Short-Sighted
(Tangents of Circles)
Mathematical Goals: Teachers will be able to
· Solve an authentic scenario involving tangents of circles and the conversion of units.
Pedagogical Goals: Teachers will be able to
· Determine the appropriate amount of information to give in a problem with the purpose of allowing students to ascertain what information is necessary, thus developing their problem solving skills.
· Determine which of the Standards of Mathematical Practice are addressed in this activity.
· Provide appropriate modifications to this activity for use in their classrooms.
Technological Goals: Teachers will be able to use a technological tool to
· Model the given situation using a DGE.
Alignment to the Common Core:
· G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them
2 – Reason abstractly and quantitatively
3 – Construct viable arguments and critique the reasoning of others
4 – Model with mathematics
5 – Use appropriate tools strategically
6 – Attend to precision
Length of Lesson:
90 minutes
Materials Needed:
GSP (optional)
Overview:
This activity involves one problem – finding the change in an arc’s length when a segment extended from the radius of a circle is extended and a tangent is drawn. Participants must ascertain the missing information in the problem, formulate a solution path, model the problem and solve. Throughout this activity and after its completion, a discussion will be held concerning its challenges and implications on pedagogy.
Activity:
The facilitator should present this problem and the participants must attempt a solution. Collaboration in groups of three or four is recommended in order to share strategies and potential ways to model the solution.
Joe is 5’8” tall. He is standing at the shore and looking out over the ocean at the horizon. He then stands on a rock 18” high. How much farther can he see to the horizon than before?
The solution to this problem is included in a separate document.
Pedagogical discussion:
· What information is necessary to solve this problem?
· What is the pedagogical benefit of providing “incomplete” problems for your students to solve?
· What strategies did you have for attempting this problem?
· What strategies might your students have for attempting this problem?
· Which of the Standards of Mathematical Practice did you use in the solving of this problem? (Considerable focus should be given to SMP#6: attending to precision).
The measurements, conversions and theorems that the participants and students may wish to use in solving this problem are:
· The average radius of the earth is 3960 miles.
· 5280 feet = 1 mile.
· 12 inches = 1 foot.
· The radius of a circle is creates a right angle with a line of tangency, thus creating a right triangle.
· The Pythagorean Theorem.
· Right Triangle Trigonometry.