Download Trig Unit Plan - Mathematics Education

Unit Plan Cover Sheet – Trigonometry Lesson Plan
Name(s): Michael Clarke, Shelley Mourtgos, Kip Saunders, & Erin Shurtz
Unit Title: Trigonometry Lesson Plan
Date: March 8, 2006
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 2
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,
2017 May 6 1:38 AM Page 1 of 21
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.
2017 May 6 1:38 AM Page 2 of 21
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
2017 May 6 1:38 AM Page 3 of 21
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
2017 May 6 1:38 AM Page 4 of 21
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.
Formative
Assessment
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)
Put up overhead.
Ask a student to read the top
paragraph.
Read out the calculations.
Time
Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.)
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
Launching Student Inquiry
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.
Pass out the student
handouts.
Ask students: Are these the
Students will frantically start typing things into calculators:
2017 May 6 1:38 AM Page 5 of 21
DAY 1
Exposure to gradians – an alternative way of measuring.
Formative
Assessment
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.)
numbers you would expect
Some will notice: The values for 30 degrees and pi over 6 are not equal, etc.
to get?
Some will notice: The values are closer for the degree measurements than for the
What is different than what
radian measurements.
you would expect?
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Tell students: Work together
15
Some students: will try basic arithmetic variations on the sine or cosine function.
with your groups to try to
min.
figure out what this device
Some students: will switch back and forth between degrees and radians
is doing, what could be
going on here.
Some students: will put the numbers into their calculator and try to get it to come
up with a function.
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
Pay attention to
which students have
graphing calculators
and which have
scientific
calculators.
Some students: will try using inverse trig. Functions to work backwards through the
listed calculations.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
Ask each group what they
5 min. Students will somewhat envy any group that figured out that it was grads.
did to try to get the numbers
given.
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.
Ask any group that figured
out it was grads if they
could determine how many
grads make up a circle.
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
2017 May 6 1:38 AM Page 6 of 21
DAY 1
Exposure to gradians – an alternative way of measuring.
Formative
Assessment
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.)
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
conclusion.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Ask what would be some of 3 min. Some will answer: It’s easier to relate angles with the same sin/cosine in different
the benefits of a system that
quadrants.
broke the circle into 400
angular units.
What system would you
use? Why?
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
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
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.
2017 May 6 1:38 AM Page 7 of 21
DAY 1
Exposure to gradians – an alternative way of measuring.
Formative
Assessment
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
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.)
mathematics).
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?
5 min.
If too many have to
think about it, ask:
Can someone give
us a quick reminder
what rational
means?
Most answer: Yes
A few: Have to think about it for a second
Some: will try to start reasoning out loud
If too many are
biasing others, say:
just think to
yourselves for a
second
2017 May 6 1:38 AM Page 8 of 21
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)
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.
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
Could be many, a few, or none.
If all think that there are others, tell them to prove it by finding as many of them as
possible.
If all think that there aren't any others, tell them to prove it and try to come up with
why that is.
If it's a good spread, tell them to explore the problem with their groups and try to
come to a consensus as a group.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Ask: them: Can you think of
15
Some: will try to use the equation x2 + y2 = 1 and try to find rational solutions, by
any ways to manipulate that
min solving, plugging in random fractions, or using their calculator.
equation to make it easier to
work with.
Ask them: Can you prove
that those points are actually
on the unit circle?
Some: will start by writing down all the irrational points they have memorized and
then stare blankly at their papers
Tell them: See how many
Some: will instantly think of using a 3, 4, triangle normalized w\ hypotenuse of 1
other points you can find.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
Say: Finish up your thought. 5 min. If no one found any, have each group go through the methods they tried. Then, skip
to lesson day 4 material on how we get the points for 30o, 45o, and 60o.
If still not quiet,
say: Ok, finish up
2017 May 6 1:38 AM Page 9 of 21
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)
Say: Finish up your
sentence.
Time
Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.)
If a group found some, have them explain how they got them and write the points
on the board. Make sure to draw a right triangle for one of the points. Have the
other groups compare their processes with what that group did.
Formative
Assessment
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
the word you were
on.
If they only found them by guess and check, have them prove that those points
work in front of the class and see if anyone else in the class can use that proof as a
springboard for developing a general method for finding rational points.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Emphasize the similarity
5 min. They will start by being confused on where to put the right angle and then will
between the Pythagorean
realize that there are two possibilities, but neither is at the origin.
theorem and the equation of
a circle. Ask students to
They will point out that the height, base length, and hypotenuse correspond to the
illustrate the relation.
sine, cosine, and radius of the point on the circle.
2017 May 6 1:38 AM Page 10 of 21
DAY 3
Starting with rational points on the Unit Circle, develop new units of Angular Measurement.
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)
Ask: Last time we found
that there were some points
on the unit circle that had
rational numbers for both
coordinates. Remember?
Ask: Why did we have to do
the division to get the
points?
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
5 min. They’ll call out 3, 4, 5 and eventually also call out 3/5, 4/5.
Some will answer that the radius needed to be one.
-- Point out that the radius corresponds to the hypotenuse of the triangle.
Introduce: Using the first
point we found, to locate
more rational points.
If not, ask if
renaming angle
measurements
affects sine/cosine
measurements any.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Pass out handouts and tell
15
At least one person in the group should be able to quickly complete the first two
students to label the points.
min. circles and night have to explain it to the other group members.
Prod them to reduce it as
much as possible.
For the third circle, second point:
- Some will think to find the angle and double it and then find sine and cosine.
Point them towards looking
at the angle.
They might think it is irrational.
- Some will try to draw in more triangles connecting to the one they know and find
geometric relationships.
If a group is stuck,
have them draw
triangles and figure
out which sides are
the same length.
2017 May 6 1:38 AM Page 11 of 21
DAY 3
Starting with rational points on the Unit Circle, develop new units of Angular Measurement.
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.)
Point them to a calculator to
get the exact angle.
- Some might try to measure with a protractor or estimate visually. If they estimate
x = 1/5 for second point, let them but make them find the y value and then have
them prove it.
Point them back towards
looking at the angle and
measuring it.
For the third circle, third point:
- Some groups will use the same first method above.
- Some will struggle to apply finding second point process to be able to find third.
Put up overhead.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
5 min.
Quickly have one student
review the first two circles.
Will talk about negatives and about switching the x and y coordinates.
Have another group explain
how they got values for the
third circle.
Will talk about finding the angle, doubling it, and using sine / cosine (or using the
angle addition property).
After explanation ask if
anyone else sees a faster
way to do it.
Formative
Assessment
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
To help reduce, ask
if they notice any
pattern in the
fractions
(denominators).
Make sure to ask
the rest of the class
if they agree with
what was labled.
If no one sees faster
way, point back to
triangles and
lengths.
2017 May 6 1:38 AM Page 12 of 21
DAY 3
Starting with rational points on the Unit Circle, develop new units of Angular Measurement.
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
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Have class name the angle.
5 min. Naming it with a word rather than a letter will confuse some.
Use a word, not
Ask how many of that angle
Some will estimate 10ish.
letter.
there are in a full circle.
Some will grab calculator and answer exactly: 9.7640629 . . .
They will complain that it’s an annoying number.
Ask how many radians are
in a full circle.
They will all answer 2 pi, but will not realize that’s just as ugly of a number.
Re-emphasize: but how
many is that?
They will eventually answer about 6 or 6.28 (might take some prodding like “more
that 10, less than 5?, etc.)
Point out that the symbol 
lets us have an angular unit
that doesn’t divide evenly
into a circle.
Suggest 8M as such a
symbol.
They will think it’s weird but slowly catch on as we label angles we’ve worked on
today.
Go back and label the angles
in circle 3, then 1, then 2.
They will answer faster if you label 4M before trying 4M-1, for example.
Ask class what they think
Some will like the fact that we divided the circle into something we can’t actually
If not, ask if
2017 May 6 1:38 AM Page 13 of 21
DAY 3
Starting with rational points on the Unit Circle, develop new units of Angular Measurement.
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)
this teaches about units of
angular measurement and
how we’ve picked them in
the past.
Time
Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.)
measure and mention it.
Some will mention that it seems we can divide it however we want, with nothing
being magical about the existing units.
Formative
Assessment
(to inform instruction
and evaluate learning
in progress)
Miscellaneous things
to remember
renaming angle
measurements
affects sine/cosine
measurements any.
Some will point out how angle names don’t affect sine and cosine
2017 May 6 1:38 AM Page 14 of 21
DAY 4
Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.
Formative
Unit Plan Sequence:
Learning activities, tasks and
Assessment
key questions (What you will do
(to inform instruction
Anticipated Student Thinking and Responses
Time
and say, what you will ask the
and evaluate learning
(What mathematics you will look for in student work and interactions.)
students to do, how you will
in progress)
Miscellaneous things
to remember
accommodate to unexpected
students’ mathematics)
Introduce Base-60
Wrap-up of day 3
5
What is the sine of 45?
Why? Let’s prove it.
5
Why were we able to prove
it?
Launching Student Inquiry
Wondering why anybody would work in any of those bases. Start to wonder why
we work with the base that we do for the unit circle.
Students will have enough of a background with trig that they will be able to
answer the question. Some might even be able to explain why and a few may be
able to prove it.
Students will see that it worked well because it was an isosceles triangle.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
What about ½ or (root 3)/2?
10
Now without the isosceles triangles explore the other angles in the hope that they
Can you prove those?
discover a way to make an equilateral triangle or some other clever way. It is
anticipated that students will try to form a new shape out of two triangles like the
example of the 45-45 triangle. Some might get caught with just one triangle. Some
may make the wrong triangle, one that is isosceles but not equilateral.
Group Presentations
Have a student
come and
demonstrate why on
the board.
Work individually
and then in groups
to find a method.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
5
Choose a group who got it or got close to present on the board.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Would any other angles
Can we use this proof method for different sized triangles?
Facilitate classroom
allow for the proof?
5
discussion.
2017 May 6 1:38 AM Page 15 of 21
DAY 4
Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.
Formative
Unit Plan Sequence:
Learning activities, tasks and
Assessment
key questions (What you will do
(to
inform
instruction
Anticipated Student Thinking and Responses
Time
and say, what you will ask the
and evaluate learning
(What mathematics you will look for in student work and interactions.)
students to do, how you will
in progress)
Miscellaneous things
to remember
accommodate to unexpected
students’ mathematics)
So why the 30-60-90
triangle?
How can we estimate the
sine of 70?
Students will see the special abilities we have with the 30, 60, and 45 degrees. Gain
appreciation for choice of base. Now know why we break it up into those
partitions. Specifically, that two 30-60-90 triangles can be combined to make an
equilateral triangle of side 1. It is now possible to find the other lengths and get
nice trig values. Also, the 45-45 is isosceles with a hypotenuse of 1, allowing us to
solve the other sides. These give easy angles and easy lengths that break up the unit
circle of 360 degrees into equal partitions.
Use other angles we know to approximate 70 degrees. Leave it open for a
discussion next class about other nice angles. i.e sin addition formula, etc…
Comprehension of
where the numbers
come from for the
trig functions of the
standard angles.
2017 May 6 1:38 AM Page 16 of 21
DAY 5
Angle measurements we already know can be used to derive the trig identity for addition of two angles
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
Review: “We’ve been
talking about different ways
to measure angles, and last
time we talked about some
particularly pretty angle
measurements – what were
they?”
7-10
min
Students will remember the special right triangles.
Draw or have a
student draw
special right
triangles and their
side lengths.
Launch: Today we want to
see if we can apply what we
learned about those angles
to find out information
about other angles.
The problem we are going
to be working with is “What
is the sine of 75 degrees?”
Some students will remember the addition identity (sin 75 = sin (45+30) =
sin45*cos30+sin30*cos45). Some may remember to use the addition identity but
not remember it correctly.
Before you start working on
that as a group, what are
some different ways we can
use what we know about the
special right triangles to
represent this problem?
Some students will draw something that represents sin75 = sin (45+30). Others
may set up something that represents sin75 = sin(30+45).
Others may set up a subtraction identity, such as sin75 = sin(90-15).
Others may set up the 75 degree angle “sideways” so that it isn’t on the unit circle.
Individual students
will offer different
ways of
representing the
problem and
present them to the
class.
2017 May 6 1:38 AM Page 17 of 21
DAY 5
Angle measurements we already know can be used to derive the trig identity for addition of two angles
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
Give a hint: “ There are
different ways of
representing this problem
and different ways for
finding an answer. To get
you started, I’m going to
show you the easiest
representation: If I extend
lines of top triangle to form
another right triangle (may
erase other line) - does this
still represent a 75 degree
angle?”
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
Students will hopefully see that this is the same angle.
Some students may want to use the triangle formed by extending the lines even
further.
Some students may want to use the inside triangle rather than the outside triangle.
If so, ask them what we can use to find out angles in the outside triangle rather than
the inside one (linear pairs).
“What are the unknown
lengths we need to
determine now to find
sin(75)?” Draw outside
little right triangle to give a
simpler visual.
Students will identify opposite side and hypotenuse. They may need help seeing
that they already know part of the other side and how the small outside triangle
drawn can be used to find the missing length. They also may be confused about
which hypotenuse they need to find.
Call on individual
student to show/tell
what the unknown
lengths are.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Hand out worksheets with
75 degree angle and right
triangles already drawn.
“Now work in your groups
to see if you can use this
10
min
Students will work
individually and in
groups to find an
answer.
2017 May 6 1:38 AM Page 18 of 21
DAY 5
Angle measurements we already know can be used to derive the trig identity for addition of two angles
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)
representation to find the
answer.”
Walk around and ask listen
to group ideas. If they get
stuck: help them see how to
use linear pairs to find the
angle measurements of the
small outer triangles, ask
them what they know about
isosceles triangles, the
Pythagorean theorem, or trig
functions to find missing
lengths.
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
Students may use any combination of trig functions, isosceles triangle theorem,
Pythagorean theorem, and information about linear pairs to determine information
about the missing hypotenuse and leg lengths.
Some students will not realize that they have an isosceles triangle with two legs
having length 1 to work with.
Some students will find the missing adjacent side first and then use that and the
Pythagorean theorem to find the missing opposite side.
Most students will not try to apply trig functions to the right triangles, but will
mostly rely on the Pythagorean theorem to determine missing sides without paying
attention to what is already known about the angles and their sines and cosines.
Some students may even introduce ideas about tangent and contangent.
Most students will probably not have enough time to finish the task.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
Have one group who got
5 min Other students will be able to follow the groups reasoning and may have some
closest to the answer present
questions as to why they did it the way they did.
their work. Let them
answer any questions from
other students.
If they did not get the
answer, ask questions about
Group presentation
Students may not have come to a final answer. One group may have come up with
one of the missing sides and another group with the other missing side.
2017 May 6 1:38 AM Page 19 of 21
DAY 5
Angle measurements we already know can be used to derive the trig identity for addition of two angles
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
isosceles triangles, linear
pairs, the Pythagorean
theorem, or trig functions to
finish the problem as a
class. Maybe have two
groups present – one on
finding the missing
hypotenuse length and
another on finding the
missing leg length.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
“What relation can we see
5-7
Students may break up addition identity and try to find where the different
Individual student
between this geometric
min components are shown in the geometric proof.
responses.
proof and the sine angle sum
identity?”
Write on the board sin(75)
= sin(30)cos(45) +
cos(30)sin(45) = (y+1/2)/h
Conclusion: We can use
what we know about special
right triangles to find
information about other
angles and to understand the
angle sum trig identities.
Similar processes may be
used to obtain other trig
2017 May 6 1:38 AM Page 20 of 21
DAY 5
Angle measurements we already know can be used to derive the trig identity for addition of two angles
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
identities. Also if there is
time, point out that these
trig identities work no
matter what angle
measurement is used.
Summative Assessment
We have discussed breaking the unit circle up into various units of angular
measurement, including 400 grads, 2 radians, 360°, and angles utilizing the 3-4-5
Pythagorean Triple. We divide a day into 24 hours. Suppose we divide the unit
circle into 24 “hours.” Determine the x and y coordinates for the following points
and plot (and label) them on the unit circle:
(a) 2 “hours”
(b) 5 “hours”
(c) 9 “hours”
What are the advantages and disadvantages of using this system to divide up a
circle?
2017 May 6 1:38 AM Page 21 of 21