. y = \frac{{1200}}{x}.

4c. y = \frac{{1200 + 5x}}{x}.

4d. If students created the equation shown in 4(c), it can be converted to standard form by dividing the numerator by the denominator. The division would look like this:

1968 division by x

Using the quotient from the division as the constant c, and using the remainder as the value of a, the equation could be written in standard form as follows:  

y = \frac{{1200}}{x} + 5.

5c. The graphs have the same shape, but the graph from Question 4 has been translated up 5 units, as a result of the extra fee.

5d. The value of b represents a vertical asymptote, and the value of c represents a horizontal asymptote.

8a. y = \frac{1}{x} + 4.

8b. y = \frac{3}{{x - 4}}.

8c. y = \frac{{ - 5}}{{x + 4}}.

8d. y = \frac{{ - 2}}{{x + 4}} - 3.

8e. The value of a cannot be determined from the graph directly. However, if the values of b and c are known, then by choosing a point on the graph and substituting its x- and y-values into the equation, the value of a can be found. (The value of a will be an approximation. Its accuracy will depend on the accuracy to which the coordinates of the point can be determined.)

9f. Assuming that the block is placed 25 cm from the wall at a height of 10 cm, and the height of the laser pointer is 150 cm, then the equation will be y = \frac{{3500}}{{x - 25}} + 10. (See the discussion above about how to obtain these values exactly from the problem situation.)

10a. The temperature of the water in the container is the average temperature of all cups of water. There are 8 cups of hot water and c cups of cold water, so there are (c + 8) cups total. The average of these is: t = \frac{{20c + 400}}{{c + 8}}.

  • Light It Up Activity Sheet 
  • Laser Pointer (or Flashlight)
  • Tape Measures
  • Tape
  • Wooden Block (at least 10 cm thick, or a thick book)
  • Graphing Calculator
  • Small, Flat Mirror
  • Carousel Cards 
  • Markers
  • Overhead Projector
  • Blank Paper
  • Timer
  • Scissors

Assessment Options
 

Play the Carousel Game. Question 10 can be used for independent assessment of student understanding.

pdficon 

 

Extensions 

Students appreciate working out a problem and seeing that their outcome reflects a real situation. After completing Question 10, students may like to see this problem in action.

After the students have algebraically determined the amount of cold water (20°C) that must be added to the container to lower the temperature of the water in the container from 50°C to 40°C, set up the experiment.

You will need:

  • Large Container (able to hold 16 cups)
  • Thermometer or Temperature Sensor
  • Calculator-Based LaboratoryTM 
  • TI-84+ Overhead Graphing Calculator
  • View Screen
  • Overhead Projector
  • 8 Cups Hot Water (50°C)
  • 8 Cups Cold Water (20°C)
  • Measuring Cup

Place the 8 cups of hot water into the container. Place the thermometer or temperature sensor in the water. Record the temperature of the 8 cups hot water; this indicates the temperature for 0 cups of cold water. Gradually pour in ¼ cup of cold water, stir, and record the temperature. Repeat with ¼ cups of cold water until the temperature reaches 40°C.

Questions for Students 

1. What was the purpose of this lesson? What did you learn?

[The purpose of the lesson was to learn about rational functions. Rational functions are quotients of two other functions, and they have a standard form.]

2. What are the connections between the values of a, b, and c in the standard form of a rational function and the graph of the function?

[The sign of a indicates whether the graph appears in the first and third or second and fourth quadrants. The value of b indicates the vertical asymptote, and the value of c indicates the horizontal asymptote.]

Teacher Reflection 

  • Was students’ level of participation high or low? Why?
  • How did your lesson address auditory, tactile and visual learning styles?
  • How did the students demonstrate understanding of the materials presented?
  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
  • Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it?
 

Learning Objectives

Students will:

  • State the domain, range and end behavior of rational functions.
  • Write rational functions that model problem situations
  • Use rational functions to solve problems