Mathematics Senior Level Capstone Course
Unit OverviewTitle of Unit: Sam, Kyle, and Kirby – A Love (of) Triangle(s) Unit Designers: Lisa Haney
Monticello High School, Albemarle County Schools
Erik Nylander
Robert E. Lee High School, Staunton City Schools
Context:
Summary of the issue, challenge, investigation, or problem.
Sam, Kyle, and Kirby are faced with several issues that can be solved using triangles.
Number of Class
Hours: 5 – 6 Hours
Unit
Design: _X_Task Based ___Project Based Other Subject
Areas/Disciplines Addressed:
None
Driving Question: How can we use concepts of trigonometry to solve practical problems? Mathematics Content
Addressed:
Pythagorean Theorem, Solve equations involving radical expressions. Law of Sines, Law of Cosines, Quadratics
MPE Addressed:
-Problem Solving, Decision Making, and Integration -Procedure and Calculation Assumption of Prior
Knowledge:
College and Career Readiness/21st Century Skills to be taught (T) during this unit or expectation (E) for student use during this unit and assessed (A):
Collaboration: Students will work in pairs or groups of 3
T & A Research: students will use internet to review and practice needed skills
E
Communication: (Oral and/or Written) Students will share their final products
E Technology: Use of Graphing calculators E
Critical Thinking/Decision Making: Students will demonstrate how they came to their conclusions
T & A Other: (Describe)
Major Products and/or Performances:
Group: Written paper will be submitted with answers to the Love (of) Triangle(s) problems.
Presentation Audience: Class School Individual: Mathematician’s Journal entries Expert
Community Other:
Launch: Event or experience used to engage the students interest and inquiry:
A big pot of spaghetti will be brought in to be shared with the class, followed by each student receiving 15 to 20 pieces of dry spaghetti.
Evaluation: Formative Assessments (During the Unit)
Interview X Practice Presentations
Mathematicians Journal X Notes
Preliminary
Plans/Outlines/Prototypes
X Checklists
Rough Drafts X Concept maps
Field Tests Other:
Summative Assessment (End of Project)
Written Products, with a rubric X 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
Resources Needed: On-site people, facilities: Facilitator/Teacher
Equipment/Technology: Internet access; graphing calculators. Materials: Spaghetti, tape, and artistic supplies Community Resources:
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal X Small/Focus Groups Whole Class Discussions X Fishbowl Discussions
Survey Other:
Material Adapted From:
Phillip’s Exeter Academy, Problems came from Exeter’s Math Department www.exeter.edu/academics/72_6539.aspx
Self and peer collaboration evaluation https://courses.worldcampus.psu.edu/public/faculty/PeerEvalForm.html
Virginia’s Senior Level Capstone Course Instructional Plan
Unit Title: Sam, Kyle, and Kirby – A Love (of) Triangle(s)
Driving Question: How can we use concepts of trigonometry to solve practical problems?
Task/Project/Problem: Sam, Kyle, and Kirby are faced with several issues that can be solved using triangles.
ENGAGE
How will student’s interested be peaked so they want to engage in the inquiry in this unit?
Number of hours _1/4_
How will the student’s interest be peaked so they want to engage in the inquiry in this unit?
Show the Videos:
Pythagoras Rap Music Video: A historical perspective.
www.youtube.com/watch?v=K3RvkV_OYE8
Student developed rap to remember the Pythagorean theorem and a demonstration of using the theorem.
http://www.youtube.com/watch?v=JvhaFCMdgeY&feature=r
elmfu
Video clips provide an introduction to Pythagoras and the Pythagorean Theorem.
Whole class discussion of the video clip – what information is provided, what questions are raised?
Mathematician Journal Prompts-
EXPLORE
Teacher provides
guidance for the explorations to prepare students with the
knowledge and skills to engage in the task.
Students will self-assess on the prior knowledge and skills assumed for the unit
Number of hours_2_
Exploration 1: Spaghetti Triangle Constructions
Students should work individually or in pairs.
Hands on Activity – Handout # 1: Use one piece of spaghetti to create two pieces so that one piece is 3 times longer than the other piece. Now if we break the longer piece of spaghetti at a random point, what is the probability that the pieces will form a triangle?
Directions for Instructor
-Let the students work for a few minutes to apply their prior
knowledge of triangles to verify that there are many different ways to construct a triangle.
-Ask the class to come up with at least one way to break the spaghetti so a triangle is not formed. Ask various students to share their thinking about this challenge.
-Class discussion about the side length conditions necessary to construct a plane triangle to bring out the Triangle Inequality theorem.
Materials and/or Resources Needed:
Mathematician Journal Prompts
Dry Spaghetti
Tape or glue
Handout # One: Spaghetti Triangle Construction
Anticipated products and artifacts if any
Triangles and non-triangle constructions
Exploration 2: Sam, Kirby and the Rowboat
Students should work in pairs or groups of 3.
See Handout # 2 for the task prompt.
Goal of activity: Use the Pythagorean Theorem to solve real world problems.
Materials/Equipment/Resources Needed:
Handout #2: Sam, Kirby, and the Rowboat handout.
Graphing calculator
Art material for creating their diagram of the situation.
Refer to Handout # 2 – Sam, Kirby, and the Rowboat.
Using the Sam and Kirby problem, the teacher will guide the students through the process of creating a diagram that accurately depicts the problem, helping them to create right triangles that can be used to derive the solution of the problem. The students learn to use their analytical reasoning to create these triangles and their knowledge of The Pythagorean Theorem to solve the triangles they have created.
The students will also use their knowledge of quadratic equations to solve the equations generated by this problem.
Students who need additional help with the mathematics in this problem can use the following websites to review the mathematics.
Pythagorean Theorem Help:
http://www.khanacademy.org/video/the-pythagorean-theorem?playlist=Geometry
http://www.khanacademy.org/video/pythagorean-theorem-ii?playlist=Geometry
http://www.khanacademy.org/exercises?exid=pythagorean_theorem_1
Law of Cosines/Sines Help
http://www.khanacademy.org/video/law-of-cosines?playlist=Trigonometry
http://www.khanacademy.org/video/proof--law-of-sines?playlist=Trigonometry
EXPLAIN
Teacher introduces the main task of the unit and
prepares students to in small group independent work...
Number of Hours_.25__
Introduce the Love Rectangle and Kyle’s Grand Adventure.
Put students in groups of two. Pairs are preferred for this activity. Student task sheet is attached; there are two versions, one with basic information and one that contains additional suggestions to get the students started. Most students should be able to use the basic information sheet at this point.
Love Rectangle and Kyle’s Grand Adventure (basic) HO #3, 4 Love Rectangle and Kyle’s Grand Adventure (with guided assistance) HO #3, 5
Checklist for students (rubric)
Explain to students that they are working in pairs. Ask them to read the task on the hand out before beginning to work together to analyze the diagrams in order to respond to the question.
Also explain to students how and when to use the Self and Peer Evaluation rubric to examine the quality and effectiveness of their team work.
Formative Assessments:
Interview of Students
Small group discussions
Rough Draft of solutions
Rubric(s) for self-evaluation of product and/or presentation, etc. Attached – Handout #6
Mathematician Journal Prompts
What challenges do you face as you begin this task? How do you plan to get started on the task? What questions do you have? ELABORATE The student groups are working independently with teacher consultations. Number of Hours__2-3__
The Love Rectangle Task Handout # 3 and Kyle’s Great
Adventure Handout #4 or Kyle’s Great Adventure (modified with hints) Handout # 5
The students work as a team to develop a response and to fully explain their answers and how they got them including all diagrams and equations that were used in developing their answers. The students are asked to draw a conclusion, connect their conclusion back to the mathematics that they used to solve the problem, and justify why this answer is their best solution.
The students submit a rough draft of each solution to the teacher for review and feedback before writing up their final solution.
During this time the teacher should be moving around the room challenging the solutions that the students are developing and asking probing questions as well as questions to advance student thinking.
Anticipated Products and Artifacts Described:
Rough draft of solutions
Mathematician Journal Prompts
Diagrams
Write up of their process for solving the problem
EVALUATE
Working groups submit products or make
presentations
Number of Hours_1_
Students will submit the team’s solutions and explanations for the Love Rectangle and Kyle’s Grand Adventure tasks, that the pair of students created. They should include any diagrams that they used in their solutions as well as the self-assessment and the peer assessment.
Map the Unit - Sam, Kyle, and Kirby – A Love (of) Triangle(s)
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:
How can we use concepts of trigonometry to solve practical problems?
KNOWLEDGE AND SKILLS NEEDED
Assumed already learned
Students will self-assess
Will be taught during the unit
1. Create diagrams using right triangles from real world problems
X X
2. Use the Pythagorean Theorem X X
3. Calculate Probabilities X X
4. Use Law of Sines X X X
5. Use of Basic Trig Functions in Right Triangle Problems
X X
6. Solve Quadratic Equations X X
7. Make Decisions about how to interpret real world problems
X X
What project tools will student’s use?
Know/need to know lists
Daily goal sheet
X Mathematician’s Journals
Briefs/Memos
Task lists
Planning Calendar
HO #1:
Spaghetti Triangle Constructions
Materials
15-20 pieces of dry spaghetti Tape or glue
1. Break a piece of spaghetti into 2 pieces so that the long piece is 3 times the length of the short piece.
2. Break the long piece again anywhere you want; you now have three pieces of spaghetti. Construct a triangle using these three pieces and attach your triangle to this sheet.
Triangle 1 Triangle 2
Triangle 3 Triangle 4
Triangle 5 Triangle 6
HO # 2
Sam, Kirby, and the Rowboat
Sam and Kirby were out in their rowboat one day, when Kirby spied a nearby water lily. Knowing that Sam liked a mathematical challenge, Kirby announced that, with the help of the plant, it was possible to
Problem Solving Self-Assessment Rubric
CATEGORY 4 3 2 1
Mathematical Reasoning
Uses complex and refined mathematical reasoning. Uses effective mathematical reasoning
Some evidence of mathematical reasoning.
Little evidence of mathematical reasoning.
Explanation Explanation is detailed and clear.
Explanation is clear.
Explanation is a little difficult to understand, but includes critical components.
Explanation is difficult to understand and is missing several components OR was not included.
Neatness and Organization
The work is presented in a neat, clear, organized fashion that is easy to read.
The work is presented in a neat and organized fashion that is usually easy to read.
The work is presented in an organized fashion but may be hard to read at times.
The work appears sloppy and unorganized. It is hard to know what information goes together.
Completion All problems are completed.
All but one of the problems are completed.
All but two of the problems are completed.
HO #3
The Love Rectangle
HO #4
Kyle’s Grand Adventure
HO #5
Kyle’s Grand Adventure (Modified)
Leaving home on a recent business trip, Kyle drove 10 miles south to reach the airport, and then boarded a plane that flew a straight course – 6 miles east and 3 miles north each minute. What was the airspeed of the plane? After two minutes of flight, Kyle was directly above the town of Greenup. How far is Greenup from Kyle’s home? A little later, the plane flew over Kyle’s birthplace, which is 50 miles from home. When did this occur?
H o m e Home
Airport
Greenup
HO #6
SELF AND PEER COLLABORATION EVALUATION
Please rate yourself and your team members on the relative contributions that were made in preparing and submitting your group paper. Your ratings will not be disclosed to other students. Be honest in this
evaluation!
In rating yourself and your peers, use a one to five point scale, where
5 = Superior; 4 = Above Average; 3 = Average;
2 = below average; and 1 = weak.
Insert your name in the first column and your peers’ names in the remaining spaces. (One name at the top of each column).
