Inquiry_Lesson_Ti_CAS Lesson Title: River Optimization Problem

Summary: This in an inquiry-based approach lesson to help students understand the different methods used for optimizing (or minimizing) a particular element of a problem. Students will use data collection, algebra, technology, and/or calculus as a means to optimize (or minimize) the elements of a particular problem.

Key Terms: Constraint, restriction, derivative, minimum, critical points, extrema.

Background Knowledge:

Using the Pythagorean Theorem

Writing an algebraic equation

Solving an equation for an unknown

Graphing functions

Concepts of the derivative

Finding graphically: Minimums & Zeroes

NCTM & Ohio Standards Addressed:

Grade 11-12:

Standard 2: Measurement.

Benchmark D: Solve problem situations involving derived measurements; e.g., density, acceleration.

Benchmark F: Write and solve real-world, multi-step problems involving money, elapsed

time and temperature, and verify reasonableness of solutions.

Grade 8-10:

Standard 4: Patterns, Functions and Algebra.

Benchmark D:Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

Grade 8-10:

Standard 6: Mathematical Processes.

Benchmark A: Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution.

Learning Objectives:

Students will be able optimize (or minimize) appropriate details of a problem using data collection, algebra, technology (TI Nspire), and/or calculus.

Materials: Picture, paper, grid paper, pencil, ruler, calculator (TI Nspire), link cable (optional)

Procedure: Students will work individually or in groups of two or three to complete the inquiry-based handout. Each student will be responsible for their own handout, but only one will be collected for each group.

Assessments: Instructor will informally assess student progress by walking around to each group, listening to students explanations, and asking questions. Instructor will formally assess student work by collecting and evaluating the written group handout.

Part 1: Teacher’s pack

RIVER OPTIMIZATION PROBLEM

Activity 1: DATA COLLECTION METHOD

Lesson plan for teachers

The problem and the extension range in difficulty from average to challenging.

Introduce the problem to the class.

Provide each student with an instruction sheet, ruler, pencil, eraser, and a grid paper.

Prior knowledge needed:

  • Measuring using rulers.
  • Pythagorean Theorem.
  • The distance formula (Distance traveled = rate of travel * time elapsed)
  • Optimization concept (minimizing, maximizing, restrictions on the domain, writing optimization equations, dependent variables, independent variables, data collection method using measuring, use of tables to draw conclusions, and extrema –maximum, minimum- concepts)
  • Knowledge of scaling and converting back to the original unit.

PROBLEM:

You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there, walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time?

This inquiry-based problem is to be assigned to students, after explaining the “optimization” concept.

  1. Introduction of the problem and discussion

Draw the picture on the board using the following steps:

  • Draw where you are (the route –lower straight line- and the exact point “you”).
  • Draw where the campground is (the route –upper straight line- and the exact point “campground”.
  • Label the points as shown in the figure.
  • Tell students that there is a slow moving river in between your location and the campground, and write down the word “river”. Draw the vertical dotted line as shown in the figure, and write down the width of the river (the distance between the 2 routes, specifically the vertical distance “1 mi.”
  • Ask students to find their options to return to the campground. Do not forget to have them justify their answers. (Students can find many possible routes will be found.) Through probing questions, lead students to realize that they should swim directly to the campground or first swim to the opposite side, and then walk straight to the campground.
  1. Summary

While verbally summarizing, write the summary on the board.

There are two options to get back to the campground:

  • Swim directly to the campground.
  • Firstswim to get to the opposite side, and then walk straight to the campground.

Swimming to the opposite side can be done through many swimming lanes, or routes.

At this point draw one of the dotted slanted segment(CD) which represents the swimming distance.

3. Introducing the restrictions

Tell students that in this problem, they are required to do both swimming and walking. What

are the restrictions? Lead them into answering that the walk distance cannot be 0.

Therefore 0 < walking distance  1.

3. Probing questions and students’ conjectures:

  • The class should discuss how long it takes to get to the campground. Students should:
  • Identify the restrictions on the walking distance
  • Guess which swimming distance (straight, etc…) they think would give them the shortest route to get back to the campground.
  • Write on the board around four or fivedifferent guesses.
  • Tell students that they will check their conjectures using a data collection activity. Show them the instruction papers.

4. Review of distance formula:

Recall: if travel is at a CONSTANT rate of speed, then

Distance traveled = rate of travel * time elapsed

Write down the formula on the board.

5. Have students solve the problem using data collection method

  • Show them the instruction papers and the grid papers that they have to use to solve the problem.
  • Divide the students intogroups of 4.
  • Distribute the instruction papers, grid papers, rulers, pencils, erasers (all of the materials to each of the students).
  • Give students a minute to look at the instruction paper.
  • Ask a student to read the directions to the class.
  • Scaling and converting back to miles:

Ask them to each reproduce the figure on their grid papers using the following scale. Have them draw 10cm to represent “1 mile”.

Teach them how to convert back to miles after measuring using their rulers.

Use the following formula:

Distance in miles = distance in cm (found by measuring using the ruler/ 10, since every 1cm represent 1 mile.)

  • Have students discuss within their groups possible routes that will lead them to the campground. Then, using the ruler, measure each of the routes that they found, and convert their measurements to miles. They should write down their answers in the table provided on the instruction sheet.
  • Using the table, deduce which of the routes gives the shortest total time to get to the campground.
  • Have students complete the instruction sheet by answering all of the questions.
  • Have students work on the extension problems.
  1. Sharing answers (on the board) with the class:
  2. Ask each group to share two or three of their findings that another group did not share.
  3. Write down their answers on the board.
  4. Check if any of the groups have found a shorter time.
  5. Using the table, deduct which route gives the shortest time to get back to the campground.

Part 1: Student’s pack

RIVER OPTIMIZATION PROBLEM

Instruction sheet for the STUDENTS

Activity 1: Data Collection Method

Date: ______Class: ______Group #: ______

Team members: ______

River Optimization Problem

Solving using data collection method

The following problem is a minimum optimization problem.

Maximize your success by following these guidelines:

GUIDELINES FOR SOLVING A MAX/MIN. OPTIMIZATION PROBLEM

Using Data Collection Method

  1. Read the problem slowly and carefully. It is important to know exactly what the problem is asking. If you misread, you have no chance of solving it correctly.
  2. Look at the sketch and figure out within your group what information you are given, and what do you have to find.
  3. Write down the restrictions on the domain.
  4. Write down what is to be optimized.
  5. Collect data and fill in the table provided.
  6. Using the data collected in the table, deduce your answer.

PROBLEM:

You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time?

Recall: If travel is at a CONSTANT rate of speed, then:

Distance traveled = rate of travel * time elapsed

Let “d” represent the distance traveled.

Let “r” represent the rate of travel.

Let “t” represent the time elapsed.

Then d = r * t t = d / r

Complete the following:

  1. What is the “SWIM rate? __ mph.
  1. What is the “TRAVEL rate”? __ mph.
  1. Recall: CD is the swim distance.

Write down the equation for the “SWIM TIME”.

______hour(s)

Recall: DB is the walk distance.

Are there any restrictions on the walk distance? If so, what are they? Justify your answer?

______

Write down the equation for the “WALK TIME”.

______hour(s)

  1. Write down in words what is to be optimized (minimized) in this problem:

______

  1. Recall: total time to get back to the campground form from your location is the total of the swim time and the walk time.

Let T = total time to get back to the campground form where you are.

Write down the equation for the total time “T”:

______ hour(s).

Data collection method to be followed

Step 1:

Scaling: reproduce the figure on the provided grid paper using the following scale: draw 10cm to represent “1 mile”.

Important note:

Below, you are asked to measure distances using a ruler. Remember that the answer that you get is in “cm” according to our scaling procedure. Since in our scale every 1cm represent 1 mile, you have to convert back your answer to miles by using the following formula:

Distance in miles = distance in cm found by measuring using the ruler / 10,

Step 2:

- Using a ruler, measure the following distances: the swim distance “CD”, and the walk distance

“DB”.

- Recall: the answer you got is in “cm”. Thus, using the conversion formula provided in the table,

convert your answer to miles, and then record it in the table.

- Compute the swim time, walk time, and total time using the formulas provided in the table and then Record your answer:

Recall: to convert your measurement with the ruler into miles use the following formula:

Distance to be recorded in table = Distance in inches found by measuring using ruler/10

Step 3:

Complete the table provided below.

Record your data for distance AD. Sort your answers, so that the shortest time is listed first, and the longest time is listed last.

AD
(mile) / CD
(Swim Distance)
= measure in cm using ruler / 10
(miles) / DB (Walk Distance)
= measure in cm using ruler / 10
(miles) / Swim Time
= swim distance in miles / 2
(hour/s) / Walk Time
= walk distance in miles / 3
(hour/s) / Total Time
= Swim Time + Walk Time
(hour/s)
0 / 10 cm / 10
= 1 mph / 10 cm / 10 = 1 / 1 / 2 = 0.5 / 1 / 3 = 0.33 / 0.83
0.1
0.2

Conclusion:

Complete the following:

  1. According to the data that you recorded in the table, what is the shortest total time to travel from where you are to the campground?

______ (hour/s).

  1. According to data that you recorded in the table, what value of x and which route gave you the shortest time?

______

  1. Record your data in the table provided on the board.

Extension 1:

Is there an advantage to record the distance AD in the table in an ascending order? Justify your answer.

______

Is there any other ways order you can use to record distance AD in the table? Are there any more advantages to using this method over the ascending order’s advantages? Give two answers and justify each one of them.

______

Summarize what you have learned in this activity including formulas:

______

Formulas learned:

______

Extension 2:

Do you think that the shortest total time to travel from where you are to the campground would be different than the one you found if you were given the choice to swim then walk, or swim only? Justify your answer.

Answer:

______

RIVER OPTIMIZATION PROBLEM

Activity 2: Solving using ALGEBRA and Ti-CAS_Nspire

Instruction sheet for the students

Date: ______Class: ______Group #: ______

Team members: ______

The following problem is a minimum optimization problem.

Maximize your success by following these guidelines:

GUIDELINES FOR SOLVING A MAX/MIN. OPTIMIZATION PROBLEM

Using Algebra

  1. Read the problem slowly and carefully. It is important to know exactly what the problem is asking. If you misread, you have no chance of solving it correctly.
  2. Look at the sketch and figure out within your group what do you have and what do you have to find.
  3. Write down the function to be optimized (minimized).
  4. Identify the domain of the function. Knowing the restrictions is a help in this step.
  5. Graph the function using your calculator and locate the minimum.

PROBLEM:

You are standing at the edge of a slow-moving river which is one mile wide, and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time?

Let x be the distance denoted in the given diagram.

Recall: If travel is at a CONSTANT rate of speed, then:

(Distance traveled) = (rate of travel) (time elapsed)

Let d represent the distance traveled.

Let r represent the rate of travel.

Let t represent the time elapsed.

Then d = rt t = d /r

Complete the following:

a)What is the SWIM rate?

___mph.

b)What is the TRAVEL rate?

___ mph.

c)Recall: DB = WALK DISTANCE

Write down the walk distance DB in terms of x:

______

Write down the walk time in terms of x:

______

d)Recall: CD = SWIM DISTANCE

Write down the swim distance CD in terms of x. Show your work.

______

Write down the swim time in terms of x:

______

e)Recall: total time elapsed is both the swim time and the walk time.

Write down the equation of the total time elapsed in terms of x:

______

f)Describe in words what is to be optimized (minimized):

______

g)Let T = total time elapsed.

Write down the optimization equations in terms of x.

______

h)Describe the restrictions (in words) to this problem:

______

i)Write down in terms of x, the domain of the function:

______

Now that the function to be minimized is determined, we need to graph the function using the graphing calculator “TI_CAS_NSpire”. We also need to determine the appropriate viewing window to observe the complete graph. Then using the minimum feature of the calculator, we should determine the minimum for the graph on the interval from [0, 1).

Graph the function using TI_CAS_NSpire. Show the complete graph.

NOTE: for additional information on how to graph a function and find its minimum using the TI_CAS_NSpire, refer to “Attachment A”.

What is the value of the minimum of the function?
______

Interpret the meaning of the minimum’s coordinates of the function T(x):
______

Conclusion:

Complete the following:

  1. According to what you have just found, what is the shortest total time to travel from where you are to the campground?
    ______
  2. According to what you have just found, what value of x and which route gave you the shortest time?

______

Extension:

Do you think that the shortest total time to travel from where you are to the campground would be different than the one you found if you were given the choice to swim then walk, or swim only? Justify your answer.

Answer:

______

Summarize the formulas you have learned in this activity:

______

Remarks
Attachment A (graphing and finding the minimum)

GRAPHING THE FUNCTION:

Follow the steps below to graph the function:

Turn the calculator on by pressing the “on” key.

Create a new document:

Press “ctrl” then “home (the key that has the home picture on). Your screen should look like the image below:

Press the right arrow key on the Nav pad (the big round button), your screen should look like the image below:

The “New Document is shaded in black; don’t change anything, Press “enter” button. Your screen should look like the image below: