HolcombMath

Friday, March 12, 2010

HL (Class 63)

Lesson Title
Lesson 21: Looking Closely (1)

Overview
Today students continue to examine functions from a very close perspective. They see that when viewed closely enough, all smooth continuous functions become essentially linear.
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 and any other previous work.
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 126)

Lesson Title
Investigation 3: Patterns of Similar Figures

Overview
In today’s class students continue to work with Rep-tile.
Textbook Sections
Problem 3.2 (Txt. p.29) Building with Rep-tiles

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 visually determine if two shapes are similar.

Turn-In (#-1)
No Homework

Handouts
No Handouts Posted

Assignment
No Homework
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 63)

Lesson Title
Lesson 19: Optimization (2)

Overview
In today’s class students continue to develop their ability to use calculus to determine the optimal solution to a variety of applied problems.
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 has to be true about the value of a derivative in order to have a maximum or minimum value?
If the derivative of a function is zero, does this always represent a maximum or minimum?
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
The derivative of a function can be used to find the optimal solution to a problem.
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 make and label a diagram to represent a situation.
I can identify the key variables in a situation.
I can mentally model what is going on in a situation.
I can create equations relating the key variables in a situation.
I can create an equation between the two main variables in a situation.
I can find and use a derivative to determine an optimal solution for a situation.
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)
PS 18

Handouts
No Handouts Posted

Assignment
PS 19 and or Past Paper
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 126)

Lesson Title
Investigation 5: Side-Angle-Shape Connections

Overview
Students continue investigating flipping and turning triangles.
Textbook Sections
Problem 5.1 (Txt. p.52) Flipping and Turning Triangles

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
How can transformations be described mathematically?

Key Knowledge
Flipping means to turn a shape over.
Turning means to rotate a shape around a given point.

Key Skills
I can determine the number of ways a shape can be turned and or flipped in order to fit into a missing hole in a pattern.

Turn-In (#-1)
ACE p.57 #1, 6
Problem 5.1 A-- How many ways do you think there really are? Get convinced!

Handouts
No Handouts Posted

Assignment
No Homework
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.