Math/Physics Workshop
Template for Group Activity
Title of Activity: Hillbilly Merry-Go-Round
Content Area: Circular Motion
Short description of the activity:
In this activity the students will use trigonometry to describe angular displacement
for an object in circular motion. They will use definitions of arc length, radius, and
circumference to define angular displacement and then convert between radians
and degrees using both a unit conversion and the unit circle. They will then use
these concepts to calculate angular speed, tangential speed, and centripetal
acceleration for a gas propelled system in circular motion.
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Connection to common core and standards: list the common core and standards
that are connected to this activity. List all specific to each discipline; can be more
than three. Are there any overarching common cores (overlapping) that fit for both
disciplines?
Overarching:
1. Analyzing and interpreting data.
2. Using mathematical and computational thinking.
3. Planning and carrying out investigations.
Math:
1. CCSS.Math.Content.HSF-TF.A.1: Understand radian measure of an angle as the
length of the arc on the unit circle subtended by the angle.
2.CCSS.Math.Content.HSF-TF.A.2: Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all real numbers, interpreted as
radian measures of angles traversed counterclockwise around the unit circle.
3.CCSS.Math.Content.HSG-C.B.5: Derive using similarity the fact that the length of the
arc intercepted by an angle is proportional to the radius, and define the radian
measure of the angle as the constant of proportionality; derive the formula for the
area of a sector.
Science:
1.MF.3.P.1: Relate radians to degrees, Δθ = Δs/r.
2.MF.3.P.2: Calculate angular speed and angular acceleration, ω = Δθ/Δt, α = Δω/Δt.
3.MF.3.P.5: Solve problems involving tangential speed, vt = rω.
4.MF.3.P.6: Solve problems involving tangential acceleration, at = rα.
5.MF.3.P.7: Calculate centripetal acceleration, ac = vt2/ r, ac = rω2.
6.MF.2.P.9: Calculate rotational motion with a constant force directed toward the
center, Fc = mv2/r.
7.MF.3.P.2: Calculate the magnitude of torque on an object, τ = Fdsinθ.
Level of complexity: Define what grade level and how complex the project will be.
Are you expecting student to develop their own experimental process or will you
provide them stop-by-step instructions? Will this be performed in one class period
or in multiple class meetings?
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This lesson is designed to be used in an 11th-12th grade general physics class to
discuss rotational motion while using trigonometric concepts with relation to the
unit circle. We have also included modifications for students with learning
disabilities who are mainstreamed into regular education classes. The activity will
be a guided inquiry activity that introduces trigonometric concepts and relates them
to the rotational motion equations the students will be using. It also includes an
extended thinking section at the end of the activity in which the students are asked
to manipulate their experimental design in order to optimize the torque exerted on
the system and increase the angular acceleration of the tube. This is an activity that
would be conducted over multiple class periods.
Description of the project: This is a longer description that includes all necessary
information for implementing the project including: pre-lab, list of equipment,
experimental process, data collection charts, data analysis, etc…
In this activity the students will use trigonometry to describe angular displacement
for an object in circular motion. They will use definitions of arc length, radius, and
circumference to define angular displacement and then convert between radians
and degrees using both a unit conversion and the unit circle. They will then use
these concepts to calculate angular speed, tangential speed, and centripetal
acceleration for a gas propelled system in circular motion. The directions for the lab
are included in the student handout for the activity. The activity can be conducted
in a variety of ways depending on the supplies and time available for the activity. It
can be done with only a ruler, pen, and balloon. It can also be done with t-shaped
glass tubing clamped to a lab station, or students can use a gas leaf blower and
propel themselves in a rotating chair.
Assessment component: This section includes the types of assessments you will
conduct during the project. This includes pre, formative, and summative
assessments. Be sure to include a grading rubric where appropriate.
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The pre-lab/ pre-assessment questions and the laboratory questions will be graded
according to the laboratory activity rubric we have attached.
Student Handouts: Create the necessary handouts you would provide to your
students while they are gathering data and performing the project. This should
include: instructions, safety statements, instructions for analysis, etc…
The students will be given two handouts. One handout is a set of pre-lab questions
that will be used as a pre-assessment. The second handout guides the students
through the lab in which the students will complete measurements and calculations
on which they will be graded as a formative assessment.
Math/Physics Concepts and Skills – Make a list of physics and math concepts and
skills necessary to complete the project. Define and explain the concepts and skills.
Physics Concepts:
Students will understand one-dimensional motion and two-dimensional motion
before addressing the following concepts with regard to rotational motion:
1.MF.3.P.1: Relate radians to degrees, Δθ = Δs/r.
2.MF.3.P.2: Calculate angular speed and angular acceleration, ω = Δθ/Δt, α =
Δω/Δt.
3.MF.3.P.5: Solve problems involving tangential speed, vt = rω.
4.MF.3.P.6: Solve problems involving tangential acceleration, at = rα.
5.MF.3.P.7: Calculate centripetal acceleration, ac = vt2/ r, ac = rω2.
6.MF.2.P.9: Calculate rotational motion with a constant force directed toward
the center, Fc = mv2/r.
7.MF.3.P.2: Calculate the magnitude of torque on an object, τ = Fdsinθ.
Math Concepts:
1. CCSS.Math.Content.HSF-TF.A.1: Understand radian measure of an angle as the
length of the arc on the unit circle subtended by the angle.
2.CCSS.Math.Content.HSF-TF.A.2: Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all real numbers, interpreted as
radian measures of angles traversed counterclockwise around the unit circle.
3.CCSS.Math.Content.HSG-C.B.5: Derive using similarity the fact that the length of the
arc intercepted by an angle is proportional to the radius, and define the radian
measure of the angle as the constant of proportionality; derive the formula for the
area of a sector.
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Math/Physics Instructional Time Line – Create an instructional timeline related
to the math/physics concepts and skills necessary to complete this activity in your
classroom. Identify when the skills and content are learned and/or introduced.
Decide if they are pre-knowledge to if they will be taught/learned in your classroom.
Math/Physics Language– Make a list of terms and definition used in this activity
that relate to each discipline (Math and Physics) individually. What are the
differences and similarities? What could you do in your teaching to make sure
students understand these differences and similarities?
Instructor Reflection: Develop a 1-page instructor’s self-reflection tool. This
should give the instructor opportunity to reflect on the instructional process and the
student action part of the project. What worked well? What should be improved?
What could be changed and why?
The instructor reflection tool is attached.
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Math/Physics Concepts and Skills – Make a list of physics and math concepts and
skills necessary to complete the tasks in the math modeling section and the physics
investigation. Define and explain the concepts and skills.
Math Concepts:
Math Skills:
Trig functions: sine, cosine, tangent
Data analysis
Unit Circle
Computational skills
Physics Concepts:
Physics Skills:
Newton’s Laws of Motion
Unit conversions
Displacement
Data analysis
Distance
Measurement
Speed
Velocity
Force
Acceleration
Angular displacement
Angular speed
Tangential speed
Centripetal acceleration
Centripetal force
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Math/Physics Instructional Time Line – Create an instructional timeline to
introduce the math and physics concepts and skills necessary to complete this
activity in your classroom.
Order of concepts and skills
Math
Physics
Algebra I, II
One-dimensional motion:
build on background
knowledge from physical
science.
Algebra I, II
Two-dimensional motion.
Trigonometry to resolve
forces in twodimensions.
Algebra I, II
Rotational motion.
Trigonometry to
describe rotational
motion.
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Math/Physics Language Sheet – Make a list of terms and definition used in this
activity that relate to each discipline (Math and Physics) individually. What are the
differences and similarities? What could you do in your teaching to make sure
students understand these differences and similarities?
Math
Physics
Degrees
Newton’s Laws of Motion
Radians
Displacement
Unit circle
Distance
Trig functions: sine, cosine, tangent
Speed, velocity, acceleration, force
Arc length
Angular displacement
Radius
Angular speed
Circumference
Tangential speed
Centripetal acceleration
Centripetal force
Differences
Similarities
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Notes:
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