LocationNotFound.doc

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Mathematics Senior Level Capstone Course Unit Overview

Title of Unit: Location not Found Unit Designers:

Denise Glassford,

Spotsylvania County Schools Rebecca Bienvenue,

Stafford County Schools Pamela Bailey, Editor

Spotsylvania County Schools Context:

Summary of the issue, challenge,

investigation, or problem.

You are lost and technology has failed, ; applying problem solving skills and proportional reasoning will help to determine your location in order to get assistance and be rescued.

Number of Class Hours:

3.5 hours

Unit

Design: _X__Task Based

___Project Based

Other Subject Areas/ Disciplines

Addressed:

Science (resources needed to survive on an island), Geography (compass directions)

Driving Question:

How are ratios and proportions useful for solving real world problems?

Mathematics Content Addressed:

Proportional reasoning; multiple representations; linear relationships; shapes in space and area.

MPE Addressed :

Problem solving, decision making, and integration

Understanding and applying functions Assumption of

Prior Knowledge:

Ability to set up and solve proportions; ability to apply the Pythagorean Theorem to calculate the hypotenuse of a right triangle; ability to solve a literal equation for an indicated variable (d = rt, A = πr2_).

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Readiness/21st Century Skills to be taught (T) during this unit or

expectation (E) for student use during this unit and assessed (A):

BIE Page 35-37

if needed)

Communication (Oral and/or Written) – poster to share their “Location not Found”

Problem and solving process

E & A Technology – use of graphing calculator

E

Critical Thinking/Decision Making – Determination of location of island

E & A Other: (Describe)

Major

Products and/ or

Performances:

Group – Poster with two methods used to solve “Location not Found” task and creation of a new “Location not Found” problem with solution.

Presentation Audience: X Class

School

Individual—Journal Entries Expert

Community Other: Launch:

Event or experience used to engage the students interest and inquiry:

The teacher will show the Gilligan’s Island Theme song video:

http://www.youtube.com/watch?v=H8Uf8Zj9W24&feature=relatedhttp://www.youtube.com/watch?v=59VRG_5A0N8

Class discussion about GPS and survival skills.

Evaluation: Formative

Assessments (During the Unit)

Interview X Practice

Presentations Mathematicians Journal X Notes

Preliminary

Plans/Outlines/Prototypes

Checklists

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Rough Drafts Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric

Peer Evaluation, with a rubric

X

Oral Presentation with a rubric

Self Evaluation, with a rubric

X Other Product(s) or

Performance(s), with a rubric

X Other:

Resources

Needed: On-site people, facilities: Facilitator/Teacher Equipment/

Technology: Internet Access, Graphing Calculators

Materials: None

Community Resources: None

Reflection Methods:

Individual, Group, and/or Whole Class

Mathematicians Journal

X Small/Focus Groups Whole Class

Discussions

X Fishbowl

Discussions

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Material Adapted From: (Provide credit for any materials or activities adapted from other sources.) Exeter’s Math Department www.exeter.edu/academics/72_6539.aspx

Gilligan’s Island Video: http://www.youtube.com/watch?v=H8Uf8Zj9W24&feature=relatedhttp://www.youtube.com/watch? v=59VRG_5A0N8

GPS Devices Articles: http://www.nps.gov/gis/gps/history.html

http://www.survivaltopics.com/survival/gps-can-fail-you/

Tutorial Sites: http://www.khanacademy.org/#browse (Specific topic links are listed on the Self Assessments and Tutorials Document)

Assessment Sites: http://www.regentsprep.org/ (Specific topic links are listed on the Self Assessments and Tutorials Document) Template adapted from Buck Institute for Education: Project Based Learning for the 21st_Century

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Virginia’s Senior Level Capstone Course Instructional Plan

Unit Title: Location Not Found

Driving Question: How are ratios and proportions useful for solving real world problems? Task/Project/Problem: You are riding in a motorboat and become shipwrecked losing all your equipment except for your radio. How will you tell the Coast Guard where you are located so you may be rescued?

ENGAGE Teacher provides an experience to engage the students. 20 minutes

The teacher will show the Gilligan’s Island Theme song video:

http://www.youtube.com/watch?

v=H8Uf8Zj9W24&feature=relatedhttp://

www.youtube.com/watch?v=59VRG_5A0N8

Teacher gives background information about the history of GPS devices (two links available below). Teacher prompts students to discuss the usefulness of a GPS and what they would do without the technology.

http://www.nps.gov/gis/gps/history.html

http://www.survivaltopics.com/survival/gps-can-fail-you/

Class discussion about survival skills, what would you do if you were stranded and did not have technology.

Mathematician Journal Prompts: Based on our

discussion if you are stranded on an island how will you get help? EXPLORE Teacher provides guidance for explorations to prepare students with knowledge and skills to

Exploration 1 (See Location not Found HO #1): Students are in groups of 2 or 3.

The goal of the exploration is for students to set up and solve proportions and discuss the methods used. Measure one step and have students determine how many steps it will take for them to walk a mile. Students compare with each other and make

necessary conversions to determine number of steps needed to walk the mile. In order to measure ONE

Mathematician Journal Prompts: Do you see any other practical situations where ratios and proportions are useful?

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engage in the task.

Students will self-assess their prior knowledge and skills needed for the unit

Number of hours—1.5

step, it is best to have the students walk about 5 or 6 steps and calculate the length of one step. Otherwise the results can be very misleading.

Teacher poses the following question: Lynn’s step measures 3 feet. She uses this

measurement in an attempt to determine the number of steps she will need to take to cover 1 mile. The result is 98 feet too long. What is the actual length of Lynn’s stride and how could she have done a more accurate job? (Exeter’s Math Department) Students solve the problem on their own, given time to discuss with a partner and then as a class the different methods they used to solve the question about the length of Lynn’s step and pacing of a mile. Teacher asks questions as needed to promote student discussion and understanding.

Exploration 2 (See Location not Found HO #1): The goal is for students to set up and solve proportions.

The teacher poses the following question: A baker makes a two- tiered wedding cake for a couple’s wedding reception. The top layer’s flat surface has a radius of 6 inches. The flat top surface area of the second bottom layer is 9 times more than the top layer. What is the radius of the larger cakelayer of the cake?

The students are given time to discuss and apply problem solving skills as they determine the solution. In whole group students discuss share different ways the problem may be solved.

Teacher asks leading questions to make sure students understand different methods to solve the problem and the role proportions play with their methods. Exploration 3 (See Location not Found HO #2): The goal is for students to self assess and complete

UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science

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tutorials as needed to ensure mastery of prerequisite skills. Websites for the self assessments and

tutorials are on the handout.

EXPLAIN

Teacher introduces the main task of the unit and prepares students for small group independent work... Number of Hours – 15 min

Students in groups of 2 or 3. They are given the peer and self evaluation rubric. Teacher goes over expectations for group work (See Location not Found HO #3).

Students are given “Location Not Found Task” handout (See Location not Found HO #4) and rubric for final product (See Location not Found HO #5). Teacher goes over expectations and answers questions.

ELABORAT E

The student groups working independently with teacher consultations. Number of Hours—1 hour

Students work with their partner on the “Location Not Found” task (See Location not Found HO #4). While students work the teacher conducts interviews and monitors progress, asking probing questions as needed.

Students complete the final product. Final product expectations:

Solve the Location not Found task using two different methods.

Create an original problem that requires using proportional reasoning when solving.

Create a poster illustrating the two methods used along with their original problem and solution.

EVALUATE Students present their original problem and solution to the class.

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Working groups submit products or make

presentations Number of Hours--30 min

Collaboration Assessment: Students complete self and peer evaluations (See Location not Found HO #3).

Teacher provides final feedback to students using the rubric (See Location not Found HO #5). Rubric also assesses communication and critical thinking skills. HO #6: Comments/Answers to Explorations and Tasks for HO #1 and HO #4.

Now that you have completed the task what would you do differently? What were the challenges that you faced? What parts did you find easy? What information would have been useful to have ahead of time?

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Map the Unit

What do students need to know and be able to do to complete the task/project/problem

successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.

Task: If you are riding a motorboat and become shipwrecked and lose all your equipment except for your radio, how would you tell the Coast Guard where you are located so you can be rescued?

KNOWLEDGE AND SKILLS NEEDED Assumed already learned

Students will self-assess

Will be taught during the unit 1. Set up and solve proportions X X

2. Apply the Pythagorean Theorem X X

3. Evaluate formulas for indicated variables

X X

4. Apply formulas to real situations X X

5. Use proportional reasoning to solve real world problems

X

6. Use linear relationships to solve real world problems

X

7. Transfer and connect between multiple representations

X

8.

9.

10.

11.

What project tools will student’s use?  Know/need to know lists  Daily goal sheet

 Mathematician’s Journals  Briefs/Memos

 Task lists

 Planning Calendar

□ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________

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HO #1

Location not Found Explorations

Exploration #1

Lynn’s step measures 3 feet. She uses this measurement in an attempt to determine the number of steps she will need to take to cover 1 mile. The result is 98 feet too long. What is the actual length of Lynn’s stride and how could she have done a more accurate job?

Exploration #2

A baker makes a two tiered wedding cake for a couple’s wedding reception. The top layer’s flat surface has a radius of 6 inches. The flat top surface area of the second bottom layer is 9 times more than the top layer. What is the radius of the larger layer of the cake cake?

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HO #2

Location not Found Exploration

Exploration #3 Self Assessments:

Proportions: http://www.regentsprep.org/Regents/math/ALGEBRA/AO3/pracProp.htm Pythagorean Theorem:

http://www.regentsprep.org/Regents/math/ALGEBRA/AT1/PracPyth.htm

Evaluating Formulas: http://www.regentsprep.org/Regents/math/ALGEBRA/AS1/PracArea.htm

Tutorials:

Proportions: http://www.khanacademy.org/video/understanding-proportions? playlist=Developmental%20Math

Pythagorean Theorem:

http://www.regentsprep.org/Regents/math/ALGEBRA/AT1/PracPyth.htm

Evaluating Formulas: http://www.khanacademy.org/video/area-and-perimeter? playlist=Geometry

http://www.khanacademy.org/video/area-of-a-circle?playlist=Geometry

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HO #3

Location not Found

Name____________________________ Partner Name_____________________ Peer Evaluation

The following is a list of statements to be answered by you about your partner. Think carefully about assigning values for each of the following statements.

Directions: Put an ‘X’ in the box that applies.

My partner… Strongly

Agree Agree Neutral Disagree

Strongly Disagree Contributed positively to discussions

Did an equal portion of the workload Helped to keep me focused on the task Was respectful of my ideas and opinions Is someone I would work with again

Self Evaluation

The following is a list of statements to be answered by you about yourself. Think carefully about assigning values for each of the following statements.

Directions: Put an ‘X’ in the box that applies.

I ,________________ , (insert name here)… Strongly_Agree Agree Neutral Disagree _DisagreeStrongly Contributed positively to discussions

Did an equal portion of the workload Helped my partner focus on the task Was respectful of my partner’s ideas and opinions

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HO #4

Location Not Found Task

At noon one day, Corey decided to take a ride in his motor boat following a straight course. After one hour of making no turns and traveling at a steady rate, the boat was 6 miles east and 8 miles north of its point of departure.

What is Corey’s position at two o’clock?

How far has Corey traveled?

What was Corey’s speed?

Assume that the fuel tank initially held 12 gallons, and that the boat gets 4 miles to the gallon. How far will Corey get before running out of fuel? When does this happen? When radioing the Coast Guard for help, how should Corey describe the boat’s position?

Final Product

1. Create a poster to demonstrate your two methods of determining the location of the boat. Be accurate and specific in your findings/explanations. Be sure to answer all questions. 2. Create a new situation and task that involves locating a person without the use of a GPS.

Incorporate the same mathematical concepts used in the problem above.

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HO #5

Location not Found Final Product Rubric

“Expert” (5 pts) “Practitioner” (3 pts) “Beginner” (1 pt) Mathematical Solution

Method #1 Accurate calculations,clear explanations, and neatly organized. Critical thinking skills and decision making evident in the

mathematics applied. Computations are correct.

Accurate calculations, weak explanations, and neatly organized. Critical thinking skills and decision making is not clear in the mathematics applied. Computations are mostly correct.

Several errors in calculations or math concepts used.

Critical thinking skills and decision making is not evident in the mathematics applied. Computations reveal numerous errors. Mathematical Solution

Method #2 Accurate calculations,clear explanations, and neatly organized. Critical thinking skills and decision making evident in the

mathematics applied. Computations are correct.

Accurate calculations, weak explanations, and neatly organized. Critical thinking skills and decision making is not clear in the mathematics applied. Computations are mostly correct.

Several errors in calculations or math concepts used.

Critical thinking skills and decision making is not evident in the mathematics applied. Computations reveal numerous errors. New Problem Scenario problem

solving reflects skills of proportional reasoning.

Critical thinking skills and decision making evident in the

mathematics applied. Computations are correct. Scenario problem solving vaguely reflects skills requested.

Critical thinking skills and decision making is not clear in the mathematics applied. Computations are mostly correct.

Skills requested will not be used to solve the problem.

Critical thinking skills and decision making is not evident in the mathematics applied. Computations reveal numerous errors. Communication Explains the problem

and solution clearly using correct

vocabulary. Answers all questions.

Explains the problem and solutions using correct vocabulary. Unable to answer all student questions.

Difficulty explaining the problem and the solution. Unable to answer all student questions.

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HO #6

Comments/Answers to Explorations and Tasks

HO #1 Comments

Exploration #1 Mile = 5280 ft.

If Lynn’s step length is 3 ft. then she takes 1760 steps in one mile. One might want to take several measurements of Lynn’s walking stride. Exploration #2

Top tier:

Bottom tier: Bottom tier = 9 (top tier) =

Be sure to inquire to determine if any additional methods were used to answer the questions.

HO #4 Comments

Corey’s position at 2:00 is 12 miles east and 16 miles north. What methods did the students use to obtain their answers? Possibilities may include graphing and tables.

At that time Corey had traveled 20 miles. Possible approaches include graphing and seeing the correlation between the 3, 4, 5 right triangle, use the Pythagorean Theorem for the first hour and doubling for two hours, or using the theorem with side lengths of 12 and 16.

Students should be able to explain why they chose their method and how they proceeded to find the answer.

Corey will run out of fuel after he has traveled 48 miles.

When does this happen? x hours = 4.8 hours = 4 hours 48 minutes

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x minutes = 48 minutes

The boat’s position when it runs out of gas is about 29.25 miles east and 39 miles north. One suggestion is to approach the problem using a table as shown below.

East North Distance Traveled

6 8 10

12 16 20

18 24 30

24 32 40

27 36 45

28.5 38 47.5

29.25 39 48.75

30 40 50

Knowing that the distance traveled is between 40 and 50 miles and the idea of midpoint a student might determine the value for east and north at 45 miles traveled and again for the distance traveled at 47.5 miles. The last step is to determine the values for the distance 48.75 which is extremely close to the 48 miles that Corey traveled.