DontGetFloggedByLog.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B Revised 12/11/12

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Mathematics Senior Level Capstone Course

Unit Overview

Title of Unit: Don’t Get Flogged by the Log Unit Designers:

Lisa Rosazza (Page) Tiffany Comer (Page) Laura Hansen (Culpeper)

Editor: Diane Leighty, UVA-SCPS Office of Mathematics Outreach Context:

Summary of the issue, challenge, investigation, or problem.

A logging company needs to purchase bands to secure logs on transport trucks. Your class has been asked to develop a method to determine the length of the bands needed based on the number of logs.

Number of Class

Hours: 3 hours

Unit

Design: _X__Task Based ___Project Based Other Subject

Areas/Disciplines Addressed:

Writing

Driving Question:

A logging company needs to purchase bands to secure logs on transport trucks. How can you determine the

length of the bands needed based on the number of logs?

Mathematics Content Addressed:

Proportions, similar figures, perimeter, circumference, right triangle relationships and patterns.

MPE Addressed:

Problem Solving, Decision Making and Integration Assumption of Prior

Knowledge:

Solve proportions, find perimeter and circumference, basic understanding of triangles and circles.

College and Career Readiness/21st Century

Collaboration- Students will work in pairs (or a group of 3 if needed)

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2 Skills to be taught (T)

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

Communication (Oral and/or Written)- written paragraph explaining strategy

A Technology- use of graphing calculator E Critical Thinking/Decision Making-

determine band length

T & A Other: (Describe)

Major Products and/or Performances:

Group

Determine the length of the band required to secure n rows of n! logs.

Presentation Audience: Class School Individual

Write a paragraph explaining how the length of the band changes as logs are added or taken away.

Expert Community Other: Launch: Event or

experience used to engage the students interest and inquiry:

Ask the following question, “Have you ever been driving up a mountain behind a logging truck and

wondered if the logs are not on securely?”

Image from Final Destination 2, sitting behind logging truck in car.

Evaluation: Formative Assessments

(During the Unit)

Interview X Practice Presentations

Mathematicians Journal Notes

Preliminary

Plans/Outlines/Prototypes

Checklists X

Rough Drafts Concept maps

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3 Summative Assessment

(End of Project)

Written Products, with a rubric X Peer Evaluation, with a rubric

Oral Presentation with a rubric Self Evaluation, with a rubric

X Other Product(s) or

Performance(s), with a rubric

Other:

Resources Needed: On-site people, facilities: Teacher Equipment/Technology: Calculators

Materials: Classroom, rulers, graph paper, textbooks, string Community Resources:

Reflection Methods: Individual, Group, and/or Whole Class

Mathematicians Journal Small/Focus Groups X

Whole Class Discussions X Fishbowl Discussions

Survey Other: X

Material Adapted From:

Problems came from Exeter’s Math Department www.exeter.edu/academics/72-6539.aspx

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Virginia’s Senior Level Capstone Course

Instructional Plan

Unit Title: Don’t Get Flogged by the Log

Driving Question: A logging company needs to purchase bands to secure logs on transport

trucks. Your class has been asked to develop a method to determine the length of the bands

needed based on the number of logs.

Task/Project/Problem: Students will need to find the length of the metal band that surrounds

the logs.

ENGAGE

How will students’ interested be piqued so they want to engage in the inquiry in this unit? Number of hours _.1__

Ask the following question, “Have you ever driven up a mountain behind a logging truck and wondered if the logs are on securely?” Images from Final Destination 2 of person sitting behind a logging truck in car.

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

The goal of this activity is to use ratios and the Pythagorean Theorem to solve problems involving similar triangles, quadrilaterals and circles.

Exploration 1:

Provide students with HO #1 and assist students through the problem by asking questions that relate this problem to similar triangles and proportions. Students can use the tools provided to assess themselves and if necessary use the recommended sources for strengthening any weakness they identify. Resources available in the classroom include textbooks, notes and a computer.

Exploration 2:

Using HO # 2, the teacher assist students through problem set 2. The teacher can lead students to make connections between problems by asking questions such as, “Are there any skills that you used in HO #1, that were necessary or helpful to use in this

Mathematician Journal Prompts: Describe the procedures you used to solve this problem. Did you make any

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5 Number of

hours_1.5__

problem set?” “What role does collinearity play in solving this problem?” Students reinforce their knowledge of ratios to find missing length numerically and algebraically. Summarize findings as a whole class before proceeding to HO #3.

Exploration 3:

Give the students HO #3, a problem showing concentric circles with a common radius that contains similar, right triangles. Ask students to find the perimeter of both triangles given the values in the diagram. [Note for teacher: Students need to find the length of the missing sides to calculate the perimeter of the triangles.] After finding the perimeter, ask students to find the distance around the larger circle (circumference.) As a challenge, they should then find the length of the portion of the total circumference that is inside the larger triangle.

Students can self-assess as needed and use any classroom resources available to review concepts.

EXPLAIN

Teacher introduces the main task of the unit and prepares students to in small group independent work...

Number of Hours_.10__

Introduce logging problem. The teacher may need to remind the students of the meaning of n! for this problem.

Put students in groups of two or three. Pairs are preferred for this activity but it may be necessary for there to be a group of three. Student task sheet is attached. There are two versions, one with basic information and one that contains additional suggestions to help students get started. Use the one with guidance only for those students you know will need extra help to get started. Most students should be able to think of what to do on their own at this point.

Logging Problem: (HO # 4a)

Logging Problem: with guided assistance (HO #4b)

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6 ELABORATE

The student groups are working independently with teacher consultations. Number of Hours_1___

Students work in groups to solve the logging problem with 3 logs. They need to find the length of the metal band that surrounds the logs. Students then find the metal band length required to secure 6 logs, by adding a third row of 3 logs. They are to look for a pattern, and write an algorithm to determine the size band for n rows of n! logs.

Mathematician Journal Prompts

EVALUATE

Working groups submit products or make

presentations Number of Hours__.4_

Directions for Instructor regarding final evaluation: Students write a paragraph in their Mathematician Journal

explaining how the length of the band changes as logs are added or taken away. All mathematical calculations should accompany this conclusion. Students complete a peer evaluation (see HO #5). The instructor grades according to a rubric attached. (see HO #6).

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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: Students find the length of a metal band that surrounds logs.

KNOWLEDGE AND SKILLS NEEDED Assumed already learned Students will self-assess Will be taught during the unit 1. Solve proportions X X

2. Use Pythagorean Theorem X X 3. Determine arc length and its relationship

to central angle

X X X 4. Find perimeter and circumference of two

dimensional figures

X

5. Recognize congruent and similar figures X X 6. A circle has 360°, a straight line 180°, and

a right angle 90°.

X

7.Analyze, interpret, predict X X 8. Transfer and connect between multiple

representations

X X

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

Problem Set 1

1. A five-foot student casts a shadow that is 40 feet long while standing 200 feet from a

streetlight. How high above the ground is the lamp? (Make a sketch of this situation.)

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HO #1b

Problem Set 1 - SOLUTIONS

1. A five-foot student casts a shadow that is 40 feet long while standing 200 feet from a

streetlight. How high above the ground is the lamp?

y = 25 ft

40 ft

Streetlight is 25 feet tall.

200 ft

2. How far from the streetlight should the student stand, in order to cast a shadow that is

exactly as long as the student is tall? Generalize this for a person with any height and

write an expression for the distance they must stand from the streetlight in terms of the

person’s height, when the shadow is the same length as the person’s height.

so the distance from the streetlight is 25 feet

Whatever the height of the person is, the distance they must stand from the streetlight in order to obtain a shadow equal to their height is represented by (25 – height of person).

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HO #2

Problem Set 2

1. Three squares are placed next to each other as shown below. The vertices A, B, and C are

collinear. Find the dimensions

n.

.

4 7 n

2. Replace the lengths 4 and 7 by

m

and

k

, respectively. Express

k

in terms of

m

and

n

.

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HO #2b

Problem Set 2 - SOLUTIONS

1. Three squares are placed next to each other as shown below. The vertices A, B, and C are

collinear. Find the dimensions

n

.

Slope of line connecting A and B is ¾

so y = 12.25…. Therefore n = 12.25

2. Replace the lengths 4 and 7 by

m

and

k

, respectively. Express

n

in terms of

m

and

k

.

Slope of line connecting A and B is

=

Solve for n….

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HO #3

Problem Set Three

Directions:

Using the given values in the diagram below, determine the perimeter of the both triangles, and

the circumference of the larger circle.

Challenge: Find the length of the arc that is inside the larger triangle (from point A to point B).

Hint: You will need to use some right triangle trigonometry to find the central angle.

B A

12 in

5 in

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13 HO #3b

Problem Set Three - Solutions

Directions:

Using the given values in the diagram below, determine the perimeter of the larger right triangle

and the circumference of the larger circle.

x = 7.2 inches

Use Pythagorean Theorem to find the hypotenuse:

2 2

5 

3 

5.83 inches

2 2

12 

7.2 

13.99 or 14 inches

P1 = 13.83 inches

P2 = 33.2 inches

C = 14.4π

= 45.24

Challenge: Find the length of the arc that is inside the larger triangle (from point A to point B).

Hint: You will need to use some right triangle trigonometry to find the central angle.

1

12 tan

7.2

12 tan

59.036

59

7.2 













_

_









Arc length =

59

45.24 7.42 inches

360 



A

B 12 in

5 in

3 in

5 3

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HO #4a

Don’t Get Flogged by the Log - A

The figure shows three circular logs, all with 12-inch diameters, that are strapped together by a

metal band. How long is the band?

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15 HO #4b

Don’t Get Flogged by the Log - B

The figure shows three circular logs, all with 12-inch diameters, that are strapped together by a

metal band. How long is the band?

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HO #4c

Don’t Get Flogged by the Log - SOLUTION

The figure shows three circular pipes, all with 12-inch diameters, that are strapped together by a

metal band. How long is the band?

The 3 segments are 12 inches long each. The 3 arcs of the circle that connect the segments are each 120 degrees or 1/3 of the circumference of the circle. (see blue arc)

3*12+3(1/3)(12π) = 73.699 inches

Challenge: If another row of logs were added, what would the length of the strap be then? Can you

generalize this for n rows of n! logs?

3 rows, 6 logs total will need a strap of length 6(12) + 12π inches

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17 HO #5

Peer/Self-Evaluation form: Don’t Get Flogged by the Log (HO #4)

The following is a list of statements to be answered and each of your group members. Think carefully about assigning rating values for each of the statements.

1- Strongly Agree 2- Agree 3- Neutral 4- Disagree 5- Strongly Disagree

Self: Teammate: Teammate: Teammate:

Was dependable in attending class Willing to accept assigned tasks Contributed positively to group discussion

Completed work on time or made alternative arrangements Helped others with their work when needed

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18 HO #6

Instructor’s Rubric

“Expert” (5 pts)

“Practitioner” (3 pts)

“Beginner” (1 pt)

Written Organized and

grammatically correct.

Organized with few grammatical errors

Lacking organization and needs

grammatical editing Mathematical

Approach

Accurate, detailed, clearly explained using a sound mathematical model including development of algorithm

One or two errors in explanation or calculators, but generally a good model.

Several errors in calculations or math concepts used.

Logging Bands All three logging band lengths correctly identified.

Two logging band lengths correctly identified.