Graphing Absolute Value Equations - January 30th

Topics for Today:

Today we took a quiz covering solving systems of equations and applications of systems.  We are going to finish up this unit with a little more work on graphing absolute value equations, using both manual (paper and pencil) method as well as through a DESMOS activity.

Sections Covered in Textbook:

None today

Resources & Tutorials:

1) Graphing Absolute Value Equations Introduction

Assignments:

1) Absolute Value Graphing Activity
2) Chapter 6 Test Corrections Due Monday
3)  Turn in missing work ASAP

Systems of Linear Inequalities - January 29th

Topics for Today:

We expanded our discussions about linear inequalities and systems to include the topic of systems of linear inequalities.  We discovered in our lesson yesterday that linear inequalities include all the points on one side of a border.  When we combine two linear inequalities, we are going to look for where both overlap.  The only way to represent this overlap region is by graphing.  (Recall that we discussed and practiced three different ways of solving linear systems - graphing, substitution method, and elimination method.)

The solution to the system

y < 2x + 1 and

y > 1/2 x -3 

looks like this:

The red region represents the overlap,

and therefore the solution to the system.

Vocabulary:  system of linear inequalities, solution of a system of linear inequalities

Sections Covered in Textbook:

7-6: Systems of Linear Inequalities (pages 377-384)

Resources & Tutorials:

1) What is a system of inequalities? 
2) How do you solve a system of inequalities by graphing?

Assignments:

1) Solving Systems of Linear Inequalities Worksheet
2) Quiz Thursday over Solving Systems of Equations only (7-1 to 7-4)

Linear Inequalities - January 28th

Addendum:

Hmm.  Sorry this did not go out last night.... 

Topics for Today:

We are still finding our way around the coordinate plane, and today we discussed how to graph the solution to a linear inequality.  A linear inequality describes a region of the coordinate plane that has a boundary line.  The solution to a linear inequality are all the coordinate points that make a linear inequality true.  The solutions to inequalities contain infinitely many more solutions than that of equations, and the same is true for linear inequalities.

Solutions to linear inequalities involve graphs.  The process for graphing linear inequalities is very similar to graphing linear equations with a few additional details.  The basic process for graphing linear inequalities is

1. Treat the inequality just like an equation.  Use the equation to graph the boundary line.
2. Determine if the boundary line is a part of the solution
• For equations that are strictly greater or less than (>  or <), the boundary is NOT included and should therefore be a dashed line.
• For equations that are greater than or equal to or less than or equal to (≥ or ≤) the boundary line IS included and should therefore be drawn as a solid line.  
3. Next, determine which side of the line the solution points fall.  The best way to accomplish this is to pick a point on either side, and test the inequality for truth.  The point that generates a true statement is on the side of the line with the solution.
4. Once the correct side of the boundary is found, shade this region to indicate where the solutions are.  

Vocabulary: linear inequality, solution of an inequality

Sections Covered in Textbook:

7-5: Linear Inequalities (pages 370-375)

Resources & Tutorials:

1. What is a linear inequality?
2. How do you figure out if the boundary is part of the graph of the inequality?

Assignments:

1) Finish Systems Applications Worksheet (2, 4, & 5)
2 )Linear Inequalities Worksheet
3)  Quiz Thursday over Solving Systems of Equations only (7-1 to 7-4)

Applications of Linear Systems - January 27th

Topics for Today:

One of the things that is most annoying about Algebra I is the focus on the processes and procedures for solving equations, inequalities, and problems.  Most of the time we are focused on process rather than application, but this foundational toolset is critical to solving problems requiring higher thinking and reasoning.

Today we used our knowledge of solving systems of equations to solve some real-world problems.  Typically students lack confidence when solving story problems, although it is these very problems where we get to use all the skills we have been building.  As I continually reinforce to our students, mastering Algebra requires repetition and practice, like any other skill we hope to master.  The only way to become competent and confident solving story problems is to do them -- LOTS of them.

The main thing to remember when solving the linear systems we have been working on is that if we have two variables, we will need two equations to solve.  The same would be true for three variables (a topic for Algebra II where you need three equations).

For these story problems, first we must identify and define our variables.   Second, we will analyze the given information and write our equations based upon the given information.  Once we have our equations, we can determine the best method to solve the system.  Finally, we must look at the question that was asked and make sure that our solution answers the question, that we have the correct units, and that our answer makes sense.

Many of these story problems follow a pattern, and identifying the pattern makes the problem easier.  For this topic, we normally have several patterns to choose from:  mixtures, distance-rate-time (these can come in many forms, and can deal with things like water and wind currents that speed up or slow down the traveler), and break-even.

Sections Covered in Textbook:

7-4: Applications of Linear Systems (pages 362-368)

Resources & Tutorials:

1) How do you solve a word problem using two equations? 
2) Simple word problem resulting in two equations (not a video)
3) Applications of Linear Systems Class Notes

Assignments:

1)  Applications of Linear Systems Worksheet (problems 1, 3, and 6)
2) Quiz Thursday over Solving Systems (7-1 to 7-4)

Solving Systems Using Elimination - January 23rd

Addendum:

Sorry; this was saved as a draft for some reason.  I thought I had it scheduled to go at 4pm yesterday....

Topics for Today:

We are still working on systems of linear equations.  Today, we discussed elimination method, and with a system of two equations, this method is really the preferred one.

Steps for Solving Using Elimination Method

1. In your original system, make sure both equations are in the same form (standard form works best!).  Line your equations up so the variables are aligned in columns.  
2. Determine which variable should be eliminated.  Look for matching numbers and opposite signs or create them using multiplication.  You may have to multiply both equations so that you can eliminate one variable.  
3. Eliminate the chosen variable.  Solve for the other variable.  
4. Take the value you found in Step 3 and substitute it into one of the original equations to solve for the other variable. 
5. Identify your solution – it will be an ordered pair!
6. Check both original equations with the solution you found.  

 Graphic Credit: http://www.hotelsrate.org/solving-systems-of-linear-equations-by-elimination-examples/

Vocabulary:  elimination method

Sections Covered in Textbook:

7-3: Solving Systems Using Elimination (pages 353-359)

Resources & Tutorials:

1. How do you solve a system of equations using the elimination by addition method? 
2. How do you solve a system of equations using the elimination by multiplication method?
3. What's another way of solving a system of equations using the elimination by multiplication method?
4. Solving Systems Using Elimination Class Notes

Assignments:

1) Solving Systems using Elimination Worksheet
2) Daily Problem Corrections

Solving Systems Using Substitution - January 22nd

Topics for Today:

We are still working on solving systems of equations, and introduced a new method today that uses algebra instead of graphing.  We have discussed the limitations of the graphing method, and why we might use algebra instead.  There are three algebraic methods used to solve systems:  substitution method, elimination method, and the matrix method.  We will learn and practice the first two; solving of matrices with systems of 2 or more equations is covered in Algebra II.

The general process for solving systems algebraically is the same.  First, we solve for one of the variables; then we substitute that solution into one of our equations to find the second variable.

For the substitution method, we follow this process:

1. In the original system, see if one variable is isolated; if not, then isolate a variable. 
2. Substitute the expression into the second equation.
3. Solve the equation for the first variable.
4. Substitute the solution found in step 3 into one of the original equations to solve for the other variable. 
5. Identify the solution as an ordered pair.
6. Check both original equations to ensure the solution works for both. 

(Graphic Credit: https://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/solve-by-substitution.php)

Vocabulary:  substitution method

Sections Covered in Textbook:

7-2: Solving Systems Using Substitution (pages 347-351)

Resources & Tutorials:

1) How to solve a system using substitution method.
2) Solving Systems of Equations by Substitution.
3) Solving Systems of Equations Using Substitution Class Notes

Assignments:

Solving Systems Using Substitution Worksheet.  **EXTRA CREDIT** - Check your solutions to problems #1-10 using DESMOS.  Class Code: 2GQ38V.

Solving Systems by Graphing - January 21st

Topics for Today:

We are still working with graphing linear equations, but we've expanded our conversation to include systems of linear equations.  A linear system of equations is simply two or more linear equations containing the same variables.  When we deal in generic equations, we almost always use the variables x and y; however, when we use systems to solve real problems, we may define our variable with different letters that better match our problem.  For instance, if we are talking about costs and revenue, we may choose to use c and r for our variables.

Systems of two linear equations have three possible types of solutions, because they are based upon where two lines intersect on a plane:  they either intersect nowhere, intersect at one point, or intersect at every point.  If there is a solution, it is represented as an ordered pair.

Summary of Systems of Equations
(Click Graphic to Enlarge)

Vocabulary:  system of linear equations, solution of a system of linear equations

Sections Covered in Textbook:

7-1: Solving Systems by Graphing (pages 340-345)

Resources & Tutorials:

1) What is a system of linear equations?
2) How do you solve a system using graphing?
3) What is a solution to a system of equations? 
4) What are the three types of solutions to a system of equations?

Assignments:

1) Solving Systems by Graphing Worksheet
2) Part 2:  Graphing Systems using DESMOS (Class Code BQVPKH)

Getting the House in Order - January 16th

Topics for Today:

Today we finished up our unit on linear equations with a unit test.  No new material was covered.

Students are reminded that they need to complete any missing or incomplete work as soon as possible. 

Sections Covered in Textbook:

None

Resources & Tutorials:

None

Assignments:

Complete and turn in any missing assignments.  Have a nice weekend!

All About Graphing Review - January 15th

Topics for Today:

No new material was covered in class today; instead, we reviewed finding slope, using the various forms of a line, and finding parallel and perpendicular lines in preparation for our test tomorrow over the chapter.  We did not cover the last 2 sections of the chapter and they will not be on the test.

To be proficient in the concepts covered in this chapter students will need to be able to move freely among the different line forms.

Students were provided a graphic organizer that includes all the main topics from this chapter.

Sections Covered in Textbook:

Test will cover sections 6-1 through 6-5
(We will cover 6-6 and 6-7 later in the school year).

Resources & Tutorials:

Blog Entries for this Chapter
1) Rate of Change and Slope
2) Slope Intercept Form
3) Standard Form
4) Point-Slope Form 1
5) Point-Slope Form Revisited
6) Parallel and Perpendicular Lines
7)  All About Linear Equations Graphic Organize
8) Chapter 6 Study Guide Answer Key

Assignments:

Study for Chapter 6 Test; Completing Chapter 6 Study Guide is highly recommended.

Parallel and Perpendicular Lines - January 14th

Topics for Today:

The slope of two lines can produce a special relationship between those lines.  Two such relationship are parallel lines and perpendicular lines.  Parallel lines exist in the same plane but will never intersect, and they always have the same slope.  Perpendicular lines are special because when they intersect, the lines form 90° angles.  The slopes of perpendicular lines are negative reciprocals of each other, and when those slopes are multiplied together, the result is -1.

We will be analyzing the slopes of two lines to determine if either relationship exists, and we will be deducing linear equations from a given point that is either parallel or perpendicular to the given line.

Parallel and perpendicular lines are always determined by the relationship of their slopes!

Vocabulary: parallel lines, perpendicular lines, negative reciprocal

Sections Covered in Textbook:

6-5: Parallel and Perpendicular Lines (pages 311 - 316)

Resources & Tutorials:

1) How do you find the slope of a line if you have a parallel line?
2)  How do you write an equation of a line in slope-intercept form if you have one point and a parallel line?
3) How do you find the slope of a line if you have a perpendicular line?
4) How do you write an equation of a line in slope-intercept form if you have one point and a perpendicular line?
5) How to tell if lines are parallel, perpendicular, or neither.

Assignments:

1) Parallel and Perpendicular Lines Worksheet
**Planning Ahead** Chapter 6 Test on Thursday (6-1 through 6-5)

Point-Slope Form Revisited - January 13th

Topics for Today:

It was evident from the collected homework covering point-slope form that the students need more more work on this topic, and to approach the concept from another angle.

Today we revisited point-slope form through a PowerPoint presentation (link below).  

The three different forms of a linear equation are all useful for different reasons.  Each one has a pattern to follow.  Once the patterns are mastered, linear graphing becomes much easier.  

Vocabulary: point-slope form

Sections Covered in Textbook:

6-4: Point-Slope Form and Writing Linear Equations (pages 304-309)

Resources & Tutorials:

1) What is the point-slope form of a linear equation?

2) How do you write an equation of a line in point-slope form if you have the slope and one point?

-----

3) Exploring Point-Slope Form (PowerPoint)

    *Support Worksheets

Assignments:

1) Redo Point-Slope Form Worksheet

2) Chapter 6 Test Thursday (Sections 6-1 through 6-5 only)

Introduction to Desmons - January 9th

Topics for Today:

Today we took a quiz covering topics associated with linear equations and their graphs.

We also began to explore the online tool of Desmos.  All students will be asked to have a Desmos account.  The easiest way to accomplish this is to login using a personal Google account.

Desmos is an excellent program and available to everyone who has a computer or other device and an internet connection.  It is a powerful tool for visualizing graphs easily.  We will be using this tool for the rest of the school year, and many of other local high schools use it regularly in their mathematics courses.

Sections Covered in Textbook:

No new sections were covered today.

Resources & Tutorials:

1) Learn Desmos - Graphing
2) Learn Desmos - Scientific

Assignments:

1) Introduction to Graphing Activity (click link)  Or go to student.desmos.com and type in code UQXTEX.  Make sure to log in before completing the activity.
2) Complete Exploring Linear Graphing Activity due Monday
***Planning Ahead***  Chapter 6 Test on Thursday

Point-Slope Form - January 8th

Topics for Today:

We discussed the point-slope form of a line today, which is the last form we'll be working with this year.  The point-slope form of a line is helpful when the given information is the slope and some coordinate point that is on that line.  We can use this form to write the equation of the line, and then change the line into a form that is more recognizable, like slope-intercept or standard form.

To work with point slope form, the slope (m) will be given, along with a specific coordinate point.

Vocabulary: point-slope form

Sections Covered in Textbook:

6-4: Point-Slope Form and Writing Linear Equations (pages 304-309)

Resources & Tutorials:

1) What is the point-slope form of a linear equation?
2) How do you write an equation of a line in point-slope form if you have the slope and one point?

Assignments:

1) Point-Slope Form Worksheet
2) Work on Exploring Linear Graphing Activity
3) Quiz Thursday (sections 6-1 through 6-3)
Point-Slope Form (6-4) WILL NOT BE ON THE QUIZ!!

Revision to Today's Post - January 7th

Topics for Today:

Our class actually did not meet today.  Despite leaving for school 1.5 hours prior to my class starting (normally it takes 25-30 minutes for me to get to school), I arrived at school late.  With the snowy road conditions it took me 2 hours to reach school, so we did not have class.

I normally prepare my blog ahead of time and have it scheduled to post at 4pm.  I was unable to intercept this entry, so what you see in the earlier post is what we will cover tomorrow. 

Assignments:

There is no new written homework.  I have instructed the boys to play in the snow and come back to school tomorrow with a good story!

Graphing Review - January 6th

Topics for Today:

Welcome back, students!  I hope everyone had a relaxing, refreshing, and fun break.  We have a lot of material to cover before our trimester ends in February, and we'll be focusing on graphing for most of that time.

Our focus is still on linear equations and their graphs.  So far we have covered finding the slope of a line given two points, slope-intercept form of a line, and standard form of a line.   We also reviewed vocabulary including topics like x- and y-intercepts.

As we further investigate the concepts of graphing, I will introduce the students to the use of Desmos, which is an online graphing calculator.  This resource is a very powerful tool that should help students visualize the "families" of graphs, and how varying one element of an equation will affect the size and shape of the graph.

Graph families will become more important as students move through their study of mathematics.  Understanding how the different components of generic equations can be changed to produce different graphs will be very helpful as students explore conic sections, graph trigonometric functions, and transition to graphing in polar coordinates.

Sections Covered in Textbook:

No new material was covered today.
We reviewed Sections 6-1 through 6-3 (pages 282-303)

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) What is a linear equation?
5) What is the y-intercept?
6) What is the slope-intercept form of a line?
7) What is the standard form of a linear equation?
8) How do you use x- and y-intercepts to graph a line in standard form?

Assignments:

1) Exploring Linear Graphing Activity
2) Slope, Slope-Intercept, and Standard Form Review Worksheet
3) Quiz Thursday (sections 6-1 through 6-4)