HolcombMath

Thursday, October 09, 2008

Intro to Calculus (Class 16)

Lesson Title
Inverse Functions

Overview
In the last class we worked through what it means for two functions to be inverses of each other: that each function undoes the other function. In today’s class we will continue our work with inverse functions by: examining what it implies in terms of the graph of the functions, by learning how we can find the inverse of a function analytically and why our process for doing this works, and by using both analytic and graphical tools for determining if an inverse function exists for a given function.
Textbook Sections
N/A

Vocabulary
function
domain
inverse function

Key Attitudes
Math is about thinking creatively.

Key Ideas
Functions can undo other functions.
If you graph a function and the function which undoes it, then the graphs will be reflections of each other across the line y = x.
An equation for a function which undoes another function can be created by working to undo each step of the original function starting with the last step and working backwards.
Functions which undo each other are called “inverses”.
The notation for inverse functions can be tricky-- f^-1 would be the inverse of the function f. The “-1” symbol’s location is crucial and looks a lot like other uses.
Key Skills
I can write an English sentence to describe what happens to a value when it is used as the input to a function.
I can write an English sentence to describe what happens to an output value of a function in order to “work it backwards” through the function.
I can evaluate a function for specific values when given the graph of the function.
I can create the graph of the inverse of a function using values from the original function when given the graph of the original function.
I can create a graph of the inverse of a function using the fact that inverse functions are reflections across the line y = x.
I can create a rule for the inverse of a function and express this rule using function notation.
Turn-In (#15)
Nothing to turn in. Homework 7 and the notes from lesson 7 will be due next class.

Handouts
Lesson for HW 7

Assignment
Notes for Lesson 7
HW 7
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, October 08, 2008

Geometry (Class 15)

Lesson Title
Using the Pythagorean Theorem

Overview
The warm-up for the day is another model drawing problem. This problem requires that students apply what they have learned about segment lengths and the Pythagorean theorem to points and segments on the coordinate plane. The lesson today focuses on applications of the Pythagorean theorem as we finish up our work, for now, on this concept.

We have a test next class.
Textbook Sections
§1.3 (Txt. p.17) Segments and their Measures

Vocabulary
length
distance
coordinates
ordered pair
diagonal
right triangle
leg(s) of a right triangle
hypotenuse
polygon

Key Attitudes
Math is about thinking creatively.

Key Ideas
The distance between points can be calculated.
The length of a line can be determined by thinking about it as a right triangle.
Key Skills
I can create a non-square rectangle, right triangle, or non-rectangular parallelogram on the coordinate plane when given the locations of two vertices of the shape.
I can determine the area of a figure drawn on dot paper.
I can create any of the possible squares whose vertices are on the dots of a 5 by 5 grid of dot paper.
I can determien the area of the square on the hypotenuse of a right triangle.
I can determine the length of the side of a square when I know the area of the square.
I can determine the length of a line segment using the locations of the endpoints of the segment.
Turn-In (#14)
Pythagorean Paths 4
Home-School Assignment 1: The Pythagorean Theorem (Due Friday)

Handouts
Model Drawing Problem: Distance and Midpoints 1

Assignment
Pythagorean Paths 5
Txt. p.124 #9, 10, 19-24, 44, 45
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, October 07, 2008

Intro to Calculus (Class 15)

Lesson Title
Inverse Functions

Overview
The warm-up for today focuses on finding a good graph to match a verbal description of a situation. The lesson for today starts us on the road for working with inverse functions.
Textbook Sections
N/A

Vocabulary
function
domain
inverse function

Key Attitudes
Math is about thinking creatively.

Key Ideas
Functions can undo other functions.
If you graph a function and the function which undoes it, then the graphs will be reflections of each other across the line y = x.
An equation for a function which undoes another function can be created by working to undo each step of the original function starting with the last step and working backwards.
Functions which undo each other are called “inverses”.
The notation for inverse functions can be tricky-- f^-1 would be the inverse of the function f. The “-1” symbol’s location is crucial and looks a lot like other uses.
Key Skills
I can write an English sentence to describe what happens to a value when it is used as the input to a function.
I can write an English sentence to describe what happens to an output value of a function in order to “work it backwards” through the function.
I can evaluate a function for specific values when given the graph of the function.
I can create the graph of the inverse of a function using values from the original function when given the graph of the original function.
I can create a graph of the inverse of a function using the fact that inverse functions are reflections across the line y = x.
I can create a rule for the inverse of a function and express this rule using function notation.
Turn-In (#14)
Homework After Test 2

Handouts
Folow-Up questions for the Lesson for HW 7
Intro to Calculus Homework 7

Assignment
Follow-up questions from the Lesson for HW 7
Homework 7 problems 1, 2
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, October 06, 2008

Geometry (Class 14)

Lesson Title
The Pythagorean Theorem

Overview
The warm-up for today is a model drawing problem which requires that the students to find the distance between two points by using the Pythagorean Theorem. The lesson today continues focusing on the Pythagorean theorem as we see another way to show that it always works.
Textbook Sections
§1.3 (Txt. p.17) Segments and their Measures

Vocabulary
length
distance
coordinates
ordered pair
diagonal
right triangle
leg(s) of a right triangle
hypotenuse
polygon

Key Attitudes
Math is about thinking creatively.

Key Ideas
The distance between points can be calculated.
The length of a line can be determined by thinking about it as a right triangle.
Key Skills
I can create a non-square rectangle, right triangle, or non-rectangular parallelogram on the coordinate plane when given the locations of two vertices of the shape.
I can determine the area of a figure drawn on dot paper.
I can create any of the possible squares whose vertices are on the dots of a 5 by 5 grid of dot paper.
I can determien the area of the square on the hypotenuse of a right triangle.
I can determine the length of the side of a square when I know the area of the square.
I can determine the length of a line segment using the locations of the endpoints of the segment.
Turn-In (#13)
Txt. p. 124 #5-8, 16-18, 31-33, 42, 43
Pythagorean Paths 3

Handouts
Model Drawing Problem: Lengths of Segments 5

Assignment
Pythagorean Paths 4
Home-School Assignment 1: The Pythagorean Theorem (Due Friday)
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, October 03, 2008

Intro to Calculus (Class 14)

Lesson Title
Reading Functions,Thinking Graphs

Overview
During class today students finish Workshop 4 and take test 2. Any additional time can be used to work on the homework for after the test.
Textbook Sections
N/A

Vocabulary
function
domain

Key Attitudes
Math is about thinking creatively.

Key Ideas
You can create functions by combining (compositing) other functions.
It is important to be able to move back and forth fluently between the graph of a function and the equation representing the function.
Key Skills
I can find the composition of two functions.
I can create a function to represent a situation involving area, perimeter, or volume.
I can use interval notation to describe the domain and range of a function (which may or may not be continuous) given the graph of the function.
I can identify expression involving a function as distances on a graph of the function and vide versa.
I can determine if the graph of a function is symmetric about the x-axis, y-axis, or origin.
I can determine if a function is odd or even when given the equation or the graph representing the function.
I can create a graph to represent the speed of a roller coaster car as a function of how far it has moved down the track when given a picture of the roller coaster track.
Turn-In (#13)
Workshop 4

Handouts
Homework After Test 2

Assignment
Homework After Test 2
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, October 02, 2008

Geometry (Class 13)

Lesson Title
Triangles and Squares

Overview
The warm-up for today continues with finding the perimeter of a dot paper shape and then using this information to solve a puzzle. Students also work on the fourth “Model Drawing Problem” relating to the lengths of segments. In the lesson for the day student will use their understanding of how to find the areas of dot paper shapes and their algebraic skills to learn about the most important theorem in high school math.
Textbook Sections
§1.3 (Txt. p.17) Segments and their Measures

Vocabulary
length
distance
coordinates
ordered pair
diagonal
right triangle
leg(s) of a right triangle
hypotenuse
polygon

Key Attitudes
Math is about thinking creatively.

Key Ideas
The distance between points can be calculated.
The length of a line can be determined by thinking about it as a right triangle.
Key Skills
I can create a non-square rectangle, right triangle, or non-rectangular parallelogram on the coordinate plane when given the locations of two vertices of the shape.
I can determine the area of a figure drawn on dot paper.
I can create any of the possible squares whose vertices are on the dots of a 5 by 5 grid of dot paper.
I can determien the area of the square on the hypotenuse of a right triangle.
I can determine the length of the side of a square when I know the area of the square.
I can determine the length of a line segment using the locations of the endpoints of the segment.
Turn-In (#12)
Model Drawing Problem 4
Txt. p. 124 #1-4, 13-15, 28-30, 40, 41

Handouts
Chapter 1- Lesson 2: Squares and Triangles
Model Drawing Problem: Lengths of Segments 4

Assignment
Txt. p. 124 #5-8, 16-18, 31-33, 42, 43
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, October 01, 2008

Intro to Calculus (Class 13)

Lesson Title
Reading Functions,Thinking Graphs

Overview
Today is a short day. During the class we will spend the majority of our time working on Workshop 4. We have a test on Friday and much of the material from Workshop 4 will help to review material that will be on the test. Here is a partial list of topics that will be addressed on the test:
Odd and even functions
“Function equations”- use a graph to solve problems like f(x+1) = f(x)
Determine if an equation represents a function
Composition of functions
Create a function to model a situation
Textbook Sections
N/A

Vocabulary
function
domain

Key Attitudes
Math is about thinking creatively.

Key Ideas
You can create functions by combining (compositing) other functions.
It is important to be able to move back and forth fluently between the graph of a function and the equation representing the function.
Key Skills
I can find the composition of two functions.
I can create a function to represent a situation involving area, perimeter, or volume.
I can use interval notation to describe the domain and range of a function (which may or may not be continuous) given the graph of the function.
I can identify expression involving a function as distances on a graph of the function and vide versa.
I can determine if the graph of a function is symmetric about the x-axis, y-axis, or origin.
I can determine if a function is odd or even when given the equation or the graph representing the function.
I can create a graph to represent the speed of a roller coaster car as a function of how far it has moved down the track when given a picture of the roller coaster track.
Turn-In (#12)
Finish Homework 6
Finish Workshop 3

Handouts
Intro to Calculus Workshop 4

Assignment
Workshop 4-- as much as possible. It is due at the end of the next class.
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.