Download Lesson Title: Beginnings of Trigonometry NCTM Standard

Lesson Title: Beginnings of Trigonometry
NCTM Standard:
PC – 5.1 Understand how angles are measured in either degrees or radians.
PC – 5.2 Carry out a procedure to convert between degree measure and
radian measure
State Standard: Common Core - None
Learning Objective(s): The student will be able to: convert from radians to
degrees, be able to define a degree, look at circular arc length, and look at some
application for motion and nautical miles.
Essential Questions (s)/ or problem: How does the definition of the radian
enable degrees to be converted to radians?
What prior knowledge or skills are needed?
Understand method of conversions
What specialized Mathematical vocabulary will you include?
Radian, Degree, angular motion, nautical miles, central angle, geometry review:
circumference/radius, arc length
What historical note will you include?
I am going to include a historical note about the father of trigonometry named,
Hipparchus of Nicaea who compiled the first trigonometric tables for simplifying
the study of astronomy. Because of his work, that same idea of mathematics allows
sound waves to be digitally stored on a CD. The first mathematician to actually
work with trigonometry was Hippocrates of Chois and Eratosthenes of Cyrene, and
these men introduced ratios that were used by Babylonians and Egyptians in
engineering. So in reality, based on the textbook trigonometry had been used about
4000 years ago and is still used today. This may be my hook for the lesson in
introducing ideas that the Egyptians had in using trigonometry.
Key Vocabulary: What words will I introduce, define, and use in the lesson?
Degree, radian, conversions, angles, central angle, arc length, angular speed/linear
speed
Activities/Procedures: Write out in relevant detail so that a substitute could
follow the lesson or attach notes, assessments, power points.
Do now or bell ringer:
For this lesson, my bell ringer will be more of introducing the history behind this
unit as a whole with trigonometry and talking about the father of trigonometry. I
will have a power point based on the father of trigonometry, Hipparchus of Nicaea,
and talk about the projects that this mathematician along with other
mathematicians before introduced trigonometry.
What will you and the students do during this lesson?
- First I will have a power point based on the history of trigonometry
- Then I will show a power point that I found on Teachers Pay Teachers,
which had a really good power point that shows the definition and meaning
behind the radian.
- Then I will do an example problem or two for the students to understand
conversions of degrees to radians and back again. I will definitely have
students comes and work some of the problems on the board as well to
check for understanding.
Radian: a central angle of a circle that cuts off an arc equal in length to the
radius of a circle
Radians to degrees: multiply by 180°/π radians
Degrees to radians: multiply by π radians/180°
Ex:
Convert to radians
a. 90°
b. 180°
c. 240°
π/2
π
4π/3
Convert to degrees
a. π/3
b. 6π/12
60°
90°
c. 5π/3
300°
- Then after working through those, I will ask students to give me a thumbs up
or down if they do or do not understand the problems. If they do not, then I
will do a few more with them reviewing the basic idea of conversions.
- Then we will go over circular arc length
s = rθ r – radius, θ – angle in radians, s – length of intercepted arc
Ex:
Find s if we know r = 2 inches and θ = 25 radians
S = 50 rad*inch
To the nearest inch, find the perimeter of a 10-degree sector from a circular
disc of radius 11 inches.
- Now we would go into applications of angular speed being part of looking at
radians.
- Ex: Miss Wilson’s truck has wheels 34 inches in diameter. If the wheels are
rotating 630 rpm(revolutions per minutes), find the truck’s speed in miles
per hour.
When students are working through conversions, many of them would not
quite understand revolutions conversions so help them see 1 revolution is 2π
- I would then work with a problem looking at the nautical mile
Ex: Point C and D are 895 statute miles apart. How far apart are C and D in
nautical miles?
1 statute mile = 0.87 nautical mile 1 nautical mile = 1.15 statute mile
How will students’ individual differences in rates of learning, styles of learning,
interests, gender, ethnic differences and needs be accommodated? How will you
show equity to all students?
Well for students to understand the history behind trigonometry will be a big help
with understanding how to apply the idea of a radian. Also, having students see
more of a visual power point helps the visual students, and repeating the definition
a few times in class with the radian helps as well.
The biggest area is helping students who do not remember how to do conversions
for units. I think also having students see how to go back and forth from radians to
degrees will be helpful.
Technology and other resources: Describe your technology use and rationale.
Technology will be mostly using the computer to show powerpoints, but also
having the students research more about angles and how certain angles affect the
structure of buildings and after the lesson is finished, I will have the students
research angles and a structure being built must have the angles of elevation are
important.
Authentic contexts: List any examples that you will use to tie the lesson to real
world problems. Explain how it will enhance the lesson.
Overall, the application problems helps students see how radians are used. Also
tying into some beginning areas of going to the next lesson in actual trig ratios.
Another thing that I will do during the lesson is have students show with their
hands where an angle would lie in looking at degrees and how radians are the same
idea as well.
Accommodations: Required by law. How will you meet the learning needs of
all students in the class?
Example:
I will definitely make sure to define certain new vocabulary words well in giving
understanding to the words in an easy way for ELL learners. Visually, power point
will be good along with having pictures of the angles and seeing what quadrant it
lies in. For the auditory leaners, constantly repeating ideas and giving the main
topics covered repeated over and over again. I will also encourage students to start
making flashcards for terms learned in trigonometry.
Gifted: For the gifted student, I would ask them to create a real life situation where
you would need to use nautical miles and how to convert in terms of radians and
degree.s
Mathematical connections
How will you make connections to other subjects or real life problems?
What mathematical processes will you include?
I will look at how degrees and angles are the basics of trigonometry and how
looking at trig identities in the next chapter will show the ideas being put into
action. Its very important also to understand translating from radians to degrees
fairly quickly especially when learning the unit circle.
Closure:
How will you check for understanding?
Vocal assessment and also givet them an exit slip.
What is your formative assessment (exit slip)?
I will give the students three problems for degrees and radians of conversions
along with an application problem.
How will you set the stage for the next lesson and make connections to the past
lessons?
Unit circle idea and show that conversions have been used before in simpler terms
and degrees and radians are the same idea.
I will have the students look at the angles on a graph and start to see how you can
draw triangles from those central angles and give a mini Segway into right triangle
trigonometry ideas.
Materials: What resources/materials will be needed during this lesson?
Computer, notes for students to be taking as well, protractor
Questioning: Develop questions on various levels of the Bloom’s Taxonomy or
Webb’s Depth of Knowledge. Identify the level that corresponds to each question.
Include at least 3 levels of questions during the lesson and make sure you ask them
during the lesson.
Level One: Find the meaning of a degree and radian
Level Two: What is the main idea of the conversion factors?
Level Three: Can you apply the concept of radians to certain geometric 3-D
objects?
Level Four: Can you distinguish between radians being used and degrees?
Level Five: Can you design a building using the idea of radians and degrees for the
layout?
Level Six: Is there a better solution for conversions? Is there a way to draw the
connection on a circle?
Assessment: How will you know that the students met the objectives (s) of the
lesson? How will assessments accommodate the differences in the students?
Return to the essential question in the wrap-up/closure.
I will go back to the essential question in asking how the definition of radian
demonstrates conversions.