Wrapping Up - December 19th

Topics for Today:

Today we reviewed standard form and discussed our homework.  For the entire unit on linear equations, we will be looking for patterns!

I hope everyone has a relaxing and fun vacation.  I made all the boys promise to not forget everything I've taught the over the break...

See you all in the new year!

Sections Covered in Textbook:

No new material was covered today.

Resources & Tutorials:

No new material was covered today.

Assignments:

None - have a wonderful break!  See you next year!

Standard Form - December 18th

Topics for Today:

Our discussion about linear equations continued today.  We have already explored slope and slope-intercept form of a line.  Today, we looked at a different form - standard form.  The standard form of a line is defined as a linear equation such that

Ax + By = C

A, B, and C must all be integers.

A must be positive.

Although it's easy to visualize a line that is in slope-intercept form (the form we worked with yesterday), it's very easy to find both the x- and y-intercepts when a line is in standard form.  These intercepts are where the line crosses the x- and y-axes, when one of our coordinates is zero.  Solving the equation when substituting a zero for a value is a quick process, because multiplying by zero removes the variable from the equation.  Once we find our intercepts, it's very easy to graph our equation.

Standard form can be nice for students who are not fond of working with fractions, and we'll be using standard form when we move to solving systems of equations in the next chapter.  In addition, many of the other graphs that students will see in later mathematics classes are written in standard form with the variables all on one side of the equation.  Comfort with standard form will help students cope when they are introduced to more complicated equations.

Vocabulary: standard form of a line, x-intercept

Sections Covered in Textbook:

6-3: Standard Form (pages 298-302)

Resources & Tutorials:

1) What is the standard form of a linear equation?
2) How do you use x- and y-intercepts to graph a line in standard form?

Assignments:

1) Standard Form worksheet

Slope-Intercept Form - December 17th

Topics for Today:

One of the most recognizable forms of a line is the slope-intercept form.  This line form is very useful because it's easy to visualize the actual line simply by looking at the equation.  From slope-intercept form, you can tell if the slope is positive or negative, and if the slope is steep or shallow, and also it demonstrates where the line crosses the y-axis (this is the y-intercept).

We talked about what an intercept is (this word sounds an awful lot like intersect!).

I reminded students of our work on solving literal equations - this skill will be especially helpful for our unit on linear equations, as we'll be looking at three different forms for a linear equation.  To put a line in slope-intercept form, simply solve for the variable "y".

Slope intercept form looks like this:  y=mx + b

• m is the slope
• b is the y-intercept 

Vocabulary: linear equation, y-intercept, slope-intercept form

Sections Covered in Textbook:

6-2: Slope-Intercept Form (pages 291-296)

Resources & Tutorials:

1) What is a linear equation?
2) What is the y-intercept?
3) What is the slope-intercept form of a line?

Assignments:

1) Slope-Intercept Form Worksheet
2) Chapter 5 Test Corrections due Wednesday

Rate of Change and Slope - December 16th

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?

Assignments:

1) Rate of Change and Slope Worksheet.
2) Chapter 5 Test corrections due Wednesday

Topics for December 11th

Today We Discussed:

We did not talk about any new topics today.  Instead, we thoroughly reviewed direct variation, how to find the constant of variation, and how to write a direct variation equation.

I also provided the student with a study guide for Chapter 5.  I will not be collecting this packet for a homework grade, but students are vehemently encouraged to complete the problems to ensure they have mastered all the concepts in Chapter 5.   Chapter 5 Test will cover sections 5-1 through 5-5.

• We did not discuss section 5-6 which deals with Arithmetic Sequences, so it 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:

• Reading Graphs
• Introduction to Relations and Functions
• Relation and Functions
• Function Rules, Tables, and Graphs
• Writing a Function Rule
• Finding Domain and Range from a Graph
• Evaluate a Function from a Graph
• Direct Variation

Assignments:

Study for Chapter 5 Test.  Completion of Study Guide Packet is HIGHLY RECOMMENDED!

Topics for December 10th

Today We Discussed:

Direct variation was the topic of today's class.  Direct variation is simply a function that can be expressed by the formula

y = kx 

where k is the constant of variation
and k ≠ 0. 

We looked at this relationship from many different perspectives: we determined if an equation represents a direct variation relationship, we derived direct variation equations, and we used the direct variation formula to solve problems.

Vocabulary: direct variation, constant of variation

Sections Covered in Textbook:

5-5: Direct Variation (pages 261-267)

Resources & Tutorials:

1) What is the formula for direct variation?
2) What is the constant of variation?
3) How do you use the formula for direct variation?

Assignments:

1) Direct Variation Worksheet
2) Quiz corrections due tomorrow
3) Chapter 5 Test on Thursday

Topics for December 9th

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 todays. 

Resources & Tutorials:

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

Assignments:

1) Function Notation and Evaluating Functions Worksheet
2) Quiz corrections due Wednesday
3) Chapter 5 Test Thursday

Topics for December 5th

Today We Discussed:

Today we took a quiz that covered interpreting graphs, relations, domain and range, functions and graphs. 

For the second part of class we looked at domain and range of graphs, and determining their elements by reading a graph.  We also talked about whether each graph represented a function.  Students will be expected to know the domain and range of functions for the rest of Algebra I, and this topic will be prevalent in Algebra II.

Domain: All Real Numbers

Range:  y <= 2

This is a function!

Sections Covered in Textbook:

No new sections were covered today.

Resources & Tutorials:

1) Find the domain and range from a graph

Assignments:

1) Finish Domain and Range of Graphs Worksheet if not finished in class.
2)  **Planning Ahead** - Chapter 5 Test Next Thursday, December 12th.

Topics for December 4th

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).

Assignments:

1) Writing a Function Rule worksheet
2)  Quiz Tomorrow over Relating Graphs to Events, Relations, Functions, and Graphs

Topics for December 3rd

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

Assignments:

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

Topics for December 2nd

Today We Discussed:

Welcome back, students!  I hope everyone had a relaxing and fun break!

Today we reviewed what relations are, and reinforced 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 is 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?

Assignments:

1) Functions and Relations Worksheet
2)  Planning Ahead:  Quiz Thursday covering graphs, relations, and functions.