Download The Tangent of an Angle

MthEd 377 Lesson Plan
Cover Sheet
Name: John Jett and Edda
Date: November 7, 2005
Section Title: 13.6 The Tangent of an Angle; 13.7 The Sine and Cosine Ratios
Big Mathematical Idea(s):
Congruent angles in similar right triangles have corresponding ratios of the sides.
These ratios will always be equal to each other for a chosen acute angle of that right triangle.
In a right triangle, if a side and an angle (other than the right angle) are given, the other side lengths can be found.
Similarly, if two sides are given, any angle other than the right angle can be found as well. This happens because all
congruent angles have the same corresponding ratios of the sides, as stated above.
Why are these BMIs important mathematically?
While similarities have mainly dealt with two triangles having equal proportions, trig functions extend this idea to
infinitely many triangles. Now we can know the ratio of any triangle’s side lengths by just choosing one angle. In other
words, the tangent, cosine, or sine of an angle will always stay the same, no matter what the side lengths are. The only
criteria is if the triangles are similar, or right triangles in this instance.
Knowing this relationship also leads to important problem-solving techniques. For example, students can find any side
of a right triangle by knowing one of its non-right (acute) angles.
How does this lesson fit in to the overall unit? (i.e., How does this lesson build
mathematically on the previous lessons and how do subsequent lessons build
mathematically on it?)
Students have just learned the side lengths and angles of special right triangles. As such, they have some previous
understanding of the minimal requirements to solve for the remaining asides of a special right triangle. They
generalized these side lengths using variables, which mean any sized special right triangle will have the same angles.
Students have dealt extensively with similar triangles. They know that similar triangles have the same side proportions
and congruent angles. This lesson builds on this idea by introducing infinitely many triangles. These triangles are
produced by creating right triangles from the same angle. Because the triangles all have congruent angles, they are
similar. This implies that they will have the same proportions one with another. However, we can now choose any acute
angle in our right triangle and know the proportions of its own sides due to the trigonometric functions.
On a simpler note, trigonometric functions allow people to solve for any side or angle of a triangle. This comes
particularly handy in elementary physics. On a broader note, this lesson will extend into the mathematical branch of
trigonometry. This will lead to theorems such as the Law of Cosines, Law of Sines, and eventually identities. We left
some sections open-ended to lead into our inverse and reciprocal trig functions. Pretty soon, a simple trig function will
turn into an Euler’s formula or be used to find polar coordinates. Let the games begin…
Grading rubric (for Keith’s use)
5 The Big Mathematical Idea addresses core mathematical concepts and is clearly
articulated
5 Description of the importance of the topic is well thought out and relevant
5 There is a clear, insightful discussion of how this lesson fits in to the
mathematical content of the overall unit
5 Lesson sequence is well thought out and detailed
5 Students' thinking is anticipated with forethought and detail
5 Reactions to students' thinking is mathematically oriented, insightful and
detailed
10 3-5 reflection paragraphs
10 Met with Dr. Leatham and made
demonstrate thoughtful
appropriate revisions based on this
reflection and are clearly
discussion
OR
articulated
30 3-5 page reflection paper
demonstrates thoughtful reflection
and is clearly articulated
Lesson Sequence: Learning
activities, tasks and key
questions (what you will do
and say, what you will ask
the students to do)
Anticipated Student
Time
Thinking and
Responses
Your response to
student responses
and thinking
Formative
Assessment,
Miscellaneous
things to
remember
Launching the Lesson
Draw two similar triangles on the
board, labeling with sides ABC and
abc.
5 min
“What do we know about these two
triangles’ sides and angles?”
Write
A B C
 
a b c
Students should
remember these
properties from
previous lessons.
Similar triangles have
congruent proportions and
angles.
So, we know that A/a =
B/b = C/c for ∆ABC and
∆abc to be similar.
on the board.
Using algebra, show that A/B=a/b
from the previous review.
“We know that these two proportions
are equal for these two triangles, but
what if we had infinitely many similar
triangles? Would it still work?”
Move onto Task 1.
Orchestrating the Task
Divide class into pairs in groups of
four (two pairs in each group, two
worksheets total.)
TASK 1: “Using a ruler and a
protractor, draw right triangles from
the angle measurements given. When
you are finished, move onto the
second part of the exploration.”
15 min
Once the class has finished the first
part, tell the class the instructions for
the second half, as written on the
worksheets. Assign groups their
angles/triangles to use.
Draw a chart on the board to
organize all the information the
groups will be presenting. Only use
the sine, cosine, and tangent ratios.
Move onto the first discussion.
TASK 2: “Now that we know the
ratios of the angles, let’s find out how
they can be useful.”
Pass out the second worksheet to the
same pairs as previously assigned.
Student should have no
problem using a
protractor and a ruler to
make right triangles.
Then again, some may not
know how to draw a right
angle.
Students may not know
how to describe their
ratios.
Encourage groups to
make whatever size
triangle they would like.
Ask what degree a right
triangle is. Show where
90 is located on the
protractor. Connect the
point of the degree mark
with the line to create a
triangle.
Write on the board
“Opposite,” “Adjacent”
and “Hypotenuse” on a
triangle with a marked
angle. Bring this to the
attention of the entire
class.
10 min
They may draw a picture
with a triangle and fill in
lengths and angle
measures they know for
each question. In each
case they will notice a
Ask “How did you know
it was a triangle?... Well
done.”
Give half the class
worksheet one. And
the other half
worksheet two.
Each group of four
will have the same
worksheet.
Provide a ruler and
a protractor for
each pair.
Draw a table on the
board that will
organize all the
information
(rows=trig ratios
and columns=angle
values.)
While walking
around, assign
groups one or two
angles each to find
the average of all
the ratios given.
Write on an
overhead
transparency.
Lesson Sequence: Learning
activities, tasks and key
questions (what you will do
and say, what you will ask
the students to do)
Anticipated Student
Time
Thinking and
Responses
right triangle.
They may not think to
draw a picture of a
triangle.
They may not realize it is
a right triangle.
Problem #1
From the previous task
they know the tangent of
an angle is equal to the
ratio of the opposite leg to
the adjacent leg in a right
triangle. They may set up
the equation:
Tan (40) = height/20.
To solve this equation they
will probably multiply
both sides of the equation
by 20. In order to
multiply 20 by
Tan (40), they will have to
know a number or ratio of
Tan (40). They will know
this from the previous
task. After multiplying
they will find the height of
the flag pole.
They may not know how
to set up the equation.
They may not see the
relation with tangent.
2.
Problem #2
Their responses will be
much like the previous
questions.
Problem #3
In this question they will
notice in the picture that
they know the legs of a
right triangle but not the
angle. They may notice
that the legs have to do
with tangent ratio. So
they may set up an
equation like this:
Tan (angle) = 60/100.
They might realize that
60/100 = .6 or 3/5, which
is approximately a number
they have used in the
previous task. So
whatever angle they used
to get that ratio is what
Your response to
student responses
and thinking
Formative
Assessment,
Miscellaneous
things to
remember
Lead them by asking
them if they could draw
a picture of the scenario.
Say, “If a flag pole is
standing straight up
what kind angle does it
make with the ground?
Ask, “Why did you use
tangent?... Well done.”
Ask, “What are you
trying to solve?”
“Is there anything you
can use that we just
learned about today?”
“How does knowing the
angle measure help?”
This question is
similar to problem
1, but instead they
will use the sine
and cosine ratios
rather than tangent.
Ask, “How did you know
to use tangent? Why did
you use that angle?”
The ratio they find
(60/100) will be a
ratio that is already
on the board. They
can look there as a
reference to find
which angle
corresponds with
this.
Bring up arc sin,
arcos, and arc tan
in the discussion.
Lesson Sequence: Learning
activities, tasks and key
questions (what you will do
and say, what you will ask
the students to do)
Anticipated Student
Time
Thinking and
Responses
Your response to
student responses
and thinking
their answer will be.
They may be confused
about what to solve for, or
in writing an equation. Or
they may not know how
they can figure out what
angle gave them that
ratio.
Lead them in the right
direction by asking,
“What do we know
about the legs in a right
triangle?”
“Can you use this
information at all?”
Formative
Assessment,
Miscellaneous
things to
remember
Have them remember
what we did in the
previous task.
Facilitating the Discussion
TASK 1: By this time, all ratios are
on the board and overhead projector.
Call the attention of the class
15 min
Some students may
wonder why I didn’t write
all the ratios on the board.
“What do you notice about all these
ratios?”
They are all the same, or
very close to each other
“Why are they all the same?”
We call the relationship between the
sides and angles of right triangles
trigonometric functions.
The triangles are all
similar, and will have
equal proportions of the
sides.
opposite leg
adjacent leg
opposite leg
Sine θ =
hypotenuse
adjacent leg
Cosine θ =
hypotenuse
Tangent θ =
Acknowledge their
observation. Explain the
names of the ratios at
this point rather than ask
the questions. Make
sure they understand
that these ratios are
reciprocals of each other
and that they have their
own special names.
Now ask the questions!
Right. These values are
approximations of the
true values, since either
out instruments or our
measuring techniques
are not always accurate.
This is why we wrote the
averages on the
overhead.
Recall the launch if
some have forgotten.
If there’s time,
bring up the fact
that the ratios we
have come up with
(and are given on
our calculators) are
only for right
triangles. Show an
example using an
equilateral triangle.
We already know
that the ratio of the
sides are all 1, but
the sin(60) ≠ 1.
Why isn’t this
working? IT’S
NOT RIGHT!!!
“Since the ratios are all the same for
a given angle, we can use calculators
to find the ratio without drawing the
triangles.”
Show them how to enter the trig
functions into their calculators.
TASK 2:
Call attention to the class. Have
assigned students come up to the
board and explain their solutions.
What did you do to figure out the side
or the angle measure?
Which trig function did you use?
How did you know to use the
tangent/cosine/sine ratio?
Could you do this for any angle?
What minimum requirements do you
10 min
We used tan/cos/sin.
We took what we knew,
(whether it was legs, or
hypotenuse) and by the
definitions that we went
over before we knew that
tan/cos/sin was the ratio
of this side to this side. So
if we knew one side and
the angle, we could figure
out the other side, because
How many ratios do we
know? Why don’t we
know more? What about
the tan/sin/cos buttons
on your calculator?
Anyone have an idea of
where they got these
numbers?
While walking
around the class,s
look for students
who have solved
the problems and
understand the
concepts well. Ask
them if they would
draw their solutions
on the board and
explain it to the
Lesson Sequence: Learning
activities, tasks and key
questions (what you will do
and say, what you will ask
the students to do)
Anticipated Student
Time
Thinking and
Responses
need to solve for the remaining sides
or angles?
Your response to
student responses
and thinking
we know every angle has
the same ratio. If we
know two of the sides we
know a ratio, and hence
can compare it to a
tan/cos/sin ratio that we
already know.
How did you figure out which angle
to use on problem three?
Formative
Assessment,
Miscellaneous
things to
remember
class during the
discussion
If there’s extra time
(haha), What can
we say about
scalene triangles?
(Draw a picture)
Could we figure out
all the sides from
knowing only an
angle and two sides
Some students may have
been stumped on this part.
Others may have looked at
eh given ratios on the
transparencies as a
reference.
Debriefing the Lesson
“What have we discovered about
angles in a right triangle?”
“How are these ratios important?”
5 min
There are different ratios
corresponding to the sides
that will always be the
same, no matter how big
or small the triangle is.
If we know an angle
measure and a side in a
right triangle we can
figure out the remaining
sides. Or if we know two
of the sides of a right
triangle we can figure out
each angle measure, along
with the third side.
That’s right. This works
because any right
triangle you choose with
the given angle will be
similar (by AA
similarity,) and thus
have equal ratios of
sides.
TASK 1: Discover
that congruent
angles in a right
triangle give the
same proportions
of the
corresponding
sides.
TASK 2: Discover
how to find other
side lengths and
angles using this
relationship.
Discover the
minimum
requirements in
all these cases.