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:
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
Field Tests Other:
(End of Project) 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
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
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)
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).
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
□ ________________________________
□ ________________________________
□ ________________________________
□ ________________________________
□ ________________________________
□ ________________________________
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.)
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).
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
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
.
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
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:
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.
A
B 12 in
5 in
Arc length =
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?
HO #4b
Don’t Get Flogged by the Log - B
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?
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
3 rows, 6 logs total will need a strap of length 6(12) + 12π inches
n rows, n! logs total will need a strap of length 3(n-1)*12 + 12π inches
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
completely Worked well with other group members Overall was a valuable member of the team
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.
