HolcombMath

Sunday, November 30, 2008

Intro to Calc Test 4

Here are Tests 4A and 4B. Answers are included.

Tuesday, November 25, 2008

Intro to Calculus (Class 32)

Lesson Title
Graphs of Polynomials (4)

Overview
During class today the main task while be Test 4 which focuses on optimization problems and polynomial functions. The remaining time will be spent on learning about ,and working on, sangaku— Japanese temple mathematics from the seventeenth century.
Textbook Sections
N/A

Vocabulary
optimization
concavity
degree
cusp
corner
polynomial
lead coefficient
end behavior

Key Attitudes
Math is about thinking creatively.

Key Ideas
A polynomial function is created when a series of terms are connected with addition such that each term is comprised of a real number coefficient and a variable raised to a whole number power. (This is not really a definition, but it captures what will be most important to us.)
The domain of a polynomial is all real numbers. Hence the graph of a polynomial will have to breaks.
Graphs of polynomials are smooth-- no pointy parts.
Graphs of polynomials have no horizontal asymptotes.
A lower bound for the degree of a polynomial can be inferred from the “bumps” on the graph of the polynomial.
The behavior at the “ends” of a polynomial (way out where x is really small or really big) can be used to determine if the degree of the polynomial is odd or even and whether the coefficient of the highest degree term (the lead coefficient) is positive or negative.
The behavior of the graph of the polynomial at the locations where it crosses the x-axis can be used to tell you the multiplicity of the real roots of the function-- tells you what degree each factored term has to be raised to.
Key Skills
I can determine if a graph could represent a polynomial function.
I can determine a lower bound on the degree of a polynomial by examining its graph.
I can determine if the degree of a polynomial is odd or even and if the leading coefficient is positive or negative by examining its graph.
I can determine if a given value is a zero of given polynomial when given a graph or the equation of the function.
I can given polynomial equation could be represented by a particular graph.
I can determine if a given polynomial could be a factor of another polynomial without using long division or graphing.
I can use a graph and long division to completely factor a polynomial (that is, to write the polynomial as a product of irreducible factors).
I can create a polynomial function to match descriptions involving the degree and the roots of the polynomial.
I can find a possible equation for a polynomial function when given its graph.
Turn-In (#31)
Workshop 9

Handouts
Sangaku 1

Assignment
Sangaku 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.

Monday, November 24, 2008

Geometry (Class 32)

Announcements
Test Monday! Logic Puzzles and Proofs

Lesson Title
The Game of Proof (4)

Overview
The warm-up for the day, Graphs with Different Sections, continues to have students examine graphs relating distance and time and interpreting them based on the shapes of different sections of the graphs. The lesson for the day continues with the focus of improving the ability to write two column proofs. During the last part of the class students will be taking Test 6 which focuses on solving logic problems using mathematics and writing two column proofs.
Textbook Sections
§3.3 (Txt. p.143) Parallel Lines and Transversals

Vocabulary
implication
deduction
syllogism
two-column proof
statement
reason

Key Attitudes
Math is about being convinced a statement is always true.

Key Ideas
Proofs can be thought of as a game where the game pieces are definitions, postulates, and theorems and the objective is to build a logical chain of conditional statements (a syllogism) using these pieces to connect the “given” to the “prove”.
Key Skills
I can build a syllogism (an If…, then… chain of reasons).
I can label a diagram with the information given in a proof.
I can use my knowledge of angle relationships to figure out a plan for proving a statement.
I can organize the angle relationships to build a logical chain of reasoning going from what is given to what is to be proved.
I can use the definitions, postulates, and theorems the class has developed to justify steps I take to solve a missing angle puzzle.
I can write a two column proof to represent the process of solving a missing angle puzzle.

Turn-In (#30)
Txt. p.146 #10, 13, 15, 16, 20, 23, 25, 26
Txt. p.149 Quiz 1 #1-8

Handouts
Chapter 2- Lesson 3: The Game of Proof
Chapter 2- Lesson 3; The Game of Proof Practice

Assignment
Txt. p.138 #7-13
Txt. p. 141 Mixed Review #29-36
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.

Friday, November 21, 2008

Intro to Calculus (Class 31)

Announcements
The next test will be on Tuesday, November 25 and will focus on optimization problems, roots of polynomials, and graphs of polynomials. (Homework 9 to 11 and workshops 7- 9).

Lesson Title
Graphs of Polynomials (3)

Overview
The focus of the class today is to finish Homework 11 and to begin work on Workshop 9. We have a test on Tuesday.
Textbook Sections
N/A

Vocabulary
optimization
concavity
degree
cusp
corner
polynomial
lead coefficient
end behavior

Key Attitudes
Math is about thinking creatively.

Key Ideas
A polynomial function is created when a series of terms are connected with addition such that each term is comprised of a real number coefficient and a variable raised to a whole number power. (This is not really a definition, but it captures what will be most important to us.)
The domain of a polynomial is all real numbers. Hence the graph of a polynomial will have to breaks.
Graphs of polynomials are smooth-- no pointy parts.
Graphs of polynomials have no horizontal asymptotes.
A lower bound for the degree of a polynomial can be inferred from the “bumps” on the graph of the polynomial.
The behavior at the “ends” of a polynomial (way out where x is really small or really big) can be used to determine if the degree of the polynomial is odd or even and whether the coefficient of the highest degree term (the lead coefficient) is positive or negative.
The behavior of the graph of the polynomial at the locations where it crosses the x-axis can be used to tell you the multiplicity of the real roots of the function-- tells you what degree each factored term has to be raised to.
Key Skills
I can determine if a graph could represent a polynomial function.
I can determine a lower bound on the degree of a polynomial by examining its graph.
I can determine if the degree of a polynomial is odd or even and if the leading coefficient is positive or negative by examining its graph.
I can determine if a given value is a zero of given polynomial when given a graph or the equation of the function.
I can given polynomial equation could be represented by a particular graph.
I can determine if a given polynomial could be a factor of another polynomial without using long division or graphing.
I can use a graph and long division to completely factor a polynomial (that is, to write the polynomial as a product of irreducible factors).
I can create a polynomial function to match descriptions involving the degree and the roots of the polynomial.
I can find a possible equation for a polynomial function when given its graph.
Turn-In (#30)
HW 11

Handouts & Sketches
Workshop 9
Workshop 9- Problem 8 Sketch
Workshop 9- Problem 10 Sketch

Assignment
Workshop 9
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, November 20, 2008

Geometry (Class 30)

Announcements
Test Monday! Logic Puzzles and Proofs

Lesson Title
The Game of Proof (3)

Overview
The warm-up today continues looking at relationships between distance and time as represented on a graph. In addition students will work a few more logic puzzles as review for the test on Monday. The lesson for the day continues building student’s ability to write two column proofs.
Textbook Sections
§3.3 (Txt. p.143) Parallel Lines and Transversals

Vocabulary
implication
deduction
syllogism
two-column proof
statement
reason

Key Attitudes
Math is about being convinced a statement is always true.

Key Ideas
Proofs can be thought of as a game where the game pieces are definitions, postulates, and theorems and the objective is to build a logical chain of conditional statements (a syllogism) using these pieces to connect the “given” to the “prove”.
Key Skills
I can build a syllogism (an If…, then… chain of reasons).
I can label a diagram with the information given in a proof.
I can use my knowledge of angle relationships to figure out a plan for proving a statement.
I can organize the angle relationships to build a logical chain of reasoning going from what is given to what is to be proved.
I can use the definitions, postulates, and theorems the class has developed to justify steps I take to solve a missing angle puzzle.
I can write a two column proof to represent the process of solving a missing angle puzzle.

Turn-In (#29)
Txt. p.146 #8, 9, 11, 12, 17-19, 21, 22, 27

Handouts
No Handouts Posted

Assignment
Txt. p.146 #10, 13, 15, 16, 20, 23, 25, 26
Txt. p.149 Quiz 1 #1-8
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.

Wednesday, November 19, 2008

Intro to Calculus (Class 30)

Announcements
The next test will be on Friday November 21 and will focus on optimization problems, roots of polynomials, and graphs of polynomials. (Homework 9 to 11 and workshops 7 and 8).

Lesson Title
Graphs of Polynomials (2)

Overview
The warm-up for today is another SAT practice question. The lesson for the day examines at what can be deduced about a polynomial function given its graph.
Textbook Sections
N/A

Vocabulary
optimization
concavity
degree
cusp
corner
polynomial
lead coefficient
end behavior

Key Attitudes
Math is about thinking creatively.

Key Ideas
A polynomial function is created when a series of terms are connected with addition such that each term is comprised of a real number coefficient and a variable raised to a whole number power. (This is not really a definition, but it captures what will be most important to us.)
The domain of a polynomial is all real numbers. Hence the graph of a polynomial will have to breaks.
Graphs of polynomials are smooth-- no pointy parts.
Graphs of polynomials have no horizontal asymptotes.
A lower bound for the degree of a polynomial can be inferred from the “bumps” on the graph of the polynomial.
The behavior at the “ends” of a polynomial (way out where x is really small or really big) can be used to determine if the degree of the polynomial is odd or even and whether the coefficient of the highest degree term (the lead coefficient) is positive or negative.
The behavior of the graph of the polynomial at the locations where it crosses the x-axis can be used to tell you the multiplicity of the real roots of the function-- tells you what degree each factored term has to be raised to.
Key Skills
I can determine if a graph could represent a polynomial function.
I can determine a lower bound on the degree of a polynomial by examining its graph.
I can determine if the degree of a polynomial is odd or even and if the leading coefficient is positive or negative by examining its graph.
I can determine if a given value is a zero of given polynomial when given a graph or the equation of the function.
I can given polynomial equation could be represented by a particular graph.
I can determine if a given polynomial could be a factor of another polynomial without using long division or graphing.
I can use a graph and long division to completely factor a polynomial (that is, to write the polynomial as a product of irreducible factors).
I can create a polynomial function to match descriptions involving the degree and the roots of the polynomial.
I can find a possible equation for a polynomial function when given its graph.
Turn-In (#29)
Homework 11 #TBA

Handouts
No Handouts Posted

Assignment
Finish HW 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.

Tuesday, November 18, 2008

Geometry (Class 29)

Lesson Title
The Game of Proof (1)

Overview
The warm-up for the day continues examining the qualitative relationships represented by a graph relating distance and time. The lesson for the day continues with “The Game of Proof” and the development of students abilities to create two column proofs.
Textbook Sections
§3.3 (Txt. p.143) Parallel Lines and Transversals

Vocabulary
implication
deduction
syllogism
two-column proof
statement
reason

Key Attitudes
Math is about being convinced a statement is always true.

Key Ideas
Proofs can be thought of as a game where the game pieces are definitions, postulates, and theorems and the objective is to build a logical chain of conditional statements (a syllogism) using these pieces to connect the “given” to the “prove”.
Key Skills
I can use symbols to represent a conditional statement (an “If…, then…” statement).
I can identify the hypothesis and conclusion of a conditional statement
I can build a syllogism (an If…, then… chain of reasons).
I can state the conclusion of a syllogism.
I can create the contrapositive of an implication.
I can create the inverse of an conditional statement.
I can write conditional statements.
Turn-In (#28)
Poster- Logic Puzzle or Missing Angle Puzzle

Handouts
No Handouts Posted

Assignment
Txt. p.146 #8, 9, 11, 12, 17-19, 21, 22, 27
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, November 17, 2008

Intro to Calculus (Class 29)

Announcements
The next test will be on Friday November 21 and will focus on optimization problems, roots of polynomials, and graphs of polynomials. (Homework 9 to 11 and workshops 7 and 8).

Lesson Title
Graphs of Polynomials

Overview
The warm-up today focuses on SAT preparation. The rest of the class time will be used for finishing work on Workshop 8, Homework 10, and Homework 11
Textbook Sections
N/A

Vocabulary
optimization
concavity

Key Attitudes
Math is about thinking creatively.

Key Ideas
The locations where the graph of a function crosses the x-axis can be used to find factors of the function.
A polynomial can have up to as many roots as its degree.
A polynomial function is smooth and has a domain of all real numbers.
The graphs of some polynomials functions do not have x-intercepts.
Key Skills
I can use a graphing calculator to locate x-intercepts of a polynomial.
I can use the x-intercepts to express a polynomial as the product of irreducible factors.
I can use long division to find factors of a polynomial.
Turn-In (#28)
Homework 10- Finish it.
Workshop 8

Handouts
Homework 11

Assignment
Homework 11 #TBA
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, November 14, 2008

Geometry (Class 28)

Lesson Title
Logic Puzzles (2)

Overview
The warm-up today is the article “You Can Grow Your Intelligence” which is based on the work of Carol Dweck. The lesson for the day continues with logic puzzles and how to use mathematics to help solve them.
Textbook Sections
§3.3 (Txt. p.143) Parallel Lines and Transversals

Vocabulary
implication
deduction
syllogism

Key Attitudes
Math is about thinking creatively.

Key Ideas
Logic problems can be solved by symbolizing the statements and building a chain of reasons (a “syllogism”).
Key Skills
I can use symbols to represent a conditional statement (an “If…, then…” statement).
I can identify the hypothesis and conclusion of a conditional statement
I can build a syllogism (an If…, then… chain of reasons).
I can state the conclusion of a syllogism.
I can create the contrapositive of an implication.
I can create the inverse of an conditional statement.
I can write conditional statements.
Turn-In (#26)
Txt. p.796 #10, 11
Txt. p. 47 #37, 38, 43, 44, 47, 48, 50, 51

Handouts
No Handouts Posted

Assignment
Test Corrections
Txt. p.796 #12, 13
Txt. p.101 #35, 36, 41, 42, 46-50

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, November 13, 2008

Intro to Calculus (Class 28)

Announcements
The next test will be on Friday November 21 and will focus on optimization problems, roots of polynomials, and graphs of polynomials. (Homework 9 to 11 and workshops 7 and 8).

Lesson Title
Roots of Polynomials (2)

Overview
The warm-up for today consists of a few SAT prep. questions. The remainder of the class is dedicated to working on Homework 10 and Workshop 8. A nice review of the concepts and skills connected to graphs of polynomials can be found at Purple Math.
Textbook Sections
N/A

Vocabulary
optimization
concavity

Key Attitudes
Math is about thinking creatively.

Key Ideas
The locations where the graph of a function crosses the x-axis can be used to find factors of the function.
A polynomial can have up to as many roots as its degree.
A polynomial function is smooth and has a domain of all real numbers.
The graphs of some polynomials functions do not have x-intercepts.
Key Skills
I can use a graphing calculator to locate x-intercepts of a polynomial.
I can use the x-intercepts to express a polynomial as the product of irreducible factors.
I can use long division to find factors of a polynomial.
Turn-In (#27)
Homework 10 #TBA

Handouts
Workshop 8

Assignment
Homework 10- Finish it.
Workshop 8 #TBA
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.

Wednesday, November 12, 2008

Geometry (Class 27)

Lesson Title
Logic Puzzles (2)

Overview
The warm-up today is the article “You Can Grow Your Intelligence” which is based on the work of Carol Dweck. The lesson for the day continues with logic puzzles and how to use mathematics to help solve them.
Textbook Sections
§2.1 (Txt. p.71) Conditional Statements

Vocabulary
implication
deduction
syllogism

Key Attitudes
Math is about thinking creatively.

Key Ideas
Logic problems can be solved by symbolizing the statements and building a chain of reasons (a “syllogism”).
Key Skills
I can use symbols to represent a conditional statement (an “If…, then…” statement).
I can identify the hypothesis and conclusion of a conditional statement
I can build a syllogism (an If…, then… chain of reasons).
I can state the conclusion of a syllogism.
I can create the contrapositive of an implication.
I can create the inverse of an conditional statement.
I can write conditional statements.
Turn-In (#26)
Txt. p.796 #10, 11
Txt. p. 47 #37, 38, 43, 44, 47, 48, 50, 51

Handouts
You Can Grow Your Intelligence

Assignment
Test Corrections
Txt. p.796 #12, 13
Txt. p.101 #35, 36, 41, 42, 46-50

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, November 10, 2008

Intro to Calculus (Class 27)

Lesson Title
Roots of Polynomials (1)

Overview
The warm-up for today focuses on research conducted by Carol Dweck focused on peoples perceptions of intelligence and how these perceptions affect learning. The lesson for the day focuses on roots of polynomials and using these roots to express a polynomial as a product of irreducible factors.
Textbook Sections
N/A

Vocabulary
optimization
concavity

Key Attitudes
Math is about thinking creatively.

Key Ideas
The locations where the graph of a function crosses the x-axis can be used to find factors of the function.
A polynomial can have up to as many roots as its degree.
A polynomial function is smooth and has a domain of all real numbers.
The graphs of some polynomials functions do not have x-intercepts.
Key Skills
I can use a graphing calculator to locate x-intercepts of a polynomial.
I can use the x-intercepts to express a polynomial as the product of irreducible factors.
I can use long division to find factors of a polynomial.
Turn-In (#26)
Homework 9

Handouts
Homework 10

Assignment
Homework 10 #TBA
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.

Sunday, November 09, 2008

Geometry Test 5

The tests are graded and the results have been posted. Here is a copy of Test 5B.

Friday, November 07, 2008

Geometry (Class 26)

Lesson Title
Logic Puzzles (1)

Overview
The warm-up for the day is our last “Fit the Units” problem where students are asked to match the correct metric units to a story. Students also have an opportunity to work another Model Drawing Problem involving circles on the coordinate plane.
In the lesson today students are introduced to logic puzzles and techniques for representing these using symbols. Students learn that the symbols can help them construct a chain of reasons to arrive at a conclusion.
Lastly, students will be taking their 5th test.
Textbook Sections
§2.1 (Txt. p.71) Conditional Statements

Vocabulary
implication
deduction
syllogism

Key Attitudes
Math is about thinking creatively.

Key Ideas
Logic problems can be solved by symbolizing the statements and building a chain of reasons (a “syllogism”).
Key Skills
I can use symbols to represent a conditional statement (an “If…, then…” statement).
I can identify the hypothesis and conclusion of a conditional statement
I can build a syllogism (an If…, then… chain of reasons).
I can state the conclusion of a syllogism.
I can create the contrapositive of an implication.
I can create the inverse of an conditional statement.
I can write conditional statements.
Turn-In (#25)
Finish Chapter 2- Lesson 1: Missing Angle Puzzles
Practice Missing Angle Puzzles
Txt. p.796 #8, 9
Txt. p. 47 #34-36, 42, 46, 49

Handouts
Chapter 2- Lesson 2: Logic Puzzles

Assignment
Txt. p.796 #10, 11
Txt. p. 47 #37, 38, 43, 44, 47, 48, 50, 51
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, November 06, 2008

Intro to Calculus (Class 26)

Lesson Title
Optimization (2)

Overview
The warm-up for the day continues with the theme of filling up bottles and creating a graph for the height as a function of time.
The lesson for the day continues to focus on finding the optimal solution to a problem. In order to accomplish this task students need to create a drawing to represent the situation, determine the variables needed, declare the variables, create a function relating the variables, and then use their graphing calculators to create a graph and locate a maximum or minimum value.
Textbook Sections
N/A

Vocabulary
optimization
concavity

Key Attitudes
Math is about thinking creatively.

Key Ideas
An optimal solution can be found by creating function, graphing the function, and then finding the maximum or minimum value.
Key Skills
I can create a function to represent a situation.
I can use a graphing calculator to find the optimal solution to a problem.

Turn-In (#25)
Nothing to turn in.

Sketches
Homework 9- Problem 9
Homework 9, Problem 10

Assignment
Homework 9- Finish it.
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.