Mathematics Senior Level Capstone Course
Unit Overview
Title of Unit: Which Path Should I Take? Unit Designers:
Blaire Conner,
Stephanie O’Brokta, and Bev Wynn
Fauquier County Schools
Diane Leighty, editor, UVA-SCPS Office of Mathematics Outreach Context:
Summary of the issue, challenge, investigation, or problem.
You are hired as a mathematician to find the optimal solution to two situations involving routing.
Number of Class Hours: 4 hours Unit
Design: _X_Task Based ___Project Based Other Subject
Areas/Disciplines Addressed:
Manufacturing, Navigation, Science
Driving Question: If I want to take the shortest and/or most efficient path, which one should I choose? Mathematics Content
Addressed:
Right triangles, Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry
MPE Addressed:
Problem Solving, Decision Making, and Integration Assumption of Prior
Knowledge:
Students will be familiar with the Pythagorean Theorem, the formula relating distance, rate, and time (d = r t), visualizing 3D figures, and converting units.
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 – E Research
Communication (Oral and/or Written) – E Technology- Students will have use of a calculator and a computer to create a typed report of their solutions.
E
Major Products and/or Performances:
Group – Written solutions with clear explanation and justification for determining the best path. The report must include all processes and procedures from preliminary investigation to final solution.
Presentation Audience: Class School
Individual – Students will be “jigsawed” into different groups where each will be responsible for presenting their groups’ findings with explanations.
Expert Community Other:
Launch: Event or experience used to engage the students interest and inquiry:
There are four different sized rectangular boxes at the front of the classroom and one garden flag. A student chosen at random is asked to determine which box would be best for packaging the flag. The class discusses, in large group, why one package might be better suited than another.
Evaluation: Formative Assessments (During the Unit)
Interview X Practice Presentations X
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 X Self Evaluation, with a rubric
X
Other Product(s) or
Performance(s), with a rubric
Other:
Resources Needed: On-site people, facilities:
Equipment/Technology: Computers, Calculators
Materials: 4 rectangular boxes, garden flag, poster board or chart paper to present solutions Community Resources:
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal X Small/Focus Groups X
Whole Class Discussions X Fishbowl Discussions
Survey Other:
Material Adapted From: Questions from Phillip Exeter Academy – www.exeter.edu/academics/72_6539.aspx
Virginia’s Senior Level Capstone Course
Instructional Plan
Unit Title:
Which Path Should I Take?Driving Question:
If I want to take the shortest and/or most efficient path, which one should I choose?Task:
You are hired as a mathematician to find the optimal solution to two situations involving routing.ENGAGE
How will
students’ interest be piqued so they want to engage in the inquiry in this unit?
Time: 15 min
There are four different sized rectangular boxes at the front of the classroom and one garden flag. A student chosen at random is asked to determine which box would be best for packaging the flag.
Each student records the factors they believe are important in determining which box is the best choice.
Discuss journal prompt after students have finished recording their thoughts. The class, in large group, discusses why one package may be better suited than another.
Materials needed: 4 rectangular boxes (different sizes) and a garden (or other type) flag (or other object such as a fishing rod that is of a fixed size).
Mathematician
Journal Prompts:
What are some
factors that will
determine which
box is the best?
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
Exploration 1:
Goin’ Fishin’ (HO #1)
Preparing to go on a fishing trip to Alaska, Sam wants
to know whether a collapsible fishing rod will fit into a
rectangular box that measures 40 in by 20 in by 3 in.
The longest section of the rod is 44.75 inches long.
Will the rod fit in the box?
Students brainstorm how to approach the task and
develop a sketch of that may help think about the
solution to the task. Refresh skills needed when using
the Pythagorean Theorem.
Mathematician
Journal Prompts:
#1: Student will
write initial
reactions to the task
and create a plan to
find the solution.
prior knowledge and skills assumed for the unit
Time: 60 min
Students respond to Journal Prompt #1 and then work
in pairs to solve the problem for about 20 minutes. The
teacher circulates around the room, asking questions to
help the students advance their thinking, giving
prompts and suggestions as necessary (without
reducing thinking level of students).
While walking around the room, the teacher needs to
observe the methods being used by the students. Ask
students to share their approaches, either using poster
paper and setting up a “wall walk” or selecting some
students with different approaches to share what they
did with the class. It is suggested that you do not
choose the most complete solution first, but leave that
to last.
HO #2
shows two solutions for teacher
guidance.
Have students respond in their Journals to prompt #2.
Materials/Equipment/Resources Needed: Calculator
Student self-assessment of skills required for unit and
recommendations for tutorial and/or practice – students
will evaluate expressions involving squares and square
roots, including simplified radical expressions. Extra
practice at
www.helpalgebra.com/onlinebook/roots.htm
Did they have to
make changes to
their plan? What
was the final
solution and how
did they arrive at
their answer?
EXPLAIN
Teacher introduces the main task of the unit and prepares students to in small group independent work...
Time: 20 min
Show video clip of Elvis the Calculus dog.
http://www.youtube.com/watch?v=yBG8SSB763w *If school does not allow YouTube, google Elvis the Calculus dog; there should be other clips available.Students work in groups of three.
Half of the groups will work on Desert Path, Task A (HO #3) and the other half will work on Spider Path , Task B (HO #4).
For possible solutions, see HO #6A and HO #6B.
Mathematician
Journal Prompts
How is the task
your group was
given similar to the
situation involving
Elvis?
Ask students to read the task in their groups first and then to use Journal Prompt to help plan their work.
Prepare students for the Formative Assessment:
Each
group completes a rough draft of their written solution
to be reviewed by the teacher before creating final draft
and presentation. (Three copies are needed for sharing
in Evaluate session.)
Go over HO #5 Which Path Should I Take Self and Peer Evaluation Rubric to prepare students for working in their small groups.
Explain/show in
detail your plan to
complete the task.
ELABORATE
The student groups are working independently with teacher consultations.Time: 90 min
The groups complete their given task and prepare to
present their solutions to another classmate. This
includes a written solution with explanations of
processes and procedures from initial reactions to final
conclusion. The work should be neatly organized for
others to view and follow during their explanation.
Students will be assessed by their peers as well as
themselves.
Formative Assessment:
Each group completes a rough
draft of their written solution to be reviewed by the
teacher before creating final draft and presentation.
(Three copies are needed for sharing in Evaluate
session.)
Mathematician
Journal Prompts
What problems are
you facing in
preparing your
presentation or with
solutions to your
problem? How will
you overcome these
problems?
EVALUATE
Working groups submit products or make presentationsTime: 30 min
Students regroup in pairs, one person with Task A:
Desert Path and one person with Task B: Spider Path.
Students take turns presenting their findings to each
other. This includes a written solution with
explanations of processes and procedures from initial
reactions to final conclusion. The work should be
neatly organized for others to view and follow during
their explanation.
Students complete self and peer evaluation of project.
Mathematician
Journal Prompts
What did you learn
as a result of this
project? What went
well? What could
have been done
differently?
Teacher evaluates presentations as well as the final
written solution for each group.
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.
KNOWLEDGE AND SKILLS
NEEDED
Assumed
already
learned
Students
will
self-assess
Will be
taught
during the
unit
1. Simplifying Radical Expressions
X
2. Finding the square root of the sum of
squares
X
3. Pythagorean Theorem
X
4. Problem Solving/Perseverance in
contexts not previously known
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
Goin’ Fishin’ Task
Preparing to go on a fishing trip to Alaska, Sam wants to know whether a collapsible fishing rod
will fit into a rectangular box that measures 40 inches by 20 inches by 3 inches. The longest
section of the rod is 44.75 inches long. Will the rod fit in the box?
1. Write your initial reactions in your journal and create a plan to find the solution.
2. Find the solution, clearly showing your method to arrive at a solution.
3. In your journal discuss the following:
a) Did your plan work?
HO #2
Solutions for Goin’ Fishin’ Task
Q. Preparing to go on a fishing trip to Alaska, Sam wants to know whether a collapsible fishing rod will fit into a rectangular box that measures 40 inches by 20 inches by 3 inches. The longest section of the rod is 44.75 inches long. Will the rod fit in the box?
Packing Method 1:
Use the Pythagorean theorem to find the distance (d) of
the red diagonal which represents the rod lying flat
on the bottom of the box.. floor the box.
Therefore, the red diagonal is not long enough to fit the fishing rod and the longest part of the rod will not lay flat on the bottom of the box.
Packing Method 2:
Use the Pythogorean Theorem to find the distance (d) of the blue diagonal which represents the rod lying on an interior plane created from a vertex at the intersection of three adjacent sides of the box to the opposite vertex at the intersection of adjacent sides of the box.
Therefore, the blue diagonal is long enough to fit the fishing rod.
Yes, the fishing rod would fit inside the box if it is placed diagonally inside the box.
UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science 10 40 inches
HO #3
Task A: Desert Path
Alex, a geologist, is in the desert, 10 km from a long, straight road. On the road, Alex’s jeep can
do 50 kph, but in the desert sands, it can only manage 30 kph. Alex is very thirsty, and knows
that there is a gas station 25 km down the road (from the nearest point N on the road) that has
ice-cold Pepsi.
Answer each question and clearly explain/show your solution method.
a. How many minutes will it take for Alex to drive to P (where the Pepsi is) through the desert?
b. Would it be faster if Alex first drove to N and then used the road to P?
HO #4
Task B: Spider Path
A spider lived in a room that measured 30 feet long by 12 feet wide by 12 feet high. One day,
the spider spied an incapacitated fly across the room, and of course wanted to crawl to it as
quickly as possible. The spider was on an end wall, one foot from the ceiling and six feet from
each of the long walls. The fly was stuck one foot from the floor on the opposite wall, also
midway between the two long walls. Knowing some geometry, the spider cleverly took the
shortest route to the fly and ate it for lunch. How far did the spider crawl?
Solve the problem. Clearly explain/show your solution method.
HO #5
Which Path Should I Take?
Self and Peer 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 (o
ne name at the top of each column).Names:
Participated in group discussions or meetings
Helped keep the group focused on the task
Contributed useful ideas
Quantity of work done
Quality of work done
Enter total scores here
https://courses.worldcampus.psu.edu/public/faculty/PeerEvalForm.html
Solution to Task B: Desert Path
a) Use the Pythagorean Theorem to determine that the distance from A to P through the desert is 26.926 km. If Alex can only drive at 30 kph through the desert, it will take him .8975 hours (26.926/30). Multiply by 60 to convert to minutes. Therefore, it will take Alex 53.9 minutes to drive directly from A to P.
b) To determine the time it will take Alex to drive from A to N and then to P, you must add (10/30) + (25/50) as he must drive 10 miles at 30 kph and 25 miles at 50 kph. Therefore, it takes
Alex .8333 hours to drive from A to N and then to P. Multiply by 60 to covert to minutes and we discover that it takes Alex 50 minutes to drive from A to N and then P, so yes, this route is faster. c) There are several possibilities here. You need to find any route (using Pythagorean Theorem)
that is faster than the one in part b. For example, you can drive 12.8 km through the desert and 17 miles on the road (25 – 8) and it will take Alex 46.01 minutes (found answer using same method as in part b).
UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science 14
26.92
HO #6B
Solution to Task B: Spider Path
The shortest path to the fly is indicated by the red lines in the first figure. The Pythagorean Theorem
will be used with the blue triangle shown in the second figure (and below).
d² = 10² + 30²
d = √1000
The distances indicated in the red are as follows:
6 + √1000 + 6 ≈ 43.62 feet
