Download explore, analyze and solve problems.

East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Enduring Understandings:
o
o
o
o
o
There are a variety of mathematical strategies that can be used to identify,
explore, analyze and solve problems.
Mathematical reasoning and logic are necessary to develop, evaluate and
defend (justify) mathematical ideas and conjectures.
The ability to coherently communicate to others (both orally and in writing)
mathematical ideas, processes and strategies is critical to demonstrating
mathematical understanding.
Mathematical concepts and ideas interconnect and have application to
contexts outside of mathematics.
Mathematical ideas and concepts can be represented in a variety of ways.
Essential Questions
Content/Skills
Unit 1: Limits
Review of Function
What is a function?
-Define relation/function
-Identify a function in its standard form.
-Graphing functions and listing its
characteristics: domain/range,
increasing/decreasing.
What are the
domain and range?
Learning Experiences
IB Topic
Resource
Assessments
ch1 (Haese)
1
Family of
Functions
(graphic
organizer)
-Graphing rational functions by hand to
understand the end behavior of a function
(vertical and horizontal asymptotes)
How are limits and
asymptotes similar
when discussing
convergence?
**<
- Informal ideas of limit and convergence
- Definition of a limit
- Finding limits by direct substitution
- Finding limits graphically
- Finding limits using the table method
- One-sided limits
- Finding limits when x approaches infinity
http://www.calculus-help.com/funstuff/phobe.html>
# of Days
2
7.1
ch21 (Haese)
Video**
Mini-quiz on
Limits
(Formative
Assessment)
3
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Written: June 2010
Content/Skills
Unit 1 cont’d
What does it mean
for a function to be
continuous?
Learning Experiences
IB Topic
Resources
- Horizontal asymptotes as limits at infinity
- Vertical asymp. as limits that are infinite.
7.7
- Definition of continuity
- Finding intervals of continuity using set
and interval notation.
7.1
Assessments
ch22 (Haese)
Supplement:
ch20 (Cirrito)
# of Days
1
Unit 1 Test
(Summative
Assessment)
1
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Content/Skills
Unit 2: Definition of a Derivative
What is the
derivative of a
parabola?
- Derivative/gradient as the slope and rate of
change of the tangent line. Notations used
for differentiation (students will discover
finding the average velocity using slope).
- Students will use the idea of a limit to get
the distance between two points closer and
closer to zero.
- Graph the f ’(x) function given f (x)
Why are limits used - Definition of derivative as
in finding the
derivative of a
 f ( x  h)  f ( x ) 
f ' ( x)  lim 

function? How is it
h 0
h


used?
(polynomial functions only)
How can the
definition of a
derivative be used
to help us find the
equation of the
tangent line?
Written: June 2010
Learning Experiences
IB Topic
Resources
7.1
Assessments
ch20 (Haese)
# of Days
1
Website of
secant line
becoming
tangent line.
http://www.calculus
help.com/funstuff/tu
torials/limits/limit06.
html
1
ch21 (Haese)
2
Supplement:
ch19 (Cirrito)
- Definition of derivative with fractions
1
- Find the equation of a tangent and normal
line.
– derivative as slope of tangent line
at a point
– normal line is perpendicular to
tangent line (negative reciprocal
slopes)
1
Unit 2 Review
& Test
(Summative
Assessment)
2
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Written: June 2010
Content/Skills
Unit 3: Rules of Differentiation
- familiarity with both forms of notation,
When taking the
derivative of a
dy

polynomial, what is dx and f (x)
the pattern between
the original
Power Rule:
function and the
- Derivatives of x n (n  Q), sin x, cos x, tan
derivative?
x, ex, and ln x
Learning Experiences
IB Topics
Resources
7.2
Assessments
Supplement:
ch19 (Cirrito)
# of Days
1
1
Chain Rule:
If y  f (u ) where u  u (x) then
dy dy du

dx du dx
1
Product Rule:
d
uv   u dv  v du
dx
dx
dx
1
Quotient Rule:
d u
 
dx  v 
v
du
dv
u
dx
dx
2
v
Review of rules all together
Quiz on Rules
of Differentiation
(Formative
Assessment)
.5
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Written: June 2010
Content/Skills
Unit 3 Rule of Differentiation cont’d
Learning Experiences
IB Topics
Resources
- Derivative of basic trigonometric
quantities (composites with the linear
function ax + b)
7.4
ch23-24
(Haese)
Knowledge and use of the following
formulas:
7.2
ch21 (Haese)
Assessments
# of Days
2.5
d
u  v   du  dv
dx
dx dx
d
u  v   du  dv
dx
dx dx
.5
d
cu   c du
dx
dx
- Further extension on finding equations of
tangent and normal lines using
differentiation rules.
ch23 (Haese)
.5
Unit 3 Review
& Test
(Summative
Assessment)
2
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Essential Questions
Content/Skills
Unit 4: Applications of Differentiation
Learning Experiences
IB Topic
Resources
Assessments
How can you use
the derivative to
determine if a
function is
increasing or
decreasing?
- Understand the meaning of a derivative as
rate of change in order to determine whether
a function is increasing or decreasing using
the first derivative test.
7.1,7.3
ch22 (Haese)
1
If the first
derivative
represents the
change in position
(or velocity) then
what would the
second derivative
represent?
– Second derivative as the derivative of the
d 2 y d  dy 
first derivative
  
dx 2 dx  dx 
7.2
Supplement:
ch20 (Cirrito)
1
ch21 (Haese)
– Familiarity with the following notation:
d2y
f (x) , y  ,
dx 2
- Using the second derivative test to find
concavity.
How do the signs of
the first and second
derivative influence
the shapes of the
graphs?
# of Days
Fully Discussed Problem:
– First Derivative Test (inc/dec test)
– Second Derivative Test (concavity)
– Points of inflection as changes in
concavity with zero and non-zero
gradients.
– Inflection points (horizontal vs. nonhorizontal inflection points)
– Combination of above knowledge and
skills to sketch graphs
1
7.1,7.3
7.7
Quiz on one
fully
discussed
question
(Formative
Assessment)
1
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Unit 4 Applications of Differentiation
cont’d
– Identification of local and global (absolute
How can calculus
extrema) max/min points.
be used to find the
max/min points of a
function?
– Use of first and second derivative in
optimization problems such as profit, area,
volume.
1
7.3
Optimization
Packet
4
Unit 4 Review
& Test
(Summative
Assessment)
2
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Essential Questions
Content/Skills
Unit 5: Kinematics
What is the
difference between
distance and
displacement?
– Kinematics problems involving
displacement, s, velocity, v, and
acceleration, a.
How would you
find the rate of
change in
displacement?
Velocity?
Acceleration?
How would you
determine if
velocity is
increasing?
Decreasing? Zero?
v
ds
dv d 2 s
, a

dt
dt dt 2
Learning Experiences
IB Topic
Resources
7.6
ch22 (Haese)
Supplement:
ch21 (Cirrito)
Assessments
Kinematics
Quiz
(Summative
Assessment)
# of Days
3-4
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Essential Questions
Content/Skills
Unit 6: Integration
What if you were
given the derivative
of a function and
you wanted to find
the original
function? Is there a
way to work
backwards?
- Area estimation using rectangles
- Left, right and midpoint
- Introduction to integral as area under
curve
Learning Experiences
Resources
Strategies
not a standard
(further
extension)
ch25 (Haese)
Supplement:
ch22 (Cirrito)
ch26 (Haese)
Assessments
# of Days
1
1
- Indefinite integration as antidifferentiation.
7.4
- Indefinite integral as a function of x with
an arbitrary constant term.
If F ( x)  f ( x) then  f ( x)dx  F ( x)  C
ch26 (Haese)
- Indefinite integral of the following basic
1
functions x n (n  Q), sin x, cos x,
and ex.
x
1
Example:  dx  ln x  C , x  0
x
ch26 (Haese)
1
- Integrating: e ax + b and (ax + b)n
ch27 (Haese)
2
- The composites ax + b of a linear function
or trigonometric function.
Example:
f ' ( x)  cos( 2 x  3)  f ( x) 
1
sin( 2 x  3)  C.
2
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Unit 6 Integration cont’d
What information
is needed to
determine the
constant term of an
indefinite integral?
What’s the
difference between
definite and
indefinite integral?
- Integration with a boundary condition to
determine the constant term.
7.5
Supplement:
ch22 (Cirrito)
Example:
dy
 3x 2  x and y=10 when
If
dx
1
x=0, then y  x 3  x 2  10
2
- Finding definite integrals using the
Fundamental Theorem of Calculus
1
Indefinite
Integration
Quiz
(Formative
Assessment)
1
7.4
b
 f ( x)dx  F (b)  F (a)
a
Pg 660
(Haese)
Supplement:
ch22 (Cirrito)
- Indefinite integral as a function vs. definite
integral as a number.
- Method of Substitution (U-Substitution)
7.5
ch26-27
(Haese)
1
.5
- Basic integration properties (rules).
Unit 6 Review
& Test
(Summative
Assessment)
2
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Essential Questions
Content/Skills
Unit 7: Applications of Integration
How can we use
integration to find
the exact area
under a curve?
What information
is needed?
– Using integration and differentiation to
solve kinematics problems involving displacement, s, velocity, v, and acceleration, a
ds
dv d 2 s
v , a

dt
dt dt 2
What does the area
under a timevelocity graph
represent?
How can we use
integration to find
the volume of
revolution of a
linear function?
What shape does it
make and what is
the geometric
formula to find the
volume of this
shape?
-Areas under curves (between the curve and
the x-axis), areas between curves.
- Volumes of revolution about the x-axis
and identifying the geometric shape.
Learning Experiences
IB Topics
Resource
7.6
ch26 (Haese)
Supplement:
ch22 (Cirrito)
7.5
ch28 (Haese)
Assessments
# of Days
Quiz on area
under a curve
& kinematic
problems
involving
integration
(Formative
Assessment)
2
3
1
Supplement:
ch22 (Cirrito)
1
- volume of revolution of one function
b
V    f ( x) 2 dx
a
1
- volumes for two defining functions
b


V     f ( x)  g ( x) dx
2
2
a
Unit 7 Review
& Test
(Summative
Assessment)
2
East Irondequoit Central School District
Course: Math SL 12 IB
Written: June 2010
Essential Questions
Content/Skills
Unit 8: Statistics
What is the
difference between
discrete and
continuous random
variables?
- Understand population, sample, and the
process of statistics.
- Discrete vs. continuous data
6.1
ch18 (Haese)
1
- Presentation of data using: frequency
tables, frequency histogram, column graphs
and stem-leaf-plot.
- Grouped data (find the mean)
- Mid-interval values
- Interval width
- Upper and lower interval boundaries
6.2
Supplement:
ch13 (Cirrito)
2-3
How do organize
data and make
predictions using
it?
Learning Experiences
IB Topics
Resources
Assessments
# of Days
- Calculate the range, mean, median and
mode (or modal) of a table and/or diagram.
- Construct and use a box and whisker plot
to find the five-number summary of a data
set.
1
- Determine if a graph is negatively or
positively skewed.
.5
- Construct a cumulative frequency table
and curve.
6.4
1
- Use raw data, tables, and/or graphs to find
the median, quartiles and percentiles.
6.3
1
- Understand the concept of standard
deviation and variance and be able to
calculate both
6.3
Unit 8 Test
(Summative
Assessment)
.5
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Content/Skills
Unit 9: Probability
How do we
mathematically
notate sets of
numbers?
- Basic introduction of probability along
with key vocabulary words.
- Concepts of trial, outcome, equally likely
outcomes, sample space (U) and event
- The probability of an event A as
n  A
P  A 
nU 
- Theoretical vs. Experimental Probability
- The complementary events A and A’ (not
A) : P(A) + P(A’) = 1
Are non/mutually
exclusive and
de/independent
events related to
each other?
What tool can you
use to find the
probability of
non/mutually
exclusive? What
about dependent
and independent
events?
Written: June 2010
Learning Experiences
IB Topics
Resources
6.5
Assessments
ch19 (Haese)
1
Supplement:
ch15(Cirrito)
- Use of Venn Diagrams, tree diagrams and
tables of outcomes to solve problems.
6.8
- Finding the probability of a combined
event with and without replacement.
6.6
- Independent events
- The definition: P A / B  P A  P A / B'
6.7
Venn & Tree
Diagram Quiz
(Formative
Assessment)
2
2
1
- Conditional probability
1
P A  B 
- The definition: P A / B  
P B 
-Mutually exclusive events: P A  B  0
- Combined events using the formula:
P A  B  P A  PB  P A  B
# of Days
6.6
Unit 9 Test
(Summative
Assessment)
1
East Irondequoit Central School District
Course: Math SL 12 IB
Essential Questions
Content/Skills
Unit 10: Statistical Distribution
How can discrete
random variables
and their
probability
distributions be
used in a game of
chance? What are
some examples of
theses games?
- Concept of discrete random variables and
their probability distributions such as:
1
P X  x   4  x  for x  {1,2,3}
18
- Applications of expectation, for example,
game of chance.
How are binomial
distributions
different than
normal
distributions? How
are binomial
distributions easily
identified?
- Calculate probabilities for the binomial
distribution.
- Find the mean of a binomial distribution.
Written: June 2010
Learning Experiences
IB Topics
Resources
6.9
Assessments
ch29 (Haese)
1
Supplement:
ch16(Cirrito)
1
- Find the expected value (mean), E(x) for
discrete data.
- Knowledge and use of the formula:
E X    xP X  x 
- Understand the properties of a normal
distribution.
-Calculate probabilities for a normal
distribution and appreciate that the
standardized value (z) gives the number of
standard deviations from the mean.
- Standardization of a normal variable.
- Use of calculator or tables to find normal
probabilities; the reverse process.
# of Days
2
6.10
.5
6.11
Supplement:
ch17(Cirrito)
Normal &
Binomial
Distribution
Quiz
(Formative
Assessment)
Unit 10 Test
(Summative
Assessment)
1.5
2