5E LESSON PLAN (PRE-AP PRE-CAL)
AUTHORS’ NAMES:
Sean Kang
TITLE OF THE LESSON:
Spaghetti Sine
DATE OF LESSON:
January 26, 2011
LENGTH OF LESSON:
80 min
NAME OF COURSE:
Pre-AP Pre-Cal
SOURCE OF THE LESSON:
Collaboration with Emily Jensen (McNeil HS) and Sean Kang,
worksheet ideas from
http://departments.jordandistrict.org/curriculum/mathematics/secondary
/impact/Algebra%20II/Ready%20for%20web%20site/Spaghetti%20Sin
e%201.doc
TEKS ADDRESSED:
§111.35. Precalculus
(c) Knowledge and skills.
(1) The student defines functions, describes characteristics of functions, and translates among
verbal, numerical, graphical, and symbolic representations of functions, including polynomial,
rational, power (including radical), exponential, logarithmic, trigonometric, and piecewise-defined
functions. The student is expected to:
(A) describe parent functions symbolically and graphically, including f(x) = xn, f(x) = 1n
x, f(x) = loga x, f(x) = 1/x, f(x) = ex, f(x) = |x|, f(x) = ax, f(x) = sin x, f(x) = arcsin x, etc.;
(B) determine the domain and range of functions using graphs, tables, and symbols;
(C) describe symmetry of graphs of even and odd functions;
(D) recognize and use connections among significant values of a function (zeros,
maximum values, minimum values, etc.), points on the graph of a function, and the
symbolic representation of a function; and
(E) investigate the concepts of continuity, end behavior, asymptotes, and limits and
connect these characteristics to functions represented graphically and numerically.
(3) The student uses functions and their properties, tools and technology, to model and solve
meaningful problems. The student is expected to:
(A) investigate properties of trigonometric and polynomial functions;
(B) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to
model real-life data;
Content Reflection: The graph of a sine or cosine function in trigonometry is directly related to the value
of the function as exhibited on the unit circle. By drawing many special right triangles within the unit
circle, we can determine the exact distance of any side of these triangles by using the sine/cosine/tangent
relationships taught in Geometry class. The beginning of this activity solely uses the sine function, so the
opposite side to the angle (centered at the origin) is the focus of all angles. As the value of the angle
increases, the students will notice a periodic waving of the height of the triangle to rise up to one and
decrease down to negative one, only to repeat indefinitely. This behavior is depicted visually using dried
pieces of spaghetti.
PERFORMANCE OBJECTIVES:
Students will be able to:

Graph the parent functions of sine and cosine

Find the domain and range of sine and cosine functions

Describe the relationship between the unit circle and both sinusoidal graphs
RESOURCES, SUPPLIES, HANDOUTS:
Vocabulary Worksheet (for student disability accommodation)
Calculators (one per group)
Spaghetti Sine Instruction Sheet (one per group)
One black/dark-colored sharpie (one per group)
Printout of blank unit circle (one per group)
One piece of yarn long enough to cover circumference of unit circle (one per group)
Scissors (one to cut yarn)
Tape or glue (one per group)
Large piece of butcher paper (or two blank sheets taped together) (one per group)
Dried spaghetti (one package total)
SAFETY CONSIDERATIONS:
Preparedness for fire/severe weather drills, protective action, and emergency medical situations
(Source: http://www.gccisd.net/default.aspx?name=od.links)
Old spaghetti can be sharp, so don’t allow students to poke each other. It is also dirty, so do not
let them eat it.
DON’T RUN WITH SCISSORS! (for the teacher)
ENGAGEMENT
What the Teacher Will Do
Discuss the behavior of a seat on a
ferris wheel with students. Lead
students to the periodic nature of
the height, and how it constantly
waves up and down.
Est. Time: ___ 10 min_____
Probing Questions and Answers
What the Student Will Do
What shape is a ferris wheel?
Answer questions and participate
[Circle]
in discussion
What is the highest height you will
go on the ferris wheel? The
lowest? [Top and bottom of wheel]
Have students get out unit circle
Pretend the ferris wheel is our unit Look at previous notes
(already completed from previous circle. What would the height be if
lesson) and discuss height of sin
the angle was 30 degrees? [1/2]
and cos
Have students get into groups of
Gather materials
~3-4 and get materials for activity
EXPLORATION
Est. Time: __35 min___
What the Teacher Will Do
Probing Questions and Answers
What the Student Will Do
Have groups complete “Spaghetti What pattern do you notice about Follow instructions on Spaghetti
Sine” instructions
the height of every special triangle? Sine sheet
[It repeats up and down from -1 to
1]
Monitor class to make sure
What variable becomes the x-axis Cut a piece of yarn for
connection is being made between on our graph? [Angle measure or circumference of circle, measure
sine function and its height on unit Ѳ]
height of sine at all special radian
circle
measures, break pieces of spaghetti
to match, glue on butcher paper
and create a wave function
Assist students as needed putting
materials together
EXPLANATION
Est. Time: __20 min____
What the Teacher Will Do
Probing Questions and Answers
Explain that each piece of spaghetti What shape does the graph of sine
represents the value of sin at a
make? [A wave]
specific radian measure, and that
there is an increasing/decreasing
pattern in its value
Have students graph f(x) = sin(x)
in their calculator to verify their
spaghetti graph is the same
Have students repeat project using What function is created from
horizontal distances instead of
using adjacent sides instead? [cos]
vertical (should create cos(x)
graph)
What the Student Will Do
Finish spaghetti graph and present
to the teacher and fellow students
Graph functions on calculator
Use more spaghetti and yarn for
cos graph
ELABORATION
What the Teacher Will Do
Have student groups think of a
real-life situation that can be
represented as a sine or cosine
graph
Have one student from each group
share their group’s poster with the
class
Est. Time: ____10 min____
Probing Questions and Answers
What the Student Will Do
What real-life situations can you Create “poster” of their real-life
think of that can be graphed using sine or cosine function
a sine or cosine function?
EVALUATION
What the Teacher Will Do
Post-assessment: Have students
determine the values of sine and
cosine beyond the 2π period.
Have one student from each group
explain why multiple answers can
be the same
Est. Time: ____5 min____
Probing Questions and Answers
What the Student Will Do
What does being periodic do to the Use spaghetti sine graph and unit
values of sine and cosine beyond circle chart to answer questions
2π?
Why are sin(0) and sin(2π) the
Present findings to the class.
same?
Present group’s poster to the class.
Vocabulary Worksheet (Print in Landscape)
Vocabulary
Definition
español
Trigonometry
(Abbreviated
trig.) The study
of the
measurement
of triangles
trigonometrÍa
Unit Circle
Wellreferenced
circle on the
Cartesian
coordinate
plane with a
radius of 1
circulo unitario
Sine
Trig. function
which is odd
and periodic;
has a range of
[-1,1] and a
domain of all
real numbers.
seno
Cosine
Trig. function
which is even
and periodic;
has a range of
[-1,1] and a
domain of all
real numbers.
coseno
Definitions in your own
words
Representation of
the vocabulary
(drawing, symbols,
examples, etc.)
Radians
A unit of
measurement
to describe the
angle between
two sides
radianes
Spaghetti Sine
Instructions
   
, , ,... 
6 4 3 
1.
Write the radian measure for each angle of the circle. 
2.
Wrap the yarn around the circle so that it is the length of the
circumference of the circle.
3.
Mark the angles on the yarn.
4.
Straighten the yarn out and tape it to the legal size papers.
5.
Label the marks on the yarn so that they correspond to the angles
   
, , ,... 
6 4 3 
on the circle. 
6.

6
Break one piece of spaghetti so that it is the vertical
distance from the initial angle 0 to the point

on
6
the circle.
7.
Tape this piece above the string at the mark labeled

.
6

6
 
4 3

2

3
8.
Continue doing this for all the points on the circle.
9.
Write the y value of sine at each radian measure.

4

6