HolcombMath

Tuesday, March 02, 2010

HL (Class 59)

Announcements
Related Rates Quiz Th.

Lesson Title
Lesson 21: Looking Closely

Overview
Today’s class begins with students working on related rate problems. Then they will be taking a look at a problem from a past IB exam concerning the transformation of function.
Textbook Sections

Vocabulary
function
independent variable
dependent variable
with respect to
rate of change
limit
derivative
explicit equation
implicit equation

Key Attitudes
Willingness to work as a group to help meet individual and group goals.

Enduring Understandings
Change is fundamental to understanding functions.
Mathematical relationships can be represented in 4 main ways: Graphical, Numerical, Algebraic, Verbal (written and oral).

Essential Question
What does a function look like when viewed very closely? Do all functions behave like this?
How can an equation for the line which approximates a function at a given point be created?

Key Knowledge
How to zoom using a GDC.
Finding the limit of a function graphically, numerically, and algebraically.

Key Skills
I can explain what it means for a function to be locally linear.
I can determine if a function will have a local linearization.
I can find the local linearization of a function, if it exists.

Turn-In (#-1)
§ (Txt. p.)

Handouts
No Handouts Posted

Assignment
PS 21
IA
Disclaimer- The assignment as stated in class is the official assignment. Every effort is made to keep this posting accurate, but you should refer to what was stated in class as the final word.

SL (Class 59)

Lesson Title
Lesson 18: Inferences from Derivatives (3)

Overview
We continue our exploration of how derivatives and second derivatives of a function can be used to uncover information about the function itself.
Textbook Sections

Vocabulary
function
independent variable
dependent variable
with respect to
rate of change
limit
derivative
difference quotient
derivative from first principals

Key Attitudes
Willingness to work as a group to help meet individual and group goals.

Enduring Understandings
Change is fundamental to understanding functions.
Mathematical relationships can be represented in 4 main ways: Graphical, Numerical, Algebraic, Verbal (written and oral).

Essential Question
What information does the first derivative give me about the original function?
What information does the second derivative give me about the original function?

Key Knowledge
If the derivative of a function changes sign around a critical point, the function is said to have a local (relative) extremum at that point.
If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point.
If the second derivative is positive at a point, the graph is concave up. If the second derivative is positive at a critical point, then the critical point is a local minimum.
If the second derivative is negative at a point, the graph is concave down. If the second derivative is negative at a critical point, then the critical point is a local maximum.
An inflection point marks the transition from concave up and concave down. The second derivative will be zero at an inflection point.
The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum here. If, however, the function has a critical point for which f′(x) = 0 and the second derivative is negative at this point, then f has local maximum here. This technique is called Second Derivative Test for Local Extrema.

Key Skills
I can use multiple representations to determine if a function has a relative maximum or a relative minimum on a given interval.
I can use multiple representations to determine if a function is concave up or concave down on a given interval.
I can use multiple representations to determine if a function has an inflection point on a given interval.

Turn-In (#-1)
Matrices Review
Vector Review

Handouts
No Handouts Posted

Assignment
PS 18
Disclaimer- The assignment as stated in class is the official assignment. Every effort is made to keep this posting accurate, but you should refer to what was stated in class as the final word.

Math 7 (Class 118)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students investigate rectangles more closely to try and figure out what is necessary for two rectangles to be similar.
Textbook Sections
Problem 2.2 (Txt. p.18) Nosing Around

Vocabulary
similar
corresponding sides
corresponding angles
segment
ratio
perimeter

Key Attitudes
Willingness to work as a group to help meet individual and group goals.

Enduring Understandings
Change is fundamental to understanding functions.
Mathematical relationships can be represented in 4 main ways: Graphical, Numerical, Algebraic, Verbal (written and oral).

Essential Question
How can transformations be described mathematically?
How do different shapes compare to each other?
What is required for two shapes to be similar?

Key Knowledge
Certain properties of a shape are maintained when a shape is enlarged or reduced.

Key Skills
I can collect data and organize data.
I can use data to make predictions and generalizations.
I can confirm or refute generalizations using data.

Turn-In (#-1)
No Homework

Handouts
No Handouts Posted

Assignment
ACE p.22 #11
Disclaimer- The assignment as stated in class is the official assignment. Every effort is made to keep this posting accurate, but you should refer to what was stated in class as the final word.

Math 6 (Class 118)

Lesson Title
Investigation 4: Polygon Properties and Tiling

Overview
In today’s class students conclude their investigation of the relationship between the number of sides of a polygon and the sum of the measures of the interior angles by developing a rationale for why the pattern they uncovered and justified for side of 3 to 9 can be extended to all polygons. They then turn their attention to “stars” and see if they can construct a formula for the sums of these shapes.
Textbook Sections
Problem 4.1 (Txt. p.42) Relating Sides to Angles

Vocabulary
tiling
regular polygon
polygon
pentagon
hexagon
octagon
angle ruler
precisely
vertex

Key Attitudes
Willingness to work as a group to help meet individual and group goals.

Enduring Understandings
Change is fundamental to understanding functions.
Mathematical relationships can be represented in 4 main ways: Graphical, Numerical, Algebraic, Verbal (written and oral).

Essential Question
Why do we compare contrast and classify objects?
How do decomposing and recomposing shapes help us build our understand of mathematics?

Key Knowledge
An angle can be measured with an angle ruler.

Key Skills
I can find the sum of the interior angles of any covnex polygon.
I can justify the sum of the interior angles of a polygon by using previously agreed on facts.
I can use what I have learned about the sum of the interior angles of a polygon to develop a method for finding the sum of the interior angles of a polygonal star.

Turn-In (#-1)
Draw an 17-gon. Find the sum of its interior angles. Prove your answer is correct by “cutting” the shape into triangles.

Handouts
No Handouts Posted

Assignment
“Interior Angles” worksheet
Disclaimer- The assignment as stated in class is the official assignment. Every effort is made to keep this posting accurate, but you should refer to what was stated in class as the final word.