Rate of Change and Slope

Topics for Today:

We will continue our discussion about functions as we explore linear functions (lines).  ALL LINES (with the exception of vertical lines) are functions.  This unit will cover many different aspects of line, beginning with rate of change, otherwise known as slope.  We associate slope with the "steepness" of a line.  Slopes can be positive, negative, zero, or undefined.

Slope is a 2-dimensional concept.  We will see how fast something rises (goes up) compared to how fast it travels in a horizontal direction.  Slope is defined as the change in the y-coordinate divided by the change in the x-coordinate.  To calculate slope, you need any two points on a line.  It does not matter where you start as long as you start in the same place for each component.

Vocabulary: rate of change, slope

Sections Covered in Textbook:

6-1: Rate of Change and Slope (pages 282-289)

Resources & Tutorials:

1) What does the slope of a line mean?
2) How do you find the slope of a line from two points?
3) How do you find the slope of a line from a graph?

4) Rate of Change and Slope class notes

5) Rate of Change and Slope notes - completed

6) Recording of Today's Class (see google classroom for password)

Assignments:

1) Rate of Change and Slope Worksheet
2) Delta Math due Friday by 8 AM

Chapter 5 Review

Today We Discussed:

We did not talk about any new topics today.  Instead, I provided the students with a study guide for Chapter 5.  This is a required assignment and will count as part of their weekly homework grade.   Chapter 5 Test will cover sections 5-1 through 5-4.

• We did not discuss section 5-5 (Direct Variation - we'll cover this later) nor 5-6 which deals with Arithmetic Sequences, so they will not be on the test. 
• The Chapter 5 Review in your textbook spans pages 275-277.

Sections Covered in Textbook:

Chapter 5 - sections 5-1 through 5-5 (pages 236-266).

Resources & Tutorials:

The blog entries for chapter 5 are:

• Relating Graphs to Events

• Relations and Functions

• Function Rules, Tables, and Graphs

• Writing a Function Rule

• Evaluating Functions

Review Sheet Answers

• 7th Grade Review Sheet Answers
• 8th Grade Review Sheet Answers

Assignments:

1) Review Packet

• 7th Grade Packet
• 8th Grade Packet

2) Chapter 5 Test tomorrow (NOTEBOOKS DUE)

3) Delta Math Assignment Due Friday by 8:00 AM

Evaluating Functions

Today We Discussed:

Relations, functions, domain and range and evaluating functions are still the topic of the day.  We worked on evaluating functions using tables, and sketching the graph.  We worked more with evaluating functions for various domain (x) values that involve both numbers and variables.  We also used function notation to translate values into coordinate points and further practiced determining the domain and range of functions and relations from a set of points and from graphs. 

Graphic credit: https://www.mathbootcamps.com/function-notation-and-evaluating-functions/

Sections Covered in Textbook:

No new sections were covered in the book today. 

Resources & Tutorials:

1) Evaluating a function from a graph. (video)
2) Evaluate a function from a graph.  (online practice)

3) More on Evaluating Functions class notes

Assignments:

1) Function Notation and Evaluating Functions Worksheet
2) Set up Delta Math account (teacher code 690046) and do equations/inequality assignment.

3) Test Tuesday over Chapter 5

Writing a Function Rule

Today We Discussed:

We continued our discussion about functions and explored how to write a function rule (basically this is an equation) from a table of values or a graph of coordinate points.  When we move deeper into linear functions, finding the slope, and graphing, we'll take a look at how to deduce a function rule that involves more than one operation.

Sections Covered in Textbook:

5-5: Writing a Function Rule (pages 254-260)

Resources & Tutorials:

1) How do you write a rule from a table? 
2) Finding the function rule from a table (more complicated examples).

3) Writing a Function Rule class notes

Assignments:

1) Writing a Function Rule worksheet

Function Rules, Tables, and Graphs

Today We Discussed:

We expanded our work on functions today with an exploration of graphing.  We discussed three different ways to view a function (as an equation, as a table, and as a graph).  We played around with linear, absolute value, and quadratic functions.

One aspect of functions that is nice is that we can pick what values to use for our domain, and this is especially helpful when dealing with functions containing fractions.  We can pick numbers for the domain that multiply to give us whole numbers, to make our graphs easier to draw.  I will continue to reinforce to students to choose wisely when picking values for the domain.

We also discussed the generic shapes of the three types of functions we talked about.  Linear functions will create a line; absolute value functions will generate a "V"; quadratic equations will generate a "U" or what is called a parabola. 

Vocabulary: independent variable, dependent variable

Sections Covered in Textbook:
5-3: Function Rules, Tables, and Graphs (pages 247 - 252)

Resources & Tutorials:

1) How do you graph a linear function using a table?
2)  Graph an absolute value function from a table.
3)  Graph quadratic function from a table.
4) 6 Mini-Coordinate Planes for Graphing

5) Function Rules, Tables, and Graphs class notes

Assignments:

1) Function Rules, Tables, and Graphs Worksheet
2)  Planning ahead - Quiz Thursday over graphs and functions.

Relations and Functions

Today We Discussed:

Today we explored what relations are, and discussed the concepts of domain and range.  We also defined functions as special relations where there is exactly one range value for each given domain value.  We modeled the different ways of representing a relation:  set of ordered pairs, table, mapping diagram, and graph, and used the vertical line test to determine if a graph is a function or not.  We also discussed why graphs that fail the vertical line test are not functions.

Many students will try to "force" a relation to be a function, or will feel like it is "bad" if a relation is not a function.  We discussed this tendency, and why it's perfectly fine to have a relation that is not a function.  Just because something does not fit a particular definition does not make it bad or good - it just is what it is!

Finally, we took a look at function notation.  Most students are intimidated by this method of representing an equation in two variables.  They are mostly comfortable with an equation of the form

y = 2x + 3 

We discussed that f(x) = 2x + 3 is just a fancy way of writing the above equation, and it can be described by saying "there is some function that uses the variable x, where the function rule is 2x+3. 

We discussed evaluating functions for given domain values, to produce range values.

DIXI-ROYD was also reinforced today.

Vocabulary:  relation, domain, range, function, vertical line test, mapping diagram, DIXI-ROYD, function rule, evaluate function, function notation

Sections Covered in Textbook:

5-2: Relations and Functions (pages 241-246)

Resources & Tutorials:

1) What is a relation?
2) What is domain?
3) What is the range of a relation?
4) How do you find the domain and range of a relation?
5) What is a function?
6) How do you figure out if a relation is a function?
7) What is function notation?
8) How do you find f(x) if given a value for x?

9) Relations and Functions class notes

Assignments:

1) Functions and Relations Worksheet
2) Dice Project due FRIDAY (7th and 8th)

Relating Graphs to Events

Today We Discussed:

We began our next unit on functions and graphs.  Today we related graphs to real events, by analyzing graphs that relate to time passing.  The shape of the graph can tell a story of what is happening as time passes.

For graphs in general, we will be talking about independent and dependent variables.  Time is considered an independent variable with other variables dependent upon it.



Projects are due tomorrow.  Check yesterday's blog for a list of all items that should be included.  NO LATE PROJECTS will be accepted.

Sections Covered in Textbook:

5-1: Relating Graphs to Events

Resources & Tutorials:

1) Relating Graphs to Events
2) Relating Graphs to Events - another video

3) Relating Graphs to Events class notes

Assignments:

1) Relating Graphs to Events Worksheet
2) Projects due TOMORROW (7th) or FRIDAY (8th)!

Dice Project - Finishing Up

Today We Discussed:

We used today's class to finish our discussion about probability, and to work on finishing projects.  All students are encouraged to type their conclusions using Google Docs.  Reminder, the finished project should include the following items:

• Rubric
• Printout of Data from 3 Trials of Rolling Dice
• Summary of Trial Data
• Frequency Tables (4)
• Theoretical Outcomes
• Histogram Containing 3 Trials
• Histogram of Trial Averages vs Theoretical Outcomes
• ThinkSheet
• Written Conclusion (Typed preferred)

Sections Covered in Textbook:

None

Resources & Tutorials:

1) Project Packet (if needed)
2) Discussion Questions (if needed)

Assignments:

Project Due THURSDAY (7th)  or FRIDAY (8th)