WhichPathShouldITake.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science 1 Partnership Grant Program NCLB Title II Part B

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

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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

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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

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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:

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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

Time: 60 min

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.

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

Mathematician

Journal Prompts:

#1: Student will

write initial

reactions to the task

and create a plan to

find the solution.

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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.

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.

Mathematician

Journal Prompts

How is the task

your group was

given similar to the

situation involving

Elvis?

In what other ways

(without calculus)

could you solve the

task you were given.

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

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EVALUATE

Working groups submit products or make

presentations

Time: 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.

Teacher evaluates presentations as well as the final

written solution for each group.

Mathematician

Journal Prompts

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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:

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

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

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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?

b) If not, what changes did you make?

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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. 40 inches

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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?

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S F

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?

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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 (one 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

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HO #6A

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).

26.92 6

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d

30 f t 10 f t

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