For Paula
MAT221: Introduction to Algebra Online
Week Overview:
Week Two Learning Outcomes
This week students will:
Utilize the logic of mathematical equality.
Execute basic equation solving techniques for both equations and inequalities.
Identify appropriate formulas for solving problems.
Implement solving techniques in real world scenarios.
Assignment Overview
| Activity | Due Date | Format | Grading Percent |
| Discussion 1: Formulas | Day 3 | Discussion | |
| Quiz: Week Two Quiz | Day 7 | Quiz | |
| Lab: ALEKS Lab | Day 7 | Lab | |
| Homework: Week Two ALEKS Homework | Day 7 | Homework |
Assignment Checklist:
Use this checklist to track when you’ve completed each assignment
____Discussion on Formulas (Due by Day 3, Thursday) Your initial post is what will be graded.
____Week Two Quiz (Due by Day 7, Monday)
____ALEKS Lab (Due by Day 7, Monday)
____ALEKS Homework (Due by Day 7, Monday)
It's a best practice to print this page and check off each assignment as it is completed.
See Assignment Details below for more information.
Office Hours
I hold office hours this week on Tuesday 4:00 – 5:00 p.m. EST. This is an opportunity to meet in real time and ask any questions you might have. We will be using e-mail for this. During my office hour, I will be online monitoring my e-mail and we can send notes back and forth to each other.
Remember that the Ask Your Instructor forum is always open for your important questions.
Intellectual Elaboration:
Hello Class: Welcome to Week 2!
Did you learn anything using the ALEKS program last week? Most students who used ALEKS will say that they learned a lot. Using ALEKS is an adaptive learning system and is almost like having your own personal tutor. You must get a concept correct a couple of times before ALEKS marks it as mastered. Getting a concept correct once, only shows the student is familiar with the work. It does not demonstrate mastery!
When you begin to solve a math equation, you start out with something like x + 3 = 7 or -3x = 9. As the work progresses you move up to harder problems such as 8x – 1 = 9+9x, 0.05r+0.4r = 27 and even (1/3)p – 5 = (1/4)p.
In the beginning you may be able to solve problems like the first two in your head. But that is not why they are there. It is essential to use those easier problems to practice solving equation using legal math methods. This is easiest to do on the shorter problems where you may be able to see the answer at a glance.
People often try to memorize what to do to solve math problems without understanding why each step is a correct solution method. This is not a strategy that will carry most college students as far in math as they need to go to earn their degree, and thus I do not recommend the memorizing method. Suing it makes math seem harder than it is.
We want to move toward the bulls eye, where the bulls eye is x = _____ or r = _____ or p = ____. We need to get the variable alone on one side with everything else on the other side. Here’s the kicker. We can only use legal math methods to move things around.
What is a legal math method? (sounds like Greek to me!)
Put simply, we can do a particular operation with a particular number (one at a time) to an equation as long as we do it to BOTH SIDES of the equation.
Example A: x + 3 = 7 we can subtract 3 from both sides. If we do, we find that x = 4.
Example B: -3x = 9 we can subtract 3 from both sides. If we do, we get that -3x-3 = 6.
What’s the challenge with subtracting 3 from both sides on example B? We can do it since it’s a legal math method, but does it make the equation more complicated or less complicated? Most would say it makes the equation more complicated since -3x = 9 appears to be simpler than -3x-3 = 6. So we can subtract 3 from both sides, but we don’t want to do that since that brings us farther away from a solution rather than closer to it.
Key question: How do we move toward the bulls eye, where the bulls eye is to get x = _____? In other words, how do we get x alone on one side of the equation and everything else on the other side?
Answer: We identify the operation that is being done to the number near the x and undo it.
Recall Example A: x + 3 = 7. The 3 is the number near the x that we need to move to the other side using our legal math methods. What operation is associated with the 3? Addition. I am adding 3 to x. So I need to undo adding 2 to x. I undo that by subtracting 3 from x. Since we have an equation, I cannot just subtract 3 from one side and be done with it. Our equation started off balanced. To keep it balanced, whenever we do an operation to one side, we must do that on the other side also. So, to solve for x, in x +3 = 7, we must subtract 3 from both sides. If we do that, we get x + 3 - 3 = 7 -3, and we find that x = 4.
Now let’s use this process on Example B. Recall Example B: -3x = 9. The -3 is the number near the x that we need to move to the other side using our legal math methods. What operation is associated with the -3? Multiplication. I am multiplying x by -3. So I need to undo multiplying x by -3. I undo that by dividing x by -3. Since we have an equation, I cannot just divide x by -3 on one side and be done with it. Our equation started off balanced. To keep it balanced, whenever we do an operation to one side, we must do that on the other side also. So, to solve for x, in -3x = 9, we must divide by -3 on both sides. If we do that, we get (-3x)/(-3) = 9/(-3), and we find that x = -3.
Don’t trust your instincts on this. Many students feel that adding 3 to both sides would be correct. Then an addition mistake is made on the left and terms that are not like terms are combined. This typically leads to an incorrect solution that will not check out.
What if we have two operations such as multiplication and subtraction, which should I undo first? Let’s look at Example C: 8x – 1 = 9+9x to address that. The first thing you may have noticed is that there is more than one x. Can I solve for x on one side and have a second x on the other side. The answer is no. When we solve for x and get x = ___. There cannot be another x in the blank, because then we’d have to know x in order to find x and that won’t work out well.
First, clear any parentheses (if there are any). This is essential. In Example C, there are no parentheses, so we can skip this step. Do all needed steps. If a step is not needed, then skip that step.
Then combine the x terms so that there is only a single x term left. In Example C, the best way to do this is to subtract 8x from both sides. Let’s see that:
8x – 1 = 9+9x
-8x -8x
---------------------
-1 = 9 + 1x Notice that 8x -8x = 0x or 0. Since 0 – 1 = -1, we need not write the 0.
-1 = 9 + x since 1x = x, we may optionally write the 1 in our scratch work. Writing the x as 1x while working out the problem is
extremely helpful to many students. For our final answer. we always drop the 1 from 1x since x is the more simplified
form.
What is the next step to solve -1 = 9 + x? We want the x alone. And in this step 9 is being added to the x.
How do I undo adding 9 to x? I subtract 9.
-1 = 9 + x
-9 -9
------------------
-10 = x
So x = -10. Let’s check that. You should ALWAYS check your answer. Seriously. Always check!
Another error to avoid:
Remember we can never add across the equals sign. So in example C, 8x – 1 = 9+9x, we can never add 8x + 9x and put 17x on the right or left side of the equation since 8x and 9x are on different sides of the equals sign. That would never be correct.
How to check: 1. “Plug in” -10 for x in the original equation. 2. Simplify, and 3. notice if it balances. 4. If it does, then x = -10 checks out. If not, then x = -10 does not check out and is not a valid solution.
Let’s see that step by step:
8x – 1 = 9+9x 1. “Plug in” -10 for x in the original equation.
How to: Replace each x with (-10). Use parentheses!
8(-10) – 1 = 9+9(-10) 2. Simplify, using the correct order of operations. *
-80 – 1 = 9-90 First I multiply.
-81 = -81 Then I subtract.
-81 = -81 is a true statement. So x = -10 is a correct solution. It checks!
What if your answer doesn’t check?
If you get a false statement, then your answer fails to check. The most likely reason an answer does not check is that you have made a math error. Start the problem over and try it again. Do not hand in an answer that does not check.
*For a review on the correct order of operations, watch this short video presentation I created. Here is the link: http://www.youtube.com/watch?v=fQc_TWK5lU0&feature=plcp To watch the video, copy and paste the link into Internet Explorer, Mozilla Firefox, or your favorite web browser.
For problems with decimals or fractions, such as 0.05r+0.4r = 27 or (1/3)p – 5 = (1/4)p, it makes the problem much easier if you multiply both sides by a convenient number so that you can work the problem out without the fractions or decimals. This makes it a LOT easier for most students.
Example D: 0.05r+0.4r = 27 Multiply both sides by 100 to clear the decimals.
Example E: (1/3)p – 5 = (1/4)p Multiply both sides by the LCD, 12, to clear the fractions. (Recall that LCD means the least common
denominator, or least common multiple, of 3 and 4.)
I suggest you watch a video or two on each of these on math tv, a terrific free math video website. Instructions on where to look are in the next session. One you see how these go, they are all very similar. One of my students says, that she forgets that with math, you have to be UBER careful. That is certainly true when it comes to solving equations.
Additional Resources (web links, videos, and articles):
Several weblinks on math:
www.mathtv.com
This site provides video explanation of many math topics, including arithmetic and algebra. Amazingly, it is free.
Follow these steps:
1. Go to www.mathtv.com,
2. Click on Algebra
3. Browse the topic list and click on linear equations in one variable.
4. Click on Addition property (like Example A above), Multiplication property (like Example B above), Both properties (like Example C above), with parentheses, with fractions (like Example E above), and with decimals (like Example D above).
5. Look through the list and click on the problem you’d like to see worked.
Many problems can be worked in English or Spanish.
http://www.wtamu.edu/academic/anns/mps/math/mathlab
This is a super text based site to help you get up to speed on basic math and algebra. It is free.
Watch Tutorial 1: How to Succeed in a Math Class, if you have not done so already. These tutorials cover the week 2 material.
Tutorial 11: Simplifying Algebraic Expressions
Tutorial 12: The Addition Property of Equality
Tutorial 13: The Multiplication Property of Equality
Tutorial 14: Solving Linear Equations (Putting it all together)
Tutorial 15: Introduction to Problem Solving
Tutorial 16: Percent and Problem Solving
Tutorial 17: Further Problem Solving
Tutorial 18: Solving Linear Inequalities
On problem solving
This site outlines George Polya’s 4 step problem solving method. Use to tackle those tough word problems.
http://teacher.scholastic.com/lessonrepro/lessonplans/steppro.htm
www.purplemath.com
A text based site with some graphics. I have looked at many pages on the purplemath website and have never seen an error. It is free.
http://www.khanacademy.org
This site provides video explanation of many math and topics from basic math to high level math and covers many other topics. Sal Khan is a terrific teacher. It is free.
Discussion Post Expectations:
Each week students will participate in online discussions with classmates, which are related to the week’s readings. These discussions replace the interactive dialogue that occurs in the traditional classroom setting. Each week, students’ initial discussion posts are due by 11:59 p.m. (of the time zone in which each student resides) on Day 3 (Thursday). Students will have until 11:59 p.m. on Day 7 (the following Monday) to make the required minimum number of response posts to classmates. Reply to each discussion question with a thoughtful and clear message. This is a good time to practice our critical thinking.
Discussion details
Discussion 1: This discussion covers an important real life problem that many adults will face in their lifetimes. When giving medication to a child, how much medication should be given to promote healing and keep the child safe? Everyone should know how to do this! Discussions represent 19.5% of the overall course grade.
In this discussion, you will be demonstrating your understanding of the math involved in this procedure. You should never use a variable without writing down what it stands for so it is clear to you and me. Carefully read the instructions in order and view the example to complete this discussion. Always read through the example. The example is like gold!
Then double check that you are working the assigned problem(s) and have answered each question that is asked. Don’t forget to use your vocabulary words in your discussion. Putting them in bold is always appreciated!
Be sure to address this part as well: Explain what the variables in the formula represent and show all steps in the computations.
Remarking on classmates’ posts can seem like a puzzle.
Since you earn valuable credit for replying to your classmates, so it is worth doing. Replies must be substantive to earn any credit. Saying "I agree" or "Your work is very neat" is not substantive and will not earn credit. So if you check a classmate’s work and cannot find any error, there may not be anything substantive to say.
Many errors are very small: dropping a negative sign, miscopying a 2 as a 3, subtracting when we should add or solving when we should simplify. Students can find these errors if they look.
If the person's calculations agree with yours, look for another classmate to reply to. You may need to look at the posts of 5 or 6 classmates to find just one to reply to.
Quizzes
There is a weekly quiz each week in this class.
We believe that watching these short videos will improve your understanding of algebra and help you succeed in the class. Thus each week, there is a quiz is to check that you’ve watched the videos. As long as class is in session, you can complete or redo your quiz. Only your highest score is saved, so you can retake each of the quizzes until you’re happy with your score. Quizzes represent 10% of the overall course grade.
Labs
It is very important that you work through every concept in each pie slice by the deadline since these grades will be automatically imported into our Gradebook at 11:59 PM Mountain Time on Monday evening. Be sure to work from pie slice 1 to pie slice 5 in numerical order. If you fail to complete the Week One pie slice, you will not be able to complete all the concepts in Week Two through Five. In order to unlock all the concepts in later weeks, you must complete all math problems in the previous weeks. The ALEKS Labs represent 40% of the overall course grade.
Be sure to use your textbook and the instructor guidance I provide each week together with ALEKS to see explanations of the problems.
Homework
To complete the following assignment, go to this week's ALEKS Homework link in the left navigation.
Week Two ALEKS Homework
The official deadline for the ALEKS homework is Monday night at the usual time. As long as class is in session, you can complete or redo your ALEKS Homework. Only your highest score is saved, so you can retake the ALEKS homework until you’re happy with your score. The homework represents 30% of the overall course grade.
