Lesson Plan
Name: Jennifer DeSieno
Unit Topic: Trig. Ratios and Functions
Lesson: Cofunctions
Date: 3/11/04
Course: Algebra II-ES
Lesson Type: Construct a Concept
NYS Mathematics, Science, and Technology Learning Standards Addressed:
Standard 1:
Standard 3:
Standard 6:
Standard 7:
Students will use mathematical analysis, scientific inquiry, and
engineering design, as appropriate, to pose questions, seek answers, and
develop solutions.
Students will understand mathematics and become mathematically
confident by communicating and reasoning mathematically, by
applying mathematics in real-world settings, and by solving problems
through the integrated study of number systems, geometry, algebra, data
analysis, probability, and trigonometry.
Students will understand the relationships and common themes that
connect mathematics, science, and technology and apply the themes to
these and other areas of learning.
Students will apply the knowledge and thinking skills of mathematics,
science, and technology to address real-life problems and make informed
decisions.
Objective:
The student will distinguish between examples and non-examples of trigonometric cofunctions.
(comprehension)
Materials:
4-page packet of cutouts (1 per group)
Pair of scissors (1 per group)
Glue stick (1 per group)
Colored pieces of large construction paper (1 per group)
Transparencies of the cutouts, already cut out and answered (teacher use)
Construct a Concept task sheet (1 per student)
Chalk (teacher use)
Transparency divided into two columns, “Column A” and “Column B”
(teacher use)
Overhead markers (teacher use)
Cofunctions Homework Assignment worksheet (1 per student)
Anticipatory Set:
I will say the following to the class:
“Good morning everyone. For today’s lesson, we will be integrating a lot of the information we
have already learned about trigonometry and use this knowledge to construct a new concept or a
new component to trigonometry that you are probably not very familiar with. To complete this
activity, we need to get into our cooperative learning groups. Please assign each member of
your four-person group a position. I hope you are all rotating positions so that each member of
the group has had a chance to be the communicator/facilitator, materials person, reporter, and
recorder…. Now that we have the positions set and our desks moved, would the communicator
please come up to the front of the room to get directions for this activity.”
Lesson Body:
Sorting and Categorizing
Hand out a packet of cutouts to each communicator. Instruct them to go back to their groups and
send the materials person to get a pair of scissors, glue stick, and piece of construction paper. In
groups they are to cut out the different pieces of the packet, even the sheet of paper that has no
writing on it. The students can either work together as a group to answer each cutout or they can
divide the work evenly among each member. The groups will be instructed to pay close
attention to whether they are using degree or radian measurement and to look for the special
angles they have previously learned. After they have answered or graphed each cutout they are
to look for differences and similarities among them. The students should then separate each
cutout according to the similarities or differences they noticed, organize them into two separate
columns, and then glue them to the construction paper.
Reflecting and Explaining
When the majority of the class is finished gluing the cutouts to the construction paper, I will
instruct them to label one column “Column A” and the other “Column B.” They will be given a
minute or two to discuss in their groups why they separated the columns the way they did. The
recorder should write down a summary of the group’s discussion on a scrap piece of paper.
As the students are working in their groups I will set up the overhead placing the transparency
version of the cutouts with the answers already written on them, in random order, on top of the
overhead.
After a few minutes when the groups are done discussing, the class will be brought together as a
whole again. A reporter from of the of the groups who volunteered will be asked to come up to
the overhead and sort the cutouts into the columns their group used. The reporter will be asked
to explain their groups’ reasoning. I will lead the discussion by asking:
o “Why did you separate the columns this way?”
o “What did you notice about the examples in each column?”
o “How do the examples in Column A differ from those in Column B?”
I will then ask the rest of the class if their group had a different answer or an alternative way of
separating the cutouts. Once I get all of the groups input and any other suggestions or reasons, I
will rearrange the transparency cutouts on the overhead the way I would like them to be
arranged, into two columns demonstrating examples and non-examples of cofunctions. (There
should be 5 in each column. See Construct a Concept task sheet for the correct answers).
Generalizing and Articulating
At this stage the students should realize the answers and values to the groups of functions in
Column A are all the same and Column B all different. I will hand out to each student the
Construct a Concept task sheet. Before I let them begin working with their groups I will
generate some thought and discussion by posing the following questions:
o “Do you think there is a reason why the pairs of functions in Column A are
the same?”
o “Is there anything else similar to the equations besides their values or graphs
being the same?”
I will give the students 2-3 minutes to discuss with their groups what they notice about the
examples in Column A and then have them write down a conjecture.
Verifying and Refining:
Once each group has developed their own conjecture I will ask them to come up with two
different examples based upon their conjecture. One example should fit under Column A and the
other under Column B. Every member of the group should write down the examples on their
task sheet, but the recorder will be given the added responsibility of writing the examples on the
blank cutouts and gluing them to the construction paper in the appropriate column.
Next, I will gather the class together as a whole again and ask a reporter from one of the groups
to state their conjecture. I will then ask for the examples they came up with and write them
down on the overhead under the appropriate column. I would ask questions to get them to
verbalize their rationale for making their conjecture and how they came up with their examples.
I will then ask the other groups if they had similar or different conjectures. For different
conjectures I would repeat the same process of having the group state their conjecture, the
examples they came up with, and their rationale. I would lead a class discussion on the
differences in conjectures, trying to get all the groups to concentrate on how the angles are
related. I will then tell them to get back into their groups and come up with a new conjecture and
more examples.
If the groups are coming up with similar conjectures then most likely they understand the
concept of cofunctions and complimentary angles. I will ask the other groups to give their
examples for each column and then write them down on the overhead. Next, with input and
assistance from the students, we will consolidate or combine the slightly different conjectures
each group developed and come up with a solid definition or a written understanding of what
cofunctions are.
Once the class has determined a definition for cofunctions I will ask them to return to their
groups once more. I will instruct each group to come up with two more examples, one that is an
example of a cofunction (fits under Column A) and one that is a non-example of a cofunction
(Column B). The recorder of the group will write the example on the extra cutouts and glue
them to the construction paper.
I will then write down formal notes on the blackboard to be sure that the students fully
comprehend the concept of cofunctions.
Formal notes:
Table of Cofunctions:
sin x = cos (90o - x)
cos x = sin (90o - x)
IMPORTANT:
tan x = cot (90o - x)
cot x = tan (90o - x)
A function of any angle is equal to the cofunction of its
complement (sum equal to 90 degrees)
Closure:
“Through our exploration of cofunctions in class today we have finally determined why
cosine is called cosine and similarly why cotangent is called cotangent. We saw that COsine is
the COfunction of sine and that COtangent is the COfunction of tangent. Furthermore, we were
able to see that a function of any angle, whether it be sine, cosine, tangent, or cotangent, is going
to be equal to the COfunction of its COmpliment. Before we leave, lets do an example…”
Find the function value of cot 60o (write on board)
“To answer this question, we need to use the cotangents cofunction, which is tangent,
to rewrite the problem.”
tan (90o - 60o)
tan 30o
(write on board)
“Therefore, the function value of cot 60o is tan 30o. Although tan 30 is a special angle
everyone should know, if a question asks for the function value of something your answer must
be in terms of another function. Do not try and give a decimal answer! Your homework for this
evening is to complete the worksheet I am handing out now. Have a good day!”
Homework/Assessment:
Cofunctions Homework Assignment worksheet.