Download UNIT 1 Prerequisites Total Number of Days: __4.50_ days__ Grade

UNIT 1 Prerequisites
Total Number of Days: __4.50_ days__ Grade/Course: _Calculus__
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ESSENTIAL QUESTIONS
How can we utilize equations to solve
problems?
Why do we want to compare rather than get
an exact answer?
How can solutions to linear inequalities be
graphically represented on a number line
Why are linear inequalities useful?
What are some types of relationships that
can be modeled by graphs?
What types of relationships can be modeled
by linear graphs
What can we do with a system of
equations/inequalities that we cannot do
with a single equation/inequality?
How can linear equations and inequalities
be applied to the solution of word
problems?
What are the characteristics of
Quadratic Functions?
How do you sketch graphs and write
equations for parabolas?
How do you sketch the graphs of polynomial
functions?
How do you divide a polynomial by another
polynomial and interpret the result?
How are real, imaginary, and complex
numbers related?
How do you perform operations with
complex number?
How do you find all zeros of a polynomial
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ENDURING UNDERSTANDINGS
Quadratic equations are necessary for an understanding of acceleration.
Systems of linear equations and/or inequalities are used to model and solve
real-world problems involving 2 variables.
The laws of integer exponents can be extended to rational exponents.
To obtain a solution to an equation, no matter how complex, always
involves the process of undoing operations.
Proportionality involves a relationship in which the ratio of two quantities
remains constant as the corresponding values of the quantities change.
Real world situations can be modeled and solved by using equations and
inequalities.
Graphs of lines are functions used to represent changes in data
Graphs of equations of parabola of y=x2.
Patterns, functions and relationships can be represented graphically,
symbolically or verbally.
Shape can be preserved during mathematical transformations.
Complex numbers guarantee and supports the Fundamental Theorem of
Algebra
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
function?
How do you sketch the graph of a rational
function?
How are the solutions to polynomial and
rational inequalities found?
PACING
CONTENT
Solving
Equations
SKILLS
 Identify different types A.SSE.1.b
A.SSE.3
of equations.
A.CED.1
 Solve linear Equation A.CED.4
A.APR.1
F.IF.6
Ex.1
A.APR.3
3x – 6 = 0
A.APR.7
Ex2.
 1  3  26 x
x2
.25
STANDARDS
(CCCS/MP)
x2
x 4
 Solve Quadratic
equations by factoring, MP.2
completing the square, MP.5
and using the
MP.7
quadratic formula.
AX2 + BX +C
Ex 1.
2x2 +9x+9 = 3
 Solve equations
involving radicals.
Ex.1
2x  7  x  2
 Solve equations with
RESOURCES
OTHER
Precalculus
(e.g., tech)
www.kutasoftware.
Pre
com/Review of basic
Calculus
Algebra.
with
Calculus I with PreLimits
6th Edition
Calculus 3rd
Pg. 496
Edition Larson,
Ronald & Robert P
Average:
24 – 40 – odd Hostetler
Smarthinking.com
Advanced:
89 – 98 even
Algebra 2,
volume1&2,
Common Core
Edition. Pearson
LEARNING
ACTIVITIES/ASSESSME
NTS
http://www.purplemath
.com/modules/solvquad
.htm
Hands on exercise- TI
84 Graphing pg. 500
Example: 3x – 6 = 0
Reasoning
Problem.(enrichment)
Standardized Test Prep:
SAT/ACT page 243
Problems of the Lesson:
pg 504 #31, 49
 www.khanacadem Exploration: pg. 497 &
y.org/math/algebr 499
a/quadratics/factor
Model It: Data Analysising_quadratics
Renewable Energy.
Pg. 505 # 71
Journal Writing:
Interpreting Points of
Intersection
Others
Written tests and
absolute values.
Ex.
Solving
Inequalities
 | x2  3x | 4 x  6
 Intervals and
Inequalities
quizzes
Daily Home work
Same As Above
Pre
Calculus
with
Limits
6th Edition
Pg. 541
Average
27 – 33 odd

 Linear Inequality in
one variable
Ex: 5x-7>3x+9
.25
 Inequalities involving
Absolute Values.
Ex: |x-5|<2
 Polynomial Inequality
Ex: X2 x - 6 <0
 Rational Inequality
2x  7
Ex:
3
x 5
 Word problem from:
http://www.sheloves
math.com/algebra/b
eginningalgebra/wordproblems-in-algebra/
Advanced:
76 – 84 odd
http://www.quickma
th.com/webMathema
tica3/quickmath/gra
phs/inequalities/bas
ic.jsp
Internet examples:
http://www.coolmath.co
m/algebra/07-solvinginequalities/03-intervalnotation-01.htm
TI-84 Graphing. Pg. 542
& 549 # 49 - 51
Standardized Test Prep:
SAT/ACT page 243
Problems of the Lesson:
pg pg # 71 , 74
Model It: Data AnalysisPrescription Drugs
Pg. 550 # 77
Journal Writing:
Creating a System
Inequalities.
Others
Written tests and
quizzes
Daily Home work
Rectangular
Coordinates
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To learn how to plot
points in the
Cartesian plane.
use the Pythagorean
Theorem a2 + b2 = c2
and the distance
formula
d  ( x2  x1 )2  ( y2  y1 )2
to find distance
between
two points.
How far is it from
(4, 3) to (15, 8)?
(provide a right triangle
figure with coordinates for
the vertices)
.25
 Use the midpoint
formula to find the
midpoint of a line
segment
Find the value of p so
that (–2, 2.5) is the
midpoint between (p,
2) and (–1, 3).
 Use a coordinate
plane and geometric
formulas to model and
solve real-life problems
At 8 AM one day, Amir
decides to walk in a
straight line on the beach.
Pre
Calculus
with
Limits
6th Edition
Pg. 2
Average:
53 – 58 odd
Advance:
66 – 70
even
 www.kutasoftwar
e.com/functions
 www.Classzone.
co
 www.khanacademy.
org/math/algebra/p
ythagorean.../dista
nce_formula
www.kutasoftware.c
om/FreeWorksheet
s/.../Midpoint%20F
ormula.pdf
Internet example:
www.purplemath.com/
modules/midpoint.html
Hands-on exercise –
graphical solution: pg. 3
Problems of the Lesson:
#31, 56
Model it: Labor Force
pg. 12 # 60
 h
‎ ttp://lessonplans. Journal Reports
Class Discussion and
fundingfactory.co
m/plan_details.asp questions
x?id=271
Others
Written tests and
quizzes
Daily Home work
After two hours of making
no turns and traveling at a
steady rate, Amir was two
mile east and four miles
north of his starting point.
How far did Amir walk and
what was his walking
speed?
Graphs of
Equations
 To sketch graphs of
equations.
f(x) = ax2
F.BF.1
F.IF.7
Pre
Calculus
with
Limits
6th Edition
Pg. 14
Average:
71 – 75 odd
Advanced:
99 – 104
even
.25
 Larger values
of a squash the curve
 Smaller values
of a expand it
 And negative values
of a flip it upside down
y = (x-4)2 + 6
 To find x and y
intercepts of graphs of
equations

http://www.rege
ntsprep.org/rege
nts/math/algebra
/ac1/pracline.ht
m
Internet example:
http://www.mathsisfun.
com/sets/graphequation.html
Hands-on exercise –
graphical solution: pg. 3
Activity-Modeling Linear
Data P. 119 – 124 , do all
application problems.
Including Data Analysis
Problems of the Lesson:
#31, 49
Model it ; Population
Statistics: pg. 24 # 75
Journal
Written tests and
quizzes
Daily Home work
Find the x- and yintercepts of
25x2 + 4y2 = 9
 To find equations of
Circles
Find the equation of the
circle with the end
points of the diameter
with (3,-4) and
(-5,12).
Linear
Equations in
Two
Variables
 To use slope to graph
linear equations in
two variables
2y = 3x -12
 Find slopes of lines
-( ) y + 7x =

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Write linear
equations in two
variables
2y = 13x -17
Use slope to identify
parallel and
perpendicular lines.
One line passes
through the points (–4,
2) and
(0, 3); another line
passes through the
points (–3, –2) and (3,
2). Are these lines
parallel, perpendicular,
or neither
Same As Above
Pre
Calculus
with
Limits
page. 25

http://www.purp
lemath.com/mod
ules/systlin1.htm
Internet example:
http://www.mathsisfun.
com/algebra/lineparallelperpendicular.html
Exploration exercise:
page 30
Average:
71 – 75 odd
Advanced:
99 – 104
even
Graphical approach:
page 37 # 101


http://www.mat
hsisfun.com/alge
bra/line-parallelperpendicular.ht
ml
Problems of the Lesson:
#31, 49
Model it ; Data Analysis
Pg. 38 # 119
Journal
Written tests and
quizzes
Daily Home work
Quadratic
Functions
and Models
 Analyze graphs of
quadratic functions
f(x)= ax2 + bx + c
F.IF.7a, 7b, 7e, 8a,
8b, 9
Average:
37 - 49 odd
Advanced:
67 - 75
even
 Write quadratic
functions in standard
form and use the results
to sketch graphs of
functions
f(x)= a(x-h)2 + k
a not equal to zero
.25
Pre
Calculus
with
Limits
 https://www.google.co
m/search?q=quadratic
+functions+in+real+life
&client=firefoxa&rls=org.mozilla:enUS:official&channel=n
p&tbm=isch&tbo=u&s
ource=univ&sa=X&ei=
D4rwUfrOHbS84AOh7
4CQBw&ved=0CD8Qs
AQ&biw=1024&bih=62
9
 http://math.about.com/
od/algebra1help/a/Qua
dratic_Formula.htm

Internet example:
http://education.ti.com/
en/us/activity/detail?id
=26347906DEF24F8F97
F089F6EFC061A1
Group ActivityExploration on P.204
P. 208 – 212
Exploration excercise:
page :129
Kangaroo Conundrum: A
Study Of A Quadratic
Function.
Graphical approach:
page 137 # 85
Problems of the Lesson:
#37 - 42
Problems of the Lesson:
#37 – 42
Model it ; Data Analysis
Page: 137 # 86
Journal
Written tests and
quizzes
Daily Home work
.5
Polynomial
Functions of
Higher
Degree
 To use transformation
to sketch graphs of
polynomial functions.
A.APR.6.
F.BF.1,
F.IF.7
Algebra 2,
volume1&2,
Common
Core Edition.
 kutasoftware.co
m
 Classzone.com
 http://hotmath.co
Internet example:
https://www.khanacade
my.org/math/algebra2/
polynomial_and_rational
The first thing to do
here is graph the
function without the
constant which by this
point should be fairly
simple for you. Then
shift accordingly.
Here is the sketch
MP4
MP5
MP7
MP8
Pearson
Pre
Calculus
with
Limits
Average:
37 - 41 odd
Advanced:
61 - 74
even
m/hotmath_help/t
opics/leadingcoefficienttest.html
 http://tutorial.mat
h.lamar.edu/Classe
s/Alg/GraphingPol
ynomials.aspx
/factoring-higher-degpolynomials/v/factoring
-5th-degree-polynomialto-find-real-zeros
Exploration excercise:
page :142
Technology Graphical
approach: page: 149 #
23 - 26
Writing about
mathematics P. 221, 222
– 225, Point of
diminishing returns and
graphical reasoning
 Use leading coefficient
test to determine the
end behavior of
graphs of polynomial
functions
Use the Leading
Coefficient Test to
determine the end
behavior of the graph of
the polynomial function
f(x) = -x3 + 5x.
Because the degree is
odd and the leading
coefficient is negative,
the graph rises to the
Problems of the Lesson:
#57 - 66
Model it ; Tree Growth
Pg.: 151, # 97
Journal
Written tests and
quizzes
Daily Home work
left and falls to the right
as shown in the figure.
 Evaluate and use zeros
of a polynomial
functions as sketching
aids.
Sketch the graph
of
.
Here is a list of the
zeroes and their
multiplicities.
Here is a sketch of the
graph.
 Use the Intermediate
Value Theorem to help
locate zeros of
polynomial functions.
A.APR.6.
Polynomial  To learn how to use
F.BF.1,
and Synthetic long division to divide
F.IF.7
Division
polynomials by other
polynomials
MP1
Divide
MP2
3
2
( 3x – 5x + 10x – 3)
MP5
by ( 3x + 1)
Pre
Calculus
with
Limits
Average:
51 – 61
odd
 kutasoftware.co
m
 Classzone.com
 http://www.purpl
emath.com/modul
es/polydiv3.htm
 Smarthinking.co
m.
Advanced:
60 - 68
even
Internet example:
http://www.wtamu.edu
/academic/anns/mps/m
ath/mathlab/col_algebra
/col_alg_tut37_syndiv.ht
m
Exploration excercise:
model it: page :160
Technology Graphical
approach: page: 160 #
65
Problems of the Lesson:
#65
.5
Use synthetic division
to divide polynomials
by binomial of the
form (x-k)
.
 Use Remainder
Theorem and the
Factor Theorem.
http://www.mathpo
rtal.org/calculators/
polynomialssolvers/syntheticdivisioncalculator.php
Group Activity Analyzing a Slant
Asymptote
P. 160 – 161 ,
Graphical Analysis, Data
Analysis, and Power of
an Engine.
Model it ; Data Analysis:
Military Personnel.
Pg.: 160 # 73
Journal
Written tests and
quizzes
Daily Home work
Given,
Use Remainder Theorem
to evaluate f(-2)
Complex
Numbers
 To use imaginary unit N.CN.1
N.CN.2
to write complex
N.CN.3
numbers.
N.CN.7
MP1
MP2
MP5
MP8
.25
Example 1
|3+4i| Resultant
value=5
 kutasoftware.co
m
 Smarthinking.co
m.
Average:
http://www.math
39 – 43 odd
sisfun.com/numb
Advanced:
ers/complex93– 104
numbers.html
even
Same as
Above
Internet example:
http://www.themathpag
e.com/alg/complexnumbers.htm
www.College.hmco.com
www.amscopub.com
Exploration excercise:
page :164
Technology Graphical
approach: page: 160 #
65
Problems of the Lesson:
#65
Problem model It:
Impedance how to find
the impedance of an
electric circuit
Page: 168
Example 2
Daily Home work
Write:
in complex form
 Add, subtract, and
multiply complex
numbers
Multiply and simplify
in the form of a+bi
 Use complex
conjugates to write
the quotient of two
complex numbers in
standard form
 Find complex
solutions of quadratic
equations
16x2 -4x+3 =0
 To use the
fundamental
theorems of Algebra
to determine the
number of zeros of
polynomial functions
Zeros of
Polynomial
Functions

 Examples
Find rational zeros of
f(x) = 3x3 -19x2+ 33x-9
.25
P(x) = x 5 + x 3 - 1 is a
5th degree polynomial
function, P(x)has exactly
5 complex zeros.
 Find the conjugate
pairs of complex
zeros.
MP1
MP2
MP7
MP8
Same as
Above
Average:
48 - 54 odd
Advanced:
99 - 105
even
 kutasoftware.co
m
 Classzone.com
 Smarthinking.co
m.
 Teacher-made
Power Points
 Graphing
calculator
 Quiz and Test
generators
 Worksheets
Internet example:
http://www.chesapeake.
edu/khennayake/MAT1
13/3.2Solution.htm
Exploration excercise:
page :183 # 121
Technology Graphical
approach: page: 183 #
128
Problems of the Lesson:
#103
Problem model It:
‎http://www.sparkno Athletics.
tes.com/math/algebr Page: 182 # 112
a2/polynomials/sect
Writing Activity ion5.rhtml
Comparing Real Zeros
and Rational Zeros,
pg. 183 # 125
http://hotmath.com/
hotmath_help/topics Daily Home work
/descartes-rule-ofsigns.html
http://academics.utep.
edu/Portals/1788/CALC
ULUS%20MATERIAL/2
_5%20ZEROS%20OF%
20POLY%20FN.pdf
Example:
https://www.khanac
ademy.org/math/alg
ebra/complexnumbers/complex_n
umbers/v/complexconjugates-example
 Find zeros of
polynomials by
factoring
Solve
x2 + 5x + 6 = 0.
 Learn how to use the
Descartes’ Rule of
signs and the Upper
and Lower Bound
Rules to find zeros of
polynomials
Find the possible
number of real roots of
the polynomial and
verify.
f(x)= x3– x2– 14 x + 24
We can verify that
there are 2 positive
roots and 1 negative
root of the given
polynomial.
Nonlinear
Inequalities
 Solve polynomial
inequalities
A.APR.3
A.APR.7
Solve and write the
solution interval of
 Solve rational
inequalities
Solve and sketch the
graph of
f(x) =
.25
≥1
 Use inequalities to
model and solve reallife problems
An 18-wheel truck stops
at a weigh station before
passing over a bridge.
The weight limit on the
bridge is 65,000 pounds.
The cab (front) of the
truck weighs 20,000
pounds, and the trailer
(back) of the truck
weighs 12,000 pounds
when empty. In pounds,
how much cargo can the
truck carry and still be
allowed to cross the
bridge?
Same As
Above
Pg.197
Average:
1 - 11 odd
MP1
MP2
MP4
MP7
MP8
Advanced:
30 0 37
even
 kutasoftware.co
m
 Classzone.com
 Smarthinking.co
m.
 Teacher-made
Power Points
 Graphing
calculator
 Quiz and Test
generators
 Worksheets
Internet example:
http://www.intmath.co
m/inequalities/3solving-non-linearinequalities.php
Exploration excercise:
page :200
Technology Graphical
approach: page: 204 #
33
Problems of the Lesson:
http://www.montere 13, 21, 25
yinstitute.org/course
s/Algebra1/COURSE_ Model It: Cable
TEXT_RESOURCE/U0 Television.
5_L1_T3_text_final.ht Page: 205 # 73
ml
Written tests and
quizzes
Daily Home work
Chapter
Review
Chapter
Wrap Up
Assessment
To review the chapter
Summative Assessment
of Polynomial and
Rational Functions
Same as Above
 kutasoftware.com
 Classzone.com
MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
Same As Above
1day
INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT
Chapter Summary with
Exercises pg. 207-211
Chapter Practice Test.
Pg. 212
Multiple Choice
Open Ended Questions
from the Instructor’s
Resource Book.
Formative/Summative:
. Written tests and
quizzes.
. Unit Test with multiple
choice and openended.
. Notebook assessments.
. Journal Reports
. Class Discussion and
Questions
Quizzess, midchapter
test and chapter test
versions A,B,C,D for
basic, Average and
Advanced group from
the Test Bank of the
Larson 6th edition
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

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

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Analyze Quadratic Functions to help students develop how to maximize area or volume (if one dimension is a constant value),
projectile motions, or building archways.
Analyze of Polynomial Functions students will help students solve polynomial equations by graphing (using technology) and
by factoring and applying the polynomial function theorems).
Polynomial and Synthetic Division
The addition of complex numbers is connected to the addition of vectors.
Zeros of Polynomial Functions
Rational Functions
Nonlinear Inequalities
 We commonly use quadratic equations in designing the shape of suspension bridge.
 We use complex numbers in electric circuits and electromagnetism
 We commonly use zeros of polynomial function in manufacturing of supplies.
 Rational numbers are used extensively in mathematics, engineering, and science.
Marzano’s Six Steps for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point)
2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
5. Ask students to discuss vocabulary words with one another (Collaborate)
6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
VOCABULARIES:
Quadratic function
Parabola
Axis
Leading Coefficient
Vertex
X-interc Equation
Linear Equation
Quadratic Equation
Intervals
Polynomial
Transformation
Vertex
Intercepts
Symmetry
X-intercepts
Y-intercepts
Intermediate
Zeroes of a polynomial function
Synthetic division
Factor theorem
Real number
Imaginary number
Complex number
Conjugate
Rational Zero
Lower and upper bound
Asymptote
Vertical and horizontal Asymptote
Slant
Connect 3, 4
Each group should draw picture(s) of a suspension bridge. Examine the shape and determine if quadratic functions is applicable.
Write a short journal on your observations of the shape. List all applicable vocabularies in your writing.
PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – PowerPoint’s/Presentations
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
Shot-Put
A shot-put throw can be modeled using the equation
(also in feet). How long was the throw?”
, where x is distance traveled (in feet) and y is the height
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
Coin- Drop
A man drops a coin from the top of a cliff that is 200 feet tall. What is the height of the coin 2 seconds after it was dropped?
Use the vertical motion model h = -16t2 + 200 where t is the time in seconds, and h is the height in feet
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.
MODIFICATIONS/ACCOMMODATIONS
Individual student learning styles would be accommodated by:
 adjusting assessment standards,
 one-to-one teacher support
 extra time testing time
 additional use of visual, auditory and other teaching methods.
 A wide range of assessments and strategies that complement the individual learning experience would be encouraged.
 Teacher directed instruction by providing students with more necessary steps in order to solve the problems
 Small Group Activities - when students are given group guided practice
 IEP/504 Modifications:
 Students will be allowed to use the graphing calculator
 Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math
 Modified assessments and assignments (classwork, homework, quizzes/tests) as needed
 Math Centers (Differentiation) – Review/Revisit topics missed by absentee students
APPENDIX
(Teacher resource extensions)
 E-Text, Interactive Digital Resourses, Teacher Resourses.
http://.Larsonsuccessnet.com/snapp/login/login.jsp?showLoginPage=true

CCSS Mathematical Practices:
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:



Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Abbreviations
o Number & Quantity
 N-RN-The Real Number System
 N-Q-Quantities
 N-CN-The Complex Number System
 N-VM-Vector and Matrix Quantities
o Algebra
 A-SSE-Seeing Structure in Equations
 A-APR-Arithmetic with Polynomials and Rational Expressions
 A-CED-Creating Equations
 A-REI-Reasoning with Equations and Inequalities
o Functions
 F-IF-Interpreting Functions
 F-BF-Building Functions
 F-LE-Linear, Quadratic and Exponential Models
 F-TF-Trigonometric Functions
o Geometry
 G-CO-Congruence
 G-SRT-Similarity, Right Triangles, & Trigonometry
 G-C-Circles
 G-GPE-Expressing Geometric Properties with Equations
 G-MG-Modeling with Geometry
o Statistics and Probability
 S-ID-Interpreting Categorical & Quantitative Data
 S-IC-Making Inferences & Justifying Conclusions
 S-CP-Conditional Probability and Rules of Probability
 S-MD-Using Probability to Make Decisions
Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets)
Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets)

MU: Measuring Up Workbook
UNIT 2
Total Number of Days: 5 days Grade/Course: Calculus
ESSENTIAL QUESTIONS
 What is a function ?
 How is the equation of a line with partial
information sketched?
 How is the domain and range of a function
ENDURING UNDERSTANDINGS
 Functions relationships are fundamental ideas in mathematics. Functions
can be represented in several ways, such as a graph generated from a table
of x and y or input and output values.
 Functions can be added, multiplied or subtracted to produce complex
determined from the graph of the function?
 How are the graphs of inverse function related?
How can functions be transformed?
 How does a function’s symmetry affect the
behavior of the graph of a function?
PACING
CONTENT
Functions
SKILLS
 To determine
whether relations
between two
variables are
functions.
functions.
 Function combinations can impact the range, domain and the graph of the
function.
 Functions model real life problems.
STANDARDS
(CCCS/MP)
F.IF.7, F.BF.1
MP1
MP2
MP5
MP8
Is it a function or
relation
.25
{(1,2),(2,4),(3,5),(2,6),(1,-3)}
 Use function
notation and
evaluate functions
A function is
represented by f(x)
= 2x + 5,Find f(-3)
 Find the domain of
functions.
Determine the
domain and range of
the given function
f(x) =
 Find the difference
RESOURCES
OTHER
calculus
(e.g., tech)
Algebra 2,
 kutasoftware.c
volume1&2,
om
Common Core

http://www.reg
Edition. Pearson
entsprep.org/Re
Pg. 60 - 65
gents/math/AL
GEBRA/AP3/LF
Pre Calculus
unction.htm
with Limits
 http://www.pur
Pg. 40 - 47
plemath.com/m
odules/fcns.htm
 http://www.pur
plemath.com/m
odules/fcns2.ht
m
 f
 https://learnzill
ion.com/lesso
ns/3545-showtherelationshipbetweenvariablesusing-a-graph
LEARNING
ACTIVITIES/ASSESSME
NTS
Internet example:
http://math.kennesaw.e
du/~sellerme/sfehtml/c
lasses/math1190/limits.
pd
Technology Graphical
approach: page: 49 # 11
- 12
Problems of the Lesson:
13, 21, 25
Model It: Wildlife .
pg: 53 # 102
Average: 72 – 79 odd
Advanced: 80 -102 even
Journal
Teacher made Quizzes
Quizzess, midchapter
test and chapter test
 www.Smarthin versions A,B,C,D for
basic,Average and
king.com
Advanced group from
the Test Bank of the
.
Larson 6th edition

quotient
Evaluate difference
of quotients for
f(x) = x2 + 2
Quiz and Test generators
Worksheets
 Use functions to
model and solve
real-life problems
Students will examine
the relationship between
the number of faces,
edges, and vertices of
various polyhedral using
function (Question will
be provided)
Analyzing
Graphs of
Functions
 To use the vertical
line test for
functions
F.BF.1, F.IF.7
MP2
MP4
MP5
.5
Use vertical line test
to verify if
y = 2x-2 a function?
=1
 Find the zeros of
 kutasoftware.c
om
 Classzone.com
 http://www.co
olmath.com/al
gebra/15functions/03vertical-linetest-01.htm
 http://www.yo
utube.com/wat
ch?v=-xvDn4FOJQ(video)
 http://calculato
r.maconstate.ed
u/findingzeroes/(calculat
or)
 http://www.yo
utube.com/watc
h?v=6YM3TrudI
zQ
Internet example:
https://www.math.ucd
avis.edu/~kouba/CalcO
ne
Technology Graphical
approach: page: 49 # 11
- 12
Group Exploration: pg.
59 and 60
Problems of the Lesson:
39, 69, 90
Model It: Data Analysis:
temperature Page: 64 #
86
Average: 35 - 45 odd
Advanced: 72 - 90 even
functions.
Example:
Teacher made Quizzes

‎
Determine the zeros of
y = x3 -5x + 1
 Determine intervals
on which functions
are increasing or
decreasing.
Determine the
decrease and increase
intervals:
f(x) = x3- 6 x2
 Determine the
relative max and
relative min. values
of functions.
An open rectangular box
http://www.bright
storm.com/math
Writing Journal
Quiz and Test generators
Worksheets
with square base is to be
made from 48 ft.2 of
material. What
dimensions will result in
a box with the largest
possible volume ?
 Determine average
rate of change of a
function.
If an object is dropped
from a tall building, then
the distance it has fallen
after t seconds is given be
2
the function d (t )  16t .
Find the average speed
over the following
intervals:
a) Between 1 and 5
seconds
b) Between t  a and
t ah
 Identify even and
odd functions.
Example:
Is f(x)
=
A Library of
 Recognize graphs of
Parent Functions
parent functions.
F.BF.1,
F.IF.7
MP4
MP5
MP6
MP7
Identify the parent
function f(x) of
.25
 Identify and graph
linear and squaring
functions.
 Identify and graph
step and other
piecewise-defined
functions
Internet example:
https://myportal.bsd40
5.org/personal/kreiling
www.Classzone. k/gprec/Shared%20Doc
com
uments/Polynomial%20
Review%20Problems.pd
https://www.google f
https://www.kuta
software.com/
.com/search?q=libra
ry+of+parent+functi
ons&client=firefoxa&rls=org.mozilla:e
nUS:official&channel
=np&tbm=isch&tbo
=u&source=univ&sa
=X&ei=LZTwUezpC5
bd4APh1YHIDA&ve
d=0CDkQsAQ&biw=
1024&bih=629
Technology Graphical
approach: page: 72 # 51,
52
Group Exploration: pg.
69
Problems of the Lesson:
61, 63
Model It: Revenue.
pg: 73 # 67
Average: 17 - 27 odd
Advanced: 18 - 28 even
Writing Journal
Teacher made Quizzes
Quiz and Test generators
Worksheets

Example:
Transformation
of Functions
 To learn how to use
vertical and
horizontal shifts in
sketching graphs of
functions.
F.BF.1,
F.IF.7
MP2
MP3
MP4
MP5
MP6
.5
 kutasoftware.c
om
 Classzone.com
 http://www.ui
owa.edu/~exa
mserv/mathm
atters/tutorial_
 You tube
http://www.yo
utube.com/wat
ch?v=3Q5Sy03
4fok
 Smarthinking.
com.
Internet example:
 http://illuminations.
nctm.org/LessonDet
ail.aspx?ID=L470
Technology Graphical
approach: page: 82 # 65
- 66
Group Exploration: pg.
75 , 76
Problems of the Lesson:
61, 63
Model It: Fuel Use.
pg: 82 # 67
Average: 9 - 12 odd
Advanced: 9 - 24 even
Writing Journal
Teacher made Quizzes
Quiz and Test generators
Worksheets
:
y =x2
y = x2 +4
y = (x-4)2 +7
 Use reflections to
sketch graphs of
functions
Reflections Across the yAxis
Consider the
following base
functions,
y = √x,
y= 
1
x
2
Reflections Across the xAxis
Consider the
following base
functions,
f (x) = x2
1
2
y=  X 2
 Use of non-rigid
transformations to
sketch graphs of
functions
Compare the
functions and the
graph of
f(x)= X2 -3
g(x)= X2 +3
Combinations of
Functions:
Composite
Functions
 Learn how to add,
subtract, multiply
and divide
functions

F.BF.1,
F.IF.7
MP1
MP2
MP4
MP8

.5
Given f(x) = 2x
g(x) = x + 4, and
h(x) = 5 – x3,
find (f + g)(2), (h – g)(2),
(f × h)(2), and (h / g)(2).
 Find the compositions
of one function with
another function
Worksheets
 kutasoftware.c
om/functions
 Classzone.com
 http://www.pur
plemath.com/m
odules/fcnops.h
tm
 Smarthinking.
com.
 Teacher-made
Power Points
 Graphing
calculator
 Quiz and Test
generators
 Worksheets
Internet example:
 http://www.purplema
th.com/modules/fcnc
omp5.htm
Technology Graphical
approach: page: 87
Problems of the Lesson:
47, 59
Model It: Health Care
Costs: page. 91 # 61
Average: 5 - 27 odd
Advanced: 6 - 28 even
Writing Journal: pg. 88
Teacher made Quizzes
Quiz and Test generators
Worksheets
:
Given the functions
p(x) = x+2 and
h(x) = x2 evaluate
(h ◦ p)(3) and (h ◦ p)(x)

Use combinations
and compositions of
functions to model
and solve real-life
problems.
 You work forty hours a
week at a furniture
store. You receive a
$220 weekly salary, plus
a 3% commission on
sales over $5000.
Assume that you
sell enough this week to
get the commission.
Given the function:
f (x) = 0.03x
and
g(x) = x – 5000
Which of
( f o g)(x)and (g o
represents your
f )(x)
commission?
Inverse
Functions
 Find inverse function
informally and verify
that two functions
are inverse functions
of each other
F.BF.1
F.IF.7
MP1
MP5
MP6

.5
Determine algebraically
whether f(x) = 3x-2 and
g(x) =
are
inverses of each other
Show that f(g(x)) and
g(f(x)) are equal to x
 Use graphs of
functions to
determine whether
functions have
inverse functions
Let f(x) = 9 - x2 for x > 0
 kutasoftware.c
om
 Classzone.com
 You tube
http://www.yo
utube.com/watc
h?v=fZxkYZw1h
48
 http://www.pur
plemath.com/m
odules/invrsfcn
7.htm
 http://home.wi
ndstream.net/o
krebs/page45.ht
ml
 Smarthinking.
com.
 Teacher-made
Power Points
 Graphing
calculator
 Quiz and Test
generators
 Worksheets
 http://www.yo
utube.com/watc
h?v=1NPvkYLjj
Internet example:
 http://home.windstre
am.net/okrebs/page4
5.html
Technology Graphical
approach: page: 100
Group Exploration: pg.
94
Amd 97
Problems of the Lesson:
24, 39, 75
Model It: U.S.
Households. page. 101 #
79
Average: 13 - 23 odd
Advanced: 29 - 45 even
Writing Journal: The
existence of an Inverse
Function
Teacher made Quizzes
Quiz and Test generators
Worksheets
Find the equation for
f -1(x)
Sketch the graph of f,
f -1, and y = x.
 Use the horizontal
line test to
determine if
functions are oneto-one
ws
http://www.math
warehouse.com/al
gebra/relation/on
e-to-onefunction.php

 Find inverse
functions
algebraically
Find the inverse of
f(x) =
Mathematical
Modeling and
variation
.5.5
F.BF.1
 Use mathematical
models to approximate F.IF.7
sets of data points
A linear model that
approximates the data
MP2
is y = 1767.0 t +
MP4
123916 where y
 kutasoftware.c Internet example:
http://www.regentspre
om
 Classzone.com p.org/Regents/math/alg
trig/ATE7/Inverse%20V
 Smarthinking. ariation.htm
com.
 Teacher-made Technology Graphical
approach: page: 110 # 9
represents the number
of employees and t = 2
represents 1992. Plot
the actual data and
model on the same set
of plane .How closely
does the model
represent the data?
 Write mathematical
models for direct
variation
If y varies directly as x ,
and the constant of
variation is k = , what
is y when x = 9 ?
 Write mathematical
models for direct
variation as an nth
power
Example using the ppt:
http://www.google.com
/#q=model+for++direct
+variation+for+nth+po
wer
 Write mathematical
model for inverse
variation
In a formula, Z varies
inversely as p.
If Z is 200 when p = 4,
find Z when p = 10.
 Write mathematical
models for joint
variation.
If y varies directly
as x and z, and y = 5
when x = 3 and z = 4,
MP5
MP6
MP7
MP8
Power Points
 Graphing
calculator
 Quiz and Test
generators
 Worksheets
https://www.kh
anacademy.org/
math/algebra/a
lgebrafunctions/direct
_inverse_variati
on/v/directvariationmodels
Problems of the Lesson:
Hooke’s Law pg. 112 #
35
Model It: DataAnalysis:
Ocean Temperature.
page. 113 # 71
Average: 13 - 27 odd
Advanced: 82 - 86 even
Teacher made Quizzes
Quizzes, mid-chapter
test and chapter test
versions A,B,C,D for
basic, Average and
Advanced group from
the Test Bank of the
Larson 6th edition
Quiz and Test generators
Worksheets
then find y when x = 2
and z = 3.
Chapter Review
General review of the
chapter.
MP1
MP3
MP4
Mp5
Chapter Wrap
Up Assessment
 Summative
Assessment of
Functions and their
graphs
MP1
MP3
MP4
Mp5
Chapter Summary with
Exercises pg. 115
Chapter Practice Test.
Pg. 123
 Graphing
Calculator
1
INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT
Formative/Summative:
. Written tests and
quizzes.
. Daily Home work.
. Worksheets.
. Project Assessment.
. Article Summaries.
. Guided practice.
. Quizzes, mid-chapter
test and chapter test
versions A,B,C,D for
basic, Average and
Advanced group from
the Test Bank of the
Larson 6th edition
.
. Notebook assessments.
. Journal Reports




Functions Analysis of function helps students in finding that the derivative and the integral are operations on functions:
Shifting, reflecting, stretching graphs
library of parent functions will help students understand what different graphs look like and then how they are transformed based on
changes to the equation.
combinations of function will help students understand that two functions can be combined to create new functions just as two real
numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers,
Marzano’s Six Steps for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point)
2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
5. Ask students to discuss vocabulary words with one another (Collaborate)
6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
VOCABULARIES:
Range
Domain
Independent
Dependent
Ordered pairs
Vertical line test
Horizontal line test
Vertical Stretch
Horizontal stretch
Vertical shrink
Horizontal shrink
Non-rigid
Rigid
Composite function
Inverse function
Regression
Variation
Connect 2, 3, 4
Each group will draw a telephone key pad for the numbers 2 through 9. Create two relations : one mapping numbers onto letters
and the other mapping letters onto numbers. Are both relations functions? Explain.
PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – PowerPoint’s/Presentations
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical,
agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
Measurement:
When buying gasoline you notice that 14 gallons is approximately the same amount of gasoline as 53 liters. Use this information to find a
linear model that relates gallons to liters. Then use the model to find the numbers of liters in 5 gallons and 25 gallons
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.
MODIFICATIONS/ACCOMMODATIONS
1. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Individual student learning styles would be accommodated by:
 adjusting assessment standards,
 one-to-one teacher support
 extra time testing time
 additional use of visual, auditory and other teaching methods.
 A wide range of assessments and strategies that complement the individual learning experience would be encouraged.






Teacher directed instruction by providing students with more necessary steps in order to solve the problems
Small Group Activities - when students are given group guided practice
IEP/504 Modifications:
Students will be allowed to use the graphing calculator
Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in
math
 Modified assessments and assignments (classwork, homework, quizzes/tests) as needed
 Math Centers (Differentiation) – Review/Revisit topics missed by absentee students
APPENDIX
(Teacher resource extensions)
CCSS Mathematical Practices:
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:

Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Abbreviations
o Number & Quantity
 N-RN-The Real Number System
 N-Q-Quantities
 N-CN-The Complex Number System
 N-VM-Vector and Matrix Quantities
o Algebra
 A-SSE-Seeing Structure in Equations
 A-APR-Arithmetic with Polynomials and Rational Expressions
 A-CED-Creating Equations
 A-REI-Reasoning with Equations and Inequalities
o Functions
 F-IF-Interpreting Functions
 F-BF-Building Functions
 F-LE-Linear, Quadratic and Exponential Models
 F-TF-Trigonometric Functions
o Geometry



 G-CO-Congruence
 G-SRT-Similarity, Right Triangles, & Trigonometry
 G-C-Circles
 G-GPE-Expressing Geometric Properties with Equations
 G-MG-Modeling with Geometry
o Statistics and Probability
 S-ID-Interpreting Categorical & Quantitative Data
 S-IC-Making Inferences & Justifying Conclusions
 S-CP-Conditional Probability and Rules of Probability
 S-MD-Using Probability to Make Decisions
Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets)
Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets)
MU: Measuring Up Workbook
UNIT 3
Total Number of Days: _4.5days__ Grade/Course: _Calculus__







ESSENTIAL QUESTIONS
How do you write and graph exponential
functions?
How do you evaluate exponential functions for
a given value?
How do you use transformations to sketch
graphs of exponential functions?
How do you evaluate logarithms with base a?
How do you use transformations to sketch
graphs of logarithmic functions?
How are patterns of change related to the
behavior of functions?
How do you recognize, evaluate and graph
logarithmic functions?
ENDURING UNDERSTANDINGS
 Exponential functions can be used to model real-life situations.
 Exponential and logarithmic functions are inverses of each other.
 Student can solve e, log and natural log.
 Exponents are used to represent complex expressions.
 Linear functions have a constant difference, whereas exponential functions have
a constant ratio.
 Real world situations can be represented symbolically and graphically
 How do you change bases in logarithmic
expressions?
 How do you use properties of logarithms to
evaluate or rewrite expressions?
PACING
CONTENT
Exponential
Functions and
their Graphs
SKILLS
 Learn how to
recognize and
evaluate
exponential
functions with
base a
STANDARDS
(CCCS/MP)
RESOURCES
OTHER
Pearson
(e.g., tech)
F.BF.5
MP1,
MP2
MP5
Algebra 2,
volume1&2,
Common Core
Edition. Pearson.
Pg. 442 -446
www.kutasoftware.
com/exponenial
functions
www.Classzone.com
‎
Pre Calculus
with Limits
pg. 217
.5
Examples of graphs
of Exponential
Functions:
LEARNING
ACTIVITIES/ASSESSME
NTS
Internet example:
http://math.arizona.edu/~
calc/Text/1.2.pdf
Standardized Test Prep:
SAT/ACT
Technology Graphical
approach: page: 227 #
33-37
http://www.purpl
emath.com/modul
es/expofcns5.htm
Problems of the Lesson:
17, 53, 65
Model It: Data Analysis:
Biology. pg: 228 # 69
Group Exploration: pg.
223, 219
Average: 15 - 30 odd
Advanced: 18 - 30 even
 Graph exponential
functions and use
the one-to-one
property.
Writing Journal
Teacher made Quizzes
Quiz and Test generators
Worksheets
Suppose Q = f (t) is an
exponential function
of t.
If:
f (20) = 88.2 and
f (23) = 91.4
Find:
(a) the base.
(b) the growth rate
Evaluate f (25).
Use Q=
 Recognize,
evaluate and
graph exponential
functions with
base e.
Graph y = e2x.
 Use exponential
functions to
model and solve
real-life
problems.
Certain bacteria, given
favorable growth
conditions, grow
continuously at a rate of
4.6% a day. Find the
bacterial population
after thirty-six hours, if
the initial population
was 250 bacteria.
Logarithmic
Functions and
their Graphs.
 Evaluate
logarithmic
functions with
base a
=x
 Graph logarithmic
functions.
Same as Above
MP1
MP4
MP5
.5
 kutasoftware.c
om
 Classzone.com
http://www.regent
sprep.org/Regents
/math/algtrig/ATP
8b/logFunction.ht
m
Internet example:
http://www.purplemath
.com/modules/graphlog
3.htm
Technology Graphical
approach: page: 237 #
61
Problems of the Lesson:
4, 15, 79
Model It: Monthly
Payment. Page: 237 #
87
Average: 13 - 27 odd
Advanced: 82 - 86 even
 Evaluate and
graph natural
logarithmic
functions.
Group Exploration: pg.
230
Teacher made Quizzes
Writing Journal:
Analyzing a Human
Memory Model
Quiz and Test generators
Worksheets
y=
 Use logarithmic
functions to model
real-life problems
In chemistry, a
solution’s pH is
defined by the
logarithmic
equation
,
where t is the
hydronium ion
concentration in
moles per liter. We
usually round pH
values to the nearest
tenth.
http://www.uiowa.
edu/~examserv/m
athmatters/tutorial
_quiz/log_exp/real
worldappslogarith
m.html
a. Find the pH of a
solution with
hydronium ion
concentration 4.5 x
10-5
b. Find the
hydronium ion
concentration of
pure water, which
has a pH of 7.
Properties of
Logarithms.
.5
 Learn how to use
the change-of –base
formula to write
and evaluate
logarithmic
expressions.
Same as Above
 kutasoftware.c
om/logarithm
MP1
MP2
MP5
 Classzone.com
 http://www.pur
plemath.com/mo
http://www.purplemath
.com/modules/graphlog
3.htm
Standardized Test Prep:
SAT/ACT
dules/logrules5.
htm
Technology Graphical
approach: page:
Problems of the Lesson:
4, 15, 79
 Use properties of
logarithms to
evaluate or rewrite
logarithmic
expressions.
Model It: Human
Memory Model. Pg.244 #
84
Group Exploration: pg.
241
4[lnz+ln(z+5)- 2l(z-5) ] =2
 Use properties of
logarithms to
expand or condense
logarithmic
expressions
Expand the
expression
Condense the
expression
3 log x + 2 log y log z.
http://dl.uncw.edu
/digilib/mathemati
cs/algebra/mat111
hb/eandl/logprop/
logprop.html
Average: 19 - 27 odd
Advanced: 62 - 72 even
Writing Journal
Quiz and Test generators
Worksheets
,
 kutasoftware.c
om
 Smarthinking.c
om.
Exponential and
Logarithmic
Equations
 Solve complicated
exponential
equations
‎
Technology Graphical
approach: page: 254#67
Problems of the Lesson:
4, 15, 79
Solve for x:
23x+1= 5x+6
.5
http://banach.millersvill
e.edu/~BobBuchanan/
math160/ExpLogModels
/main.pdf
Standardized Test Prep:
SAT/ACT
 Solve more
complicated
logarithmic
equations
+1 =3
 Use exponential
and logarithmic
equations to
model and solve
real-life problems.
The population of
Pittsburgh
from2000 to 2007 is
modeled by
http://www.chilim
ath.com/algebra/a
dvanced/log/logcondensing.html
Model It: Automobiles.
Page: 255# 117
Group Exploration: pg.
251
Average: 15 - 23 odd
Advanced: 28 - 36 even
Writing Journal:
Comparing
Mathematical Models
Teacher made Quizzes
Quiz and Test generators
Worksheets
.
P=
with t = 0
corresponding to
the year 2000.
1. Find the population
of Pittsburgh in
2005.
2 . Graph the
population as a
function of t.
3. From the graph
estimate the year in
which the
population
will reach 2:2
million.
4. Confirm this
estimates
algebraically.
Exponential and
Logarithmic
Models
.5
 Learn how to
recognize the five
most common
types of models
involving
exponential and
logarithmic
functions.
 Exponential
Growth Model
Same As Above
 kutasoftware.c
om/exponential
function
 Classzone.com
 Smarthinking.c
om.
http://academics.u
tep.edu/Portals/17
88/CALCULUS%
20MATERIAL/3_
5%20EXPO%20A
ND%20LOG%20
Internet example:
http://www.sosmath.co
m/algebra/logs/log5/lo
g54/log54.html
Technology Graphical
approach: page: 258
Problems of the Lesson:
4, 15, 35
Model It: Population
Page: 265 #36
MODELS.pdf
‎
 Exponential
Decay Model
 Gaussian Model
 Logistic Growth
Model
Group Exploration: pg.
230
Average: 51 - 59 odd
Advanced: 54 - 64 even
Writing Journal:
Comparing Population
Models
Teacher made Quizzes
Quiz and Test generators
Worksheets
 Natural
Logarithmic
model
Chapter Review
.5
To review the
chapter
Same as Above
MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
 kutasoftware.c
om
 Classzone.com
Chapter Summary with
Exercises
Chapter Practice Test.
Quizzes, mid chapter
test and chapter test
versions A,B,C,D for
basic, Average and
Advanced group from
the Test Bank of the
Larson 6th edition
Chapter
Summative
Assessment
Summative
Assessment of
Polynomial and
Rational Functions
Same As Above
Chapter Test
MP
1day
. Written tests and
quizzes.
. Daily Home work.
. Worksheets.
. Project Assessment.
. Quizzess, midchapter
test and chapter test
versions A,B,C,D for
basic, Average and
Advanced group from
the Test Bank of the
Larson 6th edition
. Notebook assessments.
. Journal Reports
. Class Discussion and
questions
INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT

Exponential and logarithmic Functions helps student develop understanding of population growth of virus, bacteri and cell phones.
Marzano’s Six Steps for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point)
2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
5. Ask students to discuss vocabulary words with one another (Collaborate)
6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
VOCABULARIES:
Transformation of graphs
Natural exponent
Natural logarithmic
Exponential growth Average
Gaussian Models
Connect1, 2 and 3
Find a partner and study the pictures below and explain which of the graph you would apply to the picture.
.
Connect 6
PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
4 Cs
Creativity: projects
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Three Part Objective
Behavior
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – PowerPoint’s/Presentations
Condition
Demonstration of Learning (DOL)
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
Biological Cell
If you start a biology experiment with 5,000,000 cells and 45% of the cells are dying every minute, how long will it
take to have less than 1,000 cells?
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
Radioactive.
Hospitals utilize the radioactive substance iodine-131 in the diagnosis of conditions of the thyroid gland. The half-life of iodine131 is eight days. a: Determine the decay constant b. (b) If a hospital acquires 2 g of iodine-131, how much of this sample will
remain after 20 days ? c: How long will it be until only 0.01 g remains?
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field
MODIFICATIONS/ACCOMMODATIONS
E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Individual student learning styles would be accommodated by:
 adjusting assessment standards,
 one-to-one teacher support
 extra time testing time
 additional use of visual, auditory and other teaching methods.
 A wide range of assessments and strategies that complement the individual learning experience would be encouraged.
 Teacher directed instruction by providing students with more necessary steps in order to solve the problems
 Small Group Activities - when students are given group guided practice
 IEP/504 Modifications:
 Students will be allowed to use the graphing calculator
 Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math
 Modified assessments and assignments (classwork, homework, quizzes/tests) as needed
 Math Centers (Differentiation) – Review/Revisit topics missed by absentee students

APPENDIX
(Teacher resource extensions) CCSS Mathematical Practices:
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:

Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Abbreviations
o Number & Quantity
 N-RN-The Real Number System
 N-Q-Quantities
 N-CN-The Complex Number System
 N-VM-Vector and Matrix Quantities
o Algebra
 A-SSE-Seeing Structure in Equations
 A-APR-Arithmetic with Polynomials and Rational Expressions
 A-CED-Creating Equations
 A-REI-Reasoning with Equations and Inequalities
o Functions
 F-IF-Interpreting Functions
 F-BF-Building Functions
 F-LE-Linear, Quadratic and Exponential Models
 F-TF-Trigonometric Functions
o Geometry
 G-CO-Congruence
 G-SRT-Similarity, Right Triangles, & Trigonometry
 G-C-Circles
 G-GPE-Expressing Geometric Properties with Equations
 G-MG-Modeling with Geometry
o Statistics and Probability
 S-ID-Interpreting Categorical & Quantitative Data
 S-IC-Making Inferences & Justifying Conclusions
 S-CP-Conditional Probability and Rules of Probability



 S-MD-Using Probability to Make Decisions
Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets)
Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets)
MU: Measuring Up Workbook
UNIT 4
Total Number of Days: _5 days__ Grade/Course: Calculus__
ESSENTIAL QUESTIONS
What is the definition of a radian?.

How can you convert from radians to
degrees and vice versa?

How do you select Radian Mode or Degree
Mode on your calculator?
The ratios of the side lengths of a triangle can be defined with trigonometric
functions.

What is the fundamental difference between
a degree and a radian?
Using radian measure, trigonometric functions can be defined on all real
numbers.

Trigonometric functions are periodic.

What is angular velocity?

The unit circle is a means of finding trig functions for any given angle.

How does it differ from linear velocity?

Any cyclic occurrence can be represented by a trig function.

How does the arc length formula allow us to
convert between angular and linear
velocity?

Trig functions can be translated and transformed.

Fundamental identities can be used to verify more complicated trig
identities.





How can the six basic trig functions be used
ENDURING UNDERSTANDINGS
The characteristics of trigonometric and circular functions and their
representations are useful in solving real-world problems.

to solve right triangles?

What is the main difference between a trig
function and its inverse?
We can use formulas to find exact value of angles that are combinations of
unit circle angles.

Any cyclic occurrence can be represented by a trig function.

Trig functions can be translated and transformed.

How can inverse trig functions be used to
calculate unknown angles in a right
triangle?

What are the two special triangles…45-4590 and 30-60-90

When discussing angles in the Cartesian
Plane, which axis is always the initial side?

Which direction of rotation is positive and
which is negative?

Why are 2π and 360o important numbers
when discussing coterminal angles?

How does the unit circle and the concept of
conterminal angles help us to generate
graphs of trig

How can you evaluate inverse trig functions
if a point not on the unit circle is included?

functions relate to the parent graphs of the
trig functions?

How can one compose a function that is
periodic but not sinusoidal.

How would one be able to compose a
function that represents damped harmonic
motion.
PACING
CONTENT
Radian and
Degree Measure
SKILLS
 Learn how to
describe angles.
 Use radian
measure
.5
STANDARDS
(CCCS/MP)
F.TF.1
T.FT. 2, 5.
F.BF.2, 3,
MP1,
MP2
MP5
RESOURCES
OTHER
Pearson
(e.g., tech)
www.kutasoftwar
Algebra 2,
e.com
volume1&2,
www.Classzone.c
Common Core
om
Edition. Pearson.
‎
Pg. 844 - 845
http://www.pur
plemath.com/m
Pre Calculus
odules/radians.
with Limits.
htm
Pg. 282
LEARNING
ACTIVITIES/ASSESSME
NTS
Internet example:
http://www.opusmath.
com/common-coreclusters/7.g.b-anglearea-surface-areavolume
Technology Graphical
approach: page: 286
Example:
Convert /6 radians
to degrees.
(/6)*(180/) =
(1/1)*(30/1) = 30
Problems of the Lesson:
18, 95
 Use degree
measure.
Average: 18 – 31 odd
Advanced: 18 – 47 even
Example:
Convert 270° to
radians. Since 180°
equates to π, then:
(270/1)*( /180)
= (3/1)*( /2) =
(3)/2
 Use angles to
model and solve
Model It: Speed of a
Bicycle. Pg: 293 #108
Group Exploration: pg.
284
Journal Writing:
Teacher made Quizzes
real-life problems.
Trigonometric
Functions: Unit
circle.
 Identify a unit
circle and describe
its relationship to
real world.
Same as Above
Worksheets
 kutasoftware.c
om
 Classzone.com
 Mathgraphs.co
m
.5
MP1
MP4
MP5
 Evaluate
trigonometric
functions using the
unit circle.
Example:
Evaluate the six trig
functions of;
t= , t=
 https://www.kha
nacademy.org/m
ath/trigonometry
/basictrigonometry/unit
_circle_tut/v/unit
-circle-definitionof-trig-functions1
Internet example:
http://coolmath.com/
precalculus-reviewcalculusintro/precalculustrigonometry/28-theunit-circle-01.htm
HSPA /SAT Prep
Technology Graphical
approach: page: 298
Problems of the Lesson:
53
Model It: Harmonic
Motion. pg: 300 #57
Group Exploration: pg.
297
Average: 43 - 53 odd
Advanced: 59 - 61 even
Writing Journal: How do
 Use the domain
and period to
evaluate sine and
cosine functions.
you evaluate
trigonometry by using
the unit circle
Teacher made Quizzes
Worksheets
t(t+c)=f(t)
 Use a calculator to
evaluate
trigonometric
functions.
Right Triangles
Trigonometry
Example:
Use calculator to
evaluate
a). sin
b) cot 1.5
 To evaluate
trigonometric
function of acute
angles
Same as Above
 kutasoftware.c
om
MP1
MP2
MP5
 Classzone.com
Internet example:
http://www.regentspr
ep.org/regents/math/
algebra/AT2/Ltrig.htm
HSPA /SAT Prep
Technology Graphical
approach: page: 303
.5
Problems of the Lesson:
66, 68
Model It: Height.
pg: 311# 71
 Use the
fundamental
trigonometric
identities
Group Exploration: pg.
305
Average: 18 – 31 odd
 Use a calculator to
evaluate
trigonometric
functions.
Advanced: 30 - 42 even
Writing Journal: Prove
that Tan=sin/cos
Teacher made Quizzes
https://www.googl
e.com/search?q=a
pplications+invol
ving+right+triang
les&client=firefox
a&rls=org.mozilla
:enUS:official&chan
nel=np&tbm=isch
&tbo=u&source=u
niv&sa=X&ei=_fn
xUf75Eqv64AOd4
YGgDA&sqi=2&v
ed=0CFEQsAQ&b
iw=1024&bih=62
9
Example:
Use calculator to
evaluate:
Cos 28o
Sec 28o
 Use trigonometric
functions to
model and solve
real-life problems.
Trigonometric
Functions of any
Angle.
.5
 Evaluate
trigonometric
functions of any
angle.
Example:
Let  be any angle
in standard position,
let (x,y) = P be a
point on the
terminal side of 
Same as Above
 kutasoftware.c
om
 Smarthinking.c
om
 Classzone.com
Internet example:
http://www.themathp
age.com/atrig/functio
ns-angle.htm
HSPA /SAT Prep
Technology Graphical
approach: page: 319# 65
- 68
http://www.thema
thpage.com/atrig/f
Problems of the Lesson:
unctions-
(other than the
origin). Let r be the
distance from P to
the origin. Then we
define:
angle.htm
59, 89
‎
Model It: Data Analysis:
Meteorology. pg: 319#
87
Group Exploration: pg.
315
Average: 37 - 49 odd
Advanced: 71 - 85 even
Writing Journal
Teacher made Quizzes
Worksheets
Use reference angles
to evaluate
trigonometric
functions.
Example:
Find the reference
angle:
a). θ=300o
b). θ= -1350
 Evaluate
trigonometric
functions of reallife numbers.
Graphs of Sine
and Cosine
Functions.
 Sketch the graphs
of basic sine and
cosine functions.
Same As Above
 Classzone.com
MP
.5
 Use amplitude
and periods to
help sketch the
graphs of sine and
cosine functions.
y=d+asin(bx-c)
y-d+acos(bx-c)
 kutasoftware.c
om
 Smarthinking.c
om.
Internet example:
http://patrickjmt.com/gra
phing-sine-and-cosinewith-differentcoefficients-amplitudeand-period-ex-1/
HSPA /SAT Prep
Interactive
Technology Graphical
Website:
http://www.purple approach: page: 329 #
math.com/module 63
s/triggrph.htm
Problems of the Lesson:
59, 89
‎
Problem model It: Data
Analysis: Astronomy.
Page: 330 # 78
Group Exploration: pg.
323 and 324
Average: 18 – 31 odd
Advanced: 18 – 47 even
 Sketch
translations of the
graphs of sine and
cosine functions.
Writing Journal: How do
you sketch the graphs of
sine and cosine
functions
Teacher made Quizzes
Worksheets
www.youtube.co
m/watch?v=dHU
M_ZgZ9Hg
 Use sine and
cosine functions
to model real-life
data.
.5
Graphs of other
Trigonometric
Functions
 To sketch the
graphs of tangent
functions.
Same as Above
MP
 kutasoftware.c
om
 Classzone.com
http://www.pur
Internet example:
http://www.intmath.
com/trigonometricgraphs/4-graphstangent-cotangentsecant-
Example:
Sketch the graph of
y= tan
plemath.com/m
odules/triggrph
2.htm
Technology Graphical
approach: page: 340 #
53, 57
 Sketch the graphs
of cotangent
functions
Problems of the Lesson:
59, 867, 89
Example: y=2cot
 Sketch the graphs
of Secant and
Cosecant
functions
Model It: Predator-Prey
Video
Model. Page: 341# 77
http://www.yout
ube.com/watch Group Exploration: pg.
?v=CMvUs338
923O8
Average: 65 - 75 odd
Advanced: 65 - 75 even
Example: y=sec2x
 Sketch the
graphs of damped
trigonometric
functions.
Writing Journal:
Combining Trig
Functions.
Teacher made Quizzes
Worksheets
Example:
f(x)=e-xsin3x
Inverse
Trigonometric
Functions
.5
 Evaluate and
graph the inverse
sine function.
cosecant.php
HSPA /SAT Prep
Same As Above
 kutasoftware.com
 Classzone.com
MP
Internet example:
http://www.mathopenr
ef.com/triginverse.html
https://www.khanaca HSPA /SAT Prep
demy.org/math/trigo
nometry/unit-circletrigTechnology Graphical
func/inverse_trig_func
approach: page: 350 #
tions/e/inverse_trig_f 69
unctions
Problems of the Lesson:
59, 92
 Evaluate and
graph the other
inverse
trigonometric
functions.
Problem model It:
Photography. Page:
351#93
Group Exploration: pg.
344
Average: 13 - 23 odd
Advanced: 116 - 120
even
Example:
Sketch a graph of
y=arcsinx
 Evaluate and
graph the
composition of
trigonometric
functions.
Writing Journal
Teacher made Quizzes
Example:
Find the exact value
tan(arccos2/3
Applications and  Solve real-life
Models
problems
involving right
triangles.
.5
Example:
Same as above.
 kutasoftware.co
 Classzone.com
MP
http://www.intmath.
com/trigonometricfunctions/4-righttriangleapplications.php
 Solve real-life
Internet example:
http://academics.utep.e
du/Portals/1788/CALC
ULUS%20MATERIAL/4_
8%20APPLICATIONS%2
0AND%20MODELS.pdfl
Technology Graphical
approach: page: 358
Problems of the Lesson:
27, 37
problems
involving
directional
bearings.
Problem model It:
Numerical and Graphical
Analysis. Page: 363 # 62
Group Exploration: pg.
355
Example:
Average: 17 - 31 odd
Advanced: 45 - 65 even
Writing Journal
Teacher made Quizzes
 Solve real-life
problems
involving
harmonic motion.
Example:
D=asinωt
Chapter Review
To review the
chapter
http://www.docstoc.
com/docs/10064014
8/006-PeriodicMotion-Unedited
Same as Above
MP1
Pg. 365-368
 kutasoftware.c
om
 Classzone.com

Chapter Summary with
Exercises pg. 364
Chapter Practice Test.
Pg. 369
Quizzess, midchapter
test and chapter test
versions A,B,C,D for
basic,Average and
Advanced group from
the Test Bank of the
Larson 6th edition
Test
Summative
Assessment of
Trigonometry
Same As Above
Chapter Test
MP
1day
. Written tests and
quizzes.
. Daily Home work.
. Worksheets.
. Project Assessment.
. Article Summaries.
. Guided practice.
. Review Games.
. Summative Test
. Unit Test with multiple
choice and openended.
. Notebook assessments.
. Journal Reports
INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT




Radian and Degree Measure-analysis of the two measures helps students in developing the conversions between them and why radian is
preferred in the real world applications like in game library
Trigonometric Functions: Analyze the Unit circle is an easy way for students to learn trigonometry, which has an incredible amount of
real life applications
Right triangle Trigonometry - students can apply Pythagorean Theorem and the six trigonometric functions
Graphs of Sine and Cosine functions will help students in developing and understanding of Fourier Analysis and electrical circuit anaysis
Marzano’s Six Steps for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point)
2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
5. Ask students to discuss vocabulary words with one another (Collaborate)
6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
VOCABULARIES:
VOCABULARIES:
angular velocity
sine
cosine
tangent
cotangent
secant
cosecant
radian
degree
radian amplitude
period
vertical translation
phase shift
horizontal asymptote
vertical asymptote
sinusoidal function
Identity
Sum and Difference formulas
Double angle formulas
Half angle formula
amplitude
period
vertical translation
phase shift
horizontal asymptote
vertical asymptote
sinusoidal function
Connect 2, 4 & 5
Students work in pairs to discover the properties of trigonometric ratios using an applet. Their task is to complete tables of trig ratios for different triangles
(the applet allows students to change the angle measure and base length of a right triangle). They start with sine of varying angles for a triangles with a
constant base, then change the base and find the sine of the same angles. In pairs students discuss their observations and particularly focus their attention
on comparing the two tables (ideally recognizing that the values are the same regardless of the length of the base). They repeat this exercise for tangent
and cosine.
PARCC/SAT/TIMMS FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – PowerPoints/Presentations
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
Transportation:
An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must the plane take off in order to avoid crashing into the
building? Assume that the airplane flies in a straight line and the angle of elevation remains constant until the airplane flies over the building.
x = arctan(
)
5.72 o . The plane must take off at an angle of elevation of about5.72 o in order to avoid hitting the building.
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.
MODIFICATIONS/ACCOMMODATIONS
2. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Individual student learning styles would be accommodated by:
 adjusting assessment standards,

one-to-one teacher support


extra time testing time

additional use of visual, auditory and other teaching methods

A wide range of assessments and strategies that complement the individual learning experience would be encouraged.

Teacher directed instruction by providing students with more necessary steps in order to solve the problems

Small Group Activities - when students are given group guided practice

IEP/504 Modifications:

Students will be allowed to use the graphing calculator

Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math

Modified assessments and assignments (classwork, homework, quizzes/tests) as needed

Math Centers (Differentiation) – Review/Revisit topics missed by absentee students
APPENDIX
(Teacher resource extensions) CCSS Mathematical Practices:
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:

Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Abbreviations
o Number & Quantity

N-RN-The Real Number System

N-Q-Quantities

N-CN-The Complex Number System

N-VM-Vector and Matrix Quantities
o Algebra

A-SSE-Seeing Structure in Equations

A-APR-Arithmetic with Polynomials and Rational Expressions

A-CED-Creating Equations

A-REI-Reasoning with Equations and Inequalities
o Functions

F-IF-Interpreting Functions

F-BF-Building Functions

F-LE-Linear, Quadratic and Exponential Models

F-TF-Trigonometric Functions
o Geometry

G-CO-Congruence

G-SRT-Similarity, Right Triangles, & Trigonometry

G-C-Circles

G-GPE-Expressing Geometric Properties with Equations

G-MG-Modeling with Geometry
o Statistics and Probability

S-ID-Interpreting Categorical & Quantitative Data

S-IC-Making Inferences & Justifying Conclusions

S-CP-Conditional Probability and Rules of Probability

S-MD-Using Probability to Make Decisions

Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets)

Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets)

MU: Measuring Up Workbook
UNIT 5
Total Number of Days: _2__ days__ Grade/Course: _Calculus__






ESENTIAL QUESTIONS
How do we use algebra and basic trig identities
to verify complicated trig identities.
How do we find trig functions of angles that
are combinations of unit circle angles
How do you rewrite trigonometric
expressions in order to simplify and evaluate
functions?
How do you verify a trigonometric identity?
How do you solve trigonometric equations
written in quadratic form or containing more
than one angle?
How do you simplify expressions and solve
equations that contain sums or differences of
angles?




ENDURING UNDERSTANDINGS
Identities are used to evaluate, simplify, and solve trigonometric expressions
and equations.
Fundamental trigonometric identities, sum and difference formulas together
with algebraic techniques are used to simplify expressions, prove identities
and solve fundamental identities and can be used to verify more complicated
trig identities.
We can use formulas to find exact value of angles that are combinations of unit
circle angles equations.
When trigonometric equations are solved an angle measure is the result.




How do you rewrite trigonometric
expressions that contain functions of multiple
or half-angles, or functions that involve square
or products trigonometric expressions?
How do you verify a trigonometric identity
graphically?
How do you use sum and difference identities
to evaluate trigonometric expressions?
How do you use double-angle and half-angle
identities to evaluate trigonometric
expressions?
PACING
CONTENT
Using
Fundamental
Identities
SKILLS
 Recognize and
write the
fundamental
trigonometric
identities.
 Use the
fundamental
trigonometric
identities to
STANDARDS
(CCCS/MP)
F.TF.8, 9
MP1,
MP2
MP5
RESOURCES
OTHER
Pearson
(e.g., tech)
Algebra 2,
www.kutasoftwar
volume1&2,
e.com
Common Core
www.Classzone.c
Edition. Pearson.
om
‎
Pre Calculus
http://coffman.du
with Limits.
blin.k12.oh.us/te
Pg. 373
achers/teacherp
ages/oconnor/se
ction5.1precalc.
cwk(WP).pdf
LEARNING
ACTIVITIES/ASSESSME
NTS
https://teacher.ocps.net
/jean.adams/media/51p
cf.pdf
Technology Graphical
approach: page: 375 and
380 # 73
Problems of the Lesson:
6, 14
Problem model It:
Analysis. Page: 363 # 62
Group Exploration: pg.
evaluate
trigonometric
functions,
simplify
trigonometric
expressions and
rewrite
trigonometric
expressions.
Example:
Evaluate:
a). sec u=-2/3
Verifying
Trigonometric
Identities.
Simplify:
sinxcos2-sinx
 Learn how verify
trigonometric
Identities.
Example:
Verify:
(tan2x + 1)(cos2x -1)
=-tan2x
374 – Fundamental Trig
Identities
Average: 15 - 34odd
Advanced: 45 - 65 even
Writing Journal
Worksheets
Teacher made Quizzes
Same as Above
MP1
MP4
MP5
 kutasoftware.c
om
 Classzone.com
 Mathgraphs.co
m
 http://www.purp
lemath.com/mo
dules/proving.h
tm

http://www.analyzemat
h.com/Trigonometry_2/
Verify_identities.html
Warm-up exercise
Technology Graphical
approach: page: 388 #
39- 44
Problems of the Lesson:
20, 34
Model It: Shadow
Length. Pg. 388 # 56
Group Exploration
Average: 21 - 27 odd
Advanced: 51 - 56 even
Writing Journal: Error
Analysis.
Worksheets
Teacher made Quizzes
Solving
Trigonometric
Equations.
 Learn how to use
standard
algebraic
techniques to
solve
trigonometric
equations.
Example:
Solve:
3tan2x-1=0
 To solve
trigonometric
equations of
quadratic type.
Example:
find all solutions of;
2sin2x – sinx – 1 =0
 To solve
trigonometric
equations
Same as Above
MP1
MP2
MP5
 kutasoftware.c
om
 Classzone.com
 mathwarehouse.
com/
 http://www.rege
ntsprep.org/Re
gents/math/algt
rig/ATT10/trige
quations2.htm

http://www.purplemath
.com/modules/solvtrig.h
tm
Warm-up exercise
HSPA /SAT Prep
Technology Graphical
approach: page: 397 #
55
Problems of the Lesson:
35, 63
Model It: Data AnalysisUnemployment Rate. Pg.
398 # 76
Group Exploration: 393
Average: 57 - 63 odd
Advanced: 75 - 86 even
involving multiple
angles.
Example:
Solve:
2cot3t-1=0
Sum and
difference
formulas
 Use inverse
trigonometric
functions to solve
trigonometric
equations.
Solve:
sec2x-2tanx=4
 Use sum and
difference
formulas to
evaluate
trigonometric
functions, verify
identities, and
solve
trigonometric
equations.
Example:
Find exact value of
(sin42cos12 –
cos42sin12)
Writing Journal:
Worksheets
Teacher made Quizzes
Same as Above
MP
 kutasoftware.c
om
 Smarthinking.c
om
 Classzone.com

http://www.regent
sprep.org/Regents
/math/algtrig/ATT
14/formulalesson.
htm
‎
http://www.analyzemat
h.com/Trigonometry_2/
Sum_diff_form_trig.html
Warm-up exercise
HSPA /SAT Prep
Technology Graphical
approach: page: 405 #
65
Problems of the Lesson:
35, 63
Model It: Harmonic
Motion. Pg. 405 # 75
Group Exploration: 400
Average: 69 - 77 odd
Advanced: 56 - 75 even
Writing Journal:
Equations with no
Solutions.
Worksheets
Teacher made Quizzes
Multiple Angle
and Product-toSum Formulas
 Use multipleangle formulas to
rewrite and
evaluate
trigonometric
functions
Same As Above
MP
Example:
Solve:
2cosx+sin2x=0
 Use powerreducing
formulas to
rewrite and
evaluate
functions
 kutasoftware.co
m
 Classzone.com
 Smarthinking.co
m.
 http://www.rege
ntsprep.org/Reg
ents/math/algtri
g/ATT14/formul
alesson.htm

http://academics.utep.e
du/Portals/1788/CALC
ULUS%20MATERIAL/5_
5%20MULTIPLE%20AN
GLE%20AND%20PROD
UCT%20TO%20SUM%2
0FORMULAS.pdf
‎
Problems of the Lesson:
35, 63
Warm-up exercise
HSPA /SAT Prep
Technology Graphical
approach: page: 416# 59
Model It: Mach Number.
Pg417 # 121
Group Exploration:
Average: 69 - 77 odd
Advanced: 56 - 75 even
Writing Journal:
Deriving an Area
Formula
 Use product-tosum and sum-toproduct formulas
to rewrite and
evaluate
trigonometric
functions.
Chapter Review
 Use trigonometric
formulas to
rewrite real-life
models
Example:
Projectile MotionR=
To review the
chapter
Worksheets
Teacher made Quizzes
 kutasoftware.c
om
 Classzone.com
 http://www.rege
ntsprep.org/Reg
ents/math/algtri
g/ATT14/formul
alesson.htm

Quizzess, midchapter
test and chapter test
versions A,B,C,D for
basic,Average and
Advanced group from
the Test Bank of the
Larson 6th edition
Test
1day
INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT

Using fundamental Identities, solving and Verifying Trigonometric Identities helps students apply their sense of algebraic equations that is
proving that the equation is true by showing that both sides equal one another
Marzano’s Six Steps for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point)
2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
5. Ask students to discuss vocabulary words with one another (Collaborate)
6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
VOCABULARIES:
Pythagorean identities
• Reciprocal identities
• Half angle formulas
• Double angle formulas
• Sum and difference formulas
• Quotient identities
• Cofunction identities
• Even/odd identities
• Power reducing formulas
• Product to sum and sum to product formula
Connect 4
Each group explore would study the pad below, make a table of frequency combination for each number. Use the sum and difference formulas
to write the sound produced by keys 3 and 7. On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of
two tones, given by
y = sin (2lt) and y = sin (2ht)
where l and h are the low and high frequencies (cycles per second) shown on the illustration.
The sound produced is thus given by
y = sin (2lt) + sin (2ht)

PARCC FRAMEWORK/ASSESSMENT
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
21ST CENTURY SKILLS
(4Cs & CTE Standards)
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
MODIFICATIONS/ACCOMMODATIONS
Individual student learning styles would be accommodated by:
 adjusting assessment standards,
 one-to-one teacher support
 extra time testing time
 additional use of visual, auditory and other teaching methods.
 A wide range of assessments and strategies that complement the individual learning experience would be encouraged.
 Teacher directed instruction by providing students with more necessary steps in order to solve the problems
 Small Group Activities - when students are given group guided practice
 IEP/504 Modifications:
 Students will be allowed to use the graphing calculator
 Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math
 Modified assessments and assignments (classwork, homework, quizzes/tests) as needed
Math Centers (Differentiation) – Review/Revisit topics missed by absentee students
APPENDIX
 (Teacher resource extensions) CCSS.Mathematical Practices:
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:

Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Abbreviations
o Number & Quantity
 N-RN-The Real Number System
 N-Q-Quantities
 N-CN-The Complex Number System
 N-VM-Vector and Matrix Quantities
o Algebra
 A-SSE-Seeing Structure in Equations
 A-APR-Arithmetic with Polynomials and Rational Expressions
 A-CED-Creating Equations
 A-REI-Reasoning with Equations and Inequalities
o Functions
 F-IF-Interpreting Functions
 F-BF-Building Functions
 F-LE-Linear, Quadratic and Exponential Models
 F-TF-Trigonometric Functions
o Geometry
 G-CO-Congruence
 G-SRT-Similarity, Right Triangles, & Trigonometry
 G-C-Circles
 G-GPE-Expressing Geometric Properties with Equations
 G-MG-Modeling with Geometry
o Statistics and Probability
 S-ID-Interpreting Categorical & Quantitative Data
 S-IC-Making Inferences & Justifying Conclusions
 S-CP-Conditional Probability and Rules of Probability
 S-MD-Using Probability to Make Decisions
 Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets)
 Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets)
MU: Measuring Up Workbook
UNIT 6
Limits and Their Properties
Total Number of Days: 21 days Grade/Course: 11-12/calculus
ESENTIAL QUESTIONS
ENDURING UNDERSTANDINGS
Students will understand that
• There are different ways of calculating limits.
• There are different types of discontinuities. The classic example is the graph of
distance versus time for a moving vehicle. In most practical applications, the
slope of the graph - which represents speed - will not be constant.
•Limits asymptotes and continuity are interconnected calculus topics.
• What is a limit?
• How is a limit calculated?
• What does continuity mean and what are the different
types of discontinuities?
• What are vertical asymptotes and how are they related to
limits?

PACING
days
I day
CONTENT
Finding Limits
Numerically and
Graphically
SKILLS
STANDARDS
(CCCS/MP)
The limit can be estimated
numerically by creating a
F.IF.6 Calculate
table and graphically by
and interpret the
drawing a graph.
Examples:
1.
Evaluate the function
f(x) =
at several
points near x=0
2.
Evaluate the limit
graphically:
Evaluating Limits
Analytically
average rate of
change of a
function
(presented
symbolically or as
a table)
over a specified
interval. Estimate
the
rate of change
from a graph.
F(x) =
3.
Evaluate
=
F.IF.4, 6,7, 7a, 8a,
9 F.LE.3
RESOURCES
LEARNING
ACTIVITIES/
ASSESSMENTS
Larson
OTHER
Larson
(e.g., tech)
Basic:
Pg 54-55#1, 3, 6.
9-12
I.
Average:
p54-55 #6, 8, 15,
16, 18
Advanced:
P 55 #14-18, 2124
Paul’s online
notes page 24
even problems
II.
III.
IV.
Thomas (12th
Ed.), Pg 74 #1250
John Rogawski
V.
https://www.khan
academy.org/math
/differentialcalculus/limits_top
ic/limits_tutorial/
v/numericallyestimating-limit
https://www.nr.ed
u/chalmeta/271D
E/Section%203.2.
pdf
http://pblpathway
s.com/calc/C10_2_
2.pdf
http://www.math
bootcamps.com/fi
nding-limits-usinga-graph/
www.kutasoftware

Teacher made
Quizzes

Quizzess,
midchapter test
and chapter test
versions A,B,C,D
for basic,Average
and Advanced
group from the
Test Bank of the
Larson 8e edition

Hand on
activities
A. http://w
ww.jame
srahn.co
m/Calcu
(2nd Ed.) p 80
#30-48
Evaluate
4.
Evaluate Limits
using properties of
limits
_________________________________
_____
Evaluate a limit
using properties of
limits
is a
polynomial, then
1.
2.
Examples:
1.
Apply the property
of substitution as
long as it do not
become undefined
3.
Describe and use a
strategy for finding
limits
Evaluate a limit
using dividing out
and rationalizing
techniques
Examples:
2.
Assume f,g,h are
functions
ES/limit
s.htm
F.IF.8 Write a
function defined
by an expression
in different but
equivalent forms
to reveal and
explain different
properties of the
function.
FIF.8a,b
Larson
Basic:
P 67 #1-21 odd,
49-55 odd, 67,
71
Average:
P 67 #10-22
even, 28, 34, 40,
50-56 even. 74,
76
Evaluate a limit
using Squeeze
Theorem (Theorem
1.8)
Apply two special
trigonometric limits
(Theorem 1.9)
VI.
http://calculus201
0.wikidot.com/sol
ving-limitsanalytically
VII.
http://tutorial.mat
h.lamar.edu/probl
ems/calci/LimitsP
roperties.aspx
Advanced:
P 67 #28, 34,
78=84 even, 105,
111, 123
Thomas
P 74 #12-50
selected
numbers
John Rogawski
Pgs. 75 #6-34
Larson
Basic:
Pgs 68 #50-60
even
3.
Squeeze Theorem
lusI/PAG
F.IF.1, 2 ,4, 7d, 8
1day
 If
.com/limits
VIII.
IX.
https://www.khan
academy.org/math
/differentialcalculus/limits_top
ic/squeeze_theore
m/v/squeezetheorem
X.
http://www.sosma
th.com/calculus/li
mcon/limcon03/li
mcon03.html
Average:
Pgs 68 #58-70
even
Advanced:
Pgs 68 #68-84
even, 101, 123
http://tutorial.mat
h.lamar.edu/probl
ems/calci/OneSide
dLimits.aspx

Understanding
the Definition of
a Limit;
Investigating
Limits as x
approaches
Infinity

Project
B. http://w
ww.proj
ectmaths
.ie/works
hops/WS
8_NR/po
werpoint
s/Session
%202.pd
f
C. http://re
alteachi
ngmeans
reallear
ning.blo
gspot.co
m/2011/
09/dawithfirstprinciple
s-andlimits.ht
ml
Examples:
4.
Show that
5.
Continuity and
One-Sided Limits
1. Determine
continuity at a
point and
continuity on an
open interval
2 block
periods
2.Apply
properties of
continuity
2. Interpret and
apply
Intermediate
Value Theorem
If 2- x2 ≤ g(x)≤ 2cosx
for all x, find the
limit as x
approaches zero in
the function g(x).
Examples:
1.
a.
b.
2.
John Rogawski
Pgs. 76 #46-74;
Pgs 99 #30-46
Calculate the limit of
as x
approaches -2 from
the right (bring in
the discussion of
STEP FUNCTION
from the ONE-
Page 57#53,56
XI.
Thomas
Pgs 74 #44-56
even; Pgs 75
#57-65
Interpret the
definition of
continuity:
Discuss the
removable and
non-removable
discontinuity in
Example 1.
Journal Writing
http://patrickjmt.c
om/calculating-alimit
I.
www.kutasoftware
.com/limits at
essential
continuity
II.
http://www.calcul
us.org/
Average:
Pgs 79 #13-19
odd, 32-42 even,
57, 58, 83, 84, 96
III.
www.kutasoftware
.com/at jump
discontinuitie and
kinks
Advanced:
Pgs 79 #23, 26,
41-53 odd, 75-81
odd, 95, 97, 100,
IV.
http://www.rootm
ath.org/calculus/c
ontinuity-and-onesided-limits
Larson
Basic:
Pgs 79 #3, 6, 10,
12- 18, 29-32
Page 81#88
Page116#67
Teacher made
Quizzes
Quizzess, midchapter
test and chapter test
versions A,B,C,D for
basic,Average and
Advanced group
from the Test Bank
of the Larson 8e
edition
Calcuator Activities
3.
SIDED LIMIT.
106
Describe the
intervals
Thomas
Pgs 101 #1, 4,
11-26
[
V.
http://tutorial.mat
h.lamar.edu/Class
es/CalcI/OneSided
Limits.aspx
I.
http://tutorial.mat
h.lamar.edu/Class
es/CalcI/InfiniteLi
mits.aspx
II.
https://www.khan
academy.org/math
Rogawski
Pgs 90 #1, 4, 5
Discuss property of
continuity (Theory
1.11) to solve the
above problem.
4.
Apply the
Intermediate Value
Theorem to show
that the polynomial
function
has a zero in the
interval
[0,1].
2 block
periods
Infinite Limits
1.Determine the
infinite Limits from
Determine infinite
limits from left to right
and Sketch the vertical
asymptotes of the graph
of a function given
below
Kuta
worksheet(creat
e for three
different levels
such as
easy,medium
and hard)
Section Project:
- Graphs and
limits of
trigonometric
functions as
given on pg 90 in
Larson
the left and the
right by creating a
table
2.Sketch the
vertical
asymptotes of the
graph of a
function
Examples
1 For
Evaluate
A.
B.
C.
Larson
Basic:
Pgs 89 #2-12
even, 24-32
even, 54
/differentialcalculus/limits_top
ic
III.
www.kutasoftware
.com/infinitelimits
IV.
https://www.math
.ucdavis.edu/~kou
ba/ProblemsList.h
tml
V.
https://www.khan
academy.org/math
/differentialcalculus/limits_top
ic/limitsinfinity/v/limitsand-infinity
VI.
http://patrickjmt.c
om/calculating-alimit-at-infinity
VII.
http://www.purpl
Average:
Pgs 88 #22-46
even, 53-57
Advanced:
Pgs 88 # 59, 62,
63
Thomas
Pg 114 #1-19
0dd, Pg 114- 115
# 34- 52 even,
80-86
2 block
periods
2. Compute the following:
a.
Rogawski
Pgs 85- 86 #1-30
selected
problems
b.
3. Determine the domain
and vertical
asymptotes(s), if any,
of the following
function:
emath.com/modul
es/asymtote.htm
4.
Determine all vertical
asymptotes of the graph of
VIII.
F(x) =
Sketch the graph to identify
the vertical asymptotes.
2block
period
Chapter Review&
Chapter Test
TIMSS/
SAT
Prep
.5
block
periods
Content Category:
 Numbers
Equations and
Functions
 Calculus
 Probability and
Statistics
Geometry
TIMSS Example1.
If xy =1 and x is greater
than 0, which of the
following statements is
true?
A. When x is
greater than
1, y is negative.
B. When x is
greater than
1, y is greater
than 1.
C.
When x is less
than 1, y is less
than 1.
D.
As x increases, y
A-REI.A.1
Explain each step
in solving a simple
equation as
following from the
equality of
numbers asserted
at the previous
step, starting from
the assumption
that the original
equation has a
solution.
Construct a viable
argument to
justify a solution
method.
Page 91 of
Larson and
Resource work
sheet
Test Form B and
C of the Resource
Book
College Board
SAT Guide
Page 227 selected
problems
TIMSS Sample
problems
I.
http://www.edinfor
matics.com/timss/p
op3/mpop3.htm?su
bmit32=Gr.+12+Adv
.+Math+Test
II.
http://www.majort
ests.com/sat/proble
m-solving-test01
III.
S-CPA 5
Understand
independence and
conditional
probability and
use them to
interpret data
http://patrickjmt.c
om/findingverticalasymptotes-ofrationalfunctions/
www.khanacadem
y.org/test-prep/sat
Other Resource
Texts
 Warm up
 Math minute
 Teacher created
hand outs
 SAT Book Second
Edition Test 1
section3 7 and
and 8 page 397 –
http://www.edinfor
matics.com/timss/ti
mss_intro.htm
Sample Tests online
increases.
Gruber’s Complete
Preparation for the New
SAT
E.
As x increases, y
decreases.
SAT Example
1) In a class of 78
students 41 are
taking French, 22
are taking German.
Of the students
taking French or
German, 9 are
taking both
courses. How many
students are not
enrolled in either
course?
4.3.C.1
Solve
equations
involving
absolute
value.
2) If‎f(x)‎=‎│(x²‎– 50)│,‎
what is the value of
f(-5) ?
The Princeton Review’s
Cracking the New SAT,
2007
Barron’s SAT 2400:
Aiming for the Perfect
Score
Maximum SAT
Kaplan SAT 2400, 2006
Edition
Up Your Score: The
Underground Guide to the
SAT, 2007-2008
INSTRUCTIONAL FOCUS OF UNIT


The limit process is a fundamental concept of Calculus. One technique to estimate limit is to graph the function and then determine the
behavior of the graph as the independent variable approaches a specific value. In this unit, limits are evaluated analytically,
graphically, and numerically. Also, identifying where a function is continuous and not continuous; identifying the different types of
discontinuity that a function may have; identify the vertical asymptotes of a graph and solving limit problems associated with those
asymptotes.
Limits are applied in real life to read the Speedometer for unlimited data Also see the statement given below
If I keep tossing a coin as long as it takes, how likely am I to never toss a head?
Rephrased as a limit problem, we might say
If I toss a coin N times, what is the probability p(N) that I have not yet tossed a head? Now what is the limit
as N→∞ of p(N)?
Academic Vocabularies By Dr Robert Marzano
limit
continuity
sandwich theorem
fundamental trig limit
infinity
tangent
normal
exist, Asymptote
Intermediate value Theorem
Marzano’s Six Steps for Teaching Vocabulary:
1.
2.
3.
4.
5.
6.
YOU provide a description, explanation or example. (Story, sketch, power point)
Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
Ask students to construct a picture, graphic or symbol for each word.
Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
Ask students to discuss vocabulary words with one another (Collaborate)
Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
1.Connected Graph(Step 1)
When we talk about a continuous function we need to make sure there are no holes or gapes in
the graph throughout its domain. So, the graph needs to be connected throughout the domain.
A connected graph on its domain is a graph you can draw without lifting the pencil/pen off the
paper. (No cheating! Don’t fold part of the paper over and draw on the fold and then unfold it
and say – hey that is not connected but I didn’t lift the pencil off the paper!)
Simple Test: Did you NEED to lift the pencil/pen off the paper to draw the graph each part of
the domain ?
Yes: It is not a connected graph
No: It is a connected graph
For example the domain of y=1/x is (-infinity, 0) and (0, infinity). From (-infinity, 0) it is
connected and from (0, infinity) it is connected.
2.Discuss the continuity of
F(x) =
with the help of a diagram in a journal form (Step2)
SAT/TIMSS FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
21ST CENTURY SKILLS
(4Cs & CTE Standards)
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
Parking Lot cost
A student parking lot at a university charges $ 2.00 for the first hour(or part of any hour) and $ 1.00 for each subsequent half hour (or any part) up to
a daily maximum of $10.00
a.Sketch a graph of cost as a function of time parked?
b. Discuss the significance of discontinuities in the graph to a student who parks there?
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics
problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical,
agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
Generator Pollution
Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering
activities to solve problems and issues in the real world.
A utility company burns coal to generate electricity.The cost C in dollars of removing p% of the air pollutants in the stack
emissions is
C=
0≤P≤ 100
Find the cost of removing 15% of the pollutants? Find the limit of C as P→100 9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.

MODIFICATIONS/ACCOMMODATIONS

Students can create games (bingo, flashcards, monopoly, etc.) to demonstrate their understanding
of classifying functions as continuous or discontinuous.
APPENDIX
(Teacher resource extensions)
3. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Notes to teacher (not to be included in your final draft):
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
UNIT 7
Differentiation
Total Number of Days: 27 Grade/Course: 11/12 Calculus
ESENTIAL QUESTIONS



ENDURING UNDERSTANDINGS
W
Students will understand that
hat is a derivative?

W
ifferentiation is the study of rates of change, which will explored by learning new
hat role do derivatives and limits play as a foundation for
methods and rules for finding derivatives of functions and then apply to find such
the calculus and in practical applications?
things as velocity, acceleration, and the rates ofHchange of related variables.
ow does the derivative represent an instantaneous rate
of change?

ow are derivatives used in optimization problems?

ow do differential equations describe rates of change?


ow do you find the derivatives of the sine function and of
the cosine functions?
How do you use derivatives to find rates of change?
• The tangent line problem leads to the formal definition of a derivative.
H

he derivative tells us the instantaneous rate of change
for a function which means
H
the same as the slope of the tangent line.

H
ifferentiation and definite integration are inverse operations.

PACI
NG
4days
CONTENT
Derivatives and
the tangent line
problem
SKILLS
Examples:
1. Evaluate the slope of the
currve y = x2 at the point (2,4),
using a numerical method
2. Estimate the derivative of the
following function using the
definition of the derivative.
F(x) = 2
Limit process to find the
derivative of a function
Examples
1.Let F(x) = 4x2 +5x+6
Determine an equation of
the tangent line to the curve
Y = f(x) at (1,1.5). Compute
f’ by the definition
2.
Numerical
Derivative of a
function
Describe that f is
differentiable at x=1, i.e.,
use the limit definition of
the derivative to
compute f'(1) .
he slope of a line in algebra is the average rate of change while the slope of the
tangent to a curve at a point in calculus is the instantaneous rate of change (the
derivative of a function).
RESOURCES
LEARNING
STANDARDS
Pearson
ACTIVITIES/ASSES
Larson
(CCCS/MP)
OTHER
SMENTS
(e.g., tech)
CMS IF-4
Demonstrate an
understanding of
the definition of the
derivative of a
function at a point,
and the notion of
differentiability.
a. Demonstrate an
understanding of
the derivative of a
function as the slope
of the tangent line to
the graph of the
function.
b. Demonstrate an
understanding of
the interpretation of
the derivative as
instantaneous rate
of change.
c. Use derivatives to
solve a variety of
problems coming
from physics,
chemistry,
economics, etc. that
involve the rate of
change of a function.
d. Demonstrate an
understanding of
the relationship
between
differentiability and
continuity.
e. Use derivative
formulas to find the
Basic
Page 103# 1-21
odd
Average
Page 103 # 12 -32
even
Thomas pg 132
13-21 odd
Advanced
Page 104 28- 38
even
Thomas page 133
#32,34 and 35
Basic:
Pgs 105 #41, 53 ,
55, Kuta
Worksheet
(simple problems)
Average:
Pgs 105 # 41-57
odd, Kuta
Worksheet
(medium
problems)
Advanced:
Examples
http://classroom.synonym.
com/equations-tangentlines-2838.html
http://www.wikihow.com/
Find-the-Equation-of-aTangent-Line
Videos:
https://www.khanacademy
.org/math/differentialcalculus/takingderivatives/derivative_intr
o/v/calculus--derivatives1--new-hd-version
work sheets
file:///C:/Users/DUKE/Do
wnloads/CalcI_Complete_Pr
actice.pdf
http://www.sosmath.com/c
alculus/diff/der01/der01.h
tml
You tube
http://www.youtube.com/
watch?v=vzDYOHETFlo
https://www.khanacademy
.org/math/differentialcalculus/limits_topic
Work sheet
https://www.math.ucdavis.
edu/~kouba/CalcOneDIREC
TORY/defderdirectory/Def
Using graphing calculators,
DI according to student
readiness
Use data and situations
relevant to student interests
Whole group and small
group cooperative learning
Trig function derivative
graphic organizer
Unit test (multiple
choice
& free response
questions)
Summative Assessment:




Quizzes
Tests
Quarterly
Assessments
Projects
Formative Assessments:



Demonstration
Class discussion
Homework
Quizzes
Larson 8e Test Bank 2.2
Mid chapter test
.Constant Rule: Theorem 2.2
Choose various Notation of
Derivatives
derivatives of
algebraic and
trigonometric
functions.

xample 1
The
intermediate
value theorem.
F(x) =5
Power Rule: Theorem 2.3

xample 1
4days

xample 2
Find the equation for the
tangent line for the graph of
F(x) = x2 when x=-2.
Constant Multiple Rule:
Theorem 2.4

xample 1
Y=
Hands on project
E
Day Light Research
project
Paul A Foerster Page
117
Larson Worksheet
2.1
Work sheet
Larson Resources 2.3
Kuta worksheets
IF-5 Apply the rules
of differentiation to
functions.
IF-3Demonstrate
knowledge of
differentiation using
algebraic functions.
a. Use the Chain
Rule and
applications to the
calculation of the
derivative of a
variety of composite
functions.
b. Find the
derivatives of
relations and use
implicit
differentiation in a
wide variety of
problems from
physics, chemistry,
economics,
Der.html
https://www.kuta
.com/limits
Thomas:
Pgs 125 #12-30

xample 2
Y= kπ2
Basic
Differentiation
Rules
http://www.cms.k1
2.nm.us/instruction
/secondary/math/
Math
Pgs 105-106 #5868 even, Kuta
Worksheet
(Difficult
problems)
Average rate change
E
project based on the
example #1 on page 129
of Jon Rogawski
E
Basic
Page 115 # 1- 15
odd
331,33,53,63
Kuta Worksheet
(easy problems)
Average
Page 115#10 -24
even,34- 44
even,62,66
Real life example
#81
E
Quiz
E
Larson 8e Test Bank
2.3Teachermade
Kuta Worksheet
(medium
problems)
Sum and Difference Rules:
Theorem 2.5
Constant rule,
Basic
Rule,product,qu
otient,chain and
implicit rule
1. Recognize the
constant Rule as
(C) = 0
2.Translate the
Power Rule
n
= nxn-1
3. Distinguish
the Product and
Quotient Rule
(fg)’ = f g’
+ f’ g
.

xample 1

xample 2
Find the derivative
of
Kuta Worksheet
(difficult
problems)
Rates of changes
Velocity and acceleration of
any moving objects
.

assessmets
E
Writing Journals
Page 128#109,110
Page 138 #97
E
Page 147#70
http://www.math.brown.ed
u/utra/derivrules.html
http://www.mathsisfun.co
m/calculus/derivativesrules.html
Page 154#11
E
xample 1
A rectangular water tank
(see figure below) is being
filled at the constant rate of
20 liters / second. The base
of the tank has dimensions w
= 1 meter and L = 2 meters.
What is the rate of change of
the height of water in the
tank?(express the answer in
cm / sec).

Chain Rule:
Explain
d/dx ( (f(x)^r))
= r * f(x)^(r-1)
* f '(x)
Advanced
Page 115-1173848
even,55,62,67,81
Real life example
(Thomas Pg 145
#77, 78)
Video
http://www.youtube.com/
watch?v=DOClHx25P4o
http://www.youtube.com/
watch?v=Yyag_L07iMg
https://www.khanacademy
.org/math/differentialcalculus
xample 2
An object is sent through the
air. Its height is modelled by
the function h(x)=5x^2+3x+65 where h(x) is
the height of the object in
meters and the x is the time
in seconds... estimate the
instantaneous rate of change
in the object's height at 3s
AREI.10.
Represent and
solve equations
and inequalities
graphically
Basic:
Pgs 117 # 83, 85,
90, 92
Average:
Pgs 117-118 # 8494 even, 102
Advanced:
Pgs 117-118 #9199 odd, 103, 104,
Real Life Example
#108
E
http://www.analyzemath.c
om/calculus/Problems/rat
e_change.html
Thomas
Page 133-134
selcted problems
Howard Anton
page 197
#37-49 0dd
3days
Product and Quotient Rule:
Product Rule
Theorem 2.7

xample 1
Y = (x³ + 5x² -6x + 9) •‎ (7x³ x² -8x + 1)

xample 2
Y=
product Rule
Understand the
concept of a
function and use
function
notation
FIF.2.
https://www.math.hmc.edu
/calculus/tutorials/prodrul
e/
Basic:
Pgs 126- 127 #117 odd, 38-49
odd, 73
Average:
Pgs #126-127 #
34-54 even, 6268, 77
Advanced:
Pgs #126-127
#48-62 even, 78,
83, 84
Real Life Problem
pg 127 #85, 87
Thomas:
Pgs 143 #29-49
odd
Angton:
video
https://www.khanacademy
.org/math/differentialcalculus/takingderivatives/product_rule/v
/quotient-rule
E
E
Pgs 203 #5-16 all
.
Quotient
Rule
2.3: Quotient Rule
http://tutorial.math.lamar.
edu/Classes/CalcI/quotient
rulef.aspx
Theorem2.8

xample 1
Differentiate
Quizzes
Larson 86 Test
E Bank 2.3
Teacher made Test
Test made with Kuta
Software
Mid chapter test
3days

xample 2
Find all points (x, y) on the
graph
Where tangent lines are
perpendicular to the line
Understand the
concept of a
function and use
function
notation
F.IF.2
E
8x+2y = 1.
3days
Section 2.4:Chain Rule
Chain Rule
Theorem 2.10
d/dx ( (f(x)^r))
= r * f(x)^(r-1)

* f '(x)
xample 1
Choose the method to
differentiate

.
xample 2
Find an equation of the line
tangent to the graph
of
x=
IF-5 Apply the rules
of differentiation to
functions.
a. Use the Chain
Rule and
applications to the
calculation of the
derivative of a
variety of composite
functions.
at
4days
Recognize
when to apply
the concept of
Guide lines for Implicit
Differentiation
1. We assume that the
equation we are given has
one independent variable,
usually x or t, and the
dependent variable,
usually y, i.e. yis a function
http://tutorial.math.lamar.
edu/Classes/CalcI/chain
rulef.aspx
Quizzes
Larson 86 Test Bank 2.4
E Kuta
Teacher made
work sheets
Average:
Pg 137 #22-28
even, 42-64 even,
74-88
Quiz- 2.4
Advanced
Pgs 137 #49- 73
odd, 81, 99, Real
Life Problem Pg
139 #101, 103,
106
E
Thomas
Pgs 167 #12-50
even
Anton:
Pgs 207 #30, 31,
32 (Word
Problem)
.
Implicit
Differentiation
Basic:
Pg 137 #8-28
even, 44, 46, 68,
74
Basic
IF-5 Apply the rules
of differentiation to
functions.
b. Find the
derivatives of
Page 146 #2-18
even,21
Kuta worksheet
easy
Average
Kuta worksheet
Work sheets
http://tutorial.math.lamar.
edu/Classes/CalcI/Implicit
DIff.aspx
http://calculator.tutorvista.
com/math/584/implicitdifferentiationcalculator.html
Home Work
Resource section 2.5
Teacher made Kuta
work sheets
f(g(x)) =
f'(g(x))g'(x)
of x or t.
2. The derivative we are
asked to determine
is dy/dx or dy/dt.
3. Since the derivative does
not automatically fall out
at the end, we usually have
extra steps where we need
to solve for it.
4. The chain rule is used
extensively and is a
required technique.
5. Implicit differentiation
expands your idea of
derivatives by requiring
you to take the derivative
of both sides of an
equation, not just one side.

xample 1
Assume that y is a
function of x . Find y'
= dy/dx for
relations and use
implicit
differentiation in a
wide variety of
problems from
physics, chemistry,
economics,
medium
Quiz – 2.5
Page 146,147 # http://17calculus.com/deri
vatives/implicit13-25
differentiation/
odd,33,41,57,67
Advanced
www.kutasoftware.com/impli
Page 146- 147 #
15-27 multiple
0f
3,42,46,48,56,68
Kuta worksheet
difficult
Thomas
Page 174 #
12,20,29,34,47
Anton
Page 207
#30,31,32
citdifferentiation
Videos
https://www.khanacademy
.org/math/differentialcalculus/takingderivatives/implicit_differe
ntiation/v/implicitderivative-of-y---cos-5x--3y#!
E
http://www.youtube.com/
watch?v=jv4gTxWqeBE
x 3 + y3 = 4 .

xample 2
Determine the slope and
concavity of the graph
of x2y + y4 = 4 + 2x at the
point (-1, 1) .
Differentiate the following
E
equations explicitly,
finding y as a function of x.
Solve for y´=dy/dx.
4days
Related Rates
Example 2
Water is being poured into a
conical reservoir at the rate of
pi cubic feet per second. The
reservoir has a radius of 6 feet
across the top and a height of
12 feet. At what rate is the
depth of the water increasing
when the depth is 6 feet?
Review for chapter Test
Chapter 2 test
2days
TIMSS/
SAT
Basic
Page 154 #
13,15,17
Kuta
worksheet
Easy
Example 1.
A ladder 10 feet long is
standing straight up against
the side of a house. The base
of the ladder is pulled away
from the side of the house at
the rate of 2 feet per
second. How high up the side
of the house will the top of
the ladder be 1 second after
the base begins being pulled
away from the house? How
high up the side of the house
will the top of the ladder be
after 2 seconds, 3 seconds, 4
seconds, 5 seconds?
Work sheets
http://www2.seminolest
ate.edu/lvosbury/Calcul
usI_Folder/RelatedRateP
roblems.htm
Average
Page 154-155 #
13,18,21,22
Kuta
worksheet
Medium
Advanced
Kuta
worksheet
Difficult
Teacher made Kuta
work sheets
Quiz – 2.6
http://tutorial.math.lam
ar.edu/problems/calci/r
elatedrates.aspx
www.kutasoftware.com/
relatedrates
Page 154 –
155#
18,19,22,27
A-REI.A.1
Explain each step
in solving a simple
equation as
following from the
equality of
numbers asserted
at the previous
step, starting from
the assumption
that the original
equation has a
solution.
Home Work
Resource section 2.6
Video
https://www.khanacade
my.org/math/differentia
lcalculus/derivative_appli
cations/rates_of_change/
v/rates-of-changebetween-radius-andarea-of-circle
Test Bank-Chapter
Test-versions A,and C
Prep
Content Category:

umbers Equations and
Functions

alculus

robability and Statistics
Geometry
Construct a viable
argument to
justify a solution
method.
N
IV.
Sample
problems from
TIMSS Example1:
http://www.ed
informatics.co
m/timss
Stu wants to wrap some
ribbon around a box as
shown and have 25 cm left
to tie a bow.
V.
ttp://www.edinfor
matics.com/timss/
pop3/mpop3.htm?
submit32=Gr.+12+
Adv.+Math+Test
SAT study guide
C
Practice Test#2
Section 2 and
P section
5(Math sections)
ttp://www.majorte
sts.com/sat/proble
m-solving-test02
4.2.B.1
VI.
How long a piece of ribbon
does he need?
A)
6 cm.
B)
2 cm
C)
Understand and
visualize
geometric
transformations
(translations,
rotations, and
reflections).
(4.2.B.1)
ww.khanacademy
.org/test-prep/sat
4
http://www.edinformati
cs.com/timss
5
6
5 cm.
D)
7
1 cm.
E)
7
7 cm.
SAT Sample problems
Example 1:
Clean soap powder is packed
in cube-shaped cartons. A
carton measures 10 cm on
each side. The company
decides to increase the length
of each edge of the carton by
10 per cent. How much does
the volume increase?
A)
0 cu.cm.
B)
1 cu. cm.
C)
00 cu. cm.
D)
31 cu. cm.
2.
A triangular prism consists of
rectangular and triangular
faces. Each rectangular face
has area r and each triangular
face has area t .What is the
total surface area of the
figure, in terms of r and t ?
A) 2r + t
Practice Test Page
452-457 and `page
463 - 468
College Board
SAT Study
Guide Page 305
- 313
Other Resource Texts
Gruber’s Complete
Preparation for the New
SAT
The Princeton Review’s
Cracking the New SAT,
2007
Barron’s SAT 2400: Aiming
for the Perfect Score
Maximum SAT
Kaplan SAT 2400, 2006
Edition
Up Your Score: The
Underground Guide to the
SAT, 2007-2008
1
2
1
3
B) 3r + 2t
C) 4r + 3t
D) 6rt
E) 32rt
INSTRUCTIONAL FOCUS OF UNIT
This unit focuses on the following: the tangent line problem largely applied to find the velocity and acceleration of any moving objects.;
using the formal definition of a derivative; finding derivatives using the power rule product rule, quotient rule, chain rule, product rule,
quotient rule, chain rule. Students will investigate the definition of the derivative and find rates of change using the limit of the difference
quotient. Students will interpret the derivative as the instantaneous rate of change or as the limit of the average rate of change. Students
will find related rates and use related rates to solve real-life problems.
Tangent LineApplication :
1. If we are traveling in a car around a corner and we hit something slippery on the road (like oil, ice, water or loose gravel) and our car starts to
skid, it will continue in a direction tangent to the curve.
Academic Vocabularies by Dr. Robert Marzano
The constant rule: The power rule
The constant multiple rule:
Product Rule:
Quotient Rule
Higher order derivatives
Second derivative
Third derivative
The Chain Rule
The General Power Rule:
The average rate of change
The instantaneous rate of change
The position function
Velocity
Average velocity
Instantaneous velocity,
An implicit function
Related rates Speed, velocity, acceleration
, instantaneous, limit, derivative,
differentiable, concavity
Marzano’s Six Steps of Teaching Academic Vocabularies
7.
8.
9.
10.
11.
explanation or example. (Story, sketch, power point)
explain meaning in their own words. (Journal, community circle, turn to your neighbor)
graphic or symbol for each word.
expand their word knowledge. (Add to their notes, use graphic organizer format)
YOU provide a description,
Ask students to restate or reAsk students to construct a picture,
Engage students in activities to
Ask students to discuss vocabulary
words with one another (Collaborate)
12.
words. (Bingo with definitions, Pictionary, Charades, etc.)
Example 1-Step 3
Have students play games with the
Calvin and Phoebe elope in a hot air balloon, which rises at a constant rate of 3 meters per second. Five seconds after they cast off, Phoebe's jilted suitor
Bonzo McTavish races up in his Porsche. He parks 50 meters from the launch pad, and runs toward the pad at 2 meters per second. At what rate is the
distance between Bonzo and the balloon changing when the balloon is 30 meters above the ground?
Example 2. Step5
Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ?
.
PARCC /SAT FRAMEWORK/ASSESSMENT

3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

solving-test01


http://www.edinformatics.com/timss/pop
http://www.majortests.com/sat/problem-
www.khanacademy.org/test-prep/sat
21ST CENTURY SKILLS
(4Cs & CTE Standards)
21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill
the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
Find the velocity and acceleration of a particle with the given position of s(t) = t3 - 2t2 - 4t + 5 at t = 2 where t is measured in seconds and s is measured in feet.
Velocity is found by taking the derivative of the position.
At 2 seconds, the velocity is 0 feet per second.
The acceleration is found by taking the derivative of the velocity function, or the second derivative of the position.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and
issues in the real world.
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
Aircraft Applications:
Suppose one airplane flies over BWI Airport at a given rate, flying east. Twelve minutes later a second plane flies over BWI at its rate, flying
north. A typical related rate application would calculate the rate at which they were separating at a later point in time. This is essentially
the problem illustrated with boats by the applet in this section.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics
impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.

MODIFICATIONS/ACCOMMODATIONS

Students can create flashcards of the
derivative graphs of the function graphs provided by the teacher. Students can then get into groups where
they can match function graphs to their derivative graphs.
APPENDIX
(Teacher resource extensions)
4.
Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
E-Text, Interactive Digital
Mathematical Practices
9.
persevere in solving them.
10.
Make sense of problems and
Reason abstractly and
quantitatively.
11.
critique the reasoning of others.
Construct viable arguments and
12.
Model with mathematics.
13.
Use appropriate tools strategically.
14.
Attend to precision.
15.
Look for and make use of
structure.
16.
in repeated reasoning.
Look for and express regularity
Notes to teacher (not to be included in your final draft):
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
UNIT 8
Application of Differentiation
.
Total Number of Days: 25 days Grade/Course: 12/Calculus
ESENTIAL QUESTIONS






What do the derivatives of a function tell us about that
function?
Why the Mean Value Theorem is considered such an
important theorem in calculus.
What is the physical interpretation of the MVT?
How do we use the first and second derivative?
What is an optimization problem and how do we
solve it?
How Can we solve real world problems using the
calculus?
PACI
NG
CONTENT
SKILLS





ENDURING UNDERSTANDINGS
Solve extrema on an interval problem applied in Highway design.
Solve problems that apply the Rolle’s Theorem and mean value theorem.
Use the first and second derivatives to identify when a function is increasing,
decreasing, concave up and/or concave down applied in Beam Deflection,Sales
growth etc. .
Combine together calculus skills with skills from previous math courses to
accurately graph a wide variety of functions.
Given a car’s velocity at two different check points, how can the police officer apply
the MVT
to determine if the car was exceeding the speed limit over the time
interval?
STANDARDS
(CCCS/MP)
RESOURCES
Larson
Pearson
OTHER
(e.g., tech)
Extrema on an
Interval
Determine the relative
maximum and minimum
using “Extreme Value
Theorem”
Examples:
1. Find the value of the
relative maximum and
minimum for
F(x) =
IF-3D
Demonstrate
knowledge of
differentiation using
algebraic functions.
a. Use differentiation
and algebraic
manipulations to
sketch, by hand,
graphs of functions.
LEARNING
ACTIVITIES/ASSESS
MENTS
Using graphing calculators,
DI according to student
readiness
Use data and situations relevant
to student interests
Whole group and small group
cooperative learning
Trig function derivative
graphic organizer
Unit test (multiple choice
2. Prove graphically that
f(x)=cos(x) has relative
maxima at all even
multiples of pi and
relative minima at all
odd multiples of pi
Definition of an
extrema of a
function on an
interval
2 days
Lesson for Goal #2:
To determine the critical
numbers on a closed interval
Examples:
1. Find the extrema of
f(x)= x2+2x-4
at interval [-1,1]
2. Find the extrema of
f(x)= x2-2-cos(x)
at interval [-1,3]
b. Identify maxima,
minima, inflection
points, and intervals
where the function is
increasing and
decreasing.
c. Use differentiation
and algebraic
manipulations to
solve optimization
(maximum-minimum
problems) in a
variety of pure and
applied contexts.
CMS IF-4f Use
formulas to find
derivatives of inverse
trigonometric
functions,
exponential functions
and logarithmic
functions.
IF-5 Apply the rules
of differentiation to
functions.
c. Demonstrate an
understanding of and
apply Rolle’s
Theorem, the Mean
Value Theorem.
& free response
questions)
Summative Assessment:
Basic:
Page 169#4,8,13 and
15
Average:
Page 169 # 5,6,8,11,17
and 18
Advanced:
Page 169 #8,10,12, 18,
20
Kuta Worksheet Easy,
Medium, or Difficult
To locate the absolute
extrema using critical
numbers
Examples:
1. Determine the




Quizzes
Tests
Quarterly
Assessments
Projects
Formative Assessments:
http://www.spar
knotes.com/math
/calcab/applicati
onsofthederivativ
e/section1.rhtml



Demonstration
Class discussion
Homework
Home Work Assignments
Larson Resource
worksheet 3.1
Teacher made
Kuta worksheet for the
section
Quizzes
Larson 8e Test Bank 3.1
Lesson for Goal #3:
The definition of
relative extrema
of a function on
an open interval
https://www.kutas
oftware.com/extre
ma
Thomas:
Page 228 #6-18 even
Anton:
Pgs 277 #4-16 even
absolute extrema for
the following
function and interval:
g(t)=2t3+ 3t2-12t+4
4 days
Kuta Worksheet Easy,
Medium, or Difficult
Extrema on a
closed interval
2. Determine the maximum
and minimum of the
function
on the interval
.
First start by finding the roots
of the function derivative:
Basic:
Page 169#19-31 odd,
40
Average:
Page 169 #25-39 odd,
42
Advanced:
Page 169 #14, 26, 34,
35, 39, 47, 48
Real Life Problems:
Pg 171 # 59. 65, 69
Thomas:
Pgs 229 #30-40 even
Now evaluate the function at
all critical points and
endpoints to find the extreme
values.
Kuta Worksheet Easy,
Medium, or Difficult
https://www.kha
nacademy.org/m
ath/differentialcalculus/derivati
ve_applications/c
ritical_points_gra
phing/v/testingcritical-pointsfor-local-extrema
Rolle’s Theorem
and the Mean
Value Theorem
To determine if Rolle’s
Theorem applies to a
function using a created
graph
3 days
Examples:
1. Determine whether
Rolle’s Theorem’s
hypotheses are satisfied
&, if so, find a number c
for which f’(c) = 0.
2. f(x) = x³ - 2x² - x + 2
on [-1,2]
Basic:
Page 176 #4-18 odd
Average:
Page 176 # 6-30 odd
Advanced:
Page 176 #10-32 even
Thomas:
Page 236 #3-30
(multiples of 3)
Kuta Worksheet Easy,
Medium, or Difficult
https://chaffeem
ath.wikispaces.co
m/file/view/Outl
ine+3.2+The+Mea
n+Value+Theore
m.pdf
http://www.cliffs
notes.com/math/
calculus/calculus
/applications-ofthederivative/meanvalue-theorem
3. f(x) = sin x on [0,π]
Instantaneous
rate of change
2 days
Mean Value Theorem
The Mean Value Theorem:
If f) is continuous on and
differentiable on (a,b ), then
there is at least one point in
(a,b ) at which:
Example:
1. Verify the conclusion of
the Mean Value Theorem
Basic:
Pg 177 #37, 40, 42, 44,
52
Average:
Pgs 177 #40-50 even,
55
Advanced:
Pgs 177 #46-50 even,
58, 59
Anton:
Pg 324 #6-14 even
Video
https://www.you
tube.com/watch?
v=fI6w2kL295Y
https://www.kha
nacademy.org/m
ath/differentialcalculus/takingderivatives/secan
t-line-slope-
Home Work Assignments
Larson Resource
worksheet 3.2
Teacher made
Kuta worksheet for the
section
Resource Quiz A,B,or D
version
for f(x)= x 2−3 x−2 on
[−2,3].
Real Life Problem:
Pgs 237 (Thomas)
#51, 56
Kuta Worksheet Easy,
Medium, or Difficult
2.
A balloon is inflated by
an electric pump.
Determine the rate of
change of
volume with respect to
radius when the radius
measures exactly 6 cm.
tangent/v/appro
ximatinginstantaneousrate-of-changeword-problem-1
Problems
http://tutorial.m
ath.lamar.edu/Cl
asses/CalcI/Tang
ents_Rates.aspx
3. A football is punted into
the air. Model the
football’s height using
the polynomial function
f (t) = -4.9t2+ 16t + 1, where
f (t) represents the height in
meters at t seconds.
Determine the instantaneous
rate of change of height at 1
s, 2 s, and 3 s.
2 days
Increasing and
Decreasing
Functions and
the First
Derivative Test
IF- 3Demonstrate
Home Work Assignments
Larson Resource
worksheet 3.3
Teacher made
Kuta worksheet for the
Definiton of Inc/Dec function
and Theorem3.5
Examples:
1. Calculate the
Intervals of Increase
and Decrease for the
Following Function
2. For f(x) = sin x +
cos x on [0,2π],
determine all
intervals where f is
increasing or
decreasing.
The First
Derivative Test
4 days
Theorem 3.6 explaining the
First derivative Test. Explain
the relative minimum and
maximum
Examples:
knowledge of
differentiation using
algebraic functions.
a. Use differentiation
and algebraic
manipulations to
sketch, by hand,
graphs of functions.
b. Identify maxima,
minima, inflection
points, and intervals
where the function is
increasing and
decreasing.
c. Use differentiation
and algebraic
manipulations to
solve optimization
(maximum-minimum
problems) in a
variety of pure and
applied contexts.
IF-4f Use formulas to
find derivatives of
inverse
trigonometric
functions,
exponential functions
and logarithmic
functions.
IF-5 Apply the rules
of differentiation to
functions.
c. Demonstrate an
understanding of and
apply Rolle’s
Theorem, the Mean
Value Theorem.
Basic:
Pgs 186- 187 #4-20
even
Average:
Pgs 186- 187 10-32
even
Advanced:
Pgs 186- 187 #14-40
even
section
Resource Quiz A,B,or D
version
https://www.kutas
oftware.com/incre
asing/decreasing
Anton:
Pgs 276 #9-21 odd
Thomas:
Pgs 228 #24, 28, 31
Kuta Worksheet Easy,
Medium, or Difficult
Video
https://www.you
tube.com/watch?
v=vQYQxpHLfoE
Basic:
Pg 186 #4-20 even,
43, 45, 80
Average:
Pg 186 #24-46 even,
Writing Journal
Page 177#56
Page 195#48
1.
Find the local
maximum and minimum
values of the function using
the first derivative test
f(x) = x4 – 2x2 + 3
82, 83, 86
Advanced:
Pg 186 #31-51 odd,
60, 74, 80, 83
Kuta Worksheet (Easy,
Medium, Difficult)
21. Determine intervals on
which a function is concave
upward or concave
downward
Thomas:
Pg 242 #39, 50, 62, 64
Anton:
Pg 287 #4, 11, 35, 60
.Find the relative extrema for
the
function
.
Using the First Derivative
Test
Find f '(x).
Work sheets
http://archives.m
ath.utk.edu/visua
l.calculus/3/grap
hing.5/index.html
https://www.kutas
oftware.com/First
video
http://www.online
mathlearning.com/
derivativetest.html
Find all critical numbers of f.
critical numbers are –1, 0
and 1.
Step 3: Determine intervals.
Concavity
The intervals are(–∞, –1), (–
1, 0), (0, 1), and (1,∞).
4 days
Theorem 3.7
IF- 3Demonstrate
knowledge of
differentiation using
algebraic functions.
a. Use differentiation
http://www.educ
ation.com/studyhelp/article/deri
vative-testderivative-testrelative/
Steps for testing concavity
The First derivative of a
function gives the slope.
When the slope
continually increases, the
function is concave upward.
When the slope
continually decreases, the
function is concave
downward.
With the second derivative.
When the second derivative
is positive, the function
is concave upward.
When the second derivative
is negative, the function
is concave downward.
Example:1.
Find the concavity of
f(x) = 5x3 + 2x2 − 3x
and algebraic
manipulations to
sketch, by hand,
graphs of functions.
b. Identify maxima,
minima, inflection
points, and intervals
where the function is
increasing and
decreasing.
Basic:
Pg 195 #1-15 odd, 27,
58
Average:
Pg 195 #12-34 even,
57, 64
Advanced:
Pg 196 #24-36 even,
63, 64, 68
Real Life Problem:
Pg 196 #68, 69
Kuta Worksheet (Easy,
Medium, Difficult)
Video
https://www.kha
nacademy.org/m
ath/differentialcalculus/derivati
ve_applications/c
oncavityinflectionpoints/v/concavit
y--concaveupwards-andconcavedownwardsintervals
Home Work Assignments
Larson Resource
worksheet 3.4
Teacher made
Kuta worksheet for the
section
Resource Quiz A,B,or D
version
http://www.yout
ube.com/watch?v
=dfyRWMEFSEU
Thomas:
Pg 242 #57, 63, 75
Finney:
Pg 215 #10-28 odd
The derivative is f'(x) =
15x2 + 4x − 3
The second derivative
is f''(x) = 30x + 4
And 30x + 4 is negative up to
x = −4/30 = −2/15, and
positive from there onwards.
So:
f(x) is concave
Notes
http://www.sosm
ath.com/calculus
/diff/der15/der1
5.html
Hands on activity/Project
http://www.math
sisfun.com/calcul
us/concave-updownconvex.html
Graphical,Numerical,Analy
tical Analysis of maximum
Volume of a box
Page 223 of Larson Text
Book
Points of
inflection
downward up to x = −2/15
f(x) is concave upward from
x = −2/15 on
Determine where the
function is concave up and
concave down:
Group Activity
Paper Folding Page 230 of
Finney #48
GO FISH
FOR
DERIVATIVES
http://www.fcps.edu/Lak
eBraddockSS/high_school
/pdfs/hands_document.pd
f
Theorem 3.8
Inflection points are where
the function changes
concavity
Example 2.
2days
The Second
Derivative Test
Determine the points of
inflection for
.y = x3− 3x2 + 3x − 1.
Since there is a change of
concavity at x = 1, and so
there is a point of
inflection at x = 1.
Second derivative test for
extrema
Theorem 3.9
https://www.kutas
oftware.com/point
of inflection
Teacher Made Kuta
Worksheet (Easy,
Medium, Difficult)
Example 1: Find any local
extrema of f(x) = x 4 −
8 x 2 using the Second
Derivative Test.
f′(x) = 0 at x = −2, 0, and 2.
Because f″(x) = 12 x 2 −16,
you find that f″(−2) = 32 > 0,
and f has a local minimum at
(−2,−16); f″(2) = 32 > 0,
and f has local maximum at
(0,0); and f″(2) = 32 > 0,
and f has a local minimum
(2,−16).
Example 2: Find any local
extrema of f(x) = sin x +
cos x on [0,2π] using the
Second Derivative Test.
f′(x) = 0 at x = π/4 and 5π/4.
Because f″(x) = −sin x −cos x,
you find
that
as a local maximum
at
Basic:
Pg 195 #27-39 odd
Average:
Pg 195 #28-32 even
https://www.kha
nacademy.org/m
ath/differentialcalculus/derivati
ve_applications/c
oncavityinflectionpoints/v/inflectio
n-points
Advanced:
Pg 195 #36-48 even
Problems
.
Also,
local minimum
at
and f h
video
. and f has a
.
http://www.cliffs
notes.com/math/
calculus/calculus
/applications-ofthederivative/secon
d-derivative-testfor-local-extrema
video
http://www.yout
ube.com/watch?v
=QtXCIxB6kW8
TIMSS Example:
1.
2 days
Chapter Review
Chapter Test
4) The value of
Page 242 of Larson
Resource Test b and
D forms
A) 0
B)
Chapter review
End of unit test
C)
D)
TIMSS/SAT
2. ) Which of the following
graphs has these
features:? f'(0) >0, f'(1) <0
and f''(x) is always negative.
Sample problems
from
http://www.edinfor
matics.com/timss
VII.
SAT practice problem
How many positive integers
less than or equal to 100 are
multiples of 3 or multiples
of 5 or multiples of
both 3 and 5?
A) 41 B) 47
C)50 D)53
SAT practice-Student
produced response
question
Page 359-361
Recap concepts Page
358
VIII.
http://ww
w.edinfor
matics.co
m/timss/p
op3/mpop
3.htm?sub
mit32=Gr.
+12+Adv.+
Math+Test
http://ww
w.majorte
sts.com/sa
t/problem
-solvingtest02
www.collegeboard.
com
SAT Practice test
3(page 513) section
2,5 and 8
I.
http://www.majo
rtests.com/sat/pr
oblem-solvingtest03
E)59
• www.takesat.com
SAT
2. There are n students in a
biology class, and only 6 of
them are seniors. If 7 juniors
are added to the class, how
many students in the class
will not be seniors?
IX.
A)n-2
B)n-3
www.kha
nacadem
y.org/test
-prep/sat
C)n-1
D)n+2
E)n+1
.5.E.
Translate words
into a
mathematical
expression or
equation.
Other
Resource
Texts
Gruber’s
Complete
Preparation for the
New SAT
The Princeton
Review’s
Cracking the New
SAT, 2007
Barron’s SAT
2400: Aiming for
the Perfect Score
Maximum SAT
Kaplan SAT 2400,
2006 Edition
Up Your Score:
The Underground
Guide to the SAT,
2007-2008
INSTRUCTIONAL FOCUS OF UNIT
This unit focuses on the following topics related rates, extrema on an interval, intermediate value related rates, extrema on an interval,
intermediate value summarizing all skills from calculus and previous math courses in curve sketching, and optimization problems.The
increasing ,decreasing functional concept can be applied to any data that can be represented as a graph. Concavity is also well explained
on graphs with bends or smooth turns
Academic Vocabularies by Dr. Marzano












Maximum
minimum
point of inflection,
tangent line
optimization
Critical numbers
Absolute Extrema
Relative Extrema
Critical Point
Increasing/decreasing functions
Anti-derivative
Point of Inflection
Marzano’s Six Steps for Teaching Academic Vocabularies
13. YOU provide a description, explanation or example. (Story, sketch, power point)
14. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor)
15. Ask students to construct a picture, graphic or symbol for each word.
16. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format)
17. Ask students to discuss vocabulary words with one another (Collaborate)
18. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Example 1. Step 1
The graph of f ', the derivative of a function f, is shown in Figure 7.2-19. Find the points of inflection of f and determine where the
function f is concave upward and where it is concave downward on [–3, 5].
Example 2—Step 3
Use the First derivative Test to determine the extrema from the table
PARCC FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01


www.khanacademy.org/test-prep/sat
21ST CENTURY SKILLS
(4Cs & CTE Standards)
st
21 Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill
the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
Using data from the UN World Population Prospects, graph the population data for three different countries from the year 1950-2010. Then,
describe how the population has changed over the past 60 years and will change for the next 90 using the Calculus concepts of
increasing/decreasing, concave up/concave down, relative extrema and inflection points. If inclined, try to connect some of the trends that you
see to history.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and
issues in the real world.
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics
impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.
Highway Design
In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6%. The top of the
filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points and The horizontal distance between the
points and is 1000 feet.
(a) Find a quadratic function Y = a x2 + bx + c, - 500≤ x ≤ 500,that describes the top of the filled region.
(b) Construct a table giving the depths d of the fill to X = -500,-400,-300,-200, -100, 0, 100, 200, 300,
400, and 500.
(c) What will be the lowest point on the completed highway?
Will it be directly over the point where the two hillsides
come together?

MODIFICATIONS/ACCOMMODATIONS


Students can create flashcards of the derivative graphs of the function graphs provided by the teacher. Students can then get into groups where
they can match function graphs to their derivative graphs.
APPENDIX
(Teacher resource extensions)
5. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Mathematical Practices
17. Make sense of problems and persevere in solving them.
18. Reason abstractly and quantitatively.
19. Construct viable arguments and critique the reasoning of others.
20. Model with mathematics.
21. Use appropriate tools strategically.
22. Attend to precision.
23. Look for and make use of structure.
24. Look for and express regularity in repeated reasoning.
6.
Notes to teacher (not to be included in your final draft):
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
UNIT 9
Integration
Total Number of Days: 26 days Grade/Course: 12/Calculus
ESENTIAL QUESTIONS
ENDURING UNDERSTANDINGS













How do you use indefinite integral notation for
antiderivatives?
How do you find a particular solution of a differential
equation?
How do you use sigma notation to write and evaluate
a sum?
How do you find the area of a plane region using
limits?
How do you use Riemann Sums to approximate area?
How do you evaluate a definite integral using limits?
How do you evaluate a definite integral using
properties of definite integrals?
How do you evaluate a definite integral using the
Fundamental Theorem of Calculus?
How do you use the Mean Value Theorem for
Integrals?
How do you find the average value of a function over a
closed interval?
How do you use the Second Fundamental Theorem of
Calculus?
How are integrals used to measure changing
quantities?
Integration is a process that is closely related to differentiation, which
will be explored by learning new methods and rules for solving definite
and indefinite integrals, including the Fundamental Theorem of
Calculus, and then applying these rules.
Integration is applied to calculate the Escape velocity, Tree and population
growth etc. Mean value Theorem is applied in medicine for treatment of
tumors for approximating the area where as average value is applied in
Blood flow and Respiratory cycle etc.
PACING
CONTENT
SKILLS
STANDARDS
(CCCS/MP)
RESOURCES
Larson
Pearson
OTHER
(e.g., tech)
Anti-derivatives
and indefinite
integration
Example 1.
1. Determine the
general solution to
the differential
equation using the
notation of AntDerivative
3 days
Write the general
solution of a
differential
equation using the
variable and
constant of
Integration
=
IF-6 Students
will apply the
rules of
integration to
functions.
a. Apply the
definition of
the integral to
model
problems in
physics,
economics,
etc., obtaining
results in
terms of
integrals.
Basic:
Pg 255 #1-13
odd, 21, 28, 30
Kuta easy
worksheet
Average:
Pg 255 #12-28
even, 36, 38
Kuta
mediumworkshee
t
Advanced:
Pg 255 #26-42
even, 68, 70
Kuta
difficultworkshee
t
Thomas:
Pg 285-286 #32,
42, 50, 66, 70
Anton:
Pg 363 #21, 29,
36, 39
Work sheet
www.kuta/different
ialequation
LEARNING
ACTIVITIES/ASSE
SSMENTS
Using graphing
calculators,
DI according to
student readiness
Use data and
situations relevant to
student interests
Whole group and
small group
cooperative learning
Trig function
derivative graphic
organizer
Unit test (multiple
choice
& free response
questions)
Summative
Assessment:


Quizzes
Tests


3 days
Indefinite integral
notation for anti derivative
Problems
2. The marginal cost
function for
producing x units
is
C = 23+16x - 3x
and the total cost
for producing 1
unit is
Rs.40. Explain the
method to
determine the
total cost function
and the average
cost function.
http://www.textboo
ksonline.tn.nic.in/b
ooks/12/std12-bmem-2.pdf
http://tutorial.math
.lamar.edu/Classes/
DE/UndeterminedC
oefficients.aspx
A-CED.
For Goals 2 and
3:
Notes
http://www.sosmat
h.com/tables/diffeq
/diffeq.html
A-REI
F--‐BF.5.
The antiderivative of a
function ƒ(x) if for all x in
the domain of ƒ,
 F'(x) = ƒ(x)
 ∫ƒ(x) dx = F(x) + C,
where C is a
constant.
Refer to page 250 for
basic integration rules
Basic:
Pg 256-257 #50,
59, 67
Kuta easy
worksheet
video
https://www.khana
cademy.org/math/d
ifferentialequations
Average:
Pg 256-257 #60,
63, 70
Kuta medium
worksheet
Advanced:
Pg 256-257 #62,
70, 82, 84
Kuta difficult
worksheet
Quarterly
Assessments
Projects
Formative
Assessments:
 Demonstration
 Class discussion
 Homework
Journals:
4.1
Page 257 #65
4.2
Page 269#73
4.3
Page280#53
4.4
Page292#54
4.5
Page 306#109
4.6
Page 314#22
Hands on activities
/projects
https://www.youtu
http://people.wallawa
lla.edu/~ken.wiggins/
examplesactivities/co
ntents.html
Examples:
Integrate
be.com/watch?v=lx
x_i0zCxrU
1.
Thomas:
Pg 288 #119, 89,
90, 92
2.
3days
Basic integration
rules to find antiderivatives of
indefinite integrals
Identify the Initial
condition
Problems
http://www.mathmagic.com/calculus
/integral.htm
Anton:
Pg 364 #41, 42,
55
Initial condition and
particular solutions
 y=F(x) is called an
initial condition
Quiz Test Bank 4.1
Teacher made
assessment using
Kuta software
Examples:
1. A ball is thrown
upward with an
initial velocity of 64
ft/sec from an initial
height of 80 feet.
Worksheet
www.kuta/basicint
egration
A. Illustrate a method to
find the position function,
giving the height, s, as a
function of the time, t
B. When does the ball hit
the ground.
2. F’’(x)=sin(x); F’(0)=1;
F(0)=6; Solve for the
differential equation.
Home Work
Larson Resource
Book
4.1-first two pages
IF-6 Students
will apply the
rules of
integration to
functions.
a. Apply the
definition of
the integral to
model
problems in
physics,
economics,
etc., obtaining
results in
terms of
integrals.
Home Work
Larson Resource
Book
4.1 pages 3 and 4
Practice problems
from Paul’s online
notes
Area
Apply sigma
notation to write
and evaluate a sum
3days
Sigma Notation
Theorem 4.2- Summation
formula
The starting point for the
summation or the lower
limit of the summation is 1
The stopping point for the
summation or the upper
limit of summation 5
Example 1:
For Goals 1-3:
You tube
Basic:
Pg 267-269 #1-11
odd, 21, 26, 27,
31, 47, 49, 53, 57
Kuta easy
worksheet
Average:
Pg 267-269 #1222 even, 26, 29,
32, 50, 54, 63
Kuta
mediumworkshee
t
https://www.khana
cademy.org/math/p
recalculus/seq_indu
ction/geometricsequenceseries/v/sigmanotation-sum
Advanced:
Pg 267-269 #20,
26, 29, 30, 51-56,
61, 66, 75
Kuta difficult
Notes
http://tutorial.math
.lamar.edu/Classes/
CalcI/SummationNo
tation.aspx
Home Work
Larson Resource
Book
4.2
Resource Quiz 4.2
Teacher made
assessment using
Kuta software
Evaluate
worksheet
Example 2.
Evaluate
Approximate the
area of a plane
region
Example 1.
Compare the upper and
lower sums to approximate
the area of the region
defined by the function
Y=
on the interval [0.2]. Use
four subintervals with equal
,2 width. (Hint: Draw a
picture)
There are four subintervals
n = 4 so the width of each
subinterval is equal to 2
=
Example 2.
Estimate the area
between
and the x-axis
using n=5 subintervals
IF-6 Students
will apply the
rules of
integration to
functions
On the
calculator,
interpret a
summation as
follows:
You Tube
http://www.youtub
e.com/watch?v=VT
WRy1VQbPI
Extra Practice
http://tutorial.math
.lamar.edu/classes/
calcI/areaproblem.
aspx
4 days
Area of a plane
region using limits
When finding the area
under a curve for a region,
it is often easiest
to approximate
area using a summation
series. This approximation
is a summation of areas of
rectangles. The rectangles
can be either left-handed
or right-handed and,
depending on
the concavity, will either
overestimate or
underestimate the true
area.
Example 1:
Use the limit definition of
Area to estimate the area of
f(x) = x3 between x=0 and
x=1
Example 2
Apply the limit process to
find the area of the region
between the graph of the
http://college.cenga
ge.com/mathematic
s/larson/calculus_a
nalytic/7e/students
/downloads/summ
aries/ex4_2_4.pdf
http://sites.csn.edu
/gcohen/181/21_ar
ea_riemann.pdf
function and the x-axis over
the given interval. y=16-x2
[-4,4]
3days
Riemann Sum and
Definite Integral
Basic:
Pg 278-279 #3-11
odd, 18, 23, 29,41
Kuta easy
worksheet
Note that the Riemann sum
when each xi is the righthand endpoint of the
subinterval [ai-1, ai] is
when each xi is the lefthand endpoint of the
subinterval [ai-1, ai] is
and when each xi is the lefthand midpoint of the
subinterval [ai-1, ai] is
Definition of
Riemann Sum
Example:
1. WORD PROBLEM
A rectangular canal,
5m wide and 100m
long has an uneven
bottom. Depth
measurements are
taken
Average:
Pg 278-279 #1329 odd, 35, 47
IF-6 Students
will apply the
rules of
integration to
functions.
a. Apply the
definition of the
integral to
model problems
in physics,
economics, etc.,
obtaining
results in terms
of integrals.
b. Demonstrate
knowledge of
the
Fundamental
Theorem of
Calculus, and
use it to
interpret
integrals as
anti-derivatives.
c. Use definite
Kuta
mediumworkshee
t
Advanced:
Pg 278-279 #2432 even, 36, 40,
47
Kuta
difficultworkshee
t
https://www.khana
cademy.org/math/i
ntegralcalculus/indefinitedefiniteintegrals/definite_i
ntegrals/v/riemann
-sums-and-integrals
http://archives.mat
h.utk.edu/visual.cal
culus/4/riemann_s
ums.4/
Home work
Resource work
sheet 4.3
Quiz 4.3 from
Larson Test Bank
Teacher made
Assessments using
Kuta software
MID CHAPTER
ASSESSMENT
http://www.lakelan
dschools.org/webpa
ges/dcox/files/Rie
mann%20sum%20
practice.pdf
Hands on activities
/projects
http://people.wallawa
lla.edu/~ken.wiggins/
examplesactivities/co
ntents.html
at every 20m along
the length of the
canal. Use these
depth
measurements to
construct a Riemann
sum using right
endpoints to
estimate the volume
of water
Dista Depth in the
nce
(m)
canal.
(m)
See table
0
2.0
to the
20
1.6
left.
40
1.8
60
2.1
80
2.1
100
1.9
2. Approximate the
area under the
curve
with a Riemann sum, using
six sub-intervals and right
endpoints.
3: Evaluate the Riemann
sum for f( x) = x 2 on [1,3]
using the four subintervals
of equal length, where x i is
the right endpoint in the ith
integrals in
problems
involving area,
velocity,
acceleration,
and the volume
of a solid.
Section project
Demonstrating the
Fundamental
Theorem
Page 294
subinterval (see Figure ) .
Examples
http://www.sosmat
h.com/calculus/inte
g/integ02/integ02.
html
http://tutorial.math
.lamar.edu/Classes/
CalcI/DefnofDefinit
eIntegral.aspx
Limits and
properties of
definite
integrals
Theorems 4.5, 4.6, 4.7, 4.8
explaining the properties of
integral limits
Example: 1
Evaluate the following
definite integral
Example 2.
S IF-6
Students will
apply the
rules of
integration
to functions.
a. Apply the
definition of
the integral
to model
problems in
physics,
economics,
etc.,
obtaining
results in
terms of
integrals.
b.
Basic:
Pg 280-281 #49,
55, 58
Kuta easy
worksheet
Average:
Pg 280-281 #56,
58, 62
Kuta
mediumworkshee
t
Problems
(Note: x must be
in radians )
4days
The Fundamental
Theorem of
Calculus Theorem 4.9
Let f (x) be continuous on
[a, b]. If F(x) is any anti
derivative of f (x), then,
Demonstrate
knowledge of
the
Fundamental
Theorem of
Calculus, and
use it to
interpret
integrals as
antiderivatives.
c. Use
definite
integrals in
problems
involving
area,
velocity,
acceleration,
and the
volume of a
solid.
d. Compute,
by hand, the
integrals of a
wide variety
of functions
using
substitution.
e. Apply the
rules of
integration
to functions.
Use definite
integrals in
problems
involving
Advanced:
Pg 280-281 #57,
62, 70
Kuta
difficultworkshee
t
Thomas:
Pgs 322 #10-40
multiples of 5
http://www.mathsi
sfun.com/calculus/i
ntegrationdefinite.html
You tube
http://www.youtub
e.com/watch?v=wyc
adSRDID4
Anton:
Pgs 394 #12-22
even
https://www.khana
cademy.org/math/i
ntegralcalculus/indefinitedefiniteintegrals/fundamen
tal-theorem-ofcalculus/v/fundame
ntal-theorem-ofcalculus
Basic:
Pg 291#6-18, 29
Average:
Pg 291 #11-27
odd, 35
Home work
http://apcentral.col
legeboard.com/apc
Larson Resource
work sheet 4.4
)dx = F(b) - F(a)
Apply FTC to evaluate
Fundamental
Theorem of
Calculus
(FTC)
Example 1.
Example 2.
area between
two f. Apply
the rules of
integration
to functions.
Use
advanced
techniques to
evaluate
integrals,
including
integration
by parts,
trigonometri
c integrals,
trigonometri
c substitution
Advanced:
Pg 291 #14-38
even
/public/repository/
AP_CurricModCalcul
usFundTheorem.pd
f
Thomas:
Pg 322 #29-41
odd, 50
Anton:
Pg 363 #5-13 odd
Kuta Software
Worksheets
crated by
Teacher
You tube
https://www.khana
cademy.org/math/d
ifferentialcalculus/derivative_
applications/mean_
value_theorem/v/m
ean-value-theorem
Mean Value Theorem(MVT)
Theorem 4.10
The Mean Value
Theorem is one of the most
important theoretical tools
in Calculus. It states that
Problems
http://tutorial.math
.lamar.edu/Classes/
CalcI/MeanValueTh
eorem.aspx
Test Bank Quiz 4.4
Teacher made
assessment using
Kuta soft ware
if f(x) is defined and
continuous on the interval
[a,b] and differentiable on
(a,b), then there is at least
one number c in the interval
(a,b) (that is a < c < b) such
that
Basic:
Pg 291#40, 44,
46, 48
Average:
Pg 291 #40-48
even
2days
Mean Value
Theorem
Example 1.
Advanced:
Pg 291 #43-51
odd, 62
Find a value of c such that
the conclusion of the mean
value theorem is satisfied
for
Real Life
Problem:
Pg 292 #62, 64
f(x) = -2x 3 + 6x – 2
Example2.
Determine a and b for
the function:
If it satisfies the hypothesis
of mean value theorem on
the interval [2, 6].

Definition for
Average Value:
Let f be a function
which is continuous
on the closed
interval [a, b].
The average
value of f from x = a
Thomas:
Pg 333 #10-30
even
Anton:
Pg 363 #31, 36,
39
http://www.vitutor.
com/calculus/lhopi
tal/mean_problems.
html
http://archives.mat
h.utk.edu/visual.cal
culus/5/average.1/
Kuta Software
Worksheets
crated by
Teacher
You tube
https://www.khana
cademy.org/averag
evalue
to x = b is the
integral
Example 1.
Assume that in a certain city
the temperature (in ◦F) t
hours after 9 A.M. is
represented by the function
T(t) = 50 + 14 sin
Find the average
temperature in that city
during the period from
9 A.M. to 9 P.M.
Second Fundamental
Theorem(4.11)
2days
Average value of a
Function
Example 1.
Calculate
Worksheet
www.kuta/average
value
Basic:
Pg 293-294 #6773 odd, 79, 83, 87
Average:
Pg 293-294 #7080 even, 84, 90
Advanced:
Pg 293-294 #7494 even
Thomas:
Pg 334 #57-65
Anton:
Pg 364 #53-57
odd
http://www.youtub
e.com/watch?v=Cz_
GWNdf_68
www.kuta/secondfu
ndamentaltheorem
Kuta Software
Worksheets
crated by
Teacher
Practice Problems
Example 2.
Calculate
http://wwwmath.mit.edu/~djk/
18_01/chapter14/e
xample04.html
Second
Fundamental
Theorem
Lesson Goal #1
Examples
Pattern recognition and
change of variables involve
a U- substitution.
http://college.cenga
ge.com/mathematic
s/larson/calculus_a
nalytic/7e/students
/downloads/summ
aries/ex4_5_1.pdf
Theorem 4.12
Example 1:
You tube
by trying to identify the pattern
https://www.khana
cademy.org/pattern
s
[ f (g(x))g'(x) ]dx
Let u=g(x)=5x2+1 and du =
g’(x) = 10 xdx
=
(
+1) + C
Example2:
FOR GOALS 1 and
2:
Basic:
Pg 304-306 #1626 even,44-54
even, 60, 64, 68,
74, 99
Theorems 4.13 and 4.14
Average:
Pg 304-306 #1628 even, 42, 4951 odd, 60, 62, 70,
74, 82, 96, 100
Example:
Advanced:
Lesson for goal #2:
1. Find
2.
4days
Integrate
Pg 304-306 #2636 even, 41, 42,
49-61 odd, 70, 71,
106
Real Life
Problem:
Pg 304-306
#111, 113
Home work
Larson Resource
work sheet 4.5
Test Bank Quiz 4.5
Teacher made
assessment using
Kuta soft ware
Kuta Software
Worksheets
crated by
Teacher
Integration
by
Substitution
Basic:
Pg 314 #2, 8, 18,
40, 45
Average:
Pg 314 #3, 14, 32,
39, 40, 45, 51
Lesson for Goal #1:
Theorem 4.16 and 4.18
Trapezoidal Rule:
1day
Use
Pattern
recognition
to find an
indefinite
integral
Simpsons Rule:
Advanced:
Pg 314 #10, 18,
34, 38, 40, 44, 51,
52
Anton:
Pg 567 #39, 40,
41(REAL LIFE
WORD
PROBLEMS)
http://tutorial.math
.lamar.edu/Classes/
CalcII/Approximati
ngDefIntegrals.aspx
Chapter
Review
2days
Chapter
Test
https://www.math.
ucdavis.edu/~koub
a/CalcTwoDIRECTO
RY/usubdirectory/
USubstitution.html
Page 316-317
Resource test
Band C versions
You tube
https://www.youtu
be.com/watch?v=R
TX-ik_8i-k
Content Category:
 Numbers Equations
and Functions
 Calculus
 Probability and
Statistics Geometry
http://www.mathw
ords.com/t/trapezo
id_rule.htm
Larson Resource
work sheet 4.6
TIMSS Example1:
Determine all complex
number z that satisfy the
equation
https://www.khana
cademy.org/math/i
ntegralcalculus/indefinitedefiniteintegrals/riemannsums/v/trapezoidal
-approximation-ofarea-under-curve
where denotes the
conjugate of z.
Change of
variable for
definite
integrals
A)( -1/3 + 5/3i)
B)( -3 -i)
C)(1 -i)
D)answer not given
SAT Example.
Amy has to visit towns B and
C in any order. The roads
connecting these towns with
her home are shown on the
diagram. How many different
Home work
X.
http://www.edinf
ormatics.com/tim
ss/pop3/mpop3.
htm?submit32=Gr
.+12+Adv.+Math+
Test5,6
http://www.erikt
hered.com/tutor/
Sample problems
from
XI.
http://www.edinfor
matics.com/timss
XII.
Test Bank Quiz 4.6
Teacher made
assessment using
Kuta soft ware
Chapter Review
Chapter Test
http://www.edinform
atics.com/timss/pop3
/mpop3.htm?submit3
2=Gr.+12+Adv.+Math
+Test
http://www.majortest
s.com/sat/problem-
Numerical
Integration
Approximation
of definite
integral using
Trapezoidal
and Simpsons
Rule
routes can she take starting
from A and returning to A,
going through both B and C
(but not more than once
through each) and not
travelling any road twice on
the same trip?
A. . 8
B. 6
C. 10
D. 4
E. 2
TIMSS/SAT
A-REI.A.1
Explain each
step in solving
a simple
equation as
following from
the equality of
numbers
asserted at the
previous step,
starting from
the
assumption
that the
original
equation has a
solution.
Construct a
viable
argument to
justify a
solution
method.
4.4.3
C.2: Represen
t all
possibilities
for a simple
counting
situation in
an organized
way and draw
conclusions
from this
representatio
n.
solving-test02
Other Resource
Texts
Gruber’s Complete
Preparation for the
New SAT
The Princeton
XIII.
Review’s Cracking
the New SAT, 2007
www.khanacademy.
org/test-prep/sat
Barron’s SAT 2400:
Aiming for the
Perfect Score
Maximum SAT
Kaplan SAT 2400,
2006 Edition
Up Your Score: The
Underground Guide
to the SAT, 20072008
http://www.edinfor
matics.com/timss
INSTRUCTIONAL FOCUS OF UNIT
 calculating an anti-derivative, solving initial value problems,
finding area under a curve by summing rectangles, using integrals to find the exact area under a curve,
total area vs. net area, mean value theorem for integrals, and u-substitution.
Academic Vocabularies By Dr. Robert Marzano
Anti-derivative
Approximating sum
Area between
Area under a curve
Constant of Integration
Definite Integral
Differential Equation
Indefinite Integral
Initial Condition:
Integrand
Upper and Lower limit
Variable of Integration
The integral sign ∫
Substitution
Trapezoidal Rule
Simpson Rule
Marzano’s Six Steps for Teaching Academic Vocabularies
19.
20.
21.
22.
explanation or example. (Story, sketch, power point)
explain meaning in their own words. (Journal, community circle, turn to your neighbor)
picture, graphic or symbol for each word.
to expand their word knowledge. (Add to their notes, use graphic organizer format)
YOU provide a description,
Ask students to restate or reAsk students to construct a
Engage students in activities
23.
vocabulary words with one another (Collaborate)
24.
with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Example 1 -Step 2
Ask students to discuss
Have students play games
A new piece of industrial equipment will depreciate in value, rapidly at first and then
less rapidly as time goes by. Suppose that the rate in dollars per year of depreciation
of a new milling machine is given by
V’(t) = 500(t − 12)
where V (t) is the value of the machine after t years. What is the total loss in value of
the machine in the first five years? In the second five years?
Example 2. – Step 6
http://www.intmath.com/integration/millionaire-calculus-game.php
PARCC FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01


www.khanacademy.org/test-prep/sat
21ST CENTURY SKILLS
(4Cs & CTE Standards)
21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and
to fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
Finding a Population Function
Since 1990, the rate of increase in the number of divorced adults (in millions)
in the United States from 1990 through 2005 can be modeled by
D= -4.9t2+ 12t+770
Where t is the year, with corresponding to 1990. The number of divorced
adults in 2005 was 22.1 million. (Source: U.S. Census Bureau)
a. Find the model for the number of divorced adults in the United States.
b. Use the model to predict the number of divorced adults in 2012.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics
problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical,
agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
http://www.youtube.com/watch?v=EBfxiKQLnJ4 Example2
9.4.12.O.(2).1
Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve
problems and issues in the real world.
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.
Volume of a pyramid using Integration
Find the volume of a pyramid with a square base that is 20 meters
tall and 20 meters on a side at the base. As with most of our applications of integration, we
begin by asking how we might approximate the volume. Since we can easily compute the
volume of a rectangular prism (that is, a “box”), we will use some boxes to approximate
the volume of the pyramid, as shown in figure 9.3.1: on the left is a cross-sectional view, on
the right is a 3D view of part of the pyramid with some of the boxes used to approximate
the volume

MODIFICATIONS/ACCOMMODATIONS

Students can create flashcards of the basic rule of integrations provided by the teacher. Students can then get into groups where
they can match the solution to their questions
APPENDIX
(Teacher resource extensions)
7. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Mathematical Practices
25. Make sense of problems and persevere in solving them.
26. Reason abstractly and quantitatively.
27. Construct viable arguments and critique the reasoning of others.
28. Model with mathematics.
29. Use appropriate tools strategically.
30. Attend to precision.
31. Look for and make use of structure.
32. Look for and express regularity in repeated reasoning.
8.
Notes to teacher (not to be included in your final draft):
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)
UNIT 10
SEQUENCE and SERIES
```
Total Number of Days: 10 days Grade/Course: 12/CALCULUS
ESENTIAL QUESTIONS

How are sequences and series used to model many
mathematical ideas and realistic situations?




How are sequences & series related?
What is an arithmetic sequence/series?
What is a geometric sequence/series?
9. How is the Principle of Mathematical Induction
used to prove statements are true for all natural
numbers?

How do we develop equations to describe
sequences and series, then use them to solve
other problems?
ENDURING UNDERSTANDINGS









A finite sequence/series contains a finite number of terms.
An infinite sequence/series contains an infinite number of terms.
Arithmetic and geometric sequences and series have a common difference or a common
ratio, respectively.
Sequence and series formulas are used to find a specific term or a total up to a specific term.
The summation symbol (sigma) can be used to quickly write sequence and series formulas.
Sequences and series can be used as prediction tools.
Use the binomial theorem and Pascal’s triangle to generate
binomial coefficients for certain types of sequences & series
Use the compound interest formula to model finance problems
PACI
NG
CONTENT
SKILLS
STANDARDS
(CCCS/MP)
RESOURCES
Larson
Pearson
OTHER
(e.g., tech)
Vocabulary
The nth term of
an Arithmetic
sequence
To find any term
of an arithmetic
sequence:
http://www.regent
sprep.org/Regents/
math/algtrig/ATP2
/ArithSeq.htm
where a1 is the first
term of the sequence,
d is the common
difference, n is the
number of the term to
find.
Problems
http://tutorial.mat
h.lamar.edu/Classe
s/CalcII/SeriesIntro
.aspx
To find the sum of a
certain number of
terms of an arithmetic
sequence:
2 Block
periods
where Sn is the sum
of n terms
(nth partial sum),
a1 is the first
term, an is
the nth term.
Example 1.
Evaluate the 25th term
of the sequence -7, -
LEARNING
ACTIVITIES/AS
SESSMENTS
You tube
https://www.khana
cademy.org/math/
precalculus/seq_in
duction
Using graphing
calculators,
DI according to
student readiness
Use data and
situations relevant
to student interests
Whole group and
small group
cooperative
learning
Trig function
derivative graphic
organizer
Unit test
(multiple choice
& free response
questions)
F.BF.2
Write arithmetic and
geometric sequences
both recursively
and with an explicit
formula, use
Class work
Page 705-707
Summative
Assessment:



Quizzes
Tests
Quarterly
The sum of an
Arithmetic
Sequence
4, -1, 2, ...
n = 25; a1 = -7, d = 3
them to model
situations, and
translate between the
two forms.
Basic
#2,4,8,2028even,45,48
Kuta basic
worksheet
Average
Example 2.
A theater has 60 seats in
the first row, 68 seats in
the second row, 76 seats
in the third row, and so
on in the same
increasing pattern. If
the theater has 20 rows
of seats, how many
seats are in the theater?
We wish to find "the
sum" of all of the seats.
n = 20, a1 = 60, d = 8
and we need a20 for the
sum.
Now, use the sum
formula:
#1638even,53,54,78,8
0
Kuta medium
worksheet

Assessme
nts
Projects
Formative
Assessments:



Demonstr
ation
Class
discussion
Homewor
k
Advanced
Kuta difficult
work sheet
# 8,20,35,50-64
even.79,81,86,88
Home work:
Resource 9,2
worksheet
Teacher made
Quiz and Test
using Kuta
software/Larso
n Test Bank
Teacher made
Mid Chapter Test
Hands on
activities
/projects
There are 2720 seats.
http://people.wall
awalla.edu/~ken.w
iggins/examplesact
ivities/contents.ht
ml
Summation
notation to
write sum
Summation notation is
used to denote a sum of
terms. Usually, the terms
follow a pattern or
formula.
Example 1:
Evaluate:
F.BF.2
Write arithmetic and
geometric sequences
both recursively
and with an explicit
formula, use
them to model
situations, and
translate between the
two forms.
https://www.khana
cademy.org/math/i
ntegralcalculus/sequences
_series_approx_calc
/calculusseries/v/writingseries-sigmanotation
Writing
Journals
Page 704
Writing about
Mathematics
Example 2.
Page 713
Writing about
Mathematics
Use sigma notation to
represent
-3 + 6 - 9 + 12 - 15 + ...
A.SSE.4
for 50 terms
term
position
1
2
3
4
term
-3
6
-9
12
When will a
geometric series
converge and
when will it
diverge?
2 block
periods
Geometric
Sequence
To find any term
of a geometric
sequence:
Class work
Basic
Kuta Easy work sheet
Page 714- 717
#26-34 even,49,64,83
Practice problems
http://www.regent
sprep.org/Regents/
math/algtrig/ATP2
/GeoSeq.htm
Average
The sum of a
Geometric series
where a1 is the first
term of the sequence,
r is the common
ratio, n is the number of
the term to find.
Example 1.:
A ball is dropped from a
height of 8 feet. The ball
bounces to 80% of its
previous height with
each bounce. How high
(to the nearest tenth of a
foot) does the ball
bounce on the fifth
bounce?
Example2:
Lesson for goal#2
To find the sum of a
certain number of
terms of a geometric
sequence:
where Sn is the sum
of n terms (nth partial
sum),
a1 is the first term, r is
the common ration.
Example 1.
Evaluate using a
Kuta medium
worksheet
# 34.39,50-64
even,84,86
Advanced
Kuta difficult
worksheet
#38,40,50-68
even,78,85,88,90,98
You tube
https://www.khana
cademy.org/math/
precalculus/seq_in
duction/precalcgeometricsequences/e/geom
etric_sequences_1
Notes and extra
practice problems
http://www.purple
math.com/modules
/series5.htm
formula:
Example2:
Evaluate S10 for 250,
100, 40, 16 ...
Mathematical Induction
is a special way of
Mathematical
Induction
proving things. It has
only 2 steps:

Step 1. Show it is
You tube
true for the first one

https://www.khanac
ademy.org/math/pre
calculus/seq_inducti
on/proof_by_inducti
on/v/proof-byinduction
Step 2. Show
that if any one is
true then the next
one is true
Example 1.
Prove that the sum
of the first n natural
numbers is given by
Page 726-727
this formula:
Basic
# 2-16
even,32,36,54,58
Average
10-28even,50,52,63
Advanced
examples
http://www.themath
page.com/aprecalc/
mathematicalinduction.htm
1+2+3+. . . +n=
#20-32 even.58,52,67
n(n + 1)
2
Example 2.
1 block
Sum of powers
of Integers.
Prove by
mathematical
induction:
http://www.math.rut
gers.edu/~erowland/
sumsofpowers.html
13 + 23 +33+………n3
=
n2(n + 1)2
4
The formulas for sums of
powers of the
first n positive integers
can be proved using the
Principle of Mathematical
Induction and, in Riemann
sums during an
introduction to definite
integration.
http://www.maa.org/
publications/periodi
cals/convergence/su
ms-of-powers-ofpositive-integersintroduction
=
n(n + 1)
.
2
Examples
http://www.mathcen
tre.ac.uk/resources/u
ploaded/mc-tysigma-2009-1.pdf
Example 1:
Evaluate using the sum of
powers
You tube
http://www.youtube.
com/watch?v=iR9B
Tx4tPN8
=
TIMSS/SAT
+
Example 2:
-REI.A.1
Explain each step in
solving a simple
equation as following
from the equality of
numbers asserted at
the previous step,
starting from the
assumption that the
original equation has
a solution. Construct a
viable argument to
justify a solution
method.
Evaluate
Chapter Test
http://www.educatio
n.gov.za/LinkClick.a
spx?fileticket=gWrc
oxvqUMI%3D&tabi
d=666&mid=1861
Example 3.
Evaluate
TIMSS
Practice Problems
Other SAT
Content Category:
 Numbers
Equations and
Functions
 Calculus
 Probability and
Statistics Geometry
Resource Texts
Gruber’s Complete
Preparation for the
http://www.edinform New SAT
atics.com/timss/pop3
The Princeton
/mpop3.htm?submit3
2=Gr.+12+Adv.+Math Review’s Cracking
the New SAT, 2007
+Test
Barron’s SAT 2400:
Aiming for the
Perfect Score
Maximum SAT
Kaplan SAT 2400,
2006 Edition
Up Your Score: The
Underground
SAT practice test#6
from SAT study guide Guide to the SAT,
2007-2008
Page689-(second
100 SAT Math Tips
Edition)
and How to Master
Section2,4,and 8)
Them Now! Charles
Gulotta
TIMSS Example1:
Example :1
Describe a method for
estimating the
perimeter of figure C.
Example 2:
Which of the following
graphs has these
features:? f'(0) >0, f'(1)
<0 and f''(x) is always
negative.
4.5.A.1
Practice algebra skills
and problem solving
strategies
4.3.B.4
Solve various other
algebraic equations
8 Practice SAT
Tests
http://www.edinf
ormatics.com/tim
ss/pop3/mpop3.h
tm?submit32=Gr.
+12+Adv.+Math+
Test
(two variables,
simultaneous
equations, quadratic
equations
4.2.C.1
8 SAT Practice
Tests
Review concepts of
geometry and apply
strategies
SAT
http://www.major
tests.com/sat/prob
lem-solving-test
4.4.B.2
SAT Example:1.
3x + y = 19 ,
and x + 3y = 1.
Find the value of
2x + 2y?
Understand
geometric notation,
lines and angles,
geometric probability.
Practice problems
A.20 B. 18
C.11 D.10
E.5
Example :2.
What is the slope of the
line l?
http://www.majortes
ts.com/sat/problemsolving-test08
A.-3 B.
C.0 D.10
E.
INSTRUCTIONAL FOCUS OF UNIT
Students will connect their prior study of algebraic patterns with the concepts in this unit and expands their understandings and skills related to
sequences and series. Students explore the basics Characteristics of arithmetic and geometric sequences and series, connecting these ideas to
functions whose domain are a subset of the integers.
Academic Vocabularies By Dr. Robert Marzano
sequence,
series,
finite,infinite terms, factorial,
recursive,
sigma notation
and summation, partial sums,
common difference & common ratio,
compound interest,
arithmetic & geometric sequences & series,
Binomial theorem,
binomial coefficients,
Pascal’s triangle,
Marzano’s Six Steps for Teaching Academic Vocabularies
25.
26.
27.
28.
explanation or example. (Story, sketch, power point)
explain meaning in their own words. (Journal, community circle, turn to your neighbor)
picture, graphic or symbol for each word.
YOU provide a description,
Ask students to restate or reAsk students to construct a
Engage students in activities
to expand their word knowledge. (Add to their notes, use graphic organizer format)
29.
30.
vocabulary words with one another (Collaborate)
with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Ask students to discuss
Have students play games
Example 1
Sum of a Geometric Series Step 1
A ball is dropped from a table that is twenty‐four inches high. The ball always rebounds three fourths of the distance fallen.
Approximately how far will the ball have traveled when it finally comes to rest?
Figure 1. Follow the bouncing ball.
Notice that this problem actually involves two infinite geometric series. One series involves the ball falling, while the other
series involves the ball rebounding.
Falling,
Rebounding,
Use the formula for an infinite geometric series with –1 < r < 1.
The ball will travel approximately 168 inches before it finally comes to rest.
Example 2
-Step 5
http://quizlet.com/6257649/sequences-and-series-unit-vocabulary-flash-cards/
PARCC FRAMEWORK/ASSESSMENT

http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test

http://www.majortests.com/sat/problem-solving-test01

www.khanacademy.org/test-prep/sat
21ST CENTURY SKILLS
(4Cs & CTE Standards)
21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
9.4.D Business, Management & Administration Career Cluster
9.4.E Education & Training Career Cluster
9.4.F Finance Career Cluster
9.4.N Marketing Career Cluster
9.4.O Science, Technology, Engineering & Mathematics Career Cluster
9.4.P Transportation, Distribution & Logistics Career Cluster
9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2
Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities.
9.4.12.D.4, 9.4.12.E.4
Solve mathematical problems and use the information to make business decisions and enhance business management duties.
You visit the Grand Canyon and drop a penny off the edge of a
cliff. The distance the penny will fall is 16 feet the first second, 48
feet the next second, 80 feet the third second, and so on in an
arithmetic sequence. What is the total distance the object will fall in
6 seconds?
Arithmetic sequence: 16, 48, 80, ...
The 6th term is 176.
Now, we are ready to find the sum:
9.4.12.F.4
Solve mathematical problems to obtain information for decision-making in financial settings.
9.4.12.N.4
Solve mathematical problems to obtain information for marketing decision-making.
9.4.12.O.15
Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to
fulfill the specific communication needs of that audience.
9.4.H(5) Biotechnology Research and Development
A mine worker discovers an ore sample containing 500 mg
of radioactive material. It is discovered that the radioactive
material has a half life of 1 day. Find the amount of
radioactive material in the sample at the beginning of the
7th day.
500 mg of ore.
Half life of one day means that half of the amount remains after 1 day.
Begin of day 1 Begin of day 2 Begin of day 3
500 mg
250 mg
125 mg
End of day 1 End of day 2 End of day 3
250 mg
125 mg
62.5 mg
...
...
Decide to either work with the "beginning" of each day, or the "end" of each day, as each can yield the answer. Only the
starting value and number of terms will differ. We will use "beginning":
9.4.12.H.(5).2
Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective
biotechnology research and development.
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7
Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural,
biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
9.4.O(2) Science and Mathematics
9.4.12.O.(2).1
Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems
and issues in the real world.
9.4.12.O.(2).2
Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
9.4.12.O.(2).3
Assess the impact that science and mathematics have on society when used to develop projects or products.
9.4.12.O.(2).4
Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and
mathematics impact problem-solving in modern society.
9.4.12.O.(2).6
Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field.

MODIFICATIONS/ACCOMMODATIONS


Students can create flashcards of the basic rule of integrations provided by the teacher. Students can then get into groups where
they can match the solution to their questions
APPENDIX
(Teacher resource extensions)
9. E-Text, Interactive Digital Resources, Teacher Resources
Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true
Mathematical Practices
33. Make sense of problems and persevere in solving them.
34. Reason abstractly and quantitatively.
35. Construct viable arguments and critique the reasoning of others.
36. Model with mathematics.
37. Use appropriate tools strategically.
38. Attend to precision.
39. Look for and make use of structure.
40. Look for and express regularity in repeated reasoning.
10.
Notes to teacher (not to be included in your final draft):
4 Cs
Creativity: projects
Critical Thinking: Math Journal
Collaboration: Teams/Groups/Stations
Communication – Powerpoints/Presentations
Three Part Objective
Behavior
Condition
Demonstration of Learning (DOL)