Advanced Math I
Unit 3: Exponential and Logarithmic Functions
Time Frame: 4.5 weeks
Unit Description
This unit focuses on the exponential and logarithmic functions using the four
representations of functions. This unit expands on the concepts taught in earlier courses
as well as provides a review of essential mathematical skills needed in this course and in
future courses. Modeling real-life problems using exponential growth and decay as well
as fitting exponential and logarithmic models to sets of data play a large part in the unit.
Student Understandings
Students recognize, evaluate, and graph exponential and logarithmic functions. The laws
of exponents and logarithms are reviewed and then used to evaluate, simplify expressions
and solve equations. They are able to use both functions to model and solve real-life
problems.
Guiding Questions
1. Can students recognize exponential functions in each of the function
representations?
2. Can students identify the growth or decay factor in each of the exponential
functions?
3. Can students graph exponential functions?
4. Can students recognize, evaluate, and graph exponential functions with base e?
5. Can students use exponential functions to model and solve real-life problems?
6. Can students recognize and graph logarithmic functions with any base?
7. Can students use logarithmic functions to model and solve real-life problems?
8. Can students use the properties of exponents and logarithms to simplify
expressions and solve equations?
9. Can students rewrite logarithmic functions with different bases?
10. Can students use exponential growth and decay functions to model and solve reallife problems?
11. Can students fit exponential and logarithmic models to sets of data?
Advanced Math IUnit 3Exponential and Logarithmic Functions
35
Unit 3 - Grade-Level Expectations (GLEs)
GLE GLE Text and Benchmarks
#
Number and Number Relations
2.
Evaluate and perform basic operations on expressions containing rational
exponents (N-2-H)
3.
Describe the relationship between exponential and logarithmic equations (N-2H)
Algebra
4.
Translate and show the relationships among non-linear graphs, related tables of
values, and algebraic symbolic representations (A-1-H)
6.
Analyze functions based on zeros, asymptotes, and local and global
characteristics for the function (A-3-H)
7.
Explain, using technology, how the graph of a function is affected by change of
degree, coefficient, and constants in polynomial, rational, radical, exponential,
and logarithmic functions. (A-3-H)
8.
Categorize non-linear graphs and their equations as quadratic, cubic,
exponential, logarithmic, step function, rational, trigonometric, or absolute value
(A-3-H) (P-5-H)
10.
Model and solve problems involving quadratic, polynomial, exponential
logarithmic, step function, rational, and absolute value equations using
technology (A-4-H)
Data Analysis, Probability, and Discrete Math
19.
Correlate/match data sets or graphs and their representations and classify them
as exponential, logarithmic, or polynomial functions (D-2-H)
Patterns, Relations, and Functions
24.
Model a given set of real-life data with a non-linear function (P-1-H) (P-5-H)
25.
Apply the concept of a function and function notation to represent and evaluate
functions (P-1-H) (P-5-H)
27.
Compare and contrast the properties of families of polynomial, rational,
exponential, and logarithmic functions, with and without technology. (P-3-H)
28.
Represent and solve problems involving the translation of functions in the
coordinate plane (P-4-H)
29.
Determine the family or families of functions that can be used to represent a
given set of real-life data, with and without technology (P-5-H)
Sample Activities
Since students have been exposed to the laws of exponents and logarithms; they
understand that y  b x and log b y  x are equivalent expressions, and are able to work
with rational exponents as well as recognize the graphs of the exponential and
logarithmic functions. However, it is usually necessary to review these concepts. Give a
Advanced Math IUnit 3Exponential and Logarithmic Functions
36
pre-test to see how much students have retained from previous courses and design the
reviews and spiral accordingly.
Ongoing: Glossary notebook
Add the following terms as they are encountered in the unit: algebraic functions,
transcendental functions, relationship of exponential and logarithmic functions, natural
base e, growth factor, growth rate, exponential growth model, exponential decay model,
natural logarithm, common logarithm, change-of-base formula.
Activity 1: The Four Representations of Exponential Functions (GLEs: 4,6,7,8, 10,
19, 29)
1. Growth is called exponential when there is a constant, called the growth factor, such
that during each unit time interval the amount present is multiplied by this factor.
Have students use this fact to decide if each of the situations below is represented by
an exponential function.
a. To attract new customers a construction company published its pre-tax profit
figures for the previous ten years. Was the growth of profits exponential?
How do you know? If exponential what is the growth factor?
Year Profit before Tax
(millions of dollars)
1992 27.0
1993 32.4
1994 38.9
1995 46.7
1996 56.0
1997 67.2
1998 80.6
1999 96.7
2000 116.1
2001 139.3
b. A second company also published its pre-tax profit figures for the previous ten
years. Its results are shown in the table below.
Year Profit before Tax
(millions of dollars)
1992 12.6
1993 13.1
1994 14.1
1995 16.2
1996 20.0
1997 29.6
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37
1998 42.7
1999 55.2
2000 71.5
2001 90.4
Was the growth of profits for this company exponential? How do you know? If
exponential what is the growth factor?
c) Determine whether each of the following table of values could correspond to a
linear function, an exponential function, or neither.
i)
x f(x)
0
1
2
3
10.5
12.7
18.9
36.7
ii)
x
0
2
4
6
f(x)
27
24
21
18
iii)
x
-1
0
1
2
f(x)
50.2
30.12
18.072
10.8432
2. Carbon dating is a technique for discovering the age of an ancient object by
measuring the amount of Carbon 14 that it contains. All plants and animals contain
Carbon 14. While they are living the amount is constant, but when they die the
amount begins to decrease. This is referred to as radioactive decay and is given by
the formula A  Ao (.886) t Ao represents the initial amount. The quantity A is the
amount remaining after t thousands of years. Let A0 = 15.3 cpm/g.
a. Fill in the table below:
Age of object
0 1 2 3 4 5 6 7 8 9 10 12 15 17
(1000’s of years)
Amount of C14
(cpm/g)
b. Sketch a graph using graph paper.
c. There are two samples of wood. One was taken from a fresh tree and
the other from Stonehenge and is 4000 years old. How much Carbon
14 does each sample contain? (answer in cpm’s)
Ask students to use the graph and the table above to answer the questions below,
then check their answers by using the given equation.
d. How long does it take for the amounts of Carbon 14 in each sample to
be halved?
e. Charcoal from the famous Lascaux Cave in France give a count of 2.34
cpm. Estimate the date of formation of the charcoal and give a date to
the paintings found in the cave.
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3. Each of the following functions gives the amount of a substance present at time t. In
each case:
 give the amount present initially
 state the growth/decay factor
 state whether or not the function represents an exponential growth model or
exponential decay model
a ) A  100(104
. )t
b) A  150(.89) t
c) A  1200(112
. )t
4. Find a possible formula for the function represented by the data below:
x
0
1
2
3
f(x) 4.30 6.02 8.43 11.80
Solutions:
32.4 38.9

 1.2 the growth factor is 1.2
27
32.4
b) The growth of the second company is not exponential. There is not a constant
growth factor.
c) i) neither ii) linear iii) exponential
1. a) yes;
2. a)
Age of 0
1
2
3
4
5
6
7
8
9
10
15 17
object
(1000’s
of
years)
Amount 15.3 13.56 12.01 10.64 9.43 8.35 7.4 6.56 5.81 5.15 4.56 2.5 2.2
of C14
(cpm/g)
c) Reading from the table: The fresh wood will contain 15.3 cpm’s of Carbon 14 while
the wood from Stonehenge will contain 9.43 cpm’s of Carbon 14.
d) Fresh wood: the amount of C14 to be halved is 7.65 cpm/g. Both the table and the
graph would put the age between 5000 and 6000 years. Wood from Stonehenge:
there would be 4.715 cpm/g present. That would put the age between 9000 and
10,000 years. Since this wood is already 4000 years old the half-life is the same as
1
the fresh wood. It is the same in each case. Using the equation solve  .886 t
2
which is ≈ 5700 years
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e) The charcoal from the caves is about 15,500 years old, so the paintings date back
to about 13,500 BC.
3. a) 100 is initial substance, 1.04 is growth factor, exponential growth
b) 150, 0.89, exponential decay
c) 1000, 1.12, exponential growth
4. f(x) = 4.3(1.4)x
Activity 2: A Study of Exponential Functions (GLEs: 4, 6, 7, 25, 27, 28)
This activity revisits some of the concepts in unit 1, this time using exponential and
logarithmic functions. Have students complete the following exercises:
1. Start with the graph of y = 3x. Write an equation for each of the conditions below and
sketch the graph labeling all asymptotes and intercepts. Verify your answers with a
graphing utility.
a) Reflect the graph through the x-axis.
b) Reflect the graph through the y-axis.
c) Shift the graph up 3 units and translate the graph 4 units to the left.
d) Reflect the graph over the x-axis then shift it 2 units to the right.
2. Fill in the table below:
f(g(x)) =
f(x) g(x) Domain of f(g(x)) Domain of f Domain of g
2
1. ln( x  4)
2. e x
3. (1  ln x)2
1
4. 2 x
3. Which of the composite functions in the table above are even, odd, or neither? How do
you know?
4. Which of the functions in the table have an inverse that is a function? Justify your
answer.
5. For those function/s that have an inverse find f 1 ( x) .
Solutions for Activity 2:
1. a) y = -3x b) y = 3-x
c) y = 3 + 3x+4 d) y  3 x 2
Advanced Math IUnit 3Exponential and Logarithmic Functions
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2.
f(g(x))
1.
ln( x 2  4)
2.
e
3.
(1  ln x)2
4.
2x
x
1
f(x)
g(x)
ln( x)
x2-4
ex
|x|
Domain of
f(g(x))
{x: x < -2 or
x > 2}
Reals
Domain of
f
x>0
Domain of g
Reals
Reals
x2
1  ln( x)
x>0
Reals
x>0
2x
1
x
{x:x≠0}
Reals
{x:x≠0}
Reals
4. 1 and 2 are even because they are symmetric over the y-axis. Numbers 3 and 4 are
neither symmetric over the y-axis nor around the origin.
4. Numbers 1, 2, and 3 do not have an inverse that is a function. Number 4 has an
inverse that is a function.
Justification:
 Numbers 1, 2, and 3 are not-one-to-one functions or
 because 1 and 2 are even functions they will not have an inverse that is a
function and number 3 decreases into a minimum then increases
 Looking at the graph or at the numerical tables of #4 the function is
strictly decreasing, but y = 1 is a horizontal asymptote so the values of the
range are never repeated. When x < 0, y < 1 and when x > 0, y > 1.
5. f 1 ( x)  log 2 x 
1
Activity 3: Growth Factors and Growth Rates (GLEs: 2, 6, 8, 19, 29)
Explain that exponential growth is usually described in terms of growth rates in percent.
For example, the growth in jobs in the last year was 2.5%. The growth factor b is 1 + .025
or 1.025. The problem may say that the population of a city has declined at the rate of
1.5% per year. The decay factor is 1 - .015 or .985.
Ask students to complete the following exercises:
1. For each of the following functions state (a) whether exponential growth or decay is
represented and (b) give the percent growth or decay rate.
a) A = 22.3(1.07)t
b) A = 10(.91)t
c) A = 1000(.85)t
2. In a recent newspaper article the 2003-04 jobs report for the Baton Rouge metro area
was given. Job changes by industry sector (since August 2003) were as follows:
 Leisure & hospitality up 4.5%
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 Education & health services down 2%
 Construction down 6%
 Financial up 3.6%
For each what is the growth/decay factor and what is the growth/decay rate?
Solutions for Activity 3:
1. a) growth at 7% b) decay at 9%
c) decay at 15%
2. Leisure has a growth factor of 1.045 and a growth rate of 4.5%. Education has a
decay factor of .98 with a decay rate of 2%. Construction has a decay factor of .94 with
a decay rate of 6%. Financial has a growth factor of 1.036 and a growth rate of 3.6%.
Activity 4: Continuous Growth and the number e (GLEs: 7, 10, 24, 25)
Prior to this activity introduce students to the irrational number e and to the function
f(x)=ex and have the students graph the three functions y1  2 x , y2  e x , y3  3x to see the
place of e on the number line. Once this is done the idea of continuous growth can be
introduced.
Thus far students have been working with the exponential function A  Ao b t where Ao
represents the initial amount, b the growth factor, and t the amount of time that has
elapsed. If the growth is continuous then b is equal to ek for some k. If b > 1
(exponential growth) then k > 0. If b < 1 (exponential decay) then k < 0. The equation
can be written as A  Ao e kt A is growing or decaying at a continuous rate of k.
Problems:
1. In each of the following equations tell whether or not there is growth or decay and
give the continuous rate of growth or decay.
a) A = 1000e0.08t
b) A  200e .2 t
c) A  2.4e  o.oo4t
2. Write the following exponential functions in the form A = bt
a) A = e.25t
b) A = e-.4t
3. The population of a city is 50,000 and it is growing at the rate of 3.5% per year. Find
a formula for the population of the city at time t years from now if the 3.5% is a) an
annual rate and b) a continuous annual rate. In each case find the population at the
end of 10 years.
4. Air pressure, P, decreases exponentially with the height above the surface of the
earth, h: P  Po e 0.00012h where Po is the air pressure at sea level and h is in meters.
a) Crested Butte ski area in Colorado is 2774 meters (about 9100 feet) high. What
is the air pressure there as a percent of sea level?
Advanced Math IUnit 3Exponential and Logarithmic Functions
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b) The maximum cruising altitude of commercial airplanes is 12,000 meters
(around 29,000 feet). At that height what is the air pressure as a percent of sea
level?
Solutions for Activity 4:
1. a) growth of 8% b) decay of 20%
2. a) b  e.25 so A = (1.28)t
c) decay of .4%
b) b  e 0.4 , A = (.67)t
3. a) A= 50000(1.035)t when t = 10 A = 70,530
b) A = 50000e.0344t when t = 10
A=70,529
4. a) 72%
b) 24%
Activity 5: A Look at Ln x Its Local and Global Behavior and Translations in the
Coordinate System (GLEs: 4, 6, 7, 8,25)
Explain that Part I is is a graphing utility activity designed to help students understand
why the range of the natural logarithmic function is the set of reals and that Part II gives
students practice in sketching the graphs of logarithmic functions that have been
translated. Have students complete both parts.
Part I
1. Graph the function f ( x)  ln x . Use a window with -1 ≤ x ≤ 10 and -10≤ y ≤ 5.
Sketch the graph. What is the domain of f(x)? For what values of x is ln x < 0? ln x =
0?
ln x > 0? Run the trace feature and find the furthest point to the left on the graph.
What is it?
2. Reset your window to 0 ≤ x ≤ 0.01 and -10 ≤ y ≤ -6. Sketch this graph. Run the trace
feature and find the furthest point to the left on the graph. What is it?
3. To get a feel for how rapidly the natural log of x is falling as the x values are getting
closer to zero, fill in the table below.
x
ln x
.01
.001
.0001
.00001
.000001
Why do you think that these points are not evident on the graph of f(x) = ln x ?
Advanced Math IUnit 3Exponential and Logarithmic Functions
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4. Look at the end behavior of the function.
a) Set your window 0 ≤ x ≤ 100 and adjust the y-values so that the graph exits at the
right. Is the graph increasing, decreasing, or constant?
b) Increase the window to -1 ≤ x ≤ 1000 and if necessary adjust the y-values. What
do you see?
5. Based on this information how would you describe the global behavior of this
function? What is its range?
Part II. Sketch the graph of each function. Give the domain, the zero, the vertical
asymptote, and the y-intercept.
1. f ( x)  ln( x  3)
2. f ( x)  ln( x  2)  1
3. f ( x)  ln(4  x)
Solutions for Activity 5 (a TI-83 was used to obtain the answers):
Part I
1) The domain is {x:x>0}. ln x = 0 at x = 1 so ln x < 0 when x < 1 and lnx > 0 when x
> 1. The furthest point to the left is (.053, -2.93)
2) (0.000106, -9.148)
3)
x
ln x
.01
-4.605
.001
-6.907
.0001
-9.210
.00001 -11.513
.000001 -13.816
The y-axis is acting as a vertical asymptote. The calculator is unable to graph the
complete function due to the limitations of the viewing screen. The TABLE function will
give a much more precise answer. Set TblStart to 1 and ΔT to increasingly small
increments.
4) The graph continues to increase no matter how large x becomes.
5) The range is the set of all real numbers. As x  0, y   and as x  , y  
Part II.
1) The domain is {x: x>-3}. The line x = -3 is a vertical asymptote. The zero is x = -2.
The y-intercept is (0, ln3)
Advanced Math IUnit 3Exponential and Logarithmic Functions
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2) The domain is {x: x > 2}. The line x = 2 is a vertical asymptote. The zero is e-1 + 2.
There is no y-intercept.
3) The domain is {x: x < 4}. The line x = 4 is a vertical asymptote. The zero is x = 3.
The y-intercept is (0, ln4).
Activity 6: Working with the Laws of Logarithms (GLEs: 2, 3)
Ask students to complete these exercises:
1. Write each expression as a rational number or as a single logarithm
a) log 8  log 5  log 3
1
b) ln 10  ln 5  ln 8
3
c) 4 log M  3 log N
1
d) 3 log M  log N 
2
2. Express y in terms of x
a) log y  2 log x
b) ln y  ln x  2 ln 7
c) log y  3  .2x
1
d) ln y  (ln 3  ln x)
4
3. Solve the following logarithmic equations:
4
a) log( x  1)  log( x )  log
3
b) log( x  3)  log( x  2)  log( x  10)
F
I
G
HJ
K
c) log 1 ( x )  log 1 ( x  2)  3
2
2
Solutions for Activity 6:
8
1. a) log  
b) 0
 15 
2.a) y = x2
b) y = 49x
3. a) 3 b) 4
c) 4
M4 
c) log  3 
N 
d) log M 3 N 
1
2
c) y  103 .2 x d) y  4 3x or 3x 
1
4
Advanced Math IUnit 3Exponential and Logarithmic Functions
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Activity 7: Solving Exponential and Logarithmic Equations (GLEs: 2, 3, 6, 10)
Part I of this activity is non-calculator based. Help students understand how to use the
laws of exponents and logarithms to write the exact answer. Part B of the activity uses a
calculator. Problems are such that students will encounter in solving real-life problems.
Part I: Solve for x and give the exact answer.
x
1
1)    9 x  2
 3
2) 16x = 8x-1
3) 2x-2 = 3
4) 2x-1 = 3x+1
Part II: Solve for x. Use your calculator to obtain the answer.
1) 1400 = 350e-.2x
2) 14.53(1.09)2x= 201
3) 2500 = 5100(.79)x
Part III:. Use a graphing utility to solve the equation ex = 5-2x
Solutions for Activity 7:
4
Part I:
1)
3
Part II:
1. -6.93
ln 3
2
ln 2
2. -12
3.
2. 15.24
3. 3.02
4.
ln 6
2
ln
3
Part III:
The two graphs intersect at x ≈ 1.06.
Activity 8: Applications of Exponential Functions: Exponential Growth and Decay
(GLEs: 10, 24)
Explain this activity and then ask students to work the problems below:
1. The number of radioactive atoms N of a particular material present at time t years
may be written in the form N = 5000e-kt, where 5000 is the number of atoms present
when t = 0, and k is a positive constant. It is found that N = 2500 when t = 5 years.
a) Determine the value of k.
b) At what value of t will N = 50?
Advanced Math IUnit 3Exponential and Logarithmic Functions
46
2. A cup of coffee contains about 100 mg of caffeine. The half-life of caffeine in the
body is about 4 hours which means that the level of caffeine in the body is decaying
at the rate of about 16% per hour.
a) Write a formula for the level of caffeine in the body as a function of the
number of hours since the coffee was drunk.
b) How long will it take until the level of caffeine reaches 20 mg?
3. A radioactive substance has a half-life of 8 years. If 200 grams are present initially,
how much will remain at the end of 12 years? How long will it be until only 10% of
the original amount remains?
4. The Angus Company has a manufacturing process that produces a radioactive waste
byproduct with a half-life of twenty years.
a) How long must the waste be stored safely to allow it to decay to one-quarter of
its original mass?
b) How long will it take to decay to10% of its original mass?
c) How long will it to decay to 1% of its original mass?
Solutions for Activity 8:
1. a) k = .1386
b) t ≈ 33.3 years
2. a) A = 100(.84)t
b) t ≈ 9.2 hours
3.a) A≈ 70.7grams
b) t ≈26.6 years
4. a) 40 years
b) 66.4 years
c) 132.9 years
Activity 9 Adding to the Function Portfolio (GLE 3, 4, 6, 7, 10, 19, 27)
Have students add the exponential and logarithmic functions to their portfolios. As usual
they should cover:
 domain and range
 local and global characteristics such as continuity, local maxima and minima,
increasing/decreasing intervals, zeros, existence of inverses, and symmetry.
 examples of translation in the coordinate plane
 a real-life example of how the function can be used
The work they have done in this unit should provide them with the necessary
information.
Sample Assessments
General Assessments

The student will perform a writing assessment which covers activities of the unit
and which uses the glossary they have created throughout this unit. The teacher
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47


will look for understanding in how the terms or concepts are used, especially with
verbs such as show, describe, justify, or compare and contrast.
The students will complete four spirals during this unit. The first spiral should
cover the material covered in the Algebra II course. Give a pretest to see how
much the students have retained and then base spirals on material that needs
review. Other spirals will be based on material learned in previous units.
The student will turn in the entries for the Library of Functions for an assessment.
The scoring rubric will include whether
 the material is thoroughly covered.
 the material presented is accurate.
 the work is neat and organized with descriptions written in complete
sentences.
 the graphs are labeled, drawn to scale, and are correct.
Activity-Specific Assessments

Activity 1: The student will demonstrate proficiency in working with data that is
exponential. The teacher will provide a set of data, ask the student to graph it and
then find the exponential functions that model the data.

Activity 7: The students will demonstrate proficiency in solving exponential and
logarithmic equations. The student will explain how to solve at least one.

Activity 8: The student will work with a group on problems such as in Activity 8.
Problems should be based on the real-life problems using exponential or
logarithmic functions.The scoring rubric will be based on:
1. teacher observation of group interaction and work
2. explanation of each group’s problem to class
3. work handed in by each member of the group
Advanced Math IUnit 3Exponential and Logarithmic Functions
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Advanced Math I
Unit 4: Trigonometry of Triangles
Time Frame: 3.5 weeks
Unit Description
This unit concentrates on triangle trigonometry. There is a review of the right triangle
ratios with an emphasis on learning how to solve real-life problems using those ratios.
The Laws of Sines and Cosines are presented so that problems involving oblique
triangles can be solved. Since these laws are used extensively in aviation and physics in
conjunction with vectors, operations with vectors are also included in the unit.
Student Understandings
Students will know how to solve triangles using various combinations of sides and
angles. They will use triangle trigonometry to model and solve real-life problems.
Vectors play a part in many of the real-life models involving triangle trigonometry.
Students will be able to perform basic vector operations and represent vectors graphically
and apply their knowledge to problems involving force and velocity.
Guiding Questions
1. Can the student solve real-life problems involving right triangles?
2. Can the student use the Law of Cosines and Law of Sines to model and solve reallife problems?
3. Can students find the areas of oblique triangles?
4. Can the student represent vectors as directed line segments?
5. Can the student write the component form of vectors?
6. Can the student add, subtract, multiply and find the magnitude of the vector
algebraically?
7. Can the student find the direction angles of vectors?
8. Can the student use vectors to model and solve real-life problems involving
quantities that have both size and direction?
Advanced Math IUnit 4Trigonometry of Triangles
49
Unit 1 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Measurement
11.
Calculate angle measures in degrees, minutes, and seconds (M-1-H)
14.
Use the Law of Sines and the Law of Cosines to solve problems involving
triangle measurements (M-4-H)
Geometry
16
Represent translations, reflections, rotations, and dilations of plane figures
using sketches, coordinates, vectors, and matrices (G-3-H)
Sample Activities
Ongoing: Glossary notebook
Students should add to their glossary the following terms as they are encountered in the
unit: sine, cosine, tangent, secant, cosecant, cotangent, angles of elevation and
depression, line of sight, oblique triangles, Law of Sines, Law of Cosines, vector, initial
point, terminal point, vector in standard position, unit vectors, zero vector, equal vectors,
magnitude of a vector, scalar, horizontal and vertical components of a vector, bearing
Activity 1: Solving Right Triangles (GLE: 11)
This activity is a review of right triangle trigonometry studied in Geometry. Students
should be able to
 identify the side opposite and the side adjacent to an angle in a right triangle
 identify an included angle
 identify and use sine, cosine and tangent
 work with the Pythagorean Theorem
This is also an ideal time to review with the students some of the properties of geometric
figures that might be used as part of a triangle trigonometry problem. Have students
review the properties of the 30-60-90 degree and 45-45-90 degree triangles. They should
know how to find the exact values of the sine, cosine, and tangent ratios for each of those
angles. Introduce students to the other three ratios – cosecant, secant, and cotangent. Be
sure to provide practice in finding the angle when the ratio is given.
Give students a handout with various geometric shapes containing right triangles and
have them solve for the missing sides or angles. Angles should be written in degrees,
minutes, and seconds and sides to the nearest hundredth.
Advanced Math IUnit 4Trigonometry of Triangles
50
Writing Assignment: What does it mean to solve a right triangle?
Possible answer: Each triangle has three sides and three angles. To solve a
triangle means to find the unknown sides and angles using the given sides and
angles.
Activity 2: Solving Right Triangles using Real-life situations (GLE: 11)
This activity helps students see how the process of solving a right triangle can be used in
real-life. Angles of depression give students some trouble. The writing exercise is
designed to help students clarify in their mind how an angle of depression is constructed.
Writing activity: Compare an angle of elevation to an angle of depression.
Problems: Students will
 Draw a picture and identify the known quantities
 Set up the problem using the desired trigonometric ratio
 Give their answer in degrees, minutes, and seconds
1. A ramp 25 feet in length rises to a loading platform that is 4 feet off of the ground.
What is the angle of elevation of the ramp?
2. Cajun Airlines Flight 111 is approaching the Baton Rouge airport at an altitude of
33,000 feet when the plane begins its descent. At this point the plane is 120 miles
from the airport. What angle will the plane’s path make with the horizontal?
3. A 35 meter line is used to tether a pilot balloon. Because of a breeze, the balloon
makes a 75 o angle with the ground. How high is the balloon?
A homeowner wants to determine the height of a tree in his backyard. He stands
some distance from the tree and determines that the angle of elevation to the top of
the tree is 40 o. He then moves 18 feet closer to the tree and discovers that the angle
of elevation is now 50 o. His line of sight is 5 feet above the ground. How tall is the
tree?
4. From a point on the North Rim of the Grand Canyon, a surveyor measures an angle of
depression of 1 o 02’. The horizontal distance between the two points is estimated to
be 11 miles. How many feet is the South Rim below the North Rim?
5. The Great Pyramid of Cheops in Egypt has a square base of 230 meters on each side.
The faces of the pyramid make an angle of 51° 50’ with the horizontal.
a. How tall is the pyramid?
b. What is the shortest distance one would have to climb up a face to reach the
top?
Advanced Math IUnit 4Trigonometry of Triangles
51
Solution for Activity 2:
Writing activity: Possible answer should include the fact that both angles are made with
the horizontal. The angle of elevation is the angle between the horizontal and the line of
sight when looking up at the object. The angle of depression is the angle between the
horizontal and the line of sight when looking down on an object.
1) 9 o 12’ 25”
2) 3o
3) 33.8 feet
4) 71.6 feet
5) 1047.6 feet
6) a) 146.31 meters; b) 136.10 meters)
Activity 3: Solving Oblique Triangles (GLEs 11, 14)
Geometry teaches that a unique triangle can be constructed with
 two sides and the included angle (SAS)
 three sides (SSS)
 two angles and the included side (ASA)
 two angles and a third side (AAS)
Having the measures of two sides and one angle (SSA) will not necessarily determine a
triangle. Given two sides and the included angle or three sides, students will use the Law
of Cosines. Given two angles and the included side or two angles and the side opposite
one of the angles, students will use the Law of Sines. The Law of Sines is also used with
what is called the ambiguous case. This applies to triangles for which two sides and the
angle opposite one of them are known. It is called the ambiguous case because the given
information can result in one triangle, two triangles, or no triangle. This exercise gives a
student practice in (1) determining whether or not the information will produce a unique
triangle and (2) which formula to use. Help students understand that they should not use
the Law of Sines to find angle measures unless they know in advance whether the angle
is obtuse or acute.
Writing activity: You are given two sides and an angle. What procedures do you use to
determine if the given information will form one unique triangle, two triangles, or no
triangle?
Problems:
Have students use the Laws of Sines or Cosines to fill in the following table. Students
will
a) sketch each triangle
b) show the setup to be used
Advanced Math IUnit 4Trigonometry of Triangles
52
Possible problems could be:
Problem # a
b
1.
30 60
2.
c
A
B
15 o
23.5
3.
4.
5.
6.
8.3
5.0
3.0
6
C
23 o 50’
18 o
10.4 14.4
7.0
75 o
9.0 4.0
56 o 20’ 64 o 30’
Solutions for Activity 3:
Writing activity: A possible answer would be: When you are given two sides and a nonincluded angle, you can use the Law of Cosines with the given information letting the side
opposite the given angle be x. This will give a quadratic equation. The quadratic
formula can be used to solve for x. Pay special attention to the discriminant. If it is
negative then there is no triangle. If it is a perfect square then there is just one side and
therefore one triangle. Otherwise there are two real values for x and therefore two
triangles.
Problem #
a
b
1.
30.0 60.0
c
A
B
C
o
o
34.74 20 27’ 135 43’ 23 o 50’
2.
81.6 23.5 28.1
161 o
3.
4.
8.3
5.0
5.
6.
3.0
6
34 o 34’ 45 o 19’
75 o
36o33’
or
143o47’
No
triangle
56 o 20’ 64 o 30’
10.4 14.4
7.0 10.37
or
2.31
9.0 4.0
5.86 6.19
15 o
18 o
100 o 07’
118o37’
or
11o13’
59 o 10’
Activity 4: Real-life Problems Involving Oblique Triangles (GLEs: 11, 14)
This activity gives students practice in solving real-life problems using the Laws of Sines
1
and Cosines. Areas of triangles are also included since the area formula A  ab sin C
2
uses the same information used in the Law of Cosines (SAS). When all three sides are
given, the student can use Heros Formula Area  s( s  a )( s  b)( s  c) where s is the
semiperimeter (one half the perimeter).
Advanced Math IUnit 4Trigonometry of Triangles
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Writing assignment: How do you determine which technique to use in solving real-life
problems involving oblique triangles?
Problems: For each problem, students should
 draw a picture and identify the known quantities
 determine whether one is given SSS, ASA, AAS, or SAS.
 apply the appropriate law.
1. Two airplanes are approaching the New Orleans airport from the east. Plane A is at
23,000 feet and plane B is at 18,000 feet. From the control tower, the angle of
elevation to plane A is 4 o while the angle of elevation to plane B is 2.5 o. How far
apart in miles are the airplanes?
2. One end of a 15 foot plank is placed on the ground at a point 11 feet from the start of
a 40o incline and the other end is allowed to rest on the incline. How far up the
incline does the plank extend?
3. At a certain point, the angle of elevation of the top of a tower which stands on level
ground is 30.0 o. At a point 100 meters nearer the tower, the angle of elevation is
58.0o. How high is the tower?
4. The Johnsons plan to fence in their back lot shown below. If fencing costs $4.50 a
foot, how much will it cost?
50 ft.
128 o
30
feet
5. From a helicopter the angles of depression of two successive milestones on a level
road below are 15.0 o and 40.0 o respectively. Find the height of the helicopter.
6. Find the area of an equilateral triangle with sides 6 cm in length.
7. Find the area of a triangle with sides 3”, 5”, and 7”.
Solutions for Activity 4:
Writing assignment: Possible answer: Set up the problem by drawing a diagram and
filling in the values that are known and labeling what needs to be found. Determine
whether SSS, ASA, AAS, or SAS is given and then use the appropriate law to solve.
Advanced Math IUnit 4Trigonometry of Triangles
54
1. 15.8 miles
2. 4.8 feet
3. 90.3 feet
4. 191.5 feet costing $861.80
5. 0.26 miles
6. 9 3cm 2 or ≈15.59cm2
7.
15 3 2
in or ≈ 6.5 in2
4
Activity 5: Practice with Vectors (GLE: 16)
This activity is designed to give students practice with vectors. Vectors have been added
to triangle trigonometry in this curriculum guide. There is no uniform order in the way
that vectors, triangle trigonometry, and parametrics are presented in the various advanced
math textbooks available to the teachers. If the teacher wants to include parametrics at
this time. see Unit 8 for the activities.
Writing activity: How does a vector differ from a line segment?
Problems:
1. Use the vectors v  2,7 and w  3, 5 to perform the given operations:
a. –v
b. 2v + 4w
c. w – v
2. For each of the following problems:
 Find the resultant of two given displacements. Express the answer as a distance
and a bearing (clockwise from the north) from the starting point to the ending
point.
 Tell the bearing from the end point back to the starting point.
 Draw the vectors on graph paper, using ruler and protractor. Show that your
answers are correct to 0.1 units of length and 1 degree of angle.
a. 9 units North followed by 6 units along a bearing of 75 degrees
b. 8 units East followed by 10 units along a bearing of 170 degrees
c. 6 units South followed by 12 units along a bearing of 300 degrees
3. An object moves 90 meters due South (bearing 180 degrees) turns and moves 50
more meters along a bearing of 240 degrees.
a. Find the resultant of these two displacement vectors
b. What is the bearing from the end point back to the starting point?


4. Vector v has a magnitude of 4 and direction 150o. Resolve v into horizontal and
vertical components.
Advanced Math IUnit 4Trigonometry of Triangles
55
Solutions for Activity 5:
Writing activity: A possible answer would be “A vector has both magnitude and direction
while a line segment has only magnitude.”
1. a) 2,7 b) -8,34 c) -5,-2
2. a) resultant 12.0; bearing 245 o
b) resultant 4.2; bearing 346 o
c) resultant 10.4; bearing 90 o
3. a) 122.9
b) 200◦


4. a  3.464i  2 j
Activity 6: Combining Vectors with Triangle Trigonometry (GLEs: 11, 14, 16)
Have students complete the following exercises to show how vectors can be used with
triangle trigonometry.
Writing activity: Below are four very common applications of vectors and their peculiar
method of indicating direction. Describe each vector quantity and sketch a picture.
1. A force f of 20 lb at 60o
2. A velocity v of 500 mph at 100 o
3. Wind velocity w of 10 knots from 200 o
4. A displacement OA from O to A of 25 km N30 oW
Problems:
1. A baseball is thrown with an initial velocity of 75 feet per second at an angle of 40o.
Find the horizontal and vertical components of velocity.
2. A pilot is flying at an airspeed of 241 mph in a wind blowing 20.4 mph from the east.
In what direction must the pilot head in order to fly due north? What is the pilot’s
speed relative to the ground?
3. A woman sets out in a row boat on a due west heading and rows at 4.8 mph. The
current is carrying the boat due south at 12 mph. What is the true course of the
rowboat and how fast is the boat traveling relative to the ground?
4. An airplane is flying due East at 190 mph. Find the resultant velocity of the airplane
if there is a tail wind of 50 mph blowing at an angle of N25oE?
Solutions for Activity 6:
Writing activity:
1. The angle is measured counterclockwise from the positive x-axis to the vector.
2. The angle, called a heading, bearing, or course is measured clockwise from the
North to the vector.
Advanced Math IUnit 4Trigonometry of Triangles
56
3. The angle is measured clockwise from North to the direction from which the
wind is blowing.
4. The angle is measured from North (or South) toward West (or East).
Problems:
1. horizontal 57.45 ft/sec, vertical 48.21 ft./sec
2. 4.9o, 240mph
3. 12.9 mph, S22 o W
4. 215.9 mph angle of N78E
Sample Assessments
General Assessments
 The student will perform a writing assessment which covers activities of the unit
and which uses the glossary they have created throughout this unit. The teacher
will look for understanding in how the terms or concepts are used, especially with
verbs such as show, describe, justify, or compare and contrast.
 The students will complete four spirals during this unit. One should be on right
triangle trigonometry (activity 1). Another should give students more practice on
vectors (activity 5). Other spirals will be based on material learned in previous
units or earlier courses.
 The student will work with a group on problems that require more work than the
textbook problems such as those put out by the National Society of Professional
Surveyors in their annual TRIG-STAR contest. This is a contest based on the
practical application of Trigonometry. The website is
http://www.acsm.net/trigstar/. Navigation problems also make excellent
problems. Give each group a problem which requires them to fly from one city to
another on a particular plane with given wind conditions. The student will
determine the heading on which to fly as well as the airspeed of that particular
plane before solving the problem. The teacher will evaluate the group’s work
using a rubric based on:
 teacher observation of group interaction and work
 explanation of each group’s problem to class
 work handed in by each member of the group
Advanced Math IUnit 4Trigonometry of Triangles
57
Activity-Specific Assessments

Activity 1: The student will demonstrate proficiency in his/her understanding and
use of the right triangle ratios. Sample questions to use:
3
1. In triangle ABC C  90o , sin B  , and side c  15. Find the length
5
of side b.
13
2. sec A 
. Find the other five trigonometric ratios.
12
3. In triangle ABC C  90 , side a = 4, and side b = 8. Find
A and B .
4. Geometric shapes containing right triangles that require students to find
the missing sides or angles.
5. Trigonometric ratios of the special right triangles (non-calculator)

Activity 3: The students will demonstrate proficiency in his/her ability to
determine (a) whether or not the given information will result in a unique triangle
and (b) which formula should be used to solve the triangle.

Activity 5: The student will demonstrate proficiency in using vector notation as
well as adding and subtracting vectors both algebraically and geometrically. It
should resemble this activity and cover questions 4 – 6 found under Guiding
Questions.
Advanced Math IUnit 4Trigonometry of Triangles
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