Backward Design Project
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School
University of Wyoming *
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Course
3550
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
Pages
12
Uploaded by brady.schaefer
Holiday Tree:
Geometry/Trigonometry Problems
Backwards Design Project
Brady Schaefer
EDST 3550
12/4/14
Lesson Title:
Holiday Tree Content Area:
Mathematics
Specific Topic:
Geometry/Trigonometry Grade Level:
9
th and 10
th
Class Composite:
17 students, 9 girls, 8 boys
Lesson Duration:
1 week
Key Words:
Pythagorean Theorem, sine, cosine, tangent, right triangle, adjacent, opposite, hypotenuse, acute angle, reference angle, angle of elevation
Standards:
Common Core State Standards (CCSS)
-
CCSS.Math.Content.HSG.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
-
CCSS.Math.Content.HSG.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.
-
CCSS.Math.Content.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
*
Unit summary:
This unit is designed for a 9
th
and 10
th
grade geometry/trigonometry class. This unit will be covered over a span of one week. The unit will first start with a quick discussion about trigonometry, and student knowledge. After discussing trigonometry, the unit will then cover the parts of a triangle and the three trigonometric ratios, sine, cosine, and tangent. Once the students understand the parts of a triangle and these ratios (SOH-CAH-TOA) they will take part in a real world performance task. Students will be working for the National Forest Service and must calculate the height of the tallest trees in Yellow Stone National Park for the national Christmas tree. This project will be graded with a teacher created rubric. Students will then take an exam at the end of the week that encompasses all the material covered throughout the unit. This final will
consist of short answer, fill in the blank, true or false, matching, and multiple choice questions.
Stage I: Identifying Desired Results
The Big Ideas:
-
The parts of a triangle
-
Trigonometric ratios (SOH-CAH-TOA) The Unit Goals:
-
Students will be able to identify the parts of a right triangle given an acute angle as a reference angle. -
Students will be able to apply trigonometric ratios to a triangle to find the missing parts of that triangle.
-
Students will be able to apply knowledge to basic trigonometry problems.
The Essential Questions:
-
What are the three parts to a triangle?
-
What are the three trigonometric ratios for right triangles?
-
What is the relationship between the three parts of a triangle and the trigonometric ratios? -
How are the three trigonometric ratios calculated?
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Stage IIa: Performance Task
Setting and Role:
-
You have just been offered a job with the National Forest Service! Congratulations! Your first task is to calculate the height of the tallest trees in Yellowstone National Park for the national Christmas tree at the White House. The President wants the tallest tree you can find!
Goal/Challenge:
-
Pick the 4 tallest trees (imaginary) in Yellowstone National Park o
Tree 1 – 26-30 feet from base of tree with a 44-48 degree angle of elevation
o
Tree 2 – 10-15 feet from base of tree with a 64-67 degree angle of elevation
o
Tree 3 – 33-38 feet from base of tree with a 33-39 degree angle of elevation
o
Tree 4 – 62-68 feet from base of tree with a 25-30 degree angle of elevation
-
Determine how you are going to calculate the exact height of those 4 trees using trigonometry (pretend you have access to all the necessary equipment)
-
Calculate the height of the 4 trees
-
Show in a short written report the height of the trees
-
Determine which tree should be used for the holiday tree
Product:
-
Create a project write up including the following:
o
Label each tree so they do not get confused with another tree (numbers work well)
o
Write the distance from the base of the tree to where you are standing for each
tree
o
Write the angle of elevation from where you are standing to the top of each tree
o
Organize your data in a table similar to the one below
Tree Number
Distance From
Base of Tree
(feet)
Angle of
Elevation to Top
of Tree (degrees)
Trigonometric
Ratio Used
(SOH-CAH_TOA)
Height of Tree
(feet)
1
?
2
?
3
?
4
?
Show calculations here:
o
Solve for the height of each tree
o
Use correct units
Audience:
-
The President of the United States of America
-
Your coworkers that you will present the tree to
-
Anyone who looks at the national Christmas tree
Criteria for Success:
-
Your write up must include
o
Each tree labeled separately
o
The distance between the base of each tree and where you are standing
o
The angle of elevation to the top of each tree from where you are standing
o
The trigonometric ratio being used
o
The height of each tree
o
All calculations presented in an organized way.
Stage IIb: Performance Task Rubric
Student Name: ________________________________________
CATEGORY
Below Basic: 1
Basic: 2
Proficient: 3
Advanced: 4
Score:
Trigonometric Ratios
Major inaccuracies in Trigonometric Ratios used, incorrect ratio used
Numerous errors made when using Trigonometric Ratios, incorrect ratio used Minor inaccuracies when using Trigonometric
Ratios, correct
ratio use
Trigonometric Ratios are completely accurate and fully
understood
Table Organization
Table is unclear and is difficult to
follow. Missing more than two elements. Table lacks clarity and is difficult to follow.
Missing two elements.
Table is generally clear and able to follow. Missing one element.
Table is exceptionally clear and easy to follow. All values
are included as well as time as a variable. Parts of a Triangle
Student rarely identifies parts of a triangle, missing three or more parts. Student sometimes identifies parts of a triangle, missing two parts.
Student frequently identifies parts of a triangle, missing one part.
Student always correctly identifies part of a
triangle, all parts are clearly shown.
Calculations Major inaccuracies were made. Calculations are either missing or cannot be followed.
Calculations are inaccurate. Big pieces of work are missing, or many computation errors were made, detracting from
the result.
Calculations are generally accurate. There are minor inaccuracies but could still be followed.
All work is completely accurate and shown in an organized manner.
Total: ____________ x 4 _____________
64 Points Possible 58-64: A (Advanced), 52-57: B (Proficient), 45-51: C (Basic), 39-44: D (Below Basic), 0-38: F (Below Basic)
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Stage IIc: Objective Assignment
Geometry / Trigonometry Exam
(20 points possible) Name:______________
Short Answer (3 points each):
1.
What theorem can be used to find the missing sides of a triangle?
Pythagorean Theorem
2.
In two sentences or less explain the circumstance(s) for using the Pythagorean Theorem.
If the triangle is a right triangle, then the Pythagorean Theorem can be used. Fill in the Blank (2 points each):
3.
The longest side of a triangle is labeled the ___
hypotenuse
_____.
4.
In order to use trigonometric on triangles, the triangle must be a _
right_
triangle.
True or False (1 point each):
5.
__
True
___ The reference angle in order to use trigonometric ratios must be an acute angle. 6.
__
False
___ The triangle side across from the reference angle is known as the adjacent side of the triangle. 7.
__
False
___ The angle of elevation is the angle between the horizontal and the line of sight looking down to an object.
Match the trigonometric function with the corresponding trigonometric ratio (1 point each):
8.
Sine ___
c
____
a. Length of oppositeside
Lengthof adjacent side
b. Lengthof hypotenuse
Lengthof opposite side
9.
Cosine ___
e
____
c. Lengthof opposite side
Lengthof hypotenuse
d. Lengthof adjacent side
Length of oppositeside
10. Tangent ___
a
____
e. Lengthof adjacent si de
Lengthof hypotenuse
Multiple Choice (2 points each):
11.
The length of the hypotenuse of a right triangle is 32 inches and the length of one of the legs is 18 inches. What is the length, to the nearest tenth
of an inch, of the other leg of the
triangle?
a.
36.7 inches
b.
28.4 inches
c.
26.5 inches
d.
25.4 inches
12. Which statement is NOT
true?
a.
The hypotenuse is the longest side in a right triangle.
b.
The Pythagorean Theorem applies to all right triangles.
c.
You can solve for the unknown side in any triangle
, if you know the lengths of the other two sides, by using the Pythagorean Theorem.
d.
You can solve for the unknown side in any triangle, if you know the lengths of the
other two sides, by using the Pythagorean Theorem.
Stage IId: Academic Prompt
A roof is shaped like an isosceles triangle. The slope of the roof makes an angle of 24
with the horizontal, and has an altitude of 3.5 m. Determine the width of the roof, to the nearest hundredth of a meter.
Solve the Problem above (10 points possible):
tan
(
24
°
)
=
3.5
w
w
∗
tan
(
24
°
)
=
3.5
w
=
3.5
tan
(
24
°
)
w
=
7.86
m
(6 points possible for showing all correct calculations)
Since this is only half the width of the roof, this must then be multiplied by 2 to get the full width of the roof, or do the appropriate trigonometric ratio twice and add the two values. (2 points possible for recognizing this is only half of the width.)
width of roof
=
2
∗
7.86
meters
width of roof
=
15.72
meters
(2 points possible for correct final answer)
Write a paragraph (3-5 sentences) explaining the steps you took to solve this problem
.
(5 points possible):
Scoring Guide: Student response must include the following steps.
o
Use the correct trigonometric ratio o
Use the correct values in the ratio
o
Solve for “w” correctly
o
Notice “w” is only half the total width of the roof (student can also use the trigonometric ratio twice and add the two values to get the total width of the roof.)
o
Multiply “w” by 2 to get the total width of the roof.
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Score: __________ (15 points possible)
Stage IIe: Other Evidence and Student Self-Assessment
Other Evidence of Understanding:
-
Informal observations
-
Participation during lecture time evaluated by student’s answers to questions and work in notes. Students Self-Assessment:
-
Able to help peers
-
Explanations during discussion (think-pair-share) time
-
Able to complete sample problems during class lecture
Stage IIf: Reflection
Phase I: Reflection/Imagination/Anticipation
-
Performance Task (Stage IIa)
o
Some students may not be intrigued by this performance task so their motivation to complete the project may be very little. o
Making an organized table may be difficult for students, so it would be a great
idea to include an example. o
A calculator in degree mode is required for this performance task, I should supply calculators and make sure they are in degree, not radian, mode. If the calculator is in radian mode the students will get incorrect answers. -
Rubric (Stage IIb)
o
Students know exactly what is expected of them. -
Teach-made test items (Stage IIc)
o
Some test items may give hints to other test items. I should either re-order my test items or swap some questions for other ones. o
The test may be too short to assess my students. I should add more questions because longer test are more reliable
-
Academic Prompt (Stage IId)
o
Students with a low reading level, English language learners, or English as a second language students may struggle with the assessment and the academic prompt. It is challenging to read and understand what exactly the problem is asking for. Although there is a visual image, it may still be difficult to extract
the correct information. If the students can not extract the correct information, then they will have difficulty completing the problem. I will tell students ahead of time that if they need help going over the question they can bring the test up to my desk and I will go over the problem with them. We can also make arrangements to take the test or complete the academic prompt outside of class. Phase II – Making up/Analysis/Use
-
Academic Prompt
Student
Response
Rationale
Colby
Colby answered the math portion correctly, but simply wrote see above for his paragraph.
Colby shows he has obtained knowledge throughout the unit; however he did not follow directions. Laura
Laura has her trigonometric ratio equation correct but arrived at the wrong answer.
Laura’s calculator may be in radian mode and not degree mode. Brenda
Brenda solved for “w” correctly and marked that as her answer.
Brenda did not recognize that “w” was only half of the total width of the roof. She may have been rushed or simply did not understand the problem. She did not fully answer the problem.
David
David wrote an excellent paragraph, but he had multiple errors in his calculations.
David knew the process for solving the problem, but he needed to read the question carefully and double check his calculations. Gil-dong
Gil-dong used the Pythagorean Theorem instead of trigonometric ratios. Gil-dong used a technique we went over in class. However, he did not recognize that not all the required criteria are present to use the Pythagorean Theorem. He assumed information that was not present in order to fit
the criteria for the Pythagorean Theorem. -
Possible Problems
o
Students reading skills and ability to follow directions may be tested more than their math skills. o
The time limit for this may be too short. This is a reasonable amount of material to fit in a week, but if students struggle, more time may be needed. o
The objective assessment may be too short. Since longer assessments are more
reliable it would be a great idea to add some items to the assessment.
-
Post-Assessment Strategy
After using this unit once, I will be able to keep what works and revise what does not. Next time, I will put more emphasis on the angle of elevation. This would be something that would be explained when they needed it, so it would be a great idea to cover this in more detail. This will require extra guided and individual practice, so stretching the unit out will be more accommodating. From here we will work on more advanced geometry and trigonometry problems and further advance the students’ knowledge. The law of cosines and law of sines would be a great topic choice for the next
unit. This is advancing student knowledge in the field of trigonometry
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