HolcombMath

Tuesday, February 09, 2010

IB Math SL (Class 54)

Lesson Title
Lesson 16: What if the variable is an exponent? (2)

Overview
In today’s lesson students continue to explore how the derivative of a function which has a variable for an exponent may be found.
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
How can an equation for the derivative of a function be created?
How can the derivative of a function which has a variable as an exponent be found?

Key Knowledge
The inverse of a function undoes a function.

Key Skills
I can find the derivative of functions which have variables as exponents.

Turn-In (#-1)
PS 15

Handouts
No Handouts Posted

Assignment
PS 16
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 109)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students return to the problem for trying to figure out the number of different arrangements that are possible with the letters MIAMI. (We’ve gotten a bit side tracked by this problem.)
Textbook Sections
Problem 2.1 (Txt. p.15) Drawing Wumps

Vocabulary
coordinate graph
quadrant
axis
axes
x-axis
y-axis
coordinates
ordered pair
origin
vertical
horizontal
plot
scale
vertices
coordinate geometry
polygon
quadrilateral
parallelogram
rhombus
annotate
rate of change
positive rate of change
negative rate of change
average rate of change
per
speed
speedometer
acceleration
distance-time graph
speed-time graph
continuous
discrete
area
definite integral
point of intersection
parallel
coincident
profit
income
expenses
cost
slope
ratio
intersection of grid lines
easy points
equilateral
regular
triangle
square
pentagon
hexagon
octagon
typical
average
horizontal
slope
gradient
rate of change
coefficient
declare your variable

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?

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

Key Skills
I can plot points accurately.
I can calculate the locations of points using an algebraic formula.

Handouts
No Handouts Posted

Assignment
Finish “City Scramble 1”
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.

IB Math HL (Class 54)

Lesson Title
Lesson 20: Related Rates (1)

Overview
In today’s class students begin to investigate how to use the rate of change of one variable with respect to another, and a relationship between variables, to calculate another rate.
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
How can the rate of change be used to find other rates of change?

Key Knowledge
The derivative of a function can be found implicitly.

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 find a related rate.
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 19

Handouts
No Handouts Posted

Assignment
PS 20
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 109)

Lesson Title
Investigation 3: Polygons and Angles

Overview
In today’s class students continue to develop their understanding of angle measures by playing the game of Four in a Row on a circular (polar) graph.
Textbook Sections
Problem 3.4 (Txt. p.29) Playing Four in a Row

Vocabulary
tiling
regular polygon
polygon
pentagon
hexagon
octagon

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 thought of as the result of a turning motion, a wedge, or two sides coming together to form a common vertex.

Key Skills
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a right angle turn.
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a given angle.
I can determine the size of the angles of the vertices of shapes A, B, D, M, R, and V in the shape set.

Turn-In (#-1)
ACE p. 35 #19-22, 27-29

Handouts
No Handouts Posted

Assignment
ACE p.35 #23-26, 36
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.

Monday, February 08, 2010

Algbera 2 (Class 53)

Lesson Title
3.1.3 How does it grow?

Overview
You may have heard the expression, “Money does not grow on trees.” However, money does, in a sense, grow in a savings account. In today’s lesson you will apply your understanding of exponential functions to solve problems involving money and interest.
Textbook Sections
3.1.3 (txt. p.125) How does it grow?

Vocabulary
interest
simple interest
compound interest

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 does it grow?
How is the rate written as a percent? As a decimal?
How is it the same or different?

Key Knowledge
Exponential growth is caused by a constant multiplication.

Key Skills
I can determine the percent by which an investment is growing when given a table of values.
I can determine if a sequence is arithmetic, geometric, or neither.
I can write and equation to represent an investment.
I can tell the difference between simple and compound interest.
I can use tables, graphs, and equations to decide on the best offer.

Turn-In (#-1)
3-26 to 3-30

Handouts
No Handouts Posted

Assignment
3-31 to 3-33
3-39 to 3-40
3-35 e
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 107)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students try to determine how many different ways they could arrange the letters “MIAMI”.
Textbook Sections
Problem 2.1 (Txt. p.15) Drawing Wumps

Vocabulary
coordinate graph
quadrant
axis
axes
x-axis
y-axis
coordinates
ordered pair
origin
vertical
horizontal
plot
scale
vertices
coordinate geometry
polygon
quadrilateral
parallelogram
rhombus
annotate
rate of change
positive rate of change
negative rate of change
average rate of change
per
speed
speedometer
acceleration
distance-time graph
speed-time graph
continuous
discrete
area
definite integral
point of intersection
parallel
coincident
profit
income
expenses
cost
slope
ratio
intersection of grid lines
easy points
equilateral
regular
triangle
square
pentagon
hexagon
octagon
typical
average
horizontal
slope
gradient
rate of change
coefficient
declare your variable

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?

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

Key Skills
I can plot points accurately.
I can calculate the locations of points using an algebraic formula.

Handouts
No Handouts Posted

Assignment
For each of the following, determine how many different 4-letter arrangements are possible;
1) ABCD
2) AACD
3) AABB
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 107)

Lesson Title
Investigation 3: Polygons and Angles

Overview
In today’s class students use the shapes in their shape set to measure angles. As time permits they also learn a new game— Polar Four-In-A-Row.
Textbook Sections
Problem 3.3 (Txt. p.27) Developing More Angle Benchmarks

Vocabulary
tiling
regular polygon
polygon
pentagon
hexagon
octagon

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 thought of as the result of a turning motion, a wedge, or two sides coming together to form a common vertex.

Key Skills
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a right angle turn.
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a given angle.
I can determine the size of the angles of the vertices of shapes A, B, D, M, R, and V in the shape set.

Turn-In (#-1)
ACE p.35 #13-18, 27-29

Handouts
No Handouts Posted

Assignment
ACE p. 35 #19-22, 27-29
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.

Friday, February 05, 2010

IB Math SL (Class 53)

Lesson Title
Lesson 16: What if the variable is an exponent? (1)

Overview
In this lesson students continue to work on problems involving implicit differentiation. In addition students will begin to investigate how to find the derivatives of exponential functions.
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
How can an equation for the derivative of a function be created?
How can the derivative of a function which has a variable as an exponent be found?

Key Knowledge
The inverse of a function undoes a function.

Key Skills
I can find the derivative of functions which have variables as exponents.

Turn-In (#-1)
PS 15

Handouts
No Handouts Posted

Assignment
PS 15
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.

IB Math HL (Class 53)

Lesson Title
Lesson 19: Optimization (3)

Overview
In today’s class students will continue to work on optimization problems.
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 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 19

Handouts
No Handouts Posted

Assignment
PS 19
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 106)

Lesson Title
Investigation 3: Polygons and Angles

Overview
In today’s class students use the shapes in their shape set to measure angles. As time permits they also learn a new game— Polar Four-In-A-Row.
Textbook Sections
Problem 3.3 (Txt. p.27) Developing More Angle Benchmarks

Vocabulary
tiling
regular polygon
polygon
pentagon
hexagon
octagon

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 thought of as the result of a turning motion, a wedge, or two sides coming together to form a common vertex.

Key Skills
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a right angle turn.
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a given angle.
I can determine the size of the angles of the vertices of shapes A, B, D, M, R, and V in the shape set.

Turn-In (#-1)
ACE p.35 #13-18, 27-29

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.

Math 7 (Class 106)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students students are introduced to the Wump family. They then asked to use coordinates to make drawings of the Wumps according to an algebraic rule.
Textbook Sections
Problem 2.1 (Txt. p.15) Drawing Wumps

Vocabulary
coordinate graph
quadrant
axis
axes
x-axis
y-axis
coordinates
ordered pair
origin
vertical
horizontal
plot
scale
vertices
coordinate geometry
polygon
quadrilateral
parallelogram
rhombus
annotate
rate of change
positive rate of change
negative rate of change
average rate of change
per
speed
speedometer
acceleration
distance-time graph
speed-time graph
continuous
discrete
area
definite integral
point of intersection
parallel
coincident
profit
income
expenses
cost
slope
ratio
intersection of grid lines
easy points
equilateral
regular
triangle
square
pentagon
hexagon
octagon
typical
average
horizontal
slope
gradient
rate of change
coefficient
declare your variable

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?

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

Key Skills
I can plot points accurately.
I can calculate the locations of points using an algebraic formula.

Turn-In (#-1)
§ (Txt. p.)

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.

Thursday, February 04, 2010

IB Math SL (Class 52)

Lesson Title
Lesson 15- Y-Not! (2)

Overview
In this lesson students get some much needed work time focusing on implicit differentiation and the chain rule. They will be having short quiz on these topics on Friday. As time permits, they will develop a method for finding the derivative of an exponential function.
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
How can an equation for the derivative of a function be created?
How can the derivative of a function which has a variable as an exponent be found?

Key Knowledge
The inverse of a function undoes a function.

Key Skills
I can find the derivative of functions which have variables as exponents.

Turn-In (#-1)
PS 15

Handouts
No Handouts Posted

Assignment
PS 15
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.

IB Math HL (Class 52)

Lesson Title
Lesson 19: Optimization (2)

Overview
In today’s class students have some much needed work time. We’ll start with considering another Triangle Function problem like in the last class (link below) and then spend the rest of the time working more optimization problems. Students also have a short quiz focusing on Lesson 18-- Inferences from Derivatives.

Students also get their next IA-- which will be due March 18.

http://holcombmath.com/sketches/I2C_geogebra_sketches/Triangle_Function_2.html

http://holcombmath.com/sketches/I2C_geogebra_sketches/Triangle_Function_3.html
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 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 19

Handouts
No Handouts Posted

Assignment
PS 19
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 105)

Lesson Title
Investigation 3: Polygons and Angles

Overview
In today’s class students work on estimating the measures of angles.
Textbook Sections
Problem 3.2 (Txt. p.25) Estimating Angle Measures
Problem 3.3 (Txt. p.27) Developing More Angle Benchmarks

Vocabulary
tiling
regular polygon
polygon
pentagon
hexagon
octagon

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 thought of as the result of a turning motion, a wedge, or two sides coming together to form a common vertex.

Key Skills
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a right angle turn.
I can sketch, and determine the angle measure, of an angle created by turning a fractional amount of a given angle.
I can determine the size of the angles of the vertices of shapes A, B, D, M, R, and V in the shape set.

Turn-In (#-1)
Problem 3.1
Find an example of a non-rectangle parallelogram
Quiz Corrections

Handouts
No Handouts Posted

Assignment
ACE p.35 #13-18, 27-29
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 105)

Lesson Title
No Class today-- Languages Project

Overview

Textbook Sections

Vocabulary
coordinate graph
quadrant
axis
axes
x-axis
y-axis
coordinates
ordered pair
origin
vertical
horizontal
plot
scale
vertices
coordinate geometry
polygon
quadrilateral
parallelogram
rhombus
annotate
rate of change
positive rate of change
negative rate of change
average rate of change
per
speed
speedometer
acceleration
distance-time graph
speed-time graph
continuous
discrete
area
definite integral
point of intersection
parallel
coincident
profit
income
expenses
cost
slope
ratio
intersection of grid lines
easy points
equilateral
regular
triangle
square
pentagon
hexagon
octagon
typical
average
horizontal
slope
gradient
rate of change
coefficient
declare your variable

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?

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

Key Skills
I can use a rubber band stretcher to enlarge a figure.
I can make a detailed list of what is the same and what is different about two shapes.

Turn-In (#-1)
ACE p.9 #1, 2
Concept Map Rough Draft (for those of you who did not do it!)
Weekly Summary

Handouts
No Handouts Posted

Assignment
§ (Txt. p.)
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.