Unit Plan Cover Sheet – Trigonometry Lesson Plan

Name(s): Michael Clarke, Shelley Mourtgos, Kip Saunders, & Erin Shurtz / Date: March 8, 2006
Unit Title: Trigonometry Lesson Plan
Fundamental Mathematical Concepts (A discussion of these concepts and the relationships between these concepts that you would like students to understand through this unit).
STUDENTS WILL UNDERSTAND:
A circle can be divided into whatever size unit of angular measurement you would like and the basic trigonometric functions still work.
1) There are several systems of measurement of an angle. Although any size unit of angular measurement will work, some units are better for particular types of problems.
1-a) Degrees and Radians are the two most common units of angular measurement in which the full rotation corresponds to 360º and 2p radians.
1-b) Gradians is another unit that has been developed in which the full rotation corresponds to 400 gradians (grads or gons) and a right angle is 100 gradians (grads or gons).
1-b-i) Gradians (a centesimal system) was first introduced by a German engineering unit to correspond to the circumference of the earth (1 grad corresponded to 100 km of the earths 40,000 km circumference).
2) Angular units of measurement are arbitrary. Some units are more useful than others. The sine and cosine of 30º, 45º, and 60º yield irrational numbers. There are angles whose sine and cosine are rational.
3) The coordinates of the points we usually label on the unit circle come from special characteristics of equilateral and isosceles triangles.
4) Angle measurements we already know can be used to derive the trig identity for addition of two angles.
Describe how state core Standards, NCTM Standards, and course readings are reflected in this unit.
Course readings:
Each day’s lesson is structured around an exploration task conducted in a small group setting, providing students with the opportunity to problem solve and communicate their ideas mathematically (Artzt & Armour-Thomas, Becoming a Reflective Mathematics Teacher, pp. 3-4
Each day’s lesson plan is structured to enhance classroom discourse by giving students the opportunity to discuss problems in small group settings prior to instructor interaction and input and then to move that discussion to the classroom setting (Artzt & Armour-Thomas, Becoming a Reflective Mathematics Teacher, pp. 16-18).
Students will discuss exploration principles as they work together in their small groups and will explain those principles as they present their findings to the class (Sherin, Mendez, & Louis, Talking about Math Talk, pp. 188-195).
This lesson plan strives to incorporate the “four faces of mathematics” by including opportunities for students to be creative as they compute, reason, and solve various problems as they come to know that a circle can be divided into a variety of angular measurements. The plan also seeks to have students comprehend the application of this understanding in daily life (Devlin, K. (2000). The four faces of mathematics. In M.J. Burke & F.R. Curcio (Eds.), Learning mathematics for a new century (2000 Yearbook). Reston, VA: National Council of Teachers of Mathematics).
Some specific places where Standards are addressed include:
NCTM Standards:
Problem Solving – build new mathematical knowledge through problem solving.
Students will use what they already know about 30 and 45 degree angles to determine sine of 75 degrees.
Reasoning and Proof – make and investigate mathematical conjectures
Analyzing the SG-3 scenario students will make conjectures that the information given must be an alternative form of measurement and through problem solving will gain an understanding that gradians are an alternative measurement of angles.
Students will conjecture about information they already know to determine and prove the addition identity.
Geometry – analyze characteristics and properties of two-and three-dimensional geometric shapes and develop mathematical arguments about geometric relationship; specify locations and describe spatial relationships using coordinate geometry and other representational systems; apply transformations and use symmetry to analyze mathematical situations; and use visualization, spatial reasoning, and geometric modeling to solve problems.
Students will analyze points within the unit circle based on drawing triangles to determine their coordinates and generalizing the geometric relationships that make this method of analysis work and using spatial reasoning.
Students will use special right triangles and geometric proofs to understand the addition identity.
Measurement – understand measurable attributes of objects and the units, systems, and processes of measurement; and apply appropriate techniques, tools, and formulas to determine measurements.
Students will identify what characteristics make a certain unit of angular measurement easy or hard to work with on the unit circle.
Process Skills –
Adapting the method of solving the problem as new elements are introduced in each new day’s task, based on reflection about the previous process of solving the problem and consolidating their mathematical thinking into statements that they can communicate with their peers (or teacher during the exploration and discussion stages.
Utah State Standards:
Standard 2: Students will represent and analyze mathematical situations and properties using patterns, relations, functions, and algebraic symbols.
Objective 2.3 Represent quantitative relationships using mathematical models and symbols.
After finding coordinates for points based on the 3-4-5 triangle, students will develop a symbol and model to refer to the angles more concisely.
Standard 3: Students will solve problems using spatial and logical reasoning, applications of geometric principles, and modeling.
Objective 3.1 Analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships.
Students will analyze properties of isosceles and equilateral triangles to develop relationships between the 30°, 45°, and 60° angles and their sines and cosines.
Students will analyze characteristics of special right triangles to develop mathematical arguments about the sines of other angles.
Standard 4: Students will understand and apply measurement tools, formulas, and techniques.
Objective 4.1 Understand measurable attributes of objects and the units, systems, and processes of measurement.
Students will understand that grads are an alternate unit of angular measurement.
Resources:
Downing, Douglas. (2001). Trigonometry the Easy Way. New York: Barron’s Educational Series, Inc.

http://standards.nctm.org/document/appendix/process.htm

Outline of Unit Plan Sequence (Anticipation of the sequencing of the unit with explication of the logical or intuitive development over the course of the unit—i.e., How might the sequence you have planned meaningfully build understanding in your students?)
DAY 1: Exposure to gradians – an alternative way of measuring.
A.  Teacher presentation of SG-3 Scenario
B.  Student problem solving of angular measurement discrepancies
C.  Student discovery of an alternative way of measuring that has 400 units in a circle.
D.  Explanation and discussion of gradians as a measurement system and the value and use of the system as an alterative measurement for angles.
DAY 2: Finding rational points on the unit circle using special right triangles and the Pythagorean theorem
A. Take vote on whether or not there are more than 4 points with rational coordinates on the unit circle.
B. Students explore to try to find more points.
C. Students share methods for finding points.
D. Discuss relationship between sine and cosine and the lengths of the right triangle.
DAY 3: Using rational points on the unit circle, develop new unit(s) of angular measurement
A. Students work on worksheet to label other coordinates on the unit circle.
B. Students share methods for finding points.
C. Give a name to the base angle for the 3-4-5 triangle.
D. Label the angles of the other points.
DAY 4: Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.
A. Re-cap on the previous day. Other triangles make for “nasty” angles.
B. What is the sin of 45 degrees? Why? Can you prove it? (classroom based discussion)
C. In groups, derive the sin of 30 and 60 degrees.
D. Many bases to choose from. Selected “easy “ one we could prove and work quickly with.
DAY 5: Angle measurements we already know can be used to derive the trig identity for addition of two angles.
A. Review briefly previous lesson.
B. Individual ideas of how to use special right triangles to represent sine of 75 degrees.
C. Group work to find the answer.
E.  Group presentations on how they thought about solving the problem.
F.  Group discussion on how the geometric proof relates to the sine addition identity.
Tools (A list of needed manipulatives, technology, and supplies, with explanations as to why these are necessary and preferable to possible alternatives).
DAY 1:
·  Stargate SG3 Overheads & handouts
·  Overhead projector
·  Each student needs calculator
DAY 2:
·  White board and marker
DAY 3:
·  Overhead & handouts of unit circles with different points based on (3/5, 4/5)
·  Overhead project
·  Each group needs calculator
DAY 4:
·  White Board (to make initial presentation)
·  Desks arranged in groups (to facilitate exploration)
DAY 5:
·  White Board and marker
·  Worksheets with 75 degree angle and right triangles drawn out.
DAY 1
Exposure to gradians – an alternative way of measuring. /
Unit Plan Sequence: Learning activities, tasks and key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected students’ mathematics) / Time / Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.) / Formative Assessment
(to inform instruction and evaluate learning in progress)
Miscellaneous things to remember /
Launching Student Inquiry
Put up overhead.
Ask a student to read the top paragraph.
Read out the calculations.
Pass out the student handouts.
Ask students: Are these the numbers you would expect to get?
What is different than what you would expect? / 5 min. / Overhead and handout say:
You are a member of Stargate SG3 team and have gated to P3X797. You have encountered an alien device that appears to be from the Ancients. Daniel Jackson has translated one of the glyphs on the device to mean TRIGONOMETRY. Your job is to identify what trigonometric function this device calculates.
Students will not pay attention to the numbers at first.
Students will frantically start typing things into calculators:
Some will notice: The values for 30 degrees and pi over 6 are not equal, etc.
Some will notice: The values are closer for the degree measurements than for the radian measurements.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Tell students: Work together with your groups to try to figure out what this device is doing, what could be going on here. / 15 min. / Some students: will try basic arithmetic variations on the sine or cosine function.
Some students: will switch back and forth between degrees and radians
Some students: will put the numbers into their calculator and try to get it to come up with a function.
Some students: will try using inverse trig. Functions to work backwards through the listed calculations. / Pay attention to which students have graphing calculators and which have scientific calculators.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
Ask each group what they did to try to get the numbers given.
Ask any group that figured out it was grads if they could determine how many grads make up a circle. / 5 min. / Students will somewhat envy any group that figured out that it was grads.
The group that figured it out will still feel a little frustrated, and like they were tricked. Most likely, they have a scientific calculator and will try to show the rest of the class how to change their calculator into grads.
They will describe how they used sines and cosines they had memorized to find their angles and set up ratios with the degree measurements to find that there are 400 total. / Ask each group for input. Include several different students in the discussion. Have one student come to the board and explain to the class how they arrived at the right conclusion.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Ask what would be some of the benefits of a system that broke the circle into 400 angular units.
What system would you use? Why? / 3 min. / Some will answer: It’s easier to relate angles with the same sin/cosine in different quadrants.
Most will say that they like degrees better but will admit it was just because they learned it first.
Some will try to suggest that it is easier to divide 360 down into parts, but they will be unsure of themselves and might benefit from some class discussion of it.
Formative Assessment
Explain how the gradian system was developed by a German engineering unit to correspond to the circumference of the earth. Tell them that 1 grad corresponds to 100 km of the earths’ 40,000 km circumference.
Explain to the students that gradians are used most often in navigation and surveying (and infrequently in mathematics). / 2 min. / Students will synthesize that 100 km x 400 gradians (the circumference of the earth) = 40,000 km. / Ask the students to explain why this was a viable system for this purpose.
Ask the students why this would be the best system for these purposes.
DAY 2
Finding rational points on the unit circle using special right triangles and the Pythagorean theorem. /
Unit Plan Sequence: Learning activities, tasks and key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected students’ mathematics) / Time / Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.) / Formative Assessment
(to inform instruction and evaluate learning in progress)
Miscellaneous things to remember /
Launching Student Inquiry
Trace unit circle onto board.
Put points at (±1, 0), (0, ±1).
Say: Someone said for these points, both coordinates are rational numbers.
Ask: Do you agree?
Take vote: Are there any other points on this circle that have rational coordinates for both x and y?
Count: and write numbers on the board.
Say: We’re going to work in groups for a little while to try to figure this out. / 5 min. / Most answer: Yes