HolcombMath

Tuesday, March 02, 2010

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.

Monday, March 01, 2010

Algebra 2 (Class 58)

Lesson Title
3.1.5: What are the Connections

Overview
In the previous lessons students started an exponential-representations web. In this class students work to develop methods for finding a rule from a graph. As they find ways to write rules based on graphs, they will build a deeper understanding of exponential functions.
Textbook Sections

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?
What’s the difference between exponential growth and exponential decay?

Key Knowledge
Exponential growth is caused by a constant multiplication.

Key Skills
I can determine if a situation is appropriately represented by exponential decay.
I can represent exponential decay using multiple representations.
I can find an equation for an exponential function when given its graph.

Turn-In (#-1)
3-56 to 3-59

Handouts
No Handouts Posted

Assignment
3-60 to 3-61, 3-64 to 3-66
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 117)

Lesson Title
Investigation 4: Polygon Properties and Tiling

Overview
In the last class students began to explore how we might be able to use our new found fact that the sum of the interior angles of any triangle might help us confirm or refute our conjecture about the sum of the interior angles of a pentagon. In today’s class this exploration resumes.
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 measure an angle with an angle ruler.
I can determine with shapes in the shape set can be used to make a tiling.
I can use facts about parallel lines to show that the sum of the interior angles of a triangle always add to 180˚

Turn-In (#-1)
No Homework

Handouts
No Handouts Posted

Assignment
Draw an 17-gon. Find the sum of its interior angles. Prove your answer is correct by “cutting” the shape into triangles.
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 117)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students explore what is meant by the term “odds” , use clues to infer the number of marbles that must be in box, and determine the probability of drawing a specific color.
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 determine the odds of something happening.
I can use odds to determine probability.
I can use information to make inferences.

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.

Tuesday, February 16, 2010

Math 6 (Class 113)

Lesson Title
Investigation 4: Polygon Properties and Tiling

Overview
In today’s class students continue to investigate how the number of sides is related to the measure of the angles of a polygon.
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 measure an angle with an angle ruler.
I can determine with shapes in the shape set can be used to make a tiling.

Turn-In (#-1)
Nothing to turn in.

Handouts
No Handouts Posted

Assignment
ACE p.35 #42
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 113)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students investigate probabilities and revisit the “ATLANTA” problem to see how a letter which is repeated three times effects the number of different possible combinations.
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
Justification of your answer to “ATLANTA”
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 56)

Lesson Title
Lesson 20: Related Rates (3)

Overview
In today’s class we wrap up our work with related rates and take a brief look at mathematical modeling.
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
IA
Review Matrices and Vectors (emailed to you).
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 SL (Class 56)

Lesson Title
Lesson 18: Inferences from Derivatives (1)

Overview
In today’s lesson, students finish up the work on developing techniques for finding the derivative of inverse trigonometric functions. They then begin turn their attention to furthering their understanding regarding information the derivative of a function can provide.

http://holcombmath.com/sketches/I2C_geogebra_sketches/Inferences_from_Derivatives_1.html

http://holcombmath.com/sketches/I2C_geogebra_sketches/Inferences_from_Derivatives_2.html

http://holcombmath.com/sketches/I2C_geogebra_sketches/Inferences_from_Derivatives_3.html

http://holcombmath.com/sketches/I2C_geogebra_sketches/Inferences_from_Derivatives_4.html
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)
PS 16

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.

Friday, February 12, 2010

Math 7 (Class 111)

Lesson Title
Investigation 2: Similar Figures

Overview
In today’s class students will analyze their drawings of the Wump family. In addition, they will be writing a quiz focusing on the “City Scramble” problems.
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)
ACE p.22 #1, 3
City Scramble 4

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.

Algebra 2 (Class 55)

Lesson Title
3.1.4 What if it does not grow?

Overview
To learn more about exponents, today students will students a new context that can be represented with an equation of the form y = ab^x.
Textbook Sections
3.1.4 (Txt. p.130) What if it does not 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?
What’s the difference between exponential growth and exponential decay?

Key Knowledge
Exponential growth is caused by a constant multiplication.

Key Skills
I can determine if a situation is appropriately represented by exponential decay.
I can represent exponential decay using multiple representations.
I can find the half-life given values.

Turn-In (#-1)
3-41 to 3-44

Handouts
No Handouts Posted

Assignment
3-45 to 3-47
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 111)

Lesson Title
Investigation 4: Polygon Properties and Tiling

Overview
In today’s class students investigate how the number of sides is related to the measure of the angles of a polygon. In addition, they write a quiz focusing on this unit so far.
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 measure an angle with an angle ruler.
I can determine with shapes in the shape set can be used to make a tiling.

Turn-In (#-1)
ACE p.35 #30 to 35
Which Region? 5

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 11, 2010

Math SL (Class 55)

Lesson Title
Lesson 18: Inferences from Derivatives (1)

Overview
In today’s lesson, students finish up the work on finding 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
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)
PS 16

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 HL (Class 55)

Lesson Title
Lesson 20: Related Rates (2)

Overview
In today’s class students continue 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.

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
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.