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Ramp it
up!
(Download
as Microsoft Word Document: Instructional
Plan)
Brought to you by
the team of
Mary,
Daniel,
Teddie,
and Shaun!
In accordance with the Pennsylvania
Governors Institute for Mathematics
Promoting a deeper understanding of
mathematics through the NCTM Process Standards
Reasoning
Representation
Communication
Connections
Problem
solving
“Ramp it Up”
Instructional Plan
Group members: Mary Cunnally, Shaun Mc Murtrie, Teddie Psitos, Dan Seif
Topic: Geometry/Measurement/Similar Triangles
Theme: The
relationship between similar triangles and their trigonometric ratios
Level: 9-11
Time
Element: 85 minutes
(1 block day or 2 regular class days)
NCTM
Standards Addressed: Geometry
and Measurement
PA Math
Standards Addressed: 2.2.11.E, 2.3.11.A,
2.3.11.C, 2.4.11.B, 2.4.11.E, 2.5.11.A, 2.5.11.B, 2.5.11.C, 2.5.11.D, 2.9.11.A,
2.9.11.B, 2.9.11.D, 2.9.11.G, 2.9.11.I.
Math
Assessment Anchors Addressed:
M11.A.2.1.3, M11.A.3.2.1, M11.B.1, M11.B.2.1.1, M11.C.1.2.1, M11.C.1.3.1,
M11.C.1.4.1
Objectives:
Ø
Students
will demonstrate mastery in the understanding of similarity.
Ø
Students
will be able to prove similarity through properties of parallel lines and
corresponding angles.
Ø
Students
will develop, demonstrate, and articulate their understanding of ratios of
corresponding sides of similar triangles.
Ø
Students
will develop, demonstrate, and articulate their understanding of the
applications of the trigonometric ratios in relation to similar triangles.
Instructional
Strategies and Plan (include strategies used to help different types of
learners, i.e. auditory, visual, etc):
Class
After
a short class discussion, the students will be placed in their groups. Then they
will research the
The
center of today’s lesson is found in the class text on page 472 <see
Materials/Resources>. The students will be given a working copy of the
problem <see Appendix A>.
The
class will read the problem out loud together. Students will then read the
problem to themselves. After they read the problem they will need to determine the
significance of the problem mathematically as well as socially; followed by one
more short discussion.
Students will work on the problem as
a group, while the teacher circulates and monitors the progress of the
cooperative groups.
As
the teacher is circulating the students with special needs will be addressed
individually. The students making slower progress will be assigned a specific
question or questions to answer or possibly a modification of the problem to
solve <see Appendix B>. Students finishing the problem very early will be
given an extension problem to present later to class <see Extensions/Enrichment
#1>.
Each group will present the solution
of the original problem (or the extension or modification where necessary).
This will be followed by a class discussion regarding the methods used and
approaches taken to solve the problem.
The teacher will then summarize the
class lesson, leading into the assignment of the homework <see
Extensions/Enrichment #2>
Materials/Resources:
McDougal
Littell Geometry
© 2007 (section 7.5 Apply the Tangent Ratio)
Website: http://www.usdoj.gov/crt/ada/
Paper
Pencils
Colored
pencils
Protractor
Ruler
Calculator
Interdisciplinary
Connections:
·
Writing: Students will develop a presentation of their findings
from the “ramp problem.” It will need to be written as a formal proposal, in
the style of a business letter, addressed to an official representative of either
the construction company or the business for which the work is being done.
·
·
Technology: Laptops, Geometers Sketchpad or Cabri Jr., TI 83 or TI 84 Graphing Calculator.
·
Sociology: Students will discuss the necessity of the
Americans with Disabilities Act.
Assessment
Strategies:
·
Formative
Evaluation (checking student understanding during the lesson): Teacher interactions with students through
probing questions, checking individual and group progress as well as individual
participation.
·
Summative
Evaluation (How will it be determined that the objectives were achieved?): The individual letters submitted by each
student following the cooperative group work.
Correctives/Remediation: Review properties of similar figures,
trigonometric ratios and setting up and solving proportions, as necessary.
Extensions/Enrichment:
1)
Our school
district is renovating one of the buildings and construction will begin soon on
the building of ramps to the entrances of the school. Groups of students will
be assigned a specific entrance to the building and will be responsible for the
design of the appropriate
2)
The students
are asked to write up a proposal to a company official in the form of a letter
stating what they believe to be the best possible solution for constructing and
designing this ramp. They should include a summary of their group’s findings
from today’s class. Based on the
question in the text, they should come up with at least one alternative
solution. They should consider the
following: with the given specifications,
is it possible to use less horizontal distance? Why or why not? Are these
solutions realistic? Why or why not? Support your answers.
Special
Accommodations (special needs students)
·
Description
of the Special Needs student selected:
Student who is legally blind.
·
Accommodations
to use with this student: Enlarge all
handouts and textbook pages. Enlarged
copies of overheads from teacher and student work may need to be given to
student upon request. Student has a wrap-around who will assist him or her as
needed.
Appendix A:
RAMP IT UP!
According to the Americans with
Disabilities Act, a ramp cannot have an incline that is greater than 5
degrees. The regulations also state that
the maximum rise of the ramp is 30 inches.
When a ramp needs to reach a height greater than 30 inches, a series of
ramps connected by 60 inch landings can be used, as shown below.
- What is the
maximum horizontal length of the base of one ramp, in feet? You should round your answer to the
nearest hundredths of a foot.
- If a doorway
is 7.5 feet above ground, what is the least number of ramps and landings
you will need to lead to the doorway?
Draw and label a diagram to justify your answer.
- To the
nearest foot, what is the total length of the base of the system of ramps
and landings in part (b)?
Appendix B:
1. Why is the
slope of the ramp 5 degrees?
2. Why do we
need landings?
3. Why can’t we
use just one continuous ramp?
4. Using the 5
degree specification, but eliminating the 30 inch requirement, what would be
the length of the ramp have to be in order to reach
the 7.5 feet door?
5. Given the
cost of a ramp at $25 per linear foot, what would be the total cost of the
original ramp your group built? What
would be the cost comparison of an alternately designed ramp?
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