7.3 Solving by Elimination

Hope you all worked hard on your quiz today.  My apologies for not being there.

As we move into solving systems of linear equations through elimination, it's important you track the signs of our terms.  Mistakes with negatives will make solving by elimination difficult.  If you need any additional practice (extra credit) with integer work, please let me know.

Please start by watching an overview of what solving linear equations through elimination is:

Solving Linear Equations by Elimination - Overview

Then, watch the videos for each section and do the related problems (#* done in video):

Solving Linear Equations by Elimination - Addition #4

HW: pg 394 #s 3, 4*, 5

Solving Linear Equations by Elimination - Subtraction #12

HW: pg 394 #s 12*, 13, 14

Solving Linear Equations by Elimination - Arrange & Add/Subtract #22

HW: pg 394 #s 22*, 23, 24

7.1 & 7.2 Quiz Tomorrow

Just reminding you that I have moved our quiz from Thursday to tomorrow.  It will not include "elimination" as a method for solving systems of linear equations.

Please be sure to do the Practice Quiz - Homework.  I am attaching videos below of the solutions for those problems.  My suggestion would be to watch the video after you attempt the homework similar to how we did the Practice Quiz - Classwork in class today.

Videos of Solutions to Practice Quiz - Homework

1. translating into algebra
2. solving systems of equations by graphing
3. solving systems of equations by substitution

If you still have questions, please feel free to email me:

jadams@a-cs.org

Good luck tomorrow!

7.2 Solving Linear Equations Word Problems through Substitution

Hope you all had a good weekend.  Your homework was to watch the video in the last post.  I know we're not supposed to give homework on weekends but this seemed pretty important;)

Anyway, I'm hoping you are feeling confident with solving linear equations by graphing.  It's a lot of work but the goal was to see how all the tools: grids, equations, tables & graphs interrelate.  

As we've begun work on solving systems of linear equations I've gotten a resounding, "Wow, this is so much quicker!"  True, but we still want to maintain a conceptual understanding of the conceptual process of finding a shared solution point.

Accordingly, your homework will include translation tasks (words to algebra) and two word problems.

Please watch this video of me solving #16 on your homework and then complete the following:

HW: pg 386 #s 8, 14, 15, 16

Then answer:

HW: pg 387 #s 25 & 26 (pay close attention to "The value of x is 4 times the value of y"
                                     as you might write this as 4x = y which is incorrect.)

Watch the second video of me solving #29 starting with a grid and then using substitution (instead of tables & graphs) once I have the equations set up.  Write down the notes from my work on # 29 and then solve your own word problem #30 using a grid to make your equations and then substitution.

HW: pg 387 #s 29 & 30

Kid Snippets: "Math Class" (Imagined by Kids)

7.2 Systems of Equations & Start of Solving by Substitution

We will wrap up some practice constructing systems of linear equations from word problems.  A hard part can be learning to translate English into Algebra and vice-versa. 

An example would be:

     Small cards cost $3 each and large cards cost $5 each.  The total amount from the
     sale of the cards is $95. 

     We can choose to make:
     x = the number of small cards sold and 
     y = the number of large cards sold.

     In this scenario 3x + 5y = 95 means $3 per small card plus $5 per large card adds up to

     a total of $95 dollars.  

     There are multiple ways of solving this first equation.  It's only when we say the total
     number of cards is 25, translated to x + y = 25 that we have a systems of equations
     with one common answer.

     In this scenario x + y = 25 translates to "the number of small cards plus the number of
     large cards is 25 cards.  Note that cost is not associated with this function.

HW: pgs 380-81 #s 31, 32, *36 (*do a grid/table/equation/graph for 36)

Today, we will also begin solving linear systems by substitution.  Please watch the following videos of me solving #3 & #4 from your HW, copy the work and answers down, and do the remaining HW problems:

AND HW: pg 386 #s 3-7

7.2 Solving Linear Systems by Substitution - Page 386, #3 - part 1

7.2 Solving Linear Systems by Substitution - Page 386, #3 - part 2
7.2 Solving Linear Systems by Substitution - Page 386, #4

7.1 Suggestions for Long-Term Success in Math

I was one of those students that had it pretty easy in math until about the time of algebra.  I could tune out the teacher most of the time and "figure out" the math I was required to do.  On some level I even passed through Algebra pretty well.

Unfortunately, instead of developing the tools I needed to succeed long-term in math I continued to look for quicker, easier ways of doing math.  By the time I got into Calculus my ability to just "figure out" math was not longer working.  I had not developed the problem-solving and reasoning I needed to continue into advanced mathematics.

Here are some simple (but time-consuming) suggestions:

• watch the videos at least once (jump ahead if it seems slow, review if you're uncertain)
• show all your work (done in columns so you can more easily track errors with negatives)
• check your answers (especially if the instructions ask you to)
• use our tools of algebra to help you:
• a grid for word problems
• a function table (x,y) with labels
• a graph with labels
• the function written as a linear equation (y=mx+b)

I'll quiz you tomorrow on these as an extra credit you can apply later.

Today, please watch the video of me solving question #3 on your homework.

HW: Practice Sheet 7.1, #s 1-4 
(For question #4, feel free to substitute non-blooming annuals cost $1.60 and the total for 24 plants is $48.)

7.1 Nice job! Now going deeper into systems of equations with word problems.

Well done on your quizzes last week!  I observed that a number of you took the time to use all three tools (function, table & graph) to solve the word problems on quiz.  Your process is getting stronger as you make mistakes, catch them and continue to work hard on your conceptual understanding of what you are doing.

You may be starting to realize that algebra is getting rather time-consuming.  This week there will be even more steps to solve our assigned tasks. As it gets more complicated it is important you understand what you are doing and why so you can remember the process for solving systems of equations.

Chapter 7 - Section 1
Basically we are combining the questions you answered on your quiz.  You will be solving word problems that have 2 linear equations instead of just one linear equation.

If this seems totally new to you, please review the videos from last week:

page 378 - Example 3
grid
function/equation
graph/table/solution

If you feel you're ready to try a more difficult problem, go to these videos:

page 380 - #34
grid
table/graph
function/solution

HW: pg 379-81 #s 18, 20, 30, 37 (show function/table/graph)  

6.5, 6.1 & 7.1 Pre-Quiz, Quiz Practice HW, Quiz

Hi, 

Don't freak out:)  We will do a pre-quiz today in class and will review it together.  You will have our pre-quiz from today to help you with tonight's practice quiz for homework.

Our quiz is tomorrow.  The format will look exactly like the pre-quiz and practice quiz. Like the pre-quiz it will have 4 questions:

1. graphing a perpendicular line through a given coordinate (chap/section 6.5)
2. graphing a linear equation on a regular graph from a word-problem (6.1)
3. graphing a linear equation on a "skewed" graph from a word-problem (6.1)
4. finding the solution to a system of linear equations (two lines) by graphing them (7.1)

If you have any areas that your are feeling uncertain review the online videos linked below and email me any specific questions:

• graphing a perpendicular line through a given coordinate (6.5)
• graphing a linear equation on a regular graph from a word-problem (6.1)
• graphing a linear equation on a "skewed" graph from a word-problem (6.1)
• finding the solution to a system of linear equations (two lines) by graphing them (7.1)

7.1 Word Problems with Graphing Linear Equations - Part Three

Hello!

I'm delighted to announce that I had three of you arguing yesterday about an algebraic solution.  Nothing warms my heart as students arguing, respectfully of course, over math.  Glad you're feeling adamant about your hard work!

As we move further into linear equations we will continue to use our tools: 

• equations like y-mx+b
• function tables (those x/y charts) and
• graphing

to helps us solve word problems.  Although it may seem cumbersome now to use all these tools, they payoff will be worth it when you can understand the concepts behind the procedures to better remember your process & computations.  It's also a great way to catch mistakes as I have frequently modeled for you:)

Please start by reviewing:

Solving Linear Equations by Graphing

and then move on to pg 378, EXAMPLE 3, shown below in the video lessons: 

Solving Word Problems with Linear Equations (make a grid) - optional
Solving Word Problems with Linear Equations (set up equation)
Solving Word Problems with Linear Equations (table, graph & test solution)

HW: pg 379, #s 13,14,16 & pg 380, #35

7.1 Word Problems with Graphing Linear Equations - Part Two

So basically there are three tools that we use in algebra to understand the work we are doing: (oh yeah, hope you had a good weekend and all that good stuff:)

1. formulaic expressions of linear equations (functions) such as our beloved slope-intercept form, y=mx+b, or standard form (used more in Chap. 7)
2. tables of coordinates (x,y) that list answers to the functions and show us patterns that helps us to identify changes like slope
3. graphs of solution sets (coordinates that work) on a cartesian plane or visuals representing the changes of variables on a scaled or skewed graph

As you read this list, it is important you know what these three tools or components are because we are working on adding a forth.  If you are not familiar with these tools please review the following video:

Systems Intro 3 - 3 Tools for Algebra

The forth area we will be working on has the actual real-world word problems that require the use of algebra to solve them, the reason we study algebra.  It's not just about getting ready for the next math class in our sequence, typically Geometry, it's about using algebra in applied settings.

Accordingly, please watch the following videos on graphing linear equations with positive and negative slopes.  Please watch them in order as the second video builds on the first and starts with a y-axis of 1970 which also ends up being 0 when we apply the equation.

Graphing Linear Equations (positive slope word problems)
Graphing Linear Equations (negative slope word problems)

Then do the homework questions as practice:

HW: pgs 327-8, #s 42, 43, 44, (write equations, show tables & graphs for each)

Challenge: pg 328, #51 (same three tasks)

7.1 Word Problems with Graphing Linear Equations - Part One

Happy Friday everyone!

Two things:

1)  Please be sure to bring your HW to class today and have it in order so I can check it.  I will be grading for effort, not correct answers.  Effort looks like organized attempts at answers (with work shown) or written questions about what you might not understand in the homework.  I plan to be pretty generous this week.

2)  We will be working on word problems with functions from Chapter 6 and graphing them to build fluency with graphing on scaled cartesian planes (similar to the graphs on the back of last night's worksheet).

classzone.com - 6.1 - example 5 - powerpoint

Our classwork will be graded today.  We will be working on:

Classwork: Practice Sheet 6.1, #s 16 & 17

Have a great weekend!

7.1 (c) Systems of Linear Equations - Solve by Graphing - Post 2

So hopefully you have a pretty good understanding of the meaning of finding the solution of two linear equations.  Tomorrow we will dive into applied real-world word problems that require the use of this type of algebra to solve them.

For today we will practice getting some fluency with finding the intersection of two lines which represents the solution for two different linear equations.  Please start by reviewing our 3 tools for doing algebra:

Systems Intro 3 - 3 Tools for Algebra

Solving Systems of Equations by Graphing 1

HW: Complete worksheet SLE#1

PLEASE MAKE SURE YOU BRING ALL HW FOR THE WEEK TOMORROW SO I CAN ENTER IT IN OUR GRADEBOOK

7.1 (b) Systems of Linear Equations - Solve by Graphing - Post 1

Looking at the Cartesian plane above you will see two distinct linear equations.  The equations are in standard form rather than slope-intercept form.  I've mentioned that we will be using standard form more as we progress through chapter 7.  Doesn't mean we have to like it (I don't) for graphing equations:)

So as you look at these two functions (equations) graphed above, you should notice that the lines represent a solution set (list of answers) for each linear equation.  I can prove this to you by plugging in and solving for one of the points:

• Looking at the blue line (4x-6y=12), I see it crosses the y-axis at (0,-2).  I will put the coordinates (x,7) for this point into the equation to show it is a correct solution to the function:
• 4x   -   6y     = 12
• 4(0) -  6(-2) = 12 (note that -6 times -2 equals +12)
• 0      +  12   = 12 
•             12   = 12  
• You can see that 12 does in fact equal 12 so this is a correct solution for this function.

Go ahead and prove that the coordinate (3,0), where the two lines intersect, is a solution for both linear equations by plugging that into both functions.

7.1 (a) System of Linear Equations (Don't worry you've already created one:)

I have three exciting videos for you today!

First, we will review

Parallel & Perpendicular Lines Review

HW: pg 357-8, #s 1*, 2*, 6 & 9 (* indicate written response questions)

Then, we will review the connect between representing equations (functions) as both lines on a cartesian plane as well as lists of coordinates.  

Systems Intro 1 - Line as set of answers (solutions)

HW: pg 325-6, #s 14, 20, 30*, 38 (* indicate written response questions)

The last video uses perpendicular lines to begin explaining how the point where two lines (equations) cross is a common answer (solution) for the two lines.  This is the definition of a solution of a system of linear equations.

Systems Intro 2 - Common Answer (Solution)

6.5 Negative Reciprocal v Negative Inverse

Great work yesterday!  I hope you were able to write some solid answers on questions #20 and #30 of the homework.  Regarding math vocabulary, I should be more specific with my own choice of words:

• When we are finding the slope of a line that is perpendicular to a given line, the textbook says to use the negative reciprocal.  I use the term negative inverse at times in the videos.  They are both ways of saying the same thing.
• The important thing is that you know to find the negative reciprocal (inverse).  

• An example would be:
• y = mx + b  where m = 5/1 , the negative reciprocal would be -1/5 (note the negative in front) not just 1/5 and not -5/1.  

• If you started with a negative slope it would become positive when you make an equation for the perpendicular line.  
• y = mx + b where m = -1/3, the perpendicular line would have a slope of 3/1 (positive)

As always, ask questions if that is unclear.  Thanks!

6.5 Equations w Parallel & Perpendicular Lines

Here are the links for 6.5:

6.5 Parallel Lines (Intro)

6.5 Write an Equation of a Parallel Line

6.5. Perpendicular Lines (Intro)

6.5  Write an Equation of a Perpendicular Line


Here is the HW assignment for 6.5:

6.5 pp 357-358  #3,5,7,20*,21,23,25,30* 

On questions #20 and #30, please explain your thinking using appropriate math vocabulary, visuals as needed and complete sentences.


#32-34 can also be done as extra credit and there will be a challenge activity for lines on a square.

6.5 Preview @HomeTutor on classzone.com

First day back was a little busier than I expected.  I have been unable to make a video for section 6.5 but there is a version at classzone.com  

I've used a screen recording to show you how to get to it.  Although I start by saying you need to set up an account to access it, you do not.  About a minute into the screen recording you can just click on @HomeTutor to access their videos:

Preview 6.5 @HomeTutor on classzone.com

There are no homework problems tonight.  We will work on them as classwork tomorrow.

Thanks for your patience as I learn all Mr. Mejia's instructional tools:)

Day 1 with Mr. Adams - Welcome Back!

Hello Algebra Students,

Hope you all had a wonderful winter vacation and enjoyed the holidays with your family and friends.

I am looking forward to teaching you while Mr. Mejia is out on paternity leave through the February break.  Although some things will be different about how I teach Algebra Concepts, I'm planning to keep more things the same for you during this time.  Using this blog and providing video lectures in advance of the classwork are two ways I hope to do this.

Today we will start by answering some questions about how you learn math: 

Concepts Survey

Then we will explore some of the basic concepts of how and why we use algebra to solve real-world math problems.  

Your homework will be to watch a video of tomorrow's lecture so we can do some problem-solving together in class. This will provide me with opportunities to help answer any questions you have.

You may choose to start with a review of slope.

Thank you for welcoming me to your classroom:)

Sincerely,

Mr. Adams